Chapter 1 NAME

The Market

Introduction,The problems in this chapter examine some variations on

the apartment market described in the text,In most of the problems we

work with the true demand curve constructed from the reservation prices

of the consumers rather than the \smoothed" demand curve that we used

in the text.

Remember that the reservation price of a consumer is that price

where he is just indi erent between renting or not renting the apartment.

At any price below the reservation price the consumer will demand one

apartment,at any price above the reservation price the consumer will de-

mand zero apartments,and exactly at the reservation price the consumer

will be indi erent between having zero or one apartment.

You should also observe that when demand curves have the \stair-

case" shape used here,there will typically be a range of prices where

supply equals demand,Thus we will ask for the the highest and lowest

price in the range.

1.1 (3) Suppose that we have 8 people who want to rent an apartment.

Their reservation prices are given below,(To keep the numbers small,

think of these numbers as being daily rent payments.)

Person=ABCDEFGH

Price = 40 25 30 35 10 18 15 5

(a) Plot the market demand curve in the following graph,(Hint,When

the market price is equal to some consumer i’s reservation price,there

will be two di erent quantities of apartments demanded,since consumer

i will be indi erent between having or not having an apartment.)

2 THE MARKET (Ch,1)

012345678

10

20

30

40

60

50

Price

Apartments

(b) Suppose the supply of apartments is xed at 5 units,In this case

there is a whole range of prices that will be equilibrium prices,What is

the highest price that would make the demand for apartments equal to 5

units? $18.

(c) What is the lowest price that would make the market demand equal

to 5 units? $15.

(d) With a supply of 4 apartments,which of the people A{H end up

getting apartments? A,B,C,D.

(e) What if the supply of apartments increases to 6 units,What is the

range of equilibrium prices? $10 to $15.

1.2 (3) Suppose that there are originally 5 units in the market and that

1 of them is turned into a condominium.

(a) Suppose that person A decides to buy the condominium,What will

be the highest price at which the demand for apartments will equal the

supply of apartments? What will be the lowest price? Enter your an-

swers in column A,in the table,Then calculate the equilibrium prices of

apartments if B,C,:::,decide to buy the condominium.

NAME 3

Person A B C D E F G H

High price 18 18 18 18 25 25 25 25

Low price 15 15 15 15 18 15 18 18

(b) Suppose that there were two people at each reservation price and 10

apartments,What is the highest price at which demand equals supply?

18,Suppose that one of the apartments was turned into a condo-

minium,Is that price still an equilibrium price? Yes.

1.3 (2) Suppose now that a monopolist owns all the apartments and that

he is trying to determine which price and quantity maximize his revenues.

(a) Fill in the box with the maximum price and revenue that the monop-

olist can make if he rents 1,2,:::,8 apartments,(Assume that he must

charge one price for all apartments.)

Number 1 2 3 4 5 6 7 8

Price 40 35 30 25 18 15 10 5

Revenue 40 70 90 100 90 90 70 40

(b) Which of the people A{F would get apartments? A,B,C,D.

(c) If the monopolist were required by law to rent exactly 5 apartments,

what price would he charge to maximize his revenue? $18.

(d) Who would get apartments? A,B,C,D,F.

(e) If this landlord could charge each individual a di erent price,and he

knew the reservation prices of all the individuals,what is the maximum

revenue he could make if he rented all 5 apartments? $148.

(f) If 5 apartments were rented,which individuals would get the apart-

ments? A,B,C,D,F.

1.4 (2) Suppose that there are 5 apartments to be rented and that the

city rent-control board sets a maximum rent of $9,Further suppose that

people A,B,C,D,and E manage to get an apartment,while F,G,and

H are frozen out.

4 THE MARKET (Ch,1)

(a) If subletting is legal|or,at least,practiced|who will sublet to whom

in equilibrium? (Assume that people who sublet can evade the city rent-

control restrictions.) E,who is willing to pay only

$10 for an apartment would sublet to F,

who is willing to pay $18.

(b) What will be the maximum amount that can be charged for the sublet

payment? $18.

(c) If you have rent control with unlimited subletting allowed,which of

the consumers described above will end up in the 5 apartments? A,

B,C,D,F.

(d) How does this compare to the market outcome? It’s the

same.

1.5 (2) In the text we argued that a tax on landlords would not get

passed along to the renters,What would happen if instead the tax was

imposed on renters?

(a) To answer this question,consider the group of people in Problem 1.1.

What is the maximum that they would be willing to pay to the landlord

if they each had to pay a $5 tax on apartments to the city? Fill in the

box below with these reservation prices.

Person A B C D E F G H

Reservation Price 35 20 25 30 5 13 10 0

(b) Using this information determine the maximum equilibrium price if

there are 5 apartments to be rented,$13.

(c) Of course,the total price a renter pays consists of his or her rent plus

the tax,This amount is $18.

(d) How does this compare to what happens if the tax is levied on the

landlords? It’s the same.

Chapter 2 NAME

Budget Constraint

Introduction,These workouts are designed to build your skills in de-

scribing economic situations with graphs and algebra,Budget sets are a

good place to start,because both the algebra and the graphing are very

easy,Where there are just two goods,a consumer who consumes x

1

units

of good 1 and x

2

units of good 2 is said to consume the consumption bun-

dle,(x

1;x

2

),Any consumption bundle can be represented by a point on

a two-dimensional graph with quantities of good 1 on the horizontal axis

and quantities of good 2 on the vertical axis,If the prices are p

1

for good 1

and p

2

for good 2,and if the consumer has income m,then she can a ord

any consumption bundle,(x

1;x

2

),such thatp

1

x

1

+p

2

x

2

m,On a graph,

the budget line is just the line segment with equation p

1

x

1

+ p

2

x

2

= m

and with x

1

and x

2

both nonnegative,The budget line is the boundary

of the budget set,All of the points that the consumer can a ord lie on

one side of the line and all of the points that the consumer cannot a ord

lie on the other.

If you know prices and income,you can construct a consumer’s bud-

get line by nding two commodity bundles that she can \just a ord" and

drawing the straight line that runs through both points.

Example,Myrtle has 50 dollars to spend,She consumes only apples and

bananas,Apples cost 2 dollars each and bananas cost 1 dollar each,You

are to graph her budget line,where apples are measured on the horizontal

axis and bananas on the vertical axis,Notice that if she spends all of her

income on apples,she can a ord 25 apples and no bananas,Therefore

her budget line goes through the point (25;0) on the horizontal axis,If

she spends all of her income on bananas,she can a ord 50 bananas and

no apples,Therfore her budget line also passes throught the point (0;50)

on the vertical axis,Mark these two points on your graph,Then draw a

straight line between them,This is Myrtle’s budget line.

What if you are not told prices or income,but you know two com-

modity bundles that the consumer can just a ord? Then,if there are just

two commodities,you know that a unique line can be drawn through two

points,so you have enough information to draw the budget line.

Example,Laurel consumes only ale and bread,If she spends all of her

income,she can just a ord 20 bottles of ale and 5 loaves of bread,Another

commodity bundle that she can a ord if she spends her entire income is

10 bottles of ale and 10 loaves of bread,If the price of ale is 1 dollar per

bottle,how much money does she have to spend? You could solve this

problem graphically,Measure ale on the horizontal axis and bread on the

vertical axis,Plot the two points,(20;5) and (10;10),that you know to

be on the budget line,Draw the straight line between these points and

extend the line to the horizontal axis,This point denotes the amount of

6 BUDGET CONSTRAINT (Ch,2)

ale Laurel can a ord if she spends all of her money on ale,Since ale costs

1 dollar a bottle,her income in dollars is equal to the largest number of

bottles she can a ord,Alternatively,you can reason as follows,Since

the bundles (20;5) and (10;10) cost the same,it must be that giving up

10 bottles of ale makes her able to a ord an extra 5 loaves of bread,So

bread costs twice as much as ale,The price of ale is 1 dollar,so the price

of bread is 2 dollars,The bundle (20;5)costsasmuchasherincome.

Therefore her income must be 20 1+5 2 = 30.

When you have completed this workout,we hope that you will be

able to do the following:

Write an equation for the budget line and draw the budget set on a

graph when you are given prices and income or when you are given

two points on the budget line.

Graph the e ects of changes in prices and income on budget sets.

Understand the concept of numeraire and know what happens to the

budget set when income and all prices are multiplied by the same

positive amount.

Know what the budget set looks like if one or more of the prices is

negative.

See that the idea of a \budget set" can be applied to constrained

choices where there are other constraints on what you can have,in

addition to a constraint on money expenditure.

NAME 7

2.1 (0) You have an income of $40 to spend on two commodities,Com-

modity 1 costs $10 per unit,and commodity 2 costs $5 per unit.

(a) Write down your budget equation,10x

1

+5x

2

=40.

(b) If you spent all your income on commodity 1,how much could you

buy? 4.

(c) If you spent all of your income on commodity 2,how much could

you buy? 8,Use blue ink to draw your budget line in the graph

below.

02468

2

4

6

x1

x2

8

Blue Line

Red Line

Black Line

Black Shading

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Blue

Shading

(d) Suppose that the price of commodity 1 falls to $5 while everything else

stays the same,Write down your new budget equation,5x

1

+5x

2

=

40,On the graph above,use red ink to draw your new budget line.

(e) Suppose that the amount you are allowed to spend falls to $30,while

the prices of both commodities remain at $5,Write down your budget

equation,5x

1

+5x

2

=30,Use black ink to draw this budget

line.

(f) On your diagram,use blue ink to shade in the area representing com-

modity bundles that you can a ord with the budget in Part (e) but could

not a ord to buy with the budget in Part (a),Use black ink or pencil to

shade in the area representing commodity bundles that you could a ord

with the budget in Part (a) but cannot a ord with the budget in Part

(e).

2.2 (0) On the graph below,draw a budget line for each case.

8 BUDGET CONSTRAINT (Ch,2)

(a) p

1

=1,p

2

=1,m = 15,(Use blue ink.)

(b) p

1

=1,p

2

=2,m = 20,(Use red ink.)

(c) p

1

=0,p

2

=1,m = 10,(Use black ink.)

(d) p

1

= p

2

,m =15p

1

,(Use pencil or black ink,Hint,How much of

good 1 could you a ord if you spend your entire budget on good 1?)

0 5 10 15 20

5

10

15

x1

x2

20

Blue Line

Red Line

Black Line

2.3 (0) Your budget is such that if you spend your entire income,you

can a ord either 4 units of good x and 6 units of good y or 12 units of x

and 2 units of y.

(a) Mark these two consumption bundles and draw the budget line in the

graph below.

0481216

4

8

12

x

y

16

NAME 9

(b) What is the ratio of the price of x to the price of y? 1/2.

(c) If you spent all of your income on x,howmuchx could you buy?

16.

(d) If you spent all of your income on y,howmuchy could you buy?

8.

(e) Write a budget equation that gives you this budget line,where the

price of x is 1,x+2y =16.

(f) Write another budget equation that gives you the same budget line,

but where the price of x is 3,3x+6y =48.

2.4 (1) Murphy was consuming 100 units of X and 50 units of Y.The

price of X rose from 2 to 3,The price of Y remained at 4.

(a) How much would Murphy’s income have to rise so that he can still

exactly a ord 100 units of X and 50 units of Y? $100.

2.5 (1) If Amy spent her entire allowance,she could a ord 8 candy bars

and 8 comic books a week,She could also just a ord 10 candy bars and

4 comic books a week,The price of a candy bar is 50 cents,Draw her

budget line in the box below,What is Amy’s weekly allowance? $6.

0 8 16 24 32

8

16

24

Candy bars

Comic books

32

12

10 BUDGET CONSTRAINT (Ch,2)

2.6 (0) In a small country near the Baltic Sea,there are only three

commodities,potatoes,meatballs,and jam,Prices have been remark-

ably stable for the last 50 years or so,Potatoes cost 2 crowns per sack,

meatballs cost 4 crowns per crock,and jam costs 6 crowns per jar.

(a) Write down a budget equation for a citizen named Gunnar who has

an income of 360 crowns per year,Let P stand for the number of sacks of

potatoes,M for the number of crocks of meatballs,and J for the number

of jars of jam consumed by Gunnar in a year,2P +4M +6J =

360.

(b) The citizens of this country are in general very clever people,but they

are not good at multiplying by 2,This made shopping for potatoes excru-

ciatingly di cult for many citizens,Therefore it was decided to introduce

a new unit of currency,such that potatoes would be the numeraire,A

sack of potatoes costs one unit of the new currency while the same rel-

ative prices apply as in the past,In terms of the new currency,what is

the price of meatballs? 2 crowns.

(c) In terms of the new currency,what is the price of jam? 3

crowns.

(d) What would Gunnar’s income in the new currency have to be for him

to be exactly able to a ord the same commodity bundles that he could

a ord before the change? 180 crowns.

(e) Write down Gunnar’s new budget equation,P +2M +3J =

180,Is Gunnar’s budget set any di erent than it was before the change?

No.

2.7 (0) Edmund Stench consumes two commodities,namely garbage and

punk rock video cassettes,He doesn’t actually eat the former but keeps

it in his backyard where it is eaten by billy goats and assorted vermin.

The reason that he accepts the garbage is that people pay him $2 per

sack for taking it,Edmund can accept as much garbage as he wishes at

that price,He has no other source of income,Video cassettes cost him

$6 each.

(a) If Edmund accepts zero sacks of garbage,how many video cassettes

can he buy? 0.

NAME 11

(b) If he accepts 15 sacks of garbage,how many video cassettes can he

buy? 5.

(c) Write down an equation for his budget line,6C?2G =0.

(d) Draw Edmund’s budget line and shade in his budget set.

0 5 10 15 20

5

10

15

Video cassettes

Garbage

20

Budget Line

Budget Set

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2.8 (0) If you think Edmund is odd,consider his brother Emmett.

Emmett consumes speeches by politicians and university administrators.

He is paid $1 per hour for listening to politicians and $2 per hour for

listening to university administrators,(Emmett is in great demand to help

ll empty chairs at public lectures because of his distinguished appearance

and his ability to refrain from making rude noises.) Emmett consumes

one good for which he must pay,We have agreed not to disclose what

that good is,but we can tell you that it costs $15 per unit and we shall

call it Good X,In addition to what he is paid for consuming speeches,

Emmett receives a pension of $50 per week.

0255075100

25

50

75

Politician speeches

Administrator speeches

100

12 BUDGET CONSTRAINT (Ch,2)

(a) Write down a budget equation stating those combinations of the three

commodities,Good X,hours of speeches by politicians (P),and hours of

speeches by university administrators (A) that Emmett could a ord to

consume per week,15X?1P?2A =50.

(b) On the graph above,draw a two-dimensional diagram showing the

locus of consumptions of the two kinds of speeches that would be possible

for Emmett if he consumed 10 units of Good X per week.

2.9 (0) Jonathan Livingstone Yuppie is a prosperous lawyer,He

has,in his own words,\outgrown those con ning two-commodity lim-

its." Jonathan consumes three goods,unblended Scotch whiskey,de-

signer tennis shoes,and meals in French gourmet restaurants,The price

of Jonathan’s brand of whiskey is $20 per bottle,the price of designer

tennis shoes is $80 per pair,and the price of gourmet restaurant meals

is $50 per meal,After he has paid his taxes and alimony,Jonathan has

$400 a week to spend.

(a) Write down a budget equation for Jonathan,where W stands for

the number of bottles of whiskey,T stands for the number of pairs of

tennis shoes,and M for the number of gourmet restaurant meals that he

consumes,20W +80T +50M = 400.

(b) Draw a three-dimensional diagram to show his budget set,Label the

intersections of the budget set with each axis.

M

TW

8

5

20

(c) Suppose that he determines that he will buy one pair of designer tennis

shoes per week,What equation must be satis ed by the combinations of

restaurant meals and whiskey that he could a ord? 20W+50M =

320.

2.10 (0) Martha is preparing for exams in economics and sociology,She

has time to read 40 pages of economics and 30 pages of sociology,In the

same amount of time she could also read 30 pages of economics and 60

pages of sociology.

NAME 13

(a) Assuming that the number of pages per hour that she can read of

either subject does not depend on how she allocates her time,how many

pages of sociology could she read if she decided to spend all of her time

on sociology and none on economics? 150 pages,(Hint,You

have two points on her budget line,so you should be able to determine

the entire line.)

(b) How many pages of economics could she read if she decided to spend

all of her time reading economics? 50 pages.

2.11 (1) Harry Hype has $5,000 to spend on advertising a new kind of

dehydrated sushi,Market research shows that the people most likely to

buy this new product are recent recipients of M.B.A,degrees and lawyers

who own hot tubs,Harry is considering advertising in two publications,

a boring business magazine and a trendy consumer publication for people

who wish they lived in California.

Fact 1,Ads in the boring business magazine cost $500 each and ads in

the consumer magazine cost $250 each.

Fact 2,Each ad in the business magazine will be read by 1,000 recent

M.B.A.’s and 300 lawyers with hot tubs.

Fact 3,Each ad in the consumer publication will be read by 300 recent

M.B.A.’s and 250 lawyers who own hot tubs.

Fact 4,Nobody reads more than one ad,and nobody who reads one

magazine reads the other.

(a) If Harry spends his entire advertising budget on the business pub-

lication,his ad will be read by 10,000 recent M.B.A.’s and by

3,000 lawyers with hot tubs.

(b) If he spends his entire advertising budget on the consumer publication,

his ad will be read by 6,000 recent M.B.A.’s and by 5,000

lawyers with hot tubs.

(c) Suppose he spent half of his advertising budget on each publication.

His ad would be read by 8,000 recent M.B.A.’s and by 4,000

lawyers with hot tubs.

(d) Draw a \budget line" showing the combinations of number of readings

by recent M.B.A.’s and by lawyers with hot tubs that he can obtain if he

spends his entire advertising budget,Does this line extend all the way

to the axes? No,Sketch,shade in,and label the budget set,which

includes all the combinations of MBA’s and lawyers he can reach if he

spends no more than his budget.

14 BUDGET CONSTRAINT (Ch,2)

(e) Let M stand for the number of instances of an ad being read by an

M.B.A,and L stand for the number of instances of an ad being read by

a lawyer,This budget line is a line segment that lies on the line with

equation M +2L =16,With a xed advertising budget,how

many readings by M.B.A.’s must he sacri ce to get an additional reading

by a lawyer with a hot tub? 2.

0481216

4

8

12

Lawyers x 1000

MBA's x 1000

16

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c

a

b

5

10

Budget

Set

Budget line

2

3

6

2.12 (0) On the planet Mungo,they have two kinds of money,blue

money and red money,Every commodity has two prices|a red-money

price and a blue-money price,Every Mungoan has two incomes|a red

income and a blue income.

In order to buy an object,a Mungoan has to pay that object’s red-

money price in red money and its blue-money price in blue money,(The

shops simply have two cash registers,and you have to pay at both registers

to buy an object.) It is forbidden to trade one kind of money for the other,

and this prohibition is strictly enforced by Mungo’s ruthless and e cient

monetary police.

There are just two consumer goods on Mungo,ambrosia and bubble

gum,All Mungoans prefer more of each good to less.

The blue prices are 1 bcu (bcu stands for blue currency unit) per

unit of ambrosia and 1 bcu per unit of bubble gum.

The red prices are 2 rcus (red currency units) per unit of ambrosia

and 6 rcus per unit of bubble gum.

(a) On the graph below,draw the red budget (with red ink) and the

blue budget (with blue ink) for a Mungoan named Harold whose blue

income is 10 and whose red income is 30,Shade in the \budget set"

containing all of the commodity bundles that Harold can a ord,given

NAME 15

its

two budget constraints,Remember,Harold has to have enough blue

money and enough red money to pay both the blue-money cost and the

red-money cost of a bundle of goods.

0 5 10 15 20

5

10

15

Ambrosia

Gum

20

Blue Lines

Red Line

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(b) Another Mungoan,Gladys,faces the same prices that Harold faces

and has the same red income as Harold,but Gladys has a blue income of

20,Explain how it is that Gladys will not spend its entire blue income

no matter what its tastes may be,(Hint,Draw Gladys’s budget lines.)

The blue budget line lies strictly outside

the red budget line,so to satisfy both

budgets,one must be strictly inside the

red budget line.

(c) A group of radical economic reformers on Mungo believe that the

currency rules are unfair,\Why should everyone have to pay two prices

for everything?" they ask,They propose the following scheme,Mungo

will continue to have two currencies,every good will have a blue price and

a red price,and every Mungoan will have a blue income and a red income.

But nobody has to pay both prices,Instead,everyone on Mungo must

declare itself to be either a Blue-Money Purchaser (a \Blue") or a Red-

Money Purchaser (a \Red") before it buys anything at all,Blues must

make all of their purchases in blue money at the blue prices,spending

only their blue incomes,Reds must make all of their purchases in red

money,spending only their red incomes.

Suppose that Harold has the same income after this reform,and that

prices do not change,Before declaring which kind of purchaser it will be,

We refer to all Mungoans by the gender-neutral pronoun,\it." Al-

though Mungo has two sexes,neither of them is remotely like either of

ours.

16 BUDGET CONSTRAINT (Ch,2)

Harold contemplates the set of commodity bundles that it could a ord

by making one declaration or the other,Let us call a commodity bundle

\attainable" if Harold can a ord it by declaring itself to be a \Blue" and

buying the bundle with blue money or if Harold can a ord the bundle

by declaring itself to be a \Red" and buying it with red money,On the

diagram below,shade in all of the attainable bundles.

0 5 10 15 20

5

10

15

Ambrosia

Gum

20

Blue Line

Red Line

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2.13 (0) Are Mungoan budgets really so fanciful? Can you think of sit-

uations on earth where people must simultaneously satisfy more than one

budget constraint? Is money the only scarce resource that people use up

when consuming? Consumption of many commodities

takes time as well as money,People have

to simultaneously satisfy a time budget

and a money budget,Other examples--people

may have a calorie budget or a cholesterol

budget or an alcohol-intake budget.

Chapter 3 NAME

Preferences

Introduction,In the previous section you learned how to use graphs to

show the set of commodity bundles that a consumer can a ord,In this

section,you learn to put information about the consumer’s preferences on

the same kind of graph,Most of the problems ask you to draw indi erence

curves.

Sometimes we give you a formula for the indi erence curve,Then

all you have to do is graph a known equation,But in some problems,we

give you only \qualitative" information about the consumer’s preferences

and ask you to sketch indi erence curves that are consistent with this

information,This requires a little more thought,Don’t be surprised or

disappointed if you cannot immediately see the answer when you look

at a problem,and don’t expect that you will nd the answers hiding

somewhere in your textbook,The best way we know to nd answers is to

\think and doodle." Draw some axes on scratch paper and label them,

then mark a point on your graph and ask yourself,\What other points on

the graph would the consumer nd indi erent to this point?" If possible,

draw a curve connecting such points,making sure that the shape of the

line you draw reflects the features required by the problem,This gives

you one indi erence curve,Now pick another point that is preferred to

the rst one you drew and draw an indi erence curve through it.

Example,Jocasta loves to dance and hates housecleaning,She has strictly

convex preferences,She prefers dancing to any other activity and never

gets tired of dancing,but the more time she spends cleaning house,the less

happy she is,Let us try to draw an indi erence curve that is consistent

with her preferences,There is not enough information here to tell us

exactly where her indi erence curves go,but there is enough information

to determine some things about their shape,Take a piece of scratch

paper and draw a pair of axes,Label the horizontal axis \Hours per day of

housecleaning." Label the vertical axis \Hours per day of dancing." Mark

a point a little ways up the vertical axis and write a 4 next to it,At this

point,she spends 4 hours a day dancing and no time housecleaning,Other

points that would be indi erent to this point would have to be points

where she did more dancing and more housecleaning,The pain of the

extra housekeeping should just compensate for the pleasure of the extra

dancing,So an indi erence curve for Jocasta must be upward sloping.

Because she loves dancing and hates housecleaning,it must be that she

prefers all the points above this indi erence curve to all of the points on

or below it,If Jocasta has strictly convex preferences,then it must be

that if you draw a line between any two points on the same indi erence

curve,all the points on the line (except the endpoints) are preferred to

the endpoints,For this to be the case,it must be that the indi erence

curve slopes upward ever more steeply as you move to the right along it.

You should convince yourself of this by making some drawings on scratch

18 PREFERENCES (Ch,3)

paper,Draw an upward-sloping curve passing through the point (0;4)

and getting steeper as one moves to the right.

When you have completed this workout,we hope that you will be

able to do the following:

Given the formula for an indi erence curve,draw this curve,and nd

its slope at any point on the curve.

Determine whether a consumer prefers one bundle to another or is

indi erent between them,given speci c indi erence curves.

Draw indi erence curves for the special cases of perfect substitutes

and perfect complements.

Draw indi erence curves for someone who dislikes one or both com-

modities.

Draw indi erence curves for someone who likes goods up to a point

but who can get \too much" of one or more goods.

Identify weakly preferred sets and determine whether these are con-

vex sets and whether preferences are convex.

Know what the marginal rate of substitution is and be able to deter-

mine whether an indi erence curve exhibits \diminishing marginal

rate of substitution."

Determine whether a preference relation or any other relation be-

tween pairs of things is transitive,whether it is reflexive,and whether

it is complete.

3.1 (0) Charlie likes both apples and bananas,He consumes nothing else.

The consumption bundle where Charlie consumes x

A

bushels of apples

per year and x

B

bushels of bananas per year is written as (x

A;x

B

),Last

year,Charlie consumed 20 bushels of apples and 5 bushels of bananas,It

happens that the set of consumption bundles (x

A;x

B

) such that Charlie

is indi erent between (x

A;x

B

)and(20;5) is the set of all bundles such

that x

B

= 100=x

A

,The set of bundles (x

A;x

B

) such that Charlie is just

indi erent between (x

A;x

B

) and the bundle (10;15) is the set of bundles

such that x

B

= 150=x

A

.

(a) On the graph below,plot several points that lie on the indi erence

curve that passes through the point (20;5),and sketch this curve,using

blue ink,Do the same,using red ink,for the indi erence curve passing

through the point (10;15).

(b) Use pencil to shade in the set of commodity bundles that Charlie

weakly prefers to the bundle (10;15),Use blue ink to shade in the set

of commodity bundles such that Charlie weakly prefers (20;5) to these

bundles.

NAME 19

010203040

10

20

30

Apples

Bananas

40

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Blue Curve

Pencil Shading

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,,,,,,,,,,,,,,,,,,,,,,,,

Red Curve

Blue Shading

For each of the following statements about Charlie’s preferences,write

\true" or \false."

(c) (30;5) (10;15),True.

(d) (10;15) (20;5),True.

(e) (20;5) (10;10),True.

(f) (24;4) (11;9:1),False.

(g) (11;14) (2;49),True.

(h) A set is convex if for any two points in the set,the line segment

between them is also in the set,Is the set of bundles that Charlie weakly

prefers to (20;5) a convex set? Yes.

(i) Is the set of bundles that Charlie considers inferior to (20;5) a convex

set? No.

(j) The slope of Charlie’s indi erence curve through a point,(x

A;x

B

),is

known as his marginal rate of substitution at that point.

20 PREFERENCES (Ch,3)

(k) Remember that Charlie’s indi erence curve through the point (10;10)

has the equation x

B

= 100=x

A

,Those of you who know calculus will

remember that the slope of a curve is just its derivative,which in this

case is?100=x

2

A

,(If you don’t know calculus,you will have to take our

word for this.) Find Charlie’s marginal rate of substitution at the point,

(10;10),?1.

(l) What is his marginal rate of substitution at the point (5;20)4.

(m) What is his marginal rate of substitution at the point (20;5)?

(?:25).

(n) Do the indi erence curves you have drawn for Charlie exhibit dimin-

ishing marginal rate of substitution? Yes.

3.2 (0) Ambrose consumes only nuts and berries,Fortunately,he likes

both goods,The consumption bundle where Ambrose consumes x

1

units

of nuts per week and x

2

units of berries per week is written as (x

1;x

2

).

The set of consumption bundles (x

1;x

2

) such that Ambrose is indi erent

between (x

1;x

2

)and(1;16) is the set of bundles such that x

1

0,x

2

0,

and x

2

=20?4

p

x

1

,The set of bundles (x

1;x

2

) such that (x

1;x

2

)

(36;0) is the set of bundles such that x

1

0,x

2

0andx

2

=24?4

p

x

1

.

(a) On the graph below,plot several points that lie on the indi erence

curve that passes through the point (1;16),and sketch this curve,using

blue ink,Do the same,using red ink,for the indi erence curve passing

through the point (36;0).

(b) Use pencil to shade in the set of commodity bundles that Ambrose

weakly prefers to the bundle (1;16),Use red ink to shade in the set of

all commodity bundles (x

1;x

2

) such that Ambrose weakly prefers (36;0)

to these bundles,Is the set of bundles that Ambrose prefers to (1;16) a

convex set? Yes.

(c) What is the slope of Ambrose’s indi erence curve at the point (9;8)?

(Hint,Recall from calculus the way to calculate the slope of a curve,If

you don’t know calculus,you will have to draw your diagram carefully

and estimate the slope.)?2=3.

NAME 21

(d) What is the slope of his indi erence curve at the point (4;12)1.

,,,,,,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,

010203040

10

20

30

Nuts

Berries

40

Pencil Shading

Red Curve

,

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,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

Red

Shading

Blue Curve

(e) What is the slope of his indi erence curve at the point (9;12)2=3

at the point (4;16)1.

(f) Do the indi erence curves you have drawn for Ambrose exhibit dimin-

ishing marginal rate of substitution? Yes.

(g) Does Ambrose have convex preferences? Yes.

3.3 (0) Shirley Sixpack is in the habit of drinking beer each evening

while watching \The Best of Bowlerama" on TV,She has a strong thumb

and a big refrigerator,so she doesn’t care about the size of the cans that

beer comes in,she only cares about how much beer she has.

(a) On the graph below,draw some of Shirley’s indi erence curves be-

tween 16-ounce cans and 8-ounce cans of beer,Use blue ink to draw these

indi erence curves.

22 PREFERENCES (Ch,3)

02468

2

4

6

16-ounce

8-ounce

8

Blue Lines

Red Lines

(b) Lorraine Quiche likes to have a beer while she watches \Masterpiece

Theatre." She only allows herself an 8-ounce glass of beer at any one

time,Since her cat doesn’t like beer and she hates stale beer,if there is

more than 8 ounces in the can she pours the excess into the sink,(She

has no moral scruples about wasting beer.) On the graph above,use red

ink to draw some of Lorraine’s indi erence curves.

3.4 (0) Elmo nds himself at a Coke machine on a hot and dusty Sunday.

The Coke machine requires exact change|two quarters and a dime,No

other combination of coins will make anything come out of the machine.

No stores are open; no one is in sight,Elmo is so thirsty that the only

thing he cares about is how many soft drinks he will be able to buy with

the change in his pocket; the more he can buy,the better,While Elmo

searches his pockets,your task is to draw some indi erence curves that

describe Elmo’s preferences about what he nds.

NAME 23

02468

2

4

6

Quarters

Dimes

8

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,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,,,,,,,,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

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,,,,,,,,,,,,

,

,

,

,,,,,,,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,,,,,,

Red

shading

Blue

shading

Black

lines

(a) If Elmo has 2 quarters and a dime in his pockets,he can buy 1 soft

drink,How many soft drinks can he buy if he has 4 quarters and 2 dimes?

2.

(b) Use red ink to shade in the area on the graph consisting of all com-

binations of quarters and dimes that Elmo thinks are just indi erent to

having 2 quarters and 1 dime,(Imagine that it is possible for Elmo to

have fractions of quarters or of dimes,but,of course,they would be use-

less in the machine.) Now use blue ink to shade in the area consisting of

all combinations that Elmo thinks are just indi erent to having 4 quarters

and 2 dimes,Notice that Elmo has indi erence \bands," not indi erence

curves.

(c) Does Elmo have convex preferences between dimes and quarters?

Yes.

(d) Does Elmo always prefer more of both kinds of money to less? No.

(e) Does Elmo have a bliss point? No.

(f) If Elmo had arrived at the Coke machine on a Saturday,the drugstore

across the street would have been open,This drugstore has a soda foun-

tain that will sell you as much Coke as you want at a price of 4 cents an

ounce,The salesperson will take any combination of dimes and quarters

in payment,Suppose that Elmo plans to spend all of the money in his

pocket on Coke at the drugstore on Saturday,On the graph above,use

pencil or black ink to draw one or two of Elmo’s indi erence curves be-

tween quarters and dimes in his pocket,(For simplicity,draw your graph

24 PREFERENCES (Ch,3)

as if Elmo’s fractional quarters and fractional dimes are accepted at the

corresponding fraction of their value.) Describe these new indi erence

curves in words,Line segments with slope?2:5.

3.5 (0) Randy Ratpack hates studying both economics and history,The

more time he spends studying either subject,the less happy he is,But

Randy has strictly convex preferences.

(a) Sketch an indi erence curve for Randy where the two commodities

are hours per week spent studying economics and hours per week spent

studying history,Will the slope of an indi erence curve be positive or

negative? Negative.

(b) Do Randy’s indi erence curves get steeper or flatter as you move from

left to right along one of them? Steeper.

02468

2

4

6

Hours studying economics

Hours studying history

8

Preference

direction

3.6 (0) Flossy Toothsome likes to spend some time studying and some

time dating,In fact her indi erence curves between hours per week spent

studying and hours per week spent dating are concentric circles around

her favorite combination,which is 20 hours of studying and 15 hours of

dating per week,The closer she is to her favorite combination,the happier

she is.

NAME 25

(a) Suppose that Flossy is currently studying 25 hours a week and dating

3hoursaweek,Wouldsheprefertobestudying30hoursaweekand

dating8hoursaweek? Yes,(Hint,Remember the formula for the

distance between two points in the plane?)

(b) On the axes below,draw a few of Flossy’s indi erence curves and

use your diagram to illustrate which of the two time allocations discussed

above Flossy would prefer.

010203040

10

20

30

Hours studying

Hours dating

40

,

(25,3)

(30,8)

(20,15)

Preference

direction

3.7 (0) Joan likes chocolate cake and ice cream,but after 10 slices of

cake,she gets tired of cake,and eating more cake makes her less happy.

Joan always prefers more ice cream to less,Joan’s parents require her to

eat everything put on her plate,In the axes below,use blue ink to draw a

set of indi erence curves that depict her preferences between plates with

di erent amounts of cake and ice cream,Be sure to label the axes.

(a) Suppose that Joan’s preferences are as before,but that her parents

allow her to leave anything on her plate that she doesn’t want,On the

graph below,use red ink to draw some indi erence curves depicting her

preferences between plates with di erent amounts of cake and ice cream.

Blue curves

Red curves

Ice cream

Chocolate cake

10

Preference

direction

26 PREFERENCES (Ch,3)

3.8 (0) Professor Goodheart always gives two midterms in his commu-

nications class,He only uses the higher of the two scores that a student

gets on the midterms when he calculates the course grade.

(a) Nancy Lerner wants to maximize her grade in this course,Let x

1

be

her score on the rst midterm and x

2

be her score on the second midterm.

Which combination of scores would Nancy prefer,x

1

=20andx

2

=70

or x

1

=60andx

2

= 60? (20,70).

(b) On the graph below,use red ink to draw an indi erence curve showing

all of the combinations of scores that Nancy likes exactly as much as

x

1

=20andx

2

= 70,Also use red ink to draw an indi erence curve

showing the combinations that Nancy likes exactly as much as x

1

=60

and x

2

= 60.

(c) Does Nancy have convex preferences over these combinations? No.

020406080

20

40

60

Grade on first midterm

Grade on second midterm

80

,

Preference

direction

Blue curves

Red

curves

(d) Nancy is also taking a course in economics from Professor Stern.

Professor Stern gives two midterms,Instead of discarding the lower grade,

Professor Stern discards the higher one,Let x

1

be her score on the rst

midterm and x

2

be her score on the second midterm,Which combination

of scores would Nancy prefer,x

1

=20andx

2

=70orx

1

=60and

x

2

= 50? (60,50).

(e) On the graph above,use blue ink to draw an indi erence curve showing

all of the combinations of scores on her econ exams that Nancy likes

exactly as well as x

1

=20andx

2

= 70,Also use blue ink to draw an

indi erence curve showing the combinations that Nancy likes exactly as

well as x

1

=60andx

2

= 50,Does Nancy have convex preferences over

these combinations? Yes.

NAME 27

3.9 (0) Mary Granola loves to consume two goods,grapefruits and

avocados.

(a) On the graph below,the slope of an indi erence curve through any

point where she has more grapefruits than avocados is?2,This means

that when she has more grapefruits than avocados,she is willing to give

up 2 grapefruit(s) to get one avocado.

(b) On the same graph,the slope of an indi erence curve at points where

she has fewer grapefruits than avocados is?1=2,This means that when

she has fewer grapefruits than avocados,she is just willing to give up

1/2 grapefruit(s) to get one avocado.

(c) On this graph,draw an indi erence curve for Mary through bundle

(10A;10G),Draw another indi erence curve through (20A;20G).

010203040

10

20

30

Avocados

Grapefruits

40

45

Slope -2

Slope -1/2

(d) Does Mary have convex preferences? Yes.

3.10 (2) Ralph Rigid likes to eat lunch at 12 noon,However,he also

likes to save money so he can buy other consumption goods by attending

the \early bird specials" and \late lunchers" promoted by his local diner.

Ralph has 15 dollars a day to spend on lunch and other stu,Lunch at

noon costs $5,If he delays his lunch until t hours after noon,he is able

to buy his lunch for a price of $5?t,Similarly if he eats his lunch t hours

before noon,he can buy it for a price of $5?t,(This is true for fractions

of hours as well as integer numbers of hours.)

(a) If Ralph eats lunch at noon,how much money does he have per day

to spend on other stu? $10.

28 PREFERENCES (Ch,3)

(b) How much money per day would he have left for other stu if he ate

at 2 P.M.? $12.

(c) On the graph below,use blue ink to draw the broken line that shows

combinations of meal time and money for other stu that Ralph can just

a ord,On this same graph,draw some indi erence curves that would be

consistent with Ralph choosing to eat his lunch at 11 A.M.

0

11 12 1 2

5

10

15

Time

Money

20

10

3.11 (0) Henry Hanover is currently consuming 20 cheeseburgers and 20

Cherry Cokes a week,A typical indi erence curve for Henry is depicted

below.

Cheeseburgers

Cherry Coke

10 20 30 400

40

30

20

10

NAME 29

(a) If someone o ered to trade Henry one extra cheeseburger for every

Coke he gave up,would Henry want to do this? No.

(b) What if it were the other way around,for every cheeseburger Henry

gave up,he would get an extra Coke,Would he accept this o er? Yes.

(c) At what rate of exchange would Henry be willing to stay put at his

current consumption level? 2 cheeseburgers for 1

Coke.

3.12 (1) Tommy Twit is happiest when he has 8 cookies and 4 glasses of

milk per day,Whenever he has more than his favorite amount of either

food,giving him still more makes him worse o,Whenever he has less

than his favorite amount of either food,giving him more makes him better

o,His mother makes him drink 7 glasses of milk and only allows him 2

cookies per day,One day when his mother was gone,Tommy’s sadistic

sister made him eat 13 cookies and only gave him 1 glass of milk,despite

the fact that Tommy complained bitterly about the last 5 cookies that she

made him eat and begged for more milk,Although Tommy complained

later to his mother,he had to admit that he liked the diet that his sister

forced on him better than what his mother demanded.

(a) Use black ink to draw some indi erence curves for Tommy that are

consistent with this story.

0

(8,4)

(13,1)

(2,7)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Cookies

1

2

3

4

5

6

7

8

9

10

11

12

Milk

30 PREFERENCES (Ch,3)

(b) Tommy’s mother believes that the optimal amount for him to consume

is 7 glasses of milk and 2 cookies,She measures deviations by absolute

values,If Tommy consumes some other bundle,say,(c;m),she measures

his departure from the optimal bundle by D = j7?mj+j2?cj.The

larger D is,the worse o she thinks Tommy is,Use blue ink in the graph

above to sketch a few of Mrs,Twit’s indi erence curves for Tommy’s

consumption,(Hint,Before you try to draw Mrs,Twit’s indi erence

curves,we suggest that you take a piece of scrap paper and draw a graph

of the locus of points (x

1;x

2

) such that jx

1

j+jx

2

j=1.)

3.13 (0) Coach Steroid likes his players to be big,fast,and obedient,If

player A is better than player B in two of these three characteristics,then

Coach Steroid prefers A to B,but if B is better than A in two of these

three characteristics,then Steroid prefers B to A,Otherwise,Steroid is

indi erent between them,Wilbur Westinghouse weighs 340 pounds,runs

very slowly,and is fairly obedient,Harold Hotpoint weighs 240 pounds,

runs very fast,and is very disobedient,Jerry Jacuzzi weighs 150 pounds,

runs at average speed,and is extremely obedient.

(a) Does Steroid prefer Westinghouse to Hotpoint or vice versa? He

prefers Westinghouse to Hotpoint.

(b) Does Steroid prefer Hotpoint to Jacuzzi or vice versa? He

prefers Hotpoint to Jacuzzi.

(c) Does Steroid prefer Westinghouse to Jacuzzi or vice versa? He

prefers Jacuzzi to Westinghouse.

(d) Does Coach Steroid have transitive preferences? No.

(e) After several losing seasons,Coach Steroid decides to change his way of

judging players,According to his new preferences,Steroid prefers player

A to player B if player A is better in all three of the characteristics that

Steroid values,and he prefers B to A if player B is better at all three

things,He is indi erent between A and B if they weigh the same,are

equally fast,and are equally obedient,In all other cases,Coach Steroid

simply says \A and B are not comparable."

(f) Are Coach Steroid’s new preferences complete? No.

(g) Are Coach Steroid’s new preferences transitive? Yes.

NAME 31

(h) Are Coach Steroid’s new preferences reflexive? Yes.

3.14 (0) The Bear family is trying to decide what to have for din-

ner,Baby Bear says that his ranking of the possibilities is (honey,grubs,

Goldilocks),Mama Bear ranks the choices (grubs,Goldilocks,honey),

while Papa Bear’s ranking is (Goldilocks,honey,grubs),They decide to

take each pair of alternatives and let a majority vote determine the family

rankings.

(a) Papa suggests that they rst consider honey vs,grubs,and then the

winner of that contest vs,Goldilocks,Which alternative will be chosen?

Goldilocks.

(b) Mama suggests instead that they consider honey vs,Goldilocks and

then the winner vs,grubs,Which gets chosen? Grubs.

(c) What order should Baby Bear suggest if he wants to get his favorite

food for dinner? Grubs versus Goldilocks,then

Honey versus the winner.

(d) Are the Bear family’s \collective preferences," as determined by vot-

ing,transitive? No.

3.15 (0) Olson likes strong co ee,the stronger the better,But he can’t

distinguish small di erences,Over the years,Mrs,Olson has discovered

that if she changes the amount of co ee by more than one teaspoon in

her six-cup pot,Olson can tell that she did it,But he cannot distinguish

di erences smaller than one teaspoon per pot,Where A and B are two

di erent cups of co ee,let us write A B if Olson prefers cup A to

cup B,Let us write A B if Olson either prefers A to B,or can’t tell

the di erence between them,Let us write A B if Olson can’t tell the

di erence between cups A and B,Suppose that Olson is o ered cups A,

B,andC all brewed in the Olsons’ six-cup pot,Cup A was brewed using

14 teaspoons of co ee in the pot,CupB was brewed using 14.75 teaspoons

of co ee in the pot and cup C was brewed using 15.5 teaspoons of co ee

in the pot,For each of the following expressions determine whether it is

true of false.

(a) A B,True.

(b) B A,True.

32 PREFERENCES (Ch,3)

(c) B C,True.

(d) A C,False.

(e) C A,False.

(f) A B,True.

(g) B A,True.

(h) B C,True.

(i) A C,False.

(j) C A,True.

(k) A B,False.

(l) B A,False.

(m) B C,False.

(n) A C,False.

(o) C A,True.

(p) Is Olson’s \at-least-as-good-as" relation,,transitive? No.

(q) Is Olson’s \can’t-tell-the-di erence" relation,,transitive? No.

(r) is Olson’s \better-than" relation,,transitive,Yes.

Chapter 4 NAME

Utility

Introduction,In the previous chapter,you learned about preferences

and indi erence curves,Here we study another way of describing prefer-

ences,the utility function,A utility function that represents a person’s

preferences is a function that assigns a utility number to each commodity

bundle,The numbers are assigned in such a way that commodity bundle

(x;y) gets a higher utility number than bundle (x

0;y

0

) if and only if the

consumer prefers (x;y)to(x

0;y

0

),If a consumer has the utility function

U(x

1;x

2

),then she will be indi erent between two bundles if they are

assigned the same utility.

If you know a consumer’s utility function,then you can nd the

indi erence curve passing through any commodity bundle,Recall from

the previous chapter that when good 1 is graphed on the horizontal axis

and good 2 on the vertical axis,the slope of the indi erence curve passing

through a point (x

1;x

2

)isknownasthemarginal rate of substitution.An

important and convenient fact is that the slope of an indi erence curve is

minus the ratio of the marginal utility of good 1 to the marginal utility of

good 2,For those of you who know even a tiny bit of calculus,calculating

marginal utilities is easy,To nd the marginal utility of either good,

you just take the derivative of utility with respect to the amount of that

good,treating the amount of the other good as a constant,(If you don’t

know any calculus at all,you can calculate an approximation to marginal

utility by the method described in your textbook,Also,at the beginning

of this section of the workbook,we list the marginal utility functions for

commonly encountered utility functions,Even if you can’t compute these

yourself,you can refer to this list when later problems require you to use

marginal utilities.)

Example,Arthur’s utility function is U(x

1;x

2

)=x

1

x

2

,Let us nd the

indi erence curve for Arthur that passes through the point (3;4),First,

calculate U(3;4) = 3 4 = 12,The indi erence curve through this

point consists of all (x

1;x

2

) such that x

1

x

2

= 12,This last equation

is equivalent to x

2

=12=x

1

,Therefore to draw Arthur’s indi erence

curve through (3;4),just draw the curve with equation x

2

=12=x

1

.At

the point (x

1;x

2

),the marginal utility of good 1 is x

2

and the marginal

utility of good 2 is x

1

,Therefore Arthur’s marginal rate of substitution

at the point (3;4) is?x

2

=x

1

=?4=3.

Example,Arthur’s uncle,Basil,has the utility function U

(x

1;x

2

)=

3x

1

x

2

10,Notice that U

(x

1;x

2

)=3U(x

1;x

2

)?10,where U(x

1;x

2

)is

Arthur’s utility function,Since U

is a positive multiple of U minus a con-

stant,it must be that any change in consumption that increases U will also

increase U

(and vice versa),Therefore we say that Basil’s utility function

is a monotonic increasing transformation of Arthur’s utility function,Let

34 UTILITY (Ch,4)

us nd Basil’s indi erence curve through the point (3;4),First we nd

that U

(3;4) = 3 3 4?10 = 26,The indi erence curve passing through

this point consists of all (x

1;x

2

) such that 3x

1

x

2

10 = 26,Simplify this

last expression by adding 10 to both sides of the equation and dividing

both sides by 3,You nd x

1

x

2

= 12,or equivalently,x

2

=12=x

1

.This

is exactly the same curve as Arthur’s indi erence curve through (3;4).

We could have known in advance that this would happen,because if two

consumers’ utility functions are monotonic increasing transformations of

each other,then these consumers must have the same preference relation

between any pair of commodity bundles.

When you have nished this workout,we hope that you will be able

to do the following:

Draw an indi erence curve through a speci ed commodity bundle

when you know the utility function.

Calculate marginal utilities and marginal rates of substitution when

you know the utility function.

Determine whether one utility function is just a \monotonic transfor-

mation" of another and know what that implies about preferences.

Find utility functions that represent preferences when goods are per-

fect substitutes and when goods are perfect complements.

Recognize utility functions for commonly studied preferences such as

perfect substitutes,perfect complements,and other kinked indi er-

ence curves,quasilinear utility,and Cobb-Douglas utility.

4.0 WarmUpExercise,This is the rst of several \warm up ex-

ercises" that you will nd in Workouts,These are here to help you see

how to do calculations that are needed in later problems,The answers to

all warm up exercises are in your answer pages,If you nd the warm up

exercises easy and boring,go ahead|skip them and get on to the main

problems,You can come back and look at them if you get stuck later.

This exercise asks you to calculate marginal utilities and marginal

rates of substitution for some common utility functions,These utility

functions will reappear in several chapters,so it is a good idea to get to

know them now,If you know calculus,you will nd this to be a breeze.

Even if your calculus is shaky or nonexistent,you can handle the rst three

utility functions just by using the de nitions in the textbook,These three

are easy because the utility functions are linear,If you do not know any

calculus,ll in the rest of the answers from the back of the workbook and

keep a copy of this exercise for reference when you encounter these utility

functions in later problems.

NAME 35

u(x

1;x

2

) MU

1

(x

1;x

2

) MU

2

(x

1;x

2

) MRS(x

1;x

2

)

2x

1

+3x

2

2 3?2=3

4x

1

+6x

2

4 6?2=3

ax

1

+bx

2

a b?a=b

2

p

x

1

+x

2

1

p

x

1

1?

1

p

x

1

lnx

1

+x

2

1=x

1

1?1=x

1

v(x

1

)+x

2

v

0

(x

1

) 1?v

0

(x

1

)

x

1

x

2

x

2

x

1

x

2

=x

1

x

a

1

x

b

2

ax

a?1

1

x

b

2

bx

a

1

x

b?1

2

ax

2

bx

1

(x

1

+2)(x

2

+1) x

2

+1 x

1

+2?

x

2

+1

x

1

+2

(x

1

+a)(x

2

+b) x

2

+b x

1

+a?

x

2

+b

x

1

+a

x

a

1

+x

a

2

ax

a?1

1

ax

a?1

2

x

1

x

2

a?1

36 UTILITY (Ch,4)

4.1 (0) Remember Charlie from Chapter 3? Charlie consumes apples and

bananas,We had a look at two of his indi erence curves,In this problem

we give you enough information so you can nd all of Charlie’s indi erence

curves,We do this by telling you that Charlie’s utility function happens

to be U(x

A;x

B

)=x

A

x

B

.

(a) Charlie has 40 apples and 5 bananas,Charlie’s utility for the bun-

dle (40;5) is U(40;5) = 200,The indi erence curve through (40;5)

includes all commodity bundles (x

A;x

B

) such that x

A

x

B

= 200,So

the indi erence curve through (40;5) has the equation x

B

=

200

x

A

,On

the graph below,draw the indi erence curve showing all of the bundles

that Charlie likes exactly as well as the bundle (40;5).

010203040

10

20

30

Apples

Bananas

40

(b) Donna o ers to give Charlie 15 bananas if he will give her 25 apples.

Would Charlie have a bundle that he likes better than (40;5) if he makes

this trade? Yes,What is the largest number of apples that Donna

could demand from Charlie in return for 15 bananas if she expects him to

be willing to trade or at least indi erent about trading? 30,(Hint,If

Donna gives Charlie 15 bananas,he will have a total of 20 bananas,If he

has 20 bananas,how many apples does he need in order to be as well-o

as he would be without trade?)

4.2 (0) Ambrose,whom you met in the last chapter,continues to thrive

on nuts and berries,You saw two of his indi erence curves,One indif-

ference curve had the equation x

2

=20?4

p

x

1

,and another indi erence

curve had the equation x

2

=24?4

p

x

1

,wherex

1

is his consumption of

NAME 37

nuts and x

2

is his consumption of berries,Now it can be told that Am-

brose has quasilinear utility,In fact,his preferences can be represented

by the utility function U(x

1;x

2

)=4

p

x

1

+x

2

.

(a) Ambrose originally consumed 9 units of nuts and 10 units of berries.

His consumption of nuts is reduced to 4 units,but he is given enough

berries so that he is just as well-o as he was before,After the change,

how many units of berries does Ambrose consume? 14.

(b) On the graph below,indicate Ambrose’s original consumption and

sketch an indi erence curve passing through this point,As you can verify,

Ambrose is indi erent between the bundle (9,10) and the bundle (25,2).

If you doubled the amount of each good in each bundle,you would have

bundles (18,20) and (50,4),Are these two bundles on the same indi er-

ence curve? No,(Hint,How do you check whether two bundles are

indi erent when you know the utility function?)

0 5 10 15 20

5

10

15

Nuts

Berries

20

(9,10)

(c) What is Ambrose’s marginal rate of substitution,MRS(x

1;x

2

),when

he is consuming the bundle (9;10)? (Give a numerical answer.)?2=3.

What is Ambrose’s marginal rate of substitution when he is consuming

the bundle (9;20)2=3.

(d) We can write a general expression for Ambrose’s marginal rate of

substitution when he is consuming commodity bundle (x

1;x

2

),This is

MRS(x

1;x

2

)=?2=

p

x

1

,Although we always write MRS(x

1;x

2

)

as a function of the two variables,x

1

and x

2

,we see that Ambrose’s utility

function has the special property that his marginal rate of substitution

does not change when the variable x

2

changes.

38 UTILITY (Ch,4)

4.3 (0) Burt’s utility function is U(x

1;x

2

)=(x

1

+2)(x

2

+ 6),where x

1

is the number of cookies and x

2

is the number of glasses of milk that he

consumes.

(a) What is the slope of Burt’s indi erence curve at the point where he

is consuming the bundle (4;6)2,Use pencil or black ink to draw

a line with this slope through the point (4;6),(Try to make this graph

fairly neat and precise,since details will matter.) The line you just drew

is the tangent line to the consumer’s indi erence curve at the point (4;6).

(b) The indi erence curve through the point (4;6) passes through the

points ( 10,0),(7,2 ),and (2,12 ),Use blue ink

to sketch in this indi erence curve,Incidentally,the equation for Burt’s

indi erence curve through the point (4;6) is x

2

= 72=(x

1

+2)?6.

0481216

4

8

12

Cookies

Glasses of milk

16

a

b

Red Line

Black Line

Blue curve

(c) Burt currently has the bundle (4;6),Ernie o ers to give Burt 9

glasses of milk if Burt will give Ernie 3 cookies,If Burt makes this trade,

he would have the bundle (1;15),Burt refuses to trade,Was this

a wise decision? Yes,U(1;15) = 63 <U(4;6) = 72.

Mark the bundle (1;15) on your graph.

(d) Ernie says to Burt,\Burt,your marginal rate of substitution is?2.

That means that an extra cookie is worth only twice as much to you as

an extra glass of milk,I o ered to give you 3 glasses of milk for every

cookie you give me,If I o er to give you more than your marginal rate

of substitution,then you should want to trade with me." Burt replies,

NAME 39

\Ernie,you are right that my marginal rate of substitution is?2,That

means that I am willing to make small trades where I get more than 2

glasses of milk for every cookie I give you,but 9 glasses of milk for 3

cookies is too big a trade,My indi erence curves are not straight lines,

you see." Would Burt be willing to give up 1 cookie for 3 glasses of milk?

Yes,U(3;9) = 75 >U(4;6) = 72,Would Burt object to

giving up 2 cookies for 6 glasses of milk? No,U(2;12) = 72 =

U(4;6).

(e) On your graph,use red ink to draw a line with slope?3 through the

point (4;6),This line shows all of the bundles that Burt can achieve by

trading cookies for milk (or milk for cookies) at the rate of 1 cookie for

every 3 glasses of milk,Only a segment of this line represents trades that

make Burt better o than he was without trade,Label this line segment

on your graph AB.

4.4 (0) Phil Rupp’s utility function is U(x;y)=maxfx;2yg.

(a) On the graph below,use blue ink to draw and label the line whose

equation is x = 10,Also use blue ink to draw and label the line whose

equation is 2y = 10.

(b) If x =10and2y<10,then U(x;y)= 10,If x<10 and 2y = 10,

then U(x;y)= 10.

(c) Now use red ink to sketch in the indi erence curve along which

U(x;y) = 10,Does Phil have convex preferences? No.

0 5 10 15 20

5

10

15

x

y

20

Red

indifference

curve

Blue

lines

x=10

2y=10

40 UTILITY (Ch,4)

4.5 (0) As you may recall,Nancy Lerner is taking Professor Stern’s

economics course,She will take two examinations in the course,and her

score for the course is the minimum of the scores that she gets on the two

exams,Nancy wants to get the highest possible score for the course.

(a) Write a utility function that represents Nancy’s preferences over al-

ternative combinations of test scores x

1

and x

2

on tests 1 and 2 re-

spectively,U(x

1;x

2

)= minfx

1;x

2

g,or any monotonic

transformation.

4.6 (0) Remember Shirley Sixpack and Lorraine Quiche from the last

chapter? Shirley thinks a 16-ounce can of beer is just as good as two

8-ounce cans,Lorraine only drinks 8 ounces at a time and hates stale

beer,so she thinks a 16-ounce can is no better or worse than an 8-ounce

can.

(a) Write a utility function that represents Shirley’s preferences between

commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer.

Let X stand for the number of 8-ounce cans and Y stand for the number

of 16-ounce cans,u(X;Y)=X +2Y.

(b) Now write a utility function that represents Lorraine’s preferences.

u(X;Y)=X +Y.

(c) Would the function utility U(X;Y) = 100X+200Y represent Shirley’s

preferences? Yes,Would the utility function U(x;y)=(5X +10Y )

2

represent her preferences? Yes,Would the utility function U(x;y)=

X +3Y represent her preferences? No.

(d) Give an example of two commodity bundles such that Shirley likes

the rst bundle better than the second bundle,while Lorraine likes the

second bundle better than the rst bundle,Shirley prefers

(0,2) to (3,0),Lorraine disagrees.

4.7 (0) Harry Mazzola has the utility function u(x

1;x

2

)=minfx

1

+

2x

2;2x

1

+ x

2

g,wherex

1

is his consumption of corn chips and x

2

is his

consumption of french fries.

(a) On the graph below,use a pencil to draw the locus of points along

which x

1

+2x

2

=2x

1

+x

2

,Use blue ink to show the locus of points for

which x

1

+2x

2

= 12,and also use blue ink to draw the locus of points

for which 2x

1

+x

2

= 12.

NAME 41

(b) On the graph you have drawn,shade in the region where both of the

following inequalities are satis ed,x

1

+2x

2

12 and 2x

1

+ x

2

12.

At the bundle (x

1;x

2

)=(8;2),one sees that 2x

1

+ x

2

= 18 and

x

1

+2x

2

= 12,Therefore u(8;2) = 12.

(c) Use black ink to sketch in the indi erence curve along which Harry’s

utility is 12,Use red ink to sketch in the indi erence curve along which

Harry’s utility is 6,(Hint,Is there anything about Harry Mazzola that

reminds you of Mary Granola?)

02468

2

4

6

Corn chips

French fries

8

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

Pencil line

Red

line

Blue

lines

Black line

Blue

lines

(d) At the point where Harry is consuming 5 units of corn chips and 2

units of french fries,how many units of corn chips would he be willing to

trade for one unit of french fries? 2.

4.8 (1) Vanna Boogie likes to have large parties,She also has a strong

preference for having exactly as many men as women at her parties,In

fact,Vanna’s preferences among parties can be represented by the utility

function U(x;y)=minf2x?y;2y?xg where x is the number of women

and y is the number of men at the party,On the graph below,let us try

to draw the indi erence curve along which Vanna’s utility is 10.

(a) Use pencil to draw the locus of points at which x = y.Whatpoint

on this gives Vanna a utility of 10? (10;10),Use blue ink to draw

the line along which 2y?x = 10,When minf2x?y;2y?xg =2y?x,

42 UTILITY (Ch,4)

there are (more men than women,more women than men)? More

women,Draw a squiggly red line over the part of the blue line for which

U(x;y)=minf2x?y;2y?xg=2y?x,This shows all the combinations

that Vanna thinks are just as good as (10;10) but where there are (more

men than women,more women than men)? More women,Now

draw a blue line along which 2x?y = 10,Draw a squiggly red line over

the part of this new blue line for which minf2x?y;2y?xg=2x?y.Use

pencil to shade in the area on the graph that represents all combinations

that Vanna likes at least as well as (10;10).

(b) Suppose that there are 9 men and 10 women at Vanna’s party,Would

Vanna think it was a better party or a worse party if 5 more men came

to her party? Worse.

(c) If Vanna has 16 women at her party and more men than women,and

if she thinks the party is exactly as good as having 10 men and 10 women,

how many men does she have at the party? 22,If Vanna has 16 women

at her party and more women than men,and if she thinks the party is

exactly as good as having 10 men and 10 women,how many men does

she have at her party? 13.

(d) Vanna’s indi erence curves are shaped like what letter of the alpha-

bet? V.

0 5 10 15 20

5

10

15

x

y

20

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

Pencil

line

Blue

lines

Squiggly

red

lines

4.9 (0) Suppose that the utility functions u(x;y)andv(x;y) are related

by v(x;y)=f(u(x;y)),In each case below,write \Yes" if the function

f is a positive monotonic transformation and \No" if it is not,(Hint for

NAME 43

calculus users,A di erentiable function f(u) is an increasing function of

u if its derivative is positive.)

(a) f(u)=3:141592u,Yes.

(b) f(u)=5;000?23u,No.

(c) f(u)=u?100;000,Yes.

(d) f(u)=log

10

u,Yes.

(e) f(u)=?e

u

,Yes.

(f) f(u)=1=u,No.

(g) f(u)=?1=u,Yes.

4.10 (0) Martha Modest has preferences represented by the utility func-

tion U(a;b)=ab=100,where a is the number of ounces of animal crackers

that she consumes and b is the number of ounces of beans that she con-

sumes.

(a) On the graph below,sketch the locus of points that Martha nds

indi erent to having 8 ounces of animal crackers and 2 ounces of beans.

Also sketch the locus of points that she nds indi erent to having 6 ounces

of animal crackers and 4 ounces of beans.

02468

2

4

6

Animal crackers

Beans

8

(8,2)

(6,4)

44 UTILITY (Ch,4)

(b) Bertha Brassy has preferences represented by the utility function

V(a;b)=1;000a

2

b

2

,wherea is the number of ounces of animal crack-

ers that she consumes and b is the number of ounces of beans that she

consumes,On the graph below,sketch the locus of points that Bertha

nds indi erent to having 8 ounces of animal crackers and 2 ounces of

beans,Also sketch the locus of points that she nds indi erent to having

6 ounces of animal crackers and 4 ounces of beans.

02468

2

4

6

Animal crackers

Beans

8

(8,2)

(6,4)

(c) Are Martha’s preferences convex? Yes,Are Bertha’s? Yes.

(d) What can you say about the di erence between the indi erence curves

you drew for Bertha and those you drew for Martha? There is no

difference.

(e) How could you tell this was going to happen without having to draw

the curves? Their utility functions only differ

by a monotonic transformation.

4.11 (0) Willy Wheeler’s preferences over bundles that contain non-

negative amounts of x

1

and x

2

are represented by the utility function

U(x

1;x

2

)=x

2

1

+x

2

2

.

(a) Draw a few of his indi erence curves,What kind of geometric g-

ure are they? Quarter circles centered at the

origin,Does Willy have convex preferences? No.

NAME 45

02468

2

4

6

x1

x2

8

Calculus 4.12 (0) Joe Bob has a utility function given by u(x

1;x

2

)=x

2

1

+2x

1

x

2

+

x

2

2

.

(a) Compute Joe Bob’s marginal rate of substitution,MRS(x

1;x

2

)=

1.

(b) Joe Bob’s straight cousin,Al,has a utility function v(x

1;x

2

)=x

2

+x

1

.

Compute Al’s marginal rate of substitution,MRS(x

1;x

2

)=?1.

(c) Do u(x

1;x

2

)andv(x

1;x

2

) represent the same preferences? Yes.

Can you show that Joe Bob’s utility function is a monotonic transforma-

tion of Al’s? (Hint,Some have said that Joe Bob is square.) Notice

that u(x

1;x

2

)=[v(x

1;x

2

)]

2

.

4.13 (0) The idea of assigning numerical values to determine a preference

ordering over a set of objects is not limited in application to commodity

bundles,The Bill James Baseball Abstract argues that a baseball player’s

batting average is not an adequate measure of his o ensive productivity.

Batting averages treat singles just the same as extra base hits,Further-

more they do not give credit for \walks," although a walk is almost as

good as a single,James argues that a double in two at-bats is better than

a single,but not as good as two singles,To reflect these considerations,

James proposes the following index,which he calls \runs created." Let A

be the number of hits plus the number of walks that a batter gets in a sea-

son,Let B be the number of total bases that the batter gets in the season.

(Thus,if a batter has S singles,W walks,D doubles,T triples,and H

46 UTILITY (Ch,4)

home runs,then A = S+D+T+H+W and B = S+W+2D+3T+4H.)

Let N be the number of times the batter bats,Then his index of runs

created in the season is de ned to be AB=N and will be called his RC.

(a) In 1987,George Bell batted 649 times,He had 39 walks,105 singles,

32 doubles,4 triples,and 47 home runs,In 1987,Wade Boggs batted 656

times,He had 105 walks,130 singles,40 doubles,6 triples,and 24 home

runs,In 1987,Alan Trammell batted 657 times,He had 60 walks,140

singles,34 doubles,3 triples,and 28 home runs,In 1987,Tony Gwynn

batted 671 times,He had 82 walks,162 singles,36 doubles,13 triples,and

7 home runs,We can calculate A,the number of hits plus walks,B the

number of total bases,and RC,the runs created index for each of these

players,For Bell,A = 227,B = 408,RC = 143,For Boggs,A = 305,

B = 429,RC = 199,For Trammell,A = 265,B = 389,RC = 157,For

Gwynn,A = 300,B = 383,RC = 171.

(b) If somebody has a preference ordering among these players,based only

on the runs-created index,which player(s) would she prefer to Trammell?

Boggs and Gwynn.

(c) The di erences in the number of times at bat for these players are

small,and we will ignore them for simplicity of calculation,On the graph

below,plot the combinations of A and B achieved by each of the players.

Draw four \indi erence curves," one through each of the four points you

have plotted,These indi erence curves should represent combinations of

A and B that lead to the same number of runs-created.

0 120 180 240 300 360

Number of hits plus walks

80

160

240

320

400

Number of total bases

480

60

Bell

Trammell

Gwynn

Boggs

NAME 47

4.14 (0) This problem concerns the runs-created index discussed in the

preceding problem,Consider a batter who bats 100 times and always

either makes an out,hits for a single,or hits a home run.

(a) Let x be the number of singles and y be the number of home runs

in 100 at-bats,Suppose that the utility function U(x;y)bywhichwe

evaluate alternative combinations of singles and home runs is the runs-

created index,Then the formula for the utility function is U(x;y)=

(x+y)(x+4y)=100.

(b) Let’s try to nd out about the shape of an indi erence curve between

singles and home runs,Hitting 10 home runs and no singles would give

him the same runs-created index as hitting 20 singles and no

home runs,Mark the points (0;10) and (x;0),where U(x;0) = U(0;10).

(c) Where x is the number of singles you solved for in the previous part,

mark the point (x=2;5) on your graph,Is U(x=2;5) greater than or less

than or equal to U(0;10)? Greater than,Is this consistent with

the batter having convex preferences between singles and home runs?

Yes.

0 5 10 15 20

5

10

15

Singles

Home runs

20

(0,10)

(20,0)

(10,5)

Preference

direction

48 UTILITY (Ch,4)

Chapter 5 NAME

Choice

Introduction,You have studied budgets,and you have studied prefer-

ences,Now is the time to put these two ideas together and do something

with them,In this chapter you study the commodity bundle chosen by a

utility-maximizing consumer from a given budget.

Given prices and income,you know how to graph a consumer’s bud-

get,If you also know the consumer’s preferences,you can graph some of

his indi erence curves,The consumer will choose the \best" indi erence

curve that he can reach given his budget,But when you try to do this,you

have to ask yourself,\How do I nd the most desirable indi erence curve

that the consumer can reach?" The answer to this question is \look in the

likely places." Where are the likely places? As your textbook tells you,

there are three kinds of likely places,These are,(i) a tangency between

an indi erence curve and the budget line; (ii) a kink in an indi erence

curve; (iii) a \corner" where the consumer specializes in consuming just

one good.

Here is how you nd a point of tangency if we are told the consumer’s

utility function,the prices of both goods,and the consumer’s income,The

budget line and an indi erence curve are tangent at a point (x

1;x

2

)ifthey

have the same slope at that point,Now the slope of an indi erence curve

at (x

1;x

2

)istheratio?MU

1

(x

1;x

2

)=MU

2

(x

1;x

2

),(This slope is also

known as the marginal rate of substitution.) The slope of the budget line

is?p

1

=p

2

,Therefore an indi erence curve is tangent to the budget line

at the point (x

1;x

2

)whenMU

1

(x

1;x

2

)=MU

2

(x

1;x

2

)=p

1

=p

2

.Thisgives

us one equation in the two unknowns,x

1

and x

2

,If we hope to solve

for the x’s,we need another equation,That other equation is the budget

equation p

1

x

1

+ p

2

x

2

= m,With these two equations you can solve for

(x

1;x

2

).

Example,A consumer has the utility function U(x

1;x

2

)=x

2

1

x

2

.The

price of good 1 is p

1

= 1,the price of good 2 is p

2

= 3,and his income

is 180,Then,MU

1

(x

1;x

2

)=2x

1

x

2

and MU

2

(x

1;x

2

)=x

2

1

.There-

fore his marginal rate of substitution is?MU

1

(x

1;x

2

)=MU

2

(x

1;x

2

)=

2x

1

x

2

=x

2

1

=?2x

2

=x

1

,This implies that his indi erence curve will be

tangent to his budget line when?2x

2

=x

1

=?p

1

=p

2

=?1=3,Simplifying

this expression,we have 6x

2

= x

1

,This is one of the two equations we

need to solve for the two unknowns,x

1

and x

2

,The other equation is

the budget equation,In this case the budget equation is x

1

+3x

2

= 180.

Solving these two equations in two unknowns,we nd x

1

= 120 and

Some people have trouble remembering whether the marginal rate

of substitution is?MU

1

=MU

2

or?MU

2

=MU

1

,It isn’t really crucial to

remember which way this goes as long as you remember that a tangency

happens when the marginal utilities of any two goods are in the same

proportion as their prices.

50 CHOICE (Ch,5)

x

2

= 20,Therefore we know that the consumer chooses the bundle

(x

1;x

2

) = (120;20).

For equilibrium at kinks or at corners,we don’t need the slope of

the indi erence curves to equal the slope of the budget line,So we don’t

have the tangency equation to work with,But we still have the budget

equation,The second equation that you can use is an equation that tells

you that you are at one of the kinky points or at a corner,You will see

exactly how this works when you work a few exercises.

Example,A consumer has the utility function U(x

1;x

2

)=minfx

1;3x

2

g.

The price of x

1

is 2,the price of x

2

is 1,and her income is 140,Her

indi erence curves are L-shaped,The corners of the L’s all lie along the

line x

1

=3x

2

,She will choose a combination at one of the corners,so this

gives us one of the two equations we need for nding the unknowns x

1

and

x

2

,The second equation is her budget equation,which is 2x

1

+x

2

= 140.

Solve these two equations to nd that x

1

=60andx

2

= 20,So we know

that the consumer chooses the bundle (x

1;x

2

)=(60;20).

When you have nished these exercises,we hope that you will be

able to do the following:

Calculate the best bundle a consumer can a ord at given prices and

income in the case of simple utility functions where the best a ord-

able bundle happens at a point of tangency.

Find the best a ordable bundle,given prices and income for a con-

sumer with kinked indi erence curves.

Recognize standard examples where the best bundle a consumer can

a ord happens at a corner of the budget set.

Draw a diagram illustrating each of the above types of equilibrium.

Apply the methods you have learned to choices made with some kinds

of nonlinear budgets that arise in real-world situations.

5.1 (0) We begin again with Charlie of the apples and bananas,Recall

that Charlie’s utility function is U(x

A;x

B

)=x

A

x

B

,Suppose that the

price of apples is 1,the price of bananas is 2,and Charlie’s income is 40.

(a) On the graph below,use blue ink to draw Charlie’s budget line,(Use

a ruler and try to make this line accurate.) Plot a few points on the

indi erence curve that gives Charlie a utility of 150 and sketch this curve

with red ink,Now plot a few points on the indi erence curve that gives

Charlie a utility of 300 and sketch this curve with black ink or pencil.

NAME 51

010203040

10

20

30

Apples

Bananas

40

a

e

Blue

budget

line

Red

curves

Black curve

Pencil line

(b) Can Charlie a ord any bundles that give him a utility of 150? Yes.

(c) Can Charlie a ord any bundles that give him a utility of 300? No.

(d) On your graph,mark a point that Charlie can a ord and that gives

him a higher utility than 150,Label that point A.

(e) Neither of the indi erence curves that you drew is tangent to Charlie’s

budget line,Let’s try to nd one that is,At any point,(x

A;x

B

),Charlie’s

marginal rate of substitution is a function of x

A

and x

B

,In fact,if you

calculate the ratio of marginal utilities for Charlie’s utility function,you

will nd that Charlie’s marginal rate of substitution is MRS(x

A;x

B

)=

x

B

=x

A

,This is the slope of his indi erence curve at (x

A;x

B

),The

slope of Charlie’s budget line is?1=2 (give a numerical answer).

(f) Write an equation that implies that the budget line is tangent to an

indi erence curve at (x

A;x

B

),?x

B

=x

A

=?1=2,There are

many solutions to this equation,Each of these solutions corresponds to

a point on a di erent indi erence curve,Use pencil to draw a line that

passes through all of these points.

52 CHOICE (Ch,5)

(g) The best bundle that Charlie can a ord must lie somewhere on the

line you just penciled in,It must also lie on his budget line,If the point

is outside of his budget line,he can’t a ord it,If the point lies inside

of his budget line,he can a ord to do better by buying more of both

goods,On your graph,label this best a ordable bundle with an E.This

happens where x

A

= 20 and x

B

= 10,Verify your answer by

solving the two simultaneous equations given by his budget equation and

the tangency condition.

(h) What is Charlie’s utility if he consumes the bundle (20;10)? 200.

(i) On the graph above,use red ink to draw his indi erence curve through

(20,10),Does this indi erence curve cross Charlie’s budget line,just touch

it,or never touch it? Just touch it.

5.2 (0) Clara’s utility function is U(X;Y)=(X +2)(Y + 1),where X

is her consumption of good X and Y is her consumption of good Y.

(a) Write an equation for Clara’s indi erence curve that goes through the

point (X;Y)=(2;8),Y =

36

X+2

1,On the axes below,sketch

Clara’s indi erence curve for U = 36.

0481216

4

8

12

Y

16

11

11

U=36

X

(b) Suppose that the price of each good is 1 and that Clara has an income

of 11,Draw in her budget line,Can Clara achieve a utility of 36 with

this budget? Yes.

NAME 53

(c) At the commodity bundle,(X;Y),Clara’s marginal rate of substitu-

tion is?

Y+1

X+2

:

(d) If we set the absolute value of the MRS equal to the price ratio,we

have the equation

Y+1

X+2

=1:

(e) The budget equation is X +Y =11.

(f) Solving these two equations for the two unknowns,X and Y,we nd

X = 5 and Y = 6.

5.3 (0) Ambrose,the nut and berry consumer,has a utility function

U(x

1;x

2

)=4

p

x

1

+x

2

,wherex

1

is his consumption of nuts and x

2

is his

consumption of berries.

(a) The commodity bundle (25;0) gives Ambrose a utility of 20,Other

points that give him the same utility are (16;4),(9,8 ),(4,

12 ),(1,16 ),and (0,20 ),Plot these points on

the axes below and draw a red indi erence curve through them.

(b) Suppose that the price of a unit of nuts is 1,the price of a unit of

berries is 2,and Ambrose’s income is 24,Draw Ambrose’s budget line

with blue ink,How many units of nuts does he choose to buy? 16

units.

(c) How many units of berries? 4 units.

(d) Find some points on the indi erence curve that gives him a utility of

25 and sketch this indi erence curve (in red).

(e) Now suppose that the prices are as before,but Ambrose’s income is

34,Draw his new budget line (with pencil),How many units of nuts will

he choose? 16 units,How many units of berries? 9 units.

54 CHOICE (Ch,5)

010152025

Nuts

5

10

15

20

Berries

5 30

Red

curve

Blue line

Red curve

Pencil line

Blue

line

(f) Now let us explore a case where there is a \boundary solution." Sup-

pose that the price of nuts is still 1 and the price of berries is 2,but

Ambrose’s income is only 9,Draw his budget line (in blue),Sketch the

indi erence curve that passes through the point (9;0),What is the slope

of his indi erence curve at the point (9;0)2=3.

(g) What is the slope of his budget line at this point1=2.

(h) Which is steeper at this point,the budget line or the indi erence

curve? Indifference curve.

(i) Can Ambrose a ord any bundles that he likes better than the point

(9;0)? No.

5.4 (1) Nancy Lerner is trying to decide how to allocate her time in

studying for her economics course,There are two examinations in this

course,Her overall score for the course will be the minimum of her scores

on the two examinations,She has decided to devote a total of 1,200

minutes to studying for these two exams,and she wants to get as high an

overall score as possible,She knows that on the rst examination if she

doesn’t study at all,she will get a score of zero on it,For every 10 minutes

that she spends studying for the rst examination,she will increase her

score by one point,If she doesn’t study at all for the second examination

she will get a zero on it,For every 20 minutes she spends studying for

the second examination,she will increase her score by one point.

NAME 55

(a) On the graph below,draw a \budget line" showing the various com-

binations of scores on the two exams that she can achieve with a total of

1,200 minutes of studying,On the same graph,draw two or three \indif-

ference curves" for Nancy,On your graph,draw a straight line that goes

through the kinks in Nancy’s indi erence curves,Label the point where

this line hits Nancy’s budget with the letter A,Draw Nancy’s indi erence

curve through this point.

0 40 60 80 100

Score on test 1

20

40

60

80

Score on test 2

20 120

a

Budget line

"L" shaped

indifference

curves

(b) Write an equation for the line passing through the kinks of Nancy’s

indi erence curves,x

1

= x

2

.

(c) Write an equation for Nancy’s budget line,10x

1

+20x

2

=

1;200.

(d) Solve these two equations in two unknowns to determine the intersec-

tion of these lines,This happens at the point (x

1;x

2

)= (40;40).

(e) Given that she spends a total of 1,200 minutes studying,Nancy will

maximize her overall score by spending 400 minutes studying for the

rst examination and 800 minutes studying for the second examina-

tion.

5.5 (1) In her communications course,Nancy also takes two examina-

tions,Her overall grade for the course will be the maximum of her scores

on the two examinations,Nancy decides to spend a total of 400 minutes

studying for these two examinations,If she spends m

1

minutes studying

56 CHOICE (Ch,5)

for the rst examination,her score on this exam will be x

1

= m

1

=5,If

she spends m

2

minutes studying for the second examination,her score on

this exam will be x

2

= m

2

=10.

(a) On the graph below,draw a \budget line" showing the various combi-

nations of scores on the two exams that she can achieve with a total of 400

minutes of studying,On the same graph,draw two or three \indi erence

curves" for Nancy,On your graph,nd the point on Nancy’s budget line

that gives her the best overall score in the course.

(b) Given that she spends a total of 400 minutes studying,Nancy will

maximize her overall score by achieving a score of 80 on the rst

examination and 0 on the second examination.

(c) Her overall score for the course will then be 80.

020406080

20

40

60

Score on test 1

Score on test 2

80

,

Preference

direction

Max,

overall

score

5.6 (0) Elmer’s utility function is U(x;y)=minfx;y

2

g.

(a) If Elmer consumes 4 units of x and 3 units of y,his utility is 4.

(b) If Elmer consumes 4 units of x and 2 units of y,his utility is 4.

(c) If Elmer consumes 5 units of x and 2 units of y,his utility is 4.

(d) On the graph below,use blue ink to draw the indi erence curve for

Elmer that contains the bundles that he likes exactly as well as the bundle

(4;2).

NAME 57

(e) On the same graph,use blue ink to draw the indi erence curve for

Elmer that contains bundles that he likes exactly as well as the bundle

(1;1) and the indi erence curve that passes through the point (16;5).

(f) On your graph,use black ink to show the locus of points at which

Elmer’s indi erence curves have kinks,What is the equation for this

curve? x = y

2

.

(g) On the same graph,use black ink to draw Elmer’s budget line when

the price of x is 1,the price of y is 2,and his income is 8,What bundle

does Elmer choose in this situation? (4,2).

0 8 12 16 20

x

4

8

12

16

y

4 24

Black

line

(16,5)

Blue

curves

Blue curve

Black curve

Chosen

bundle

(h) Suppose that the price of x is 10 and the price of y is 15 and Elmer

buys 100 units of x,What is Elmer’s income? 1,150,(Hint,At rst

you might think there is too little information to answer this question.

But think about how much y he must be demanding if he chooses 100

units of x.)

5.7 (0) Linus has the utility function U(x;y)=x+3y.

(a) On the graph below,use blue ink to draw the indi erence curve passing

through the point (x;y)=(3;3),Use black ink to sketch the indi erence

curve connecting bundles that give Linus a utility of 6.

58 CHOICE (Ch,5)

0481216

4

8

12

Y

16

X

(3,3)

Blue

curve

Black

curve

Red line

(b) On the same graph,use red ink to draw Linus’s budget line if the

price of x is 1 and the price of y is 2 and his income is 8,What bundle

does Linus choose in this situation? (0,4).

(c) What bundle would Linus choose if the price of x is 1,the price of y

is 4,and his income is 8? (8,0).

5.8 (2) Remember our friend Ralph Rigid from Chapter 3? His favorite

diner,Food for Thought,has adopted the following policy to reduce the

crowds at lunch time,if you show up for lunch t hours before or after

12 noon,you get to deduct t dollars from your bill,(This holds for any

fraction of an hour as well.)

NAME 59

0

11 12 1 2

5

10

15

Time

Money

20

10

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

Red curves

Blue budget set

(a) Use blue ink to show Ralph’s budget set,On this graph,the horizontal

axis measures the time of day that he eats lunch,and the vertical axis

measures the amount of money that he will have to spend on things other

than lunch,Assume that he has $20 total to spend and that lunch at

noon costs $10,(Hint,How much money would he have left if he ate at

noon?at1P.M.?at11A.M.?)

(b) Recall that Ralph’s preferred lunch time is 12 noon,but that he is

willing to eat at another time if the food is su ciently cheap,Draw

some red indi erence curves for Ralph that would be consistent with his

choosing to eat at 11 A.M.

5.9 (0) Joe Grad has just arrived at the big U,He has a fellowship that

covers his tuition and the rent on an apartment,In order to get by,Joe

has become a grader in intermediate price theory,earning $100 a month.

Out of this $100 he must pay for his food and utilities in his apartment.

His utilities expenses consist of heating costs when he heats his apartment

and air-conditioning costs when he cools it,To raise the temperature of

his apartment by one degree,it costs $2 per month (or $20 per month

to raise it ten degrees),To use air-conditioning to cool his apartment by

a degree,it costs $3 per month,Whatever is left over after paying the

utilities,he uses to buy food at $1 per unit.

60 CHOICE (Ch,5)

02030405060

Temperature

20

40

60

80

100

Food

120

10 70 80 90 100

December September August

Black budget constraint

Blue budget constraint

Red budget constraint

(a) When Joe rst arrives in September,the temperature of his apartment

is 60 degrees,If he spends nothing on heating or cooling,the temperature

in his room will be 60 degrees and he will have $100 left to spend on food.

If he heated the room to 70 degrees,he would have $80 left to spend

on food,If he cooled the room to 50 degrees,he would have $70 left

to spend on food,On the graph below,show Joe’s September budget

constraint (with black ink),(Hint,You have just found three points that

Joe can a ord,Apparently,his budget set is not bounded by a single

straight line.)

(b) In December,the outside temperature is 30 degrees and in August

poor Joe is trying to understand macroeconomics while the temperature

outside is 85 degrees,On the same graph you used above,draw Joe’s

budget constraints for the months of December (in blue ink) and August

(in red ink).

(c) Draw a few smooth (unkinky) indi erence curves for Joe in such a way

that the following are true,(i) His favorite temperature for his apartment

would be 65 degrees if it cost him nothing to heat it or cool it,(ii)Joe

chooses to use the furnace in December,air-conditioning in August,and

neither in September,(iii) Joe is better o in December than in August.

(d) In what months is the slope of Joe’s budget constraint equal to the

slope of his indi erence curve? August and December.

NAME 61

(e) In December Joe’s marginal rate of substitution between food and

degrees Fahrenheit is -2,In August,his MRS is 3.

(f) Since Joe neither heats nor cools his apartment in September,we

cannot determine his marginal rate of substitution exactly,but we do

know that it must be no smaller than -2 and no larger than

3,(Hint,Look carefully at your graph.)

5.10 (0) Central High School has $60,000 to spend on computers and

other stu,so its budget equation is C + X =60;000,where C is ex-

penditure on computers and X is expenditures on other things,C.H.S.

currently plans to spend $20,000 on computers.

The State Education Commission wants to encourage \computer lit-

eracy" in the high schools under its jurisdiction,The following plans have

been proposed.

Plan A,This plan would give a grant of $10,000 to each high school in

the state that the school could spend as it wished.

Plan B,This plan would give a $10,000 grant to any high school,so

long as the school spent at least $10,000 more than it currently spends on

computers,Any high school can choose not to participate,in which case it

does not receive the grant,but it doesn’t have to increase its expenditure

on computers.

Plan C,Plan C is a \matching grant." For every dollar’s worth of

computers that a high school orders,the state will give the school 50

cents.

Plan D,This plan is like plan C,except that the maximum amount of

matching funds that any high school could get from the state would be

limited to $10,000.

(a) Write an equation for Central High School’s budget if plan A is

adopted,C + X =70;000,Use black ink to draw the bud-

get line for Central High School if plan A is adopted.

(b) If plan B is adopted,the boundary of Central High School’s budget set

has two separate downward-sloping line segments,One of these segments

describes the cases where C.H.S,spends at least $30,000 on computers.

This line segment runs from the point (C;X)=(70;000;0) to the point

(C;X)= (30,000,40,000).

(c) Another line segment corresponds to the cases where C.H.S,spends

less than $30,000 on computers,This line segment runs from (C;X)=

(30,000,30,000) to the point (C;X)=(0;60;000),Use red

ink to draw these two line segments.

62 CHOICE (Ch,5)

(d) If plan C is adopted and Central High School spendsC dollars on com-

puters,then it will have X =60;000?:5C dollars left to spend on other

things,Therefore its budget line has the equation,5C+X=60,000.

Use blue ink to draw this budget line.

(e) If plan D is adopted,the school district’s budget consists of two

line segments that intersect at the point where expenditure on comput-

ers is 20,000 and expenditure on other instructional materials is

50,000.

(f) The slope of the flatter line segment is?:5,The slope of the

steeper segment is?1,Use pencil to draw this budget line.

0 20 30 40 50 60

Thousands of dollars worth of computers

10

20

30

40

50

Thousands of dollars worth of other things

60

10

Red budget line

(plan B)

Black budget line (plan A)

Pencil budget line

(plan D)

Blue

budget

line

(plan C)

5.11 (0) Suppose that Central High School has preferences that can

be represented by the utility function U(C;X)=CX

2

,Let us try to

determine how the various plans described in the last problem will a ect

the amount that C.H.S,spends on computers.

NAME 63

(a) If the state adopts none of the new plans,nd the expenditure on

computers that maximizes the district’s utility subject to its budget con-

straint,20,000.

(b) If plan A is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,23,333.

(c) On your graph,sketch the indi erence curve that passes through the

point (30,000,40,000) if plan B is adopted,At this point,which is steeper,

the indi erence curve or the budget line? The budget line.

(d) If plan B is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,(Hint,Look

at your graph.) 30,000.

(e) If plan C is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,40,000.

(f) If plan D is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,23,333.

5.12 (0) The telephone company allows one to choose between two

di erent pricing plans,For a fee of $12 per month you can make as

many local phone calls as you want,at no additional charge per call.

Alternatively,you can pay $8 per month and be charged 5 cents for each

local phone call that you make,Suppose that you have a total of $20 per

month to spend.

(a) On the graph below,use black ink to sketch a budget line for someone

who chooses the rst plan,Use red ink to draw a budget line for someone

who chooses the second plan,Where do the two budget lines cross?

(80;8).

64 CHOICE (Ch,5)

0 40 60 80 100

Local phone calls

4

8

12

16

Other goods

20 120

Black line

Red line

Pencil curve

Blue curve

(b) On the graph above,use pencil to draw indi erence curves for some-

one who prefers the second plan to the rst,Use blue ink to draw an

indi erence curve for someone who prefers the rst plan to the second.

5.13 (1) This is a puzzle|just for fun,Lewis Carroll (1832-1898),

author of Alice in Wonderland and Through the Looking Glass,was a

mathematician,logician,and political scientist,Carroll loved careful rea-

soning about puzzling things,Here Carroll’s Alice presents a nice bit

of economic analysis,At rst glance,it may seem that Alice is talking

nonsense,but,indeed,her reasoning is impeccable.

\I should like to buy an egg,please." she said timidly,\How do you

sell them?"

\Fivepence farthing for one|twopence for two," the Sheep replied.

\Then two are cheaper than one?" Alice said,taking out her purse.

\Only you must eat them both if you buy two," said the Sheep.

\Then I’ll have one please," said Alice,as she put the money down

on the counter,For she thought to herself,\They mightn’t be at all nice,

you know."

(a) Let us try to draw a budget set and indi erence curves that are

consistent with this story,Suppose that Alice has a total of 8 pence to

spend and that she can buy either 0,1,or 2 eggs from the Sheep,but no

fractional eggs,Then her budget set consists of just three points,The

point where she buys no eggs is (0;8),Plot this point and label it A.On

your graph,the point where she buys 1 egg is (1;2

3

4

),(A farthing is 1/4

of a penny.) Plot this point and label it B.

(b) The point where she buys 2 eggs is (2;6),Plot this point and

label it C,If Alice chooses to buy 1 egg,she must like the bundleB better

than either the bundle A or the bundle C,Draw indi erence curves for

Alice that are consistent with this behavior.

NAME 65

01234

2

4

6

Eggs

Other goods

8

b

a

c

66 CHOICE (Ch,5)

Chapter 6 NAME

Demand

Introduction,In the previous chapter,you found the commodity bundle

that a consumer with a given utility function would choose in a speci c

price-income situation,In this chapter,we take this idea a step further.

We nd demand functions,which tell us for any prices and income you

might want to name,how much of each good a consumer would want,In

general,the amount of each good demanded may depend not only on its

own price,but also on the price of other goods and on income,Where

there are two goods,we write demand functions for Goods 1 and 2 as

x

1

(p

1;p

2;m)andx

2

(p

1;p

2;m).

When the consumer is choosing positive amounts of all commodities

and indi erence curves have no kinks,the consumer chooses a point of

tangency between her budget line and the highest indi erence curve that

it touches.

Example,Consider a consumer with utility function U(x

1;x

2

)=(x

1

+

2)(x

2

+ 10),To nd x

1

(p

1;p

2;m)andx

2

(p

1;p

2;m),we need to nd a

commodity bundle (x

1;x

2

) on her budget line at which her indi erence

curve is tangent to her budget line,The budget line will be tangent to

the indi erence curve at (x

1;x

2

) if the price ratio equals the marginal

rate of substitution,For this utility function,MU

1

(x

1;x

2

)=x

2

+10 and

MU

2

(x

1;x

2

)=x

1

+ 2,Therefore the \tangency equation" is p

1

=p

2

=

(x

2

+ 10)=(x

1

+ 2),Cross-multiplying the tangency equation,one nds

p

1

x

1

+2p

1

= p

2

x

2

+10p

2

.

The bundle chosen must also satisfy the budget equation,p

1

x

1

+

p

2

x

2

= m,This gives us two linear equations in the two unknowns,x

1

and x

2

,You can solve these equations yourself,using high school algebra.

You will nd that the solution for the two \demand functions" is

x

1

=

m?2p

1

+10p

2

2p

1

x

2

=

m+2p

1

10p

2

2p

2

:

There is one thing left to worry about with the \demand functions" we

just found,Notice that these expressions will be positive only if m?2p

1

+

10p

2

> 0andm+2p

1

10p

2

> 0,If either of these expressions is negative,

then it doesn’t make sense as a demand function,What happens in this

For some utility functions,demand for a good may not be a ected by

all of these variables,For example,with Cobb-Douglas utility,demand

for a good depends on the good’s own price and on income but not on the

other good’s price,Still,there is no harm in writing demand for Good

1 as a function of p

1

,p

2

,andm,It just happens that the derivative of

x

1

(p

1;p

2;m) with respect to p

2

is zero.

68 DEMAND (Ch,6)

case is that the consumer will choose a \boundary solution" where she

consumes only one good,At this point,her indi erence curve will not be

tangent to her budget line.

When a consumer has kinks in her indi erence curves,she may choose

a bundle that is located at a kink,In the problems with kinks,you

will be able to solve for the demand functions quite easily by looking

at diagrams and doing a little algebra,Typically,instead of nding a

tangency equation,you will nd an equation that tells you \where the

kinks are." With this equation and the budget equation,you can then

solve for demand.

You might wonder why we pay so much attention to kinky indi er-

ence curves,straight line indi erence curves,and other \funny cases."

Our reason is this,In the funny cases,computations are usually pretty

easy,But often you may have to draw a graph and think about what

you are doing,That is what we want you to do,Think and ddle with

graphs,Don’t just memorize formulas,Formulas you will forget,but the

habit of thinking will stick with you.

When you have nished this workout,we hope that you will be able

to do the following:

Find demand functions for consumers with Cobb-Douglas and other

similar utility functions.

Find demand functions for consumers with quasilinear utility func-

tions.

Find demand functions for consumers with kinked indi erence curves

and for consumers with straight-line indi erence curves.

Recognize complements and substitutes from looking at a demand

curve.

Recognize normal goods,inferior goods,luxuries,and necessities from

looking at information about demand.

Calculate the equation of an inverse demand curve,given a simple

demand equation.

6.1 (0) Charlie is back|still consuming apples and bananas,His util-

ity function is U(x

A;x

B

)=x

A

x

B

,We want to nd his demand func-

tion for apples,x

A

(p

A;p

B;m),and his demand function for bananas,

x

B

(p

A;p

B;m).

(a) When the prices arep

A

andp

B

and Charlie’s income ism,the equation

for Charlie’s budget line isp

A

x

A

+p

B

x

B

= m,The slope of Charlie’s indif-

ference curve at the bundle (x

A;x

B

)is?MU

1

(x

A;x

B

)=MU

2

(x

A;x

B

)=

x

B

=x

A

,The slope of Charlie’s budget line is?p

A

=p

B

,Char-

lie’s indi erence curve will be tangent to his budget line at the point

(x

A;x

B

) if the following equation is satis ed,p

A

=p

B

= x

B

=x

A

.

NAME 69

(b) You now have two equations,the budget equation and the tan-

gency equation,that must be satis ed by the bundle demanded,Solve

these two equations for x

A

and x

B

,Charlie’s demand function for ap-

ples is x

A

(p

A;p

B;m)=

m

2p

A

,and his demand function for bananas is

x

B

(p

A;p

B;m)=

m

2p

B

.

(c) In general,the demand for both commodities will depend on the price

of both commodities and on income,But for Charlie’s utility function,

the demand function for apples depends only on income and the price

of apples,Similarly,the demand for bananas depends only on income

and the price of bananas,Charlie always spends the same fraction of his

income on bananas,What fraction is this? 1=2.

6.2 (0) Douglas Corn eld’s preferences are represented by the utility

function u(x

1;x

2

)=x

2

1

x

3

2

,The prices of x

1

and x

2

are p

1

and p

2

.

(a) The slope of Corn eld’s indi erence curve at the point (x

1;x

2

)is

2x

2

=3x

1

.

(b) If Corn eld’s budget line is tangent to his indi erence curve at (x

1;x

2

),

then

p

1

x

1

p

2

x

2

= 2/3,(Hint,Look at the equation that equates the slope

of his indi erence curve with the slope of his budget line.) When he is

consuming the best bundle he can a ord,what fraction of his income does

Douglas spend on x

1

2/5.

(c) Other members of Doug’s family have similar utility functions,but

the exponents may be di erent,or their utilities may be multiplied by a

positive constant,If a family member has a utility function U(x;y)=

cx

a

1

x

b

2

where a,b,andc are positive numbers,what fraction of his or her

income will that family member spend on x

1

a/(a+b).

6.3 (0) Our thoughts return to Ambrose and his nuts and berries,Am-

brose’s utility function is U(x

1;x

2

)=4

p

x

1

+ x

2

,wherex

1

is his con-

sumption of nuts and x

2

is his consumption of berries.

(a) Let us nd his demand function for nuts,The slope of Ambrose’s

indi erence curve at (x

1;x

2

)is?

2

p

x

1

,Setting this slope equal to

the slope of the budget line,you can solve for x

1

without even using the

budget equation,The solution is x

1

=

2p

2

p

1

2

.

70 DEMAND (Ch,6)

(b) Let us nd his demand for berries,Now we need the budget equation.

In Part (a),you solved for the amount of x

1

that he will demand,The

budget equation tells us that p

1

x

1

+ p

2

x

2

= M,Plug the solution that

you found for x

1

into the budget equation and solve for x

2

as a function

of income and prices,The answer is x

2

=

M

p

2

4

p

2

p

1

.

(c) When we visited Ambrose in Chapter 5,we looked at a \boundary

solution," where Ambrose consumed only nuts and no berries,In that

example,p

1

=1,p

2

=2,andM = 9,If you plug these numbers into the

formulas we found in Parts (a) and (b),you nd x

1

= 16,and

x

2

=?3:5,Since we get a negative solution for x

2

,it must be that

the budget line x

1

+2x

2

= 9 is not tangent to an indi erence curve when

x

2

0,The best that Ambrose can do with this budget is to spend all

of his income on nuts,Looking at the formulas,we see that at the prices

p

1

=1andp

2

= 2,Ambrose will demand a positive amount of both goods

if and only if M> 16.

6.4 (0) Donald Fribble is a stamp collector,The only things other

than stamps that Fribble consumes are Hostess Twinkies,It turns out

that Fribble’s preferences are represented by the utility function u(s;t)=

s +lnt where s is the number of stamps he collects and t is the number

of Twinkies he consumes,The price of stamps is p

s

and the price of

Twinkies is p

t

,Donald’s income is m.

(a) Write an expression that says that the ratio of Fribble’s marginal

utility for Twinkies to his marginal utility for stamps is equal to the ratio

of the price of Twinkies to the price of stamps,1=t = p

t

=p

s

,(Hint:

The derivative of lnt with respect to t is 1=t,and the derivative of s with

respect to s is 1.)

(b) You can use the equation you found in the last part to show that if he

buys both goods,Donald’s demand function for Twinkies depends only

on the price ratio and not on his income,Donald’s demand function for

Twinkies is t(p

s;p

t;m)=p

s

=p

t

.

(c) Notice that for this special utility function,if Fribble buys both goods,

then the total amount of money that he spends on Twinkies has the

peculiar property that it depends on only one of the three variables m,

p

t

,andp

s

,namely the variable p

s

,(Hint,The amount of money that

he spends on Twinkies is p

t

t(p

s;p

t;m).)

NAME 71

(d) Since there are only two goods,any money that is not spent on

Twinkies must be spent on stamps,Use the budget equation and Don-

ald’s demand function for Twinkies to nd an expression for the number

of stamps he will buy if his income is m,the price of stamps is p

s

and the

price of Twinkies is p

t

,s =

m

p

s

1.

(e) The expression you just wrote down is negative if m<p

s

,Surely

it makes no sense for him to be demanding negative amounts of postage

stamps,If m<p

s

,what would Fribble’s demand for postage stamps be?

s =0 What would his demand for Twinkies be? t = m=p

t

.

(Hint,Recall the discussion of boundary optimum.)

(f) Donald’s wife complains that whenever Donald gets an extra dollar,

he always spends it all on stamps,Is she right? (Assume that m>p

s

.)

Yes.

(g) Suppose that the price of Twinkies is $2 and the price of stamps is $1.

On the graph below,draw Fribble’s Engel curve for Twinkies in red ink

and his Engel curve for stamps in blue ink,(Hint,First draw the Engel

curves for incomes greater than $1,then draw them for incomes less than

$1.)

02468

2

4

6

Quantities

Income

8

Blue

line

Red line

1

0.5

6.5 (0) Shirley Sixpack,as you will recall,thinks that two 8-ounce cans

of beer are exactly as good as one 16-ounce can of beer,Suppose that

these are the only sizes of beer available to her and that she has $30 to

spend on beer,Suppose that an 8-ounce beer costs $.75 and a 16-ounce

beer costs $1,On the graph below,draw Shirley’s budget line in blue ink,

and draw some of her indi erence curves in red.

72 DEMAND (Ch,6)

010203040

10

20

30

16-ounce cans

8-ounce cans

40

Blue

budget

line

Red

curves

Red curve

(a) At these prices,which size can will she buy,or will she buy some of

each? 16-ounce cans.

(b) Suppose that the price of 16-ounce beers remains $1 and the price of

8-ounce beers falls to $.55,Will she buy more 8-ounce beers? No.

(c) What if the price of 8-ounce beers falls to $.40? How many 8-ounce

beers will she buy then? 75 cans.

(d) If the price of 16-ounce beers is $1 each and if Shirley chooses some

8-ounce beers and some 16-ounce beers,what must be the price of 8-ounce

beers? $.50.

(e) Now let us try to describe Shirley’s demand function for 16-ounce beers

as a function of general prices and income,Let the prices of 8-ounce and

16-ounce beers be p

8

and p

16

,and let her income be m.Ifp

16

< 2p

8

,then

the number of 16-ounce beers she will demand is m=p

16

,If p

16

> 2p

8

,

then the number of 16-ounce beers she will demand is 0,If p

16

=

2 p

8

,she will be indi erent between any a ordable combinations.

6.6 (0) Miss Mu et always likes to have things \just so." In fact the

only way she will consume her curds and whey is in the ratio of 2 units of

whey per unit of curds,She has an income of $20,Whey costs $.75 per

unit,Curds cost $1 per unit,On the graph below,draw Miss Mu et’s

budget line,and plot some of her indi erence curves,(Hint,Have you

noticed something kinky about Miss Mu et?)

NAME 73

(a) How many units of curds will Miss Mu et demand in this situation?

8 units,How many units of whey? 16 units.

0 8 16 24 32

8

16

24

Curds

Whey

32

w = 2c

Budget

line

Indifference

curves

(b) Write down Miss Mu et’s demand function for whey as a function

of the prices of curds and whey and of her income,where p

c

is the price

of curds,p

w

is the price of whey,and m is her income,D(p

c;p

w;m)=

m

p

w

+p

c

=2

,(Hint,You can solve for her demands by solving two equa-

tions in two unknowns,One equation tells you that she consumes twice

as much whey as curds,The second equation is her budget equation.)

6.7 (1) Mary’s utility function is U(b;c)=b+ 100c?c

2

,whereb is the

number of silver bells in her garden and c is the number of cockle shells.

She has 500 square feet in her garden to allocate between silver bells and

cockle shells,Silver bells each take up 1 square foot and cockle shells each

take up 4 square feet,She gets both kinds of seeds for free.

(a) To maximize her utility,given the size of her garden,Mary should

plant 308 silver bells and 48 cockle shells,(Hint,Write down

her \budget constraint" for space,Solve the problem as if it were an

ordinary demand problem.)

(b) If she suddenly acquires an extra 100 square feet for her garden,how

much should she increase her planting of silver bells? 100 extra

silver bells,How much should she increase her planting of

cockle shells? Not at all.

74 DEMAND (Ch,6)

(c) If Mary had only 144 square feet in her garden,how many cockle

shells would she grow? 36.

(d) If Mary grows both silver bells and cockle shells,then we know that

the number of square feet in her garden must be greater than 192.

6.8 (0) Casper consumes cocoa and cheese,He has an income of $16.

Cocoa is sold in an unusual way,There is only one supplier and the more

cocoa one buys from him,the higher the price one has to pay per unit.

In fact,x units of cocoa will cost Casper a total of x

2

dollars,Cheese is

sold in the usual way at a price of $2 per unit,Casper’s budget equation,

therefore,is x

2

+2y =16wherex is his consumption of cocoa and y is

his consumption of cheese,Casper’s utility function is U(x;y)=3x+y.

(a) On the graph below,draw the boundary of Casper’s budget set in

blue ink,Use red ink to sketch two or three of his indi erence curves.

0481216

4

8

12

Cheese

16

Cocoa

Red

indifference

curves

Blue budget line

(b) Write an equation that says that at the point (x;y),the slope

of Casper’s budget \line" equals the slope of his indi erence \curve."

2x=2=3=1,Casper demands 3 units of cocoa and 3.5

units of cheese.

6.9 (0) Perhaps after all of the problems with imaginary people and

places,you would like to try a problem based on actual fact,The U.S.

government’s Bureau of Labor Statistics periodically makes studies of

family budgets and uses the results to compile the consumer price index.

These budget studies and a wealth of other interesting economic data can

be found in the annually published Handbook of Labor Statistics,The

NAME 75

tables below report total current consumption expenditures and expendi-

tures on certain major categories of goods for 5 di erent income groups

in the United States in 1961,People within each of these groups all had

similar incomes,Group A is the lowest income group and Group E is the

highest.

Table 6.1

Expenditures by Category for Various Income Groups in 1961

Income Group A B C D E

Food Prepared at Home 465 783 1078 1382 1848

Food Away from Home 68 171 213 384 872

Housing 626 1090 1508 2043 4205

Clothing 119 328 508 830 1745

Transportation 139 519 826 1222 2048

Other 364 745 1039 1554 3490

Total Expenditures 1781 3636 5172 7415 14208

Table 6.2

Percentage Allocation of Family Budget

Income Group A B C D E

Food Prepared at Home 26 22 21 19 13

Food Away from Home 3.8 4.7 4.1 5.2 6.1

Housing 35 30 29 28 30

Clothing 6.7 9.0 9.8 11 12

Transportation 7.8 14 16 17 14

(a) Complete Table 6.2.

(b) Which of these goods are normal goods? All of them.

(c) Which of these goods satisfy your textbook’s de nition of luxury

goods at most income levels? Food away from home,

clothing,transportation.

76 DEMAND (Ch,6)

(d) Which of these goods satisfy your textbook’s de nition of necessity

goods at most income levels? Food prepared at home,

housing.

(e) On the graph below,use the information from Table 6.1 to draw

\Engel curves." (Use total expenditure on current consumption as income

for purposes of drawing this curve.) Use red ink to draw the Engel curve

for food prepared at home,Use blue ink to draw an Engel curve for food

away from home,Use pencil to draw an Engel curve for clothing,How

does the shape of an Engel curve for a luxury di er from the shape of

an Engel curve for a necessity? The curve for a luxury

gets flatter as income rises,the curve for

a necessity gets steeper.

0 750 1500 2250 3000

3

6

9

Total expenditures (thousands of dollars)

12

Expenditure on specific goods

Red lineBlue

line

Pencil

line

6.10 (0) Percy consumes cakes and ale,His demand function for cakes

is q

c

= m?30p

c

+20p

a

,wherem is his income,p

a

is the price of ale,p

c

is the price of cakes,and q

c

is his consumption of cakes,Percy’s income

is $100,and the price of ale is $1 per unit.

(a) Is ale a substitute for cakes or a complement? Explain,A

substitute,An increase in the price of

ale increases demand for cakes.

NAME 77

(b) Write an equation for Percy’s demand function for cakes where income

and the price of ale are held xed at $100 and $1,q

c

= 120?30p

c

.

(c) Write an equation for Percy’s inverse demand function for cakes where

income is $100 and the price of ale remains at $1,p

c

=4?q

c

=30.

At what price would Percy buy 30 cakes? $3,Use blue ink to draw

Percy’s inverse demand curve for cakes.

(d) Suppose that the price of ale rises to $2.50 per unit and remains

there,Write an equation for Percy’s inverse demand for cakes,p

c

=

5?q

c

=30,Use red ink to draw in Percy’s new inverse demand curve

for cakes.

0306090120

1

2

3

Number of cakes

Price

4

Blue Line

Red Line

6.11 (0) Richard and Mary Stout have fallen on hard times,but remain

rational consumers,They are making do on $80 a week,spending $40 on

food and $40 on all other goods,Food costs $1 per unit,On the graph

below,use black ink to draw a budget line,Label their consumption

bundle with the letter A.

(a) The Stouts suddenly become eligible for food stamps,This means

that they can go to the agency and buy coupons that can be exchanged

for $2 worth of food,Each coupon costs the Stouts $1,However,the

maximum number of coupons they can buy per week is 10,On the graph,

draw their new budget line with red ink.

78 DEMAND (Ch,6)

(b) If the Stouts have homothetic preferences,how much more food will

they buy once they enter the food stamp program? 5 units.

0 40 60 80 100 120

20

40

60

80

100

Dollars worth of other things

120

20

a

New consumption point

45

Red budget line

Black budget line

Food

Calculus 6.12 (2) As you may remember,Nancy Lerner is taking an economics

course in which her overall score is the minimum of the number of correct

answers she gets on two examinations,For the rst exam,each correct

answer costs Nancy 10 minutes of study time,For the second exam,each

correct answer costs her 20 minutes of study time,In the last chapter,

you found the best way for her to allocate 1200 minutes between the two

exams,Some people in Nancy’s class learn faster and some learn slower

than Nancy,Some people will choose to study more than she does,and

some will choose to study less than she does,In this section,we will nd

a general solution for a person’s choice of study times and exam scores as

a function of the time costs of improving one’s score.

(a) Suppose that if a student does not study for an examination,he or

she gets no correct answers,Every answer that the student gets right

on the rst examination costs P

1

minutes of studying for the rst exam.

Every answer that he or she gets right on the second examination costs

P

2

minutes of studying for the second exam,Suppose that this student

spends a total of M minutes studying for the two exams and allocates

the time between the two exams in the most e cient possible way,Will

the student have the same number of correct answers on both exams?

NAME 79

Yes,Write a general formula for this student’s overall score for the

course as a function of the three variables,P

1

,P

2

,andM,S =

M

P

1

+P

2

.

If this student wants to get an overall score of S,with the smallest pos-

sible total amount of studying,this student must spend P

1

S minutes

studying for the rst exam and P

2

S studying for the second exam.

(b) Suppose that a student has the utility function

U(S;M)=S?

A

2

M

2;

where S is the student’s overall score for the course,M is the number

of minutes the student spends studying,and A is a variable that reflects

how much the student dislikes studying,In Part (a) of this problem,you

found that a student who studies for M minutes and allocates this time

wisely between the two exams will get an overall score of S =

M

P

1

+P

2

.

Substitute

M

P

1

+P

2

for S in the utility function and then di erentiate with

respect to M to nd the amount of study time,M,that maximizes the

student’s utility,M =

1

A(P

1

+P

2

)

,Your answer will be a function of

the variables P

1

,P

2

,andA,If the student chooses the utility-maximizing

amount of study time and allocates it wisely between the two exams,he

or she will have an overall score for the course of S =

1

A(P

1

+P

2

)

2

.

(c) Nancy Lerner has a utility function like the one presented above,She

chose the utility-maximizing amount of study time for herself,For Nancy,

P

1

=10andP

2

= 20,She spent a total of M =1;200 minutes studying

for the two exams,This gives us enough information to solve for the

variable A in Nancy’s utility function,In fact,for Nancy,A =

1

36;000

.

(d) Ed Fungus is a student in Nancy’s class,Ed’s utility function is just

like Nancy’s,with the same value of A,But Ed learns more slowly than

Nancy,In fact it takes Ed exactly twice as long to learn anything as it

takes Nancy,so that for him,P

1

=20andP

2

= 40,Ed also chooses his

amount of study time so as to maximize his utility,Find the ratio of the

amount of time Ed spends studying to the amount of time Nancy spends

studying,1/2,Will his score for the course be greater than half,

equal to half,or less than half of Nancy’s? Less than half.

6.13 (1) Here is a puzzle for you,At rst glance,it would appear that

there is not nearly enough information to answer this question,But when

you graph the indi erence curve and think about it a little,you will see

that there is a neat,easily calculated solution.

80 DEMAND (Ch,6)

Kinko spends all his money on whips and leather jackets,Kinko’s

utility function is U(x;y)=minf4x;2x+yg,wherex is his consumption

of whips and y is his consumption of leather jackets,Kinko is consuming

15 whips and 10 leather jackets,The price of whips is $10,You are to

nd Kinko’s income.

(a) Graph the indi erence curve for Kinko that passes through the point

(15;10),What is the slope of this indi erence curve at (15;10)2.

What must be the price of leather jackets if Kinko chooses this point?

$5,Now,what is Kinko’s income? 15 10 + 10 5 = 200.

010203040

10

20

30

Whips

Leather jackets

40

(15,10)

Indifference

curve

2x + y = 40

4x = 40

Chapter 7 NAME

Revealed Preference

Introduction,In the last section,you were given a consumer’s pref-

erences and then you solved for his or her demand behavior,In this

chapter we turn this process around,you are given information about a

consumer’s demand behavior and you must deduce something about the

consumer’s preferences,The main tool is the weak axiom of revealed pref-

erence,This axiom says the following,If a consumer chooses commodity

bundle A when she can a ord bundleB,then she will never choose bundle

B from any budget in which she can also a ord A,The idea behind this

axiomisthatifyouchooseA when you could have had B,you must like

A better than B,But if you like A better than B,then you will never

choose B when you can have A,If somebody chooses A when she can

a ord B,we say that for her,A is directly revealed preferred to B.The

weak axiom says that if A is directly revealed preferred to B,thenB is

not directly revealed preferred to A.

Example,Let us look at an example of how you check whether one bundle

is revealed preferred to another,Suppose that a consumer buys the bundle

(x

A

1;x

A

2

)=(2;3) at prices (p

A

1;p

A

2

)=(1;4),The cost of bundle (x

A

1;x

A

2

)

at these prices is (2 1) + (3 4) = 14,Bundle (2;3) is directly revealed

preferred to all the other bundles that she can a ord at prices (1;4),when

she has an income of 14,For example,the bundle (5;2) costs only 13 at

prices (1;4),so we can say that for this consumer (2;3) is directly revealed

preferred to (1;4).

You will also have some problems about price and quantity indexes.

A price index is a comparison of average price levels between two di erent

times or two di erent places,If there is more than one commodity,it is not

necessarily the case that all prices changed in the same proportion,Let us

suppose that we want to compare the price level in the \current year" with

the price level in some \base year." One way to make this comparison

is to compare the costs in the two years of some \reference" commodity

bundle,Two reasonable choices for the reference bundle come to mind.

One possibility is to use the current year’s consumption bundle for the

reference bundle,The other possibility is to use the bundle consumed

in the base year,Typically these will be di erent bundles,If the base-

year bundle is the reference bundle,the resulting price index is called the

Laspeyres price index,If the current year’s consumption bundle is the

reference bundle,then the index is called the Paasche price index.

Example,Suppose that there are just two goods,In 1980,the prices

were (1;3) and a consumer consumed the bundle (4;2),In 1990,the

prices were (2;4) and the consumer consumed the bundle (3;3),The cost

of the 1980 bundle at 1980 prices is (1 4)+ (3 2) = 10,The cost of this

same bundle at 1990 prices is (2 4) + (4 2) = 16,If 1980 is treated

as the base year and 1990 as the current year,the Laspeyres price ratio

82 REVEALED PREFERENCE (Ch,7)

is 16=10,To calculate the Paasche price ratio,you nd the ratio of the

cost of the 1990 bundle at 1990 prices to the cost of the same bundle at

1980 prices,The 1990 bundle costs (2 3) + (4 3) = 18 at 1990 prices.

The same bundle cost (1 3) + (3 3) = 12 at 1980 prices,Therefore

the Paasche price index is 18=12,Notice that both price indexes indicate

that prices rose,but because the price changes are weighted di erently,

the two approaches give di erent price ratios.

Making an index of the \quantity" of stu consumed in the two

periods presents a similar problem,How do you weight changes in the

amount of good 1 relative to changes in the amount of good 2? This time

we could compare the cost of the two periods’ bundles evaluated at some

reference prices,Again there are at least two reasonable possibilities,the

Laspeyres quantity index and the Paasche quantity index,The Laspeyres

quantity index uses the base-year prices as the reference prices,and the

Paasche quantity index uses current prices as reference prices.

Example,In the example above,the Laspeyres quantity index is the ratio

of the cost of the 1990 bundle at 1980 prices to the cost of the 1980 bundle

at 1980 prices,The cost of the 1990 bundle at 1980 prices is 12 and the

cost of the 1980 bundle at 1980 prices is 10,so the Laspeyres quantity

index is 12/10,The cost of the 1990 bundle at 1990 prices is 18 and

the cost of the 1980 bundle at 1990 prices is 16,Therefore the Paasche

quantity index is 18/16.

When you have completed this section,we hope that you will be able

to do the following:

Decide from given data about prices and consumption whether one

commodity bundle is preferred to another.

Given price and consumption data,calculate Paasche and Laspeyres

price and quantity indexes.

Use the weak axiom of revealed preferences to make logical deduc-

tions about behavior.

Use the idea of revealed preference to make comparisons of well-being

across time and across countries.

7.1 (0) When prices are (4;6),Goldie chooses the bundle (6;6),and

when prices are (6;3),she chooses the bundle (10;0).

(a) On the graph below,show Goldie’s rst budget line in red ink and

her second budget line in blue ink,Mark her choice from the rst budget

with the label A,and her choice from the second budget with the label

B.

(b) Is Goldie’s behavior consistent with the weak axiom of revealed pref-

erence? No.

NAME 83

0 5 10 15 20

5

10

15

Good 1

Good 2

20

a

b

Blue line

Red line

7.2 (0) Freddy Frolic consumes only asparagus and tomatoes,which are

highly seasonal crops in Freddy’s part of the world,He sells umbrellas for

a living,which provides a fluctuating income depending on the weather.

But Freddy doesn’t mind; he never thinks of tomorrow,so each week he

spends as much as he earns,One week,when the prices of asparagus and

tomatoes were each $1 a pound,Freddy consumed 15 pounds of each,Use

blue ink to show the budget line in the diagram below,Label Freddy’s

consumption bundle with the letter A.

(a) What is Freddy’s income? $30.

(b) The next week the price of tomatoes rose to $2 a pound,but the price

of asparagus remained at $1 a pound,By chance,Freddy’s income had

changed so that his old consumption bundle of (15,15) was just a ordable

at the new prices,Use red ink to draw this new budget line on the graph

below,Does your new budget line go through the point A? Yes.

What is the slope of this line1=2.

(c) How much asparagus can he a ord now if he spent all of his income

on asparagus? 45 pounds.

(d) What is Freddy’s income now? $45.

84 REVEALED PREFERENCE (Ch,7)

(e) Use pencil to shade the bundles of goods on Freddy’s new red budget

line that he de nitely will not purchase with this budget,Is it possible

that he would increase his consumption of tomatoes when his budget

changes from the blue line to the red one? No.

010203040

10

20

30

Asparagus

Tomatoes

40

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

a

Blue line

Red line

Pencil

shading

7.3 (0) Pierre consumes bread and wine,For Pierre,the price of bread

is 4 francs per loaf,and the price of wine is 4 francs per glass,Pierre has

an income of 40 francs per day,Pierre consumes 6 glasses of wine and 4

loaves of bread per day.

Bob also consumes bread and wine,For Bob,the price of bread is

1/2 dollar per loaf and the price of wine is 2 dollars per glass,Bob has

an income of $15 per day.

(a) If Bob and Pierre have the same tastes,can you tell whether Bob is

better o than Pierre or vice versa? Explain,Bob is better

off,He can afford Pierre’s bundle and

still have income left.

(b) Suppose prices and incomes for Pierre and Bob are as above and that

Pierre’s consumption is as before,Suppose that Bob spends all of his

income,Give an example of a consumption bundle of wine and bread such

that,if Bob bought this bundle,we would know that Bob’s tastes are not

the same as Pierre’s tastes,7.5 wine and 0 bread,for

example,If they had the same preferences,

NAME 85

this violates WARP,since each can afford

but rejects the other’s bundle.

7.4 (0) Here is a table of prices and the demands of a consumer named

Ronald whose behavior was observed in 5 di erent price-income situa-

tions.

Situation p

1

p

2

x

1

x

2

A 1 1 5 35

B 1 2 35 10

C 1 1 10 15

D 3 1 5 15

E 1 2 10 10

(a) Sketch each of his budget lines and label the point chosen in each case

by the letters A,B,C,D,and E.

(b) Is Ronald’s behavior consistent with the Weak Axiom of Revealed

Preference? Yes.

(c) Shade lightly in red ink all of the points that you are certain are worse

for Ronald than the bundle C.

(d) Suppose that you are told that Ronald has convex and monotonic

preferences and that he obeys the strong axiom of revealed preference.

Shade lightly in blue ink all of the points that you are certain are at least

as good as the bundle C.

010203040

10

20

30

x1

x2

40

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,

,

,,

,,

e

d c

b

a

Red shading

Blue shading

86 REVEALED PREFERENCE (Ch,7)

7.5 (0) Horst and Nigel live in di erent countries,Possibly they have

di erent preferences,and certainly they face di erent prices,They each

consume only two goods,x and y,Horst has to pay 14 marks per unit of

x and 5 marks per unit of y,Horst spends his entire income of 167 marks

on 8 units of x and 11 units of y,Good x costs Nigel 9 quid per unit and

good y costs him 7 quid per unit,Nigel buys 10 units of x and 9 units of

y.

(a) Which prices and income would Horst prefer,Nigel’s income and prices

or his own,or is there too little information to tell? Explain your answer.

Horst prefers Nigel’s budget to his own.

With Nigel’s budget,he can afford his own

bundle with money left over.

(b) Would Nigel prefer to have Horst’s income and prices or his own,or

is there too little information to tell? There is too little

information to tell.

7.6 (0) Here is a table that illustrates some observed prices and choices

for three di erent goods at three di erent prices in three di erent situa-

tions.

Situation p

1

p

2

p

3

x

1

x

2

x

3

A 1 2 8 2 1 3

B 4 1 8 3 4 2

C 3 1 2 2 6 2

(a) We will ll in the table below as follows,Where i and j stand for any

of the letters A,B,and C in Row i and Column j of the matrix,write

the value of the Situation-j bundle at the Situation-i prices,For example,

in Row A and Column A,we put the value of the bundle purchased in

Situation A at Situation A prices,From the table above,we see that in

Situation A,the consumer bought bundle (2;1;3) at prices (1;2;8),The

cost of this bundle A at prices A is therefore (1 2)+(2 1)+(8 3) = 28,

so we put 28 in Row A,Column A,In Situation B the consumer bought

bundle (3;4;2),The value of the Situation-B bundle,evaluated at the

situation-A prices is (1 3) + (2 4) + (8 2) = 27,so put 27 in Row

A,Column B,We have lled in some of the boxes,but we leave a few for

you to do.

NAME 87

Prices=Quantities A B C

A 28 27 30

B 33 32 30

C 13 17 16

(b) Fill in the entry in Row i and Column j of the table below with a D if

the Situation-i bundle is directly revealed preferred to the Situation-j bun-

dle,For example,in Situation A the consumer’s expenditure is $28,We

see that at Situation-A prices,he could also a ord the Situation-B bun-

dle,which cost 27,Therefore the Situation-A bundle is directly revealed

preferred to the Situation-B bundle,so we put a D in Row A,Column

B,Now let us consider Row B,Column A,The cost of the Situation-B

bundle at Situation-B prices is 32,The cost of the Situation-A bundle

at Situation-B prices is 33,So,in Situation B,the consumer could not

a ord the Situation-A bundle,Therefore Situation B is not directly re-

vealed preferred to Situation A,So we leave the entry in Row B,Column

A blank,Generally,there is a D in Row i Column j if the number in the

ij entry of the table in part (a) is less than or equal to the entry in Row

i,Columni,There will be a violation of WARP if for some i and j,there

is a D in Row i Column j and also a D in Row j,Columni.Dothese

observations violate WARP? No.

Situation A B C

A | D I

B I | D

C D I |

(c) Now ll in Row i,Columnj with an I if observation i is indirectly

revealed preferred to j,Do these observations violate the Strong Axiom

of Revealed Preference? Yes.

7.7 (0) It is January,and Joe Grad,whom we met in Chapter 5,is

shivering in his apartment when the phone rings,It is Mandy Manana,

one of the students whose price theory problems he graded last term.

Mandy asks if Joe would be interested in spending the month of February

in her apartment,Mandy,who has switched majors from economics to

political science,plans to go to Aspen for the month and so her apartment

will be empty (alas),All Mandy asks is that Joe pay the monthly service

charge of $40 charged by her landlord and the heating bill for the month

of February,Since her apartment is much better insulated than Joe’s,

it only costs $1 per month to raise the temperature by 1 degree,Joe

88 REVEALED PREFERENCE (Ch,7)

thanks her and says he will let her know tomorrow,Joe puts his earmu s

back on and muses,If he accepts Mandy’s o er,he will still have to pay

rent on his current apartment but he won’t have to heat it,If he moved,

heating would be cheaper,but he would have the $40 service charge,The

outdoor temperature averages 20 degrees Fahrenheit in February,and it

costs him $2 per month to raise his apartment temperature by 1 degree.

Joe is still grading homework and has $100 a month left to spend on food

and utilities after he has paid the rent on his apartment,The price of

food is still $1 per unit.

(a) Draw Joe’s budget line for February if he moves to Mandy’s apartment

and on the same graph,draw his budget line if he doesn’t move.

(b) After drawing these lines himself,Joe decides that he would be better

o not moving,From this,we can tell,using the principle of revealed

preference that Joe must plan to keep his apartment at a temperature of

less than 60 degrees.

(c) Joe calls Mandy and tells her his decision,Mandy o ers to pay half

the service charge,Draw Joe’s budget line if he accepts Mandy’s new

o er,Joe now accepts Mandy’s o er,From the fact that Joe accepted

this o er we can tell that he plans to keep the temperature in Mandy’s

apartment above 40 degrees.

0 10 20 30 40 50 60 70 80

20

40

60

80

100

120

Food

Don't move budget line

Move budget line

'New offer'

budget line

Temperature

7.8 (0) Lord Peter Pommy is a distinguished criminologist,schooled

in the latest techniques of forensic revealed preference,Lord Peter is in-

vestigating the disappearance of Sir Cedric Pinchbottom who abandoned

his aging mother on a street corner in Liverpool and has not been seen

NAME 89

since,Lord Peter has learned that Sir Cedric left England and is living

under an assumed name somewhere in the Empire,There are three sus-

pects,R,Preston McAfee of Brass Monkey,Ontario,Canada,Richard

Manning of North Shag,New Zealand,and Richard Stevenson of Gooey

Shoes,Falkland Islands,Lord Peter has obtained Sir Cedric’s diary,which

recorded his consumption habits in minute detail,By careful observation,

he has also discovered the consumption behavior of McAfee,Manning,and

Stevenson,All three of these gentlemen,like Sir Cedric,spend their entire

incomes on beer and sausage,Their dossiers reveal the following:

Sir Cedric Pinchbottom | In the year before his departure,Sir

Cedric consumed 10 kilograms of sausage and 20 liters of beer per

week,At that time,beer cost 1 English pound per liter and sausage

cost 1 English pound per kilogram.

R,Preston McAfee | McAfee is known to consume 5 liters of beer

and 20 kilograms of sausage,In Brass Monkey,Ontario beer costs 1

Canadian dollar per liter and sausage costs 2 Canadian dollars per

kilogram.

Richard Manning | Manning consumes 5 kilograms of sausage

and 10 liters of beer per week,In North Shag,a liter of beer costs

2 New Zealand dollars and sausage costs 2 New Zealand dollars per

kilogram.

Richard Stevenson | Stevenson consumes 5 kilograms of sausage

and 30 liters of beer per week,In Gooey Shoes,a liter of beer costs 10

Falkland Island pounds and sausage costs 20 Falkland Island pounds

per kilogram.

(a) Draw the budget line for each of the three fugitives,using a di erent

color of ink for each one,Label the consumption bundle that each chooses.

On this graph,superimpose Sir Cedric’s budget line and the bundle he

chose.

90 REVEALED PREFERENCE (Ch,7)

010203040

10

20

30

Beer

Sausage

40

McAfee

Manning

Pinchbottom

Stevenson

(b) After pondering the dossiers for a few moments,Lord Peter an-

nounced,\Unless Sir Cedric has changed his tastes,I can eliminate one

of the suspects,Revealed preference tells me that one of the suspects is

innocent." Which one? McAfee.

(c) After thinking a bit longer,Lord Peter announced,\If Sir Cedric

left voluntarily,then he would have to be better o than he was before.

Therefore if Sir Cedric left voluntarily and if he has not changed his tastes,

he must be living in Falklands.

7.9 (1) The McCawber family is having a tough time making ends meet.

They spend $100 a week on food and $50 on other things,A new welfare

program has been introduced that gives them a choice between receiving

a grant of $50 per week that they can spend any way they want,and

buying any number of $2 food coupons for $1 apiece,(They naturally

are not allowed to resell these coupons.) Food is a normal good for the

McCawbers,As a family friend,you have been asked to help them decide

on which option to choose,Drawing on your growing fund of economic

knowledge,you proceed as follows.

(a) On the graph below,draw their old budget line in red ink and label

their current choice C,Now use black ink to draw the budget line that

they would have with the grant,If they chose the coupon option,how

much food could they buy if they spent all their money on food coupons?

$300,How much could they spend on other things if they bought

NAME 91

no food? $150,Use blue ink to draw their budget line if they choose

the coupon option.

0 30 60 90 120 150 180 210 240

30

60

90

120

150

180

Other things

Food

c

a

b

Black budget line

Blue budget line

Red budget line

(b) Using the fact that food is a normal good for the McCawbers,and

knowing what they purchased before,darken the portion of the black

budget line where their consumption bundle could possibly be if they

chose the lump-sum grant option,Label the ends of this line segment A

and B.

(c) After studying the graph you have drawn,you report to the McCaw-

bers,\I have enough information to be able to tell you which choice to

make,You should choose the coupon because you can

get more food even when other expenditure

is constant.

(d) Mr,McCawber thanks you for your help and then asks,\Would you

have been able to tell me what to do if you hadn’t known whether food

was a normal good for us?" On the axes below,draw the same budget

lines you drew on the diagram above,but draw indi erence curves for

which food is not a normal good and for which the McCawbers would be

better o with the program you advised them not to take.

92 REVEALED PREFERENCE (Ch,7)

0 30 60 90 120 150 180 210 240

30

60

90

120

150

180

Other things

Food

c

a

b

Black budget

line

Blue budget line

Red budget line

7.10 (0) In 1933,the Swedish economist Gunnar Myrdal (who later won

a Nobel prize in economics) and a group of his associates at Stockholm

University collected a fantastically detailed historical series of prices and

price indexes in Sweden from 1830 until 1930,This was published in a

book called The Cost of Living in Sweden,In this book you can nd

100 years of prices for goods such as oat groats,hard rye bread,salted

cod sh,beef,reindeer meat,birchwood,tallow candles,eggs,sugar,and

co ee,There are also estimates of the quantities of each good consumed

by an average working-class family in 1850 and again in 1890.

The table below gives prices in 1830,1850,1890,and 1913,for flour,

meat,milk,and potatoes,In this time period,these four staple foods

accounted for about 2/3 of the Swedish food budget.

Prices of Staple Foods in Sweden

Prices are in Swedish kronor per kilogram,except for milk,which is in

Swedish kronor per liter.

1830 1850 1890 1913

Grain Flour,14,14,16,19

Meat,28,34,66,85

Milk,07,08,10,13

Potatoes,032,044,051,064

Based on the tables published in Myrdal’s book,typical consump-

tion bundles for a working-class Swedish family in 1850 and 1890 are

listed below,(The reader should be warned that we have made some

NAME 93

approximations and simpli cations to draw these simple tables from the

much more detailed information in the original study.)

Quantities Consumed by a Typical Swedish Family

Quantities are measured in kilograms per year,except for milk,which is

measured in liters per year.

1850 1890

Grain Flour 165 220

Meat 22 42

Milk 120 180

Potatoes 200 200

(a) Complete the table below,which reports the annual cost of the 1850

and 1890 bundles of staple foods at various years’ prices.

Cost of 1850 and 1890 Bundles at Various Years’ Prices

Cost 1850 bundle 1890 bundle

Cost at 1830 Prices 44.1 61.6

Cost at 1850 Prices 49.0 68.3

Cost at 1890 Prices 63.1 91.1

Cost at 1913 Prices 78.5 113.7

(b) Is the 1890 bundle revealed preferred to the 1850 bundle? Yes.

(c) The Laspeyres quantity index for 1890 with base year 1850 is the ratio

of the value of the 1890 bundle at 1850 prices to the value of the 1850

bundle at 1850 prices,Calculate the Laspeyres quantity index of staple

food consumption for 1890 with base year 1850,1.39.

(d) The Paasche quantity index for 1890 with base year 1850 is the ratio

of the value of the 1890 bundle at 1890 prices to the value of the 1850

bundle at 1890 prices,Calculate the Paasche quantity index for 1890 with

base year 1850,1.44.

(e) The Laspeyres price index for 1890 with base year 1850 is calculated

using 1850 quantities for weights,Calculate the Laspeyres price index for

1890 with base year 1850 for this group of four staple foods,1.29.

94 REVEALED PREFERENCE (Ch,7)

(f) If a Swede were rich enough in 1850 to a ord the 1890 bundle of staple

foods in 1850,he would have to spend 1.39 timesasmuchonthese

foods as does the typical Swedish worker of 1850.

(g) If a Swede in 1890 decided to purchase the same bundle of food staples

that was consumed by typical 1850 workers,he would spend the fraction

.69 of the amount that the typical Swedish worker of 1890 spends on

these goods.

7.11 (0) This question draws from the tables in the previous question.

Let us try to get an idea of what it would cost an American family at

today’s prices to purchase the bundle consumed by an average Swedish

family in 1850,In the United States today,the price of flour is about $.40

per kilogram,the price of meat is about $3.75 per kilogram,the price of

milk is about $.50 per liter,and the price of potatoes is about $1 per

kilogram,We can also compute a Laspeyres price index across time and

across countries and use it to estimate the value of a current US dollar

relative to the value of an 1850 Swedish kronor.

(a) How much would it cost an American at today’s prices to buy the bun-

dle of staple food commodities purchased by an average Swedish working-

class family in 1850? $408.

(b) Myrdal estimates that in 1850,about 2=3 of the average family’s

budget was spent on food,In turn,the four staples discussed in the last

question constitute about 2=3 of the average family’s food budget,If the

prices of other goods relative to the price of the food staples are similar

in the United States today to what they were in Sweden in 1850,about

how much would it cost an American at current prices to consume the

same overall consumption bundle consumed by a Swedish working-class

family in 1850? $919.

(c) Using the Swedish consumption bundle of staple foods in 1850 as

weights,calculate a Laspeyres price index to compare prices in current

American dollars relative to prices in 1850 Swedish kronor,8.35,If

we use this to estimate the value of current dollars relative to 1850 Swedish

kronor,we would say that a U.S,dollar today is worth about,12 1850

Swedish kronor.

7.12 (0) Suppose that between 1960 and 1985,the price of all goods

exactly doubled while every consumer’s income tripled.

NAME 95

(a) Would the Laspeyres price index for 1985,with base year 1960 be less

than 2,greater than 2,or exactly equal to 2? Exactly 2,What

about the Paasche price index? Exactly 2.

(b) If bananas are a normal good,will total banana consumption in-

crease? Yes,If everybody has homothetic preferences,can you de-

termine by what percentage total banana consumption must have in-

creased? Explain,Yes,by 50%,Everybody’s budget

line shifted out by 50%,With homothetic

preferences,the consumption of each good

increases in the same proportion.

7.13 (1) Norm and Sheila consume only meat pies and beer,Meat pies

used to cost $2 each and beer was $1 per can,Their gross income used

to be $60 per week,but they had to pay an income tax of $10,Use red

ink to sketch their old budget line for meat pies and beer.

02030405060

10

20

30

40

50

60

10

Pies

Beer

Black

budget

line

Red budget line

Blue budget line

96 REVEALED PREFERENCE (Ch,7)

(a) They used to buy 30 cans of beer per week and spent the rest of their

income on meat pies,How many meat pies did they buy? 10.

(b) The government decided to eliminate the income tax and to put a

sales tax of $1 per can on beer,raising its price to $2 per can,Assuming

that Norm and Sheila’s pre-tax income and the price of meat pies did not

change,draw their new budget line in blue ink.

(c) The sales tax on beer induced Norm and Sheila to reduce their beer

consumption to 20 cans per week,What happened to their consumption

of meat pies? Stayed the same--10,How much revenue

did this tax raise from Norm and Sheila? $20.

(d) This part of the problem will require some careful thinking,Suppose

that instead of just taxing beer,the government decided to tax both beer

andmeatpiesatthesame percentage rate,and suppose that the price

of beer and the price of meat pies each went up by the full amount of

the tax,The new tax rate for both goods was set high enough to raise

exactly the same amount of money from Norm and Sheila as the tax on

beer used to raise,This new tax collects $,50 for every bottle of beer

sold and $ 1 for every meat pie sold,(Hint,If both goods are

taxed at the same rate,the e ect is the same as an income tax.) How

large an income tax would it take to raise the same revenue as the $1 tax

on beer? $20,Now you can gure out how big a tax on each good

is equivalent to an income tax of the amount you just found.

(e) Use black ink to draw the budget line for Norm and Sheila that cor-

responds to the tax in the last section,Are Norm and Sheila better o

having just beer taxed or having both beer and meat pies taxed if both

sets of taxes raise the same revenue? Both,(Hint,Try to use the

principle of revealed preference.)

Chapter 8 NAME

Slutsky Equation

Introduction,It is useful to think of a price change as having two dis-

tinct e ects,a substitution e ect and an income e ect,The substitution

e ect of a price change is the change that would have happened if in-

come changed at the same time in such a way that the consumer could

exactly a ord her old consumption bundle,The rest of the change in the

consumer’s demand is called the income e ect,Why do we bother with

breaking a real change into the sum of two hypothetical changes? Because

we know things about the pieces that we wouldn’t know about the whole

without taking it apart,In particular,we know that the substitution ef-

fect of increasing the price of a good must reduce the demand for it,We

also know that the income e ect of an increase in the price of a good is

equivalent to the e ect of a loss of income,Therefore if the good whose

price has risen is a normal good,then both the income and substitution

e ect operate to reduce demand,But if the good is an inferior good,

income and substitution e ects act in opposite directions.

Example,A consumer has the utility function U(x

1;x

2

)=x

1

x

2

and an

income of $24,Initially the price of good 1 was $1 and the price of good 2

was $2,Then the price of good 2 rose to $3 and the price of good 1 stayed

at $1,Using the methods you learned in Chapters 5 and 6,you will nd

that this consumer’s demand function for good 1 is D

1

(p

1;p

2;m)=m=2p

1

and her demand function for good 2 is D

2

(p

1;p

2;m)=m=2p

2

,Therefore

initially she will demand 12 units of good 1 and 6 units of good 2,If,

when the price of good 2 rose to $3,her income had changed enough so

that she could exactly a ord her old bundle,her new income would have

to be (1 12)+ (3 6) = $30,At an income of $30,at the new prices,she

would demand D

2

(1;3;30) = 5 units of good 2,Before the change she

bought 6 units of 2,so the substitution e ect of the price change on her

demand for good 2 is 5?6=?1 units,Our consumer’s income didn’t

really change,Her income stayed at $24,Her actual demand for good 2

after the price change was D

2

(1;3;24) = 4,The di erence between what

she actually demanded after the price change and what she would have

demanded if her income had changed to let her just a ord the old bundle

is the income e ect,In this case the income e ect is 4?5=?1 units

of good 2,Notice that in this example,both the income e ect and the

substitution e ect of the price increase worked to reduce the demand for

good 2.

When you have completed this workout,we hope that you will be

able to do the following:

Find Slutsky income e ect and substitution e ect of a speci c price

change if you know the demand function for a good.

Show the Slutsky income and substitution e ects of a price change

98 SLUTSKY EQUATION (Ch,8)

on an indi erence curve diagram.

Show the Hicks income and substitution e ects of a price change on

an indi erence curve diagram.

Find the Slutsky income and substitution e ects for special util-

ity functions such as perfect substitutes,perfect complements,and

Cobb-Douglas.

Use an indi erence-curve diagram to show how the case of a Gi en

good might arise.

Show that the substitution e ect of a price increase unambiguously

decreases demand for the good whose price rose.

Apply income and substitution e ects to draw some inferences about

behavior.

8.1 (0) Gentle Charlie,vegetarian that he is,continues to consume

apples and bananas,His utility function is U(x

A;x

B

)=x

A

x

B

,The price

of apples is $1,the price of bananas is $2,and Charlie’s income is $40 a

day,The price of bananas suddenly falls to $1.

(a) Before the price change,Charlie consumed 20 apples and

10 bananas per day,On the graph below,use black ink to draw

Charlie’s original budget line and put the label A on his chosen consump-

tion bundle.

(b) If,after the price change,Charlie’s income had changed so that he

could exactly a ord his old consumption bundle,his new income would

have been 30,With this income and the new prices,Charlie would

consume 15 apples and 15 bananas,Use red ink to draw

the budget line corresponding to this income and these prices,Label the

bundle that Charlie would choose at this income and the new prices with

the letter B.

(c) Does the substitution e ect of the fall in the price of bananas make

him buy more bananas or fewer bananas? More bananas,How

many more or fewer? 5 more.

(d) After the price change,Charlie actually buys 20 apples and

20 bananas,Use blue ink to draw Charlie’s actual budget line

after the price change,Put the label C on the bundle that he actually

chooses after the price change,Draw 3 horizontal lines on your graph,one

from A to the vertical axis,one from B to the vertical axis,and one from

C to the vertical axis,Along the vertical axis,label the income e ect,the

substitution e ect,and the total e ect on the demand for bananas,Is the

NAME 99

blue line parallel to the red line or the black line that you drew before?

Red line.

010203040

10

20

30

Apples

Bananas

40

a

b

c

Total

Substitution

Income

Red line

Blue line

Black line

(e) The income e ect of the fall in the price of bananas on Charlie’s

demand for bananas is the same as the e ect of an (increase,decrease)

increase in his income of $ 10 per day,Does the income

e ect make him consume more bananas or fewer? More,How many

more or how many fewer? 5 more.

(f) Does the substitution e ect of the fall in the price of bananas make

Charlie consume more apples or fewer? Fewer,How many more or

fewer? 5 fewer,Does the income e ect of the fall in the price of

bananas make Charlie consume more apples or fewer? More,What

is the total e ect of the change in the price of bananas on the demand for

apples? Zero.

8.2 (0) Neville’s passion is ne wine,When the prices of all other

goods are xed at current levels,Neville’s demand function for high-

quality claret is q =,02m?2p,wherem is his income,p is the price

of claret (in British pounds),and q is the number of bottles of claret that

he demands,Neville’s income is 7,500 pounds,and the price of a bottle

of suitable claret is 30 pounds.

100 SLUTSKY EQUATION (Ch,8)

(a) How many bottles of claret will Neville buy? 90.

(b) If the price of claret rose to 40 pounds,how much income would Neville

have to have in order to be exactly able to a ord the amount of claret

and the amount of other goods that he bought before the price change?

8,400 pounds,At this income,and a price of 40 pounds,how

many bottles would Neville buy? 88 bottles.

(c) At his original income of 7,500 and a price of 40,how much claret

would Neville demand? 70 bottles.

(d) When the price of claret rose from 30 to 40,the number of bottles

that Neville demanded decreased by 20,The substitution e ect (in-

creased,reduced) reduced his demand by 2 bottles and

the income e ect (increased,reduced) reduced his demand by 18

bottles.

8.3 (0) Note,Do this problem only if you have read the section entitled

\Another Substitution E ect" that describes the \Hicks substitution ef-

fect",Consider the gure below,which shows the budget constraint and

the indi erence curves of good King Zog,Zog is in equilibrium with an

income of $300,facing prices p

X

=$4andp

Y

= $10.

C

E

F

Y

X

30

22.5

30 35 43 9075 120

NAME 101

(a) How much X does Zog consume? 30.

(b) If the price of X falls to $2.50,while income and the price of Y stay

constant,how much X will Zog consume? 35.

(c) How much income must be taken away from Zog to isolate the Hicksian

income and substitution e ects (i.e.,to make him just able to a ord to

reach his old indi erence curve at the new prices)? $75.

(d) The total e ect of the price change is to change consumption from

the point E to the point C.

(e) The income e ect corresponds to the movement from the point

F to the point C while the substitution e ect corre-

sponds to the movement from the point E to the point F.

(f) Is X a normal good or an inferior good? An inferior

good.

(g) On the axes below,sketch an Engel curve and a demand curve for

Good X that would be reasonable given the information in the graph

above,Be sure to label the axes on both your graphs.

Income

x

300

225

4330

102 SLUTSKY EQUATION (Ch,8)

Price

x

1

2

3

4

30

2.5

35

8.4 (0) Maude spends all of her income on delphiniums and hollyhocks.

She thinks that delphiniums and hollyhocks are perfect substitutes; one

delphinium is just as good as one hollyhock,Delphiniums cost $4 a unit

and hollyhocks cost $5 a unit.

(a) If the price of delphiniums decreases to $3 a unit,will Maude buy

more of them? Yes,What part of the change in consumption is due

to the income e ect and what part is due to the substitution e ect?

All due to income effect.

(b) If the prices of delphiniums and hollyhocks are respectively p

d

=$4

and p

h

= $5 and if Maude has $120 to spend,draw her budget line in

blue ink,Draw the highest indi erence curve that she can attain in red

ink,and label the point that she chooses as A.

NAME 103

010203040

10

20

30

Hollyhocks

Delphiniums

40

a

b

Red

curves

Black line

Blue line

(c) Now let the price of hollyhocks fall to $3 a unit,while the price of

delphiniums does not change,Draw her new budget line in black ink.

Draw the highest indi erence curve that she can now reach with red ink.

Label the point she chooses now as B.

(d) How much would Maude’s income have to be after the price of holly-

hocks fell,so that she could just exactly a ord her old commodity bundle

A? $120.

(e) When the price of hollyhocks fell to $3,what part of the change in

Maude’s demand was due to the income e ect and what part was due to

the substitution e ect? All substitution effect.

8.5 (1) Suppose that two goods are perfect complements,If the price

of one good changes,what part of the change in demand is due to the

substitution e ect,and what part is due to the income e ect? All

income effect.

8.6 (0) Douglas Corn eld’s demand function for good x is x(p

x;p

y;m)=

2m=5p

x

,His income is $1,000,the price of x is $5,and the price of y is

$20,If the price of x falls to $4,then his demand for x will change from

80 to 100.

(a) If his income were to change at the same time so that he could exactly

a ord his old commodity bundle at p

x

=4andp

y

= 20,what would his

new income be? 920,What would be his demand for x at this new

level of income,at prices p

x

=4andp

y

= 20? 92.

104 SLUTSKY EQUATION (Ch,8)

(b) The substitution e ect is a change in demand from 80 to

92,The income e ect of the price change is a change in demand from

92 to 100.

(c) On the axes below,use blue ink to draw Douglas Corn eld’s budget

line before the price change,Locate the bundle he chooses at these prices

on your graph and label this point A,Use black ink to draw Douglas

Corn eld’s budget line after the price change,Label his consumption

bundle after the change by B.

0 40 80 120 160 200 240 280 320

20

40

60

80

y

x

a

b

c

Blue line

Black line

Black line

(d) On the graph above,use black ink to draw a budget line with the new

prices but with an income that just allows Douglas to buy his old bundle,

A,Find the bundle that he would choose with this budget line and label

this bundle C.

8.7 (1) Mr,Consumer allows himself to spend $100 per month on

cigarettes and ice cream,Mr,C’s preferences for cigarettes and ice cream

are una ected by the season of the year.

(a) In January,the price of cigarettes was $1 per pack,while ice cream

cost $2 per pint,Faced with these prices,Mr,C bought 30 pints of ice

cream and 40 packs of cigarettes,Draw Mr,C’s January budget line with

blue ink and label his January consumption bundle with the letter J.

NAME 105

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

J

F

Blue

budget

line

Red budget line

Black budget line

A

Pencil

budget

line

Cigarettes

Ice cream

0

(b) In February,Mr,C again had $100 to spend and ice cream still cost

$2 per pint,but the price of cigarettes rose to $1.25 per pack,Mr,C

consumed 30 pints of ice cream and 32 packs of cigarettes,Draw Mr,C’s

February budget line with red ink and mark his February bundle with

the letter F,The substitution e ect of this price change would make him

buy (less,more,the same amount of) less cigarettes and (less,more,

thesameamountof) more ice cream,Since this is true and the total

change in his ice cream consumption was zero,it must be that the income

e ect of this price change on his consumption of ice cream makes him buy

(more,less,the same amount of) less ice cream,The income

e ect of this price change is like the e ect of an (increase,decrease)

106 SLUTSKY EQUATION (Ch,8)

decrease in his income,Therefore the information we have suggests

that ice cream is a(n) (normal,inferior,neutral) normal good.

(c) In March,Mr,C again had $100 to spend,Ice cream was on sale for $1

per pint,Cigarette prices,meanwhile,increased to $1.50 per pack,Draw

his March budget line with black ink,Is he better o than in January,

worse o,or can you not make such a comparison? Better off.

How does your answer to the last question change if the price of cigarettes

had increased to $2 per pack? Now you can’t tell.

8.8 (1) This problem continues with the adventures of Mr,Consumer

from the previous problem.

(a) In April,cigarette prices rose to $2 per pack and ice cream was still

on sale for $1 per pint,Mr,Consumer bought 34 packs of cigarettes and

32 pints of ice cream,Draw his April budget line with pencil and label

his April bundle with the letter A,Was he better o or worse o than

in January? Worse off,Was he better o or worse o than in

February,or can’t one tell? Better off.

(b) In May,cigarettes stayed at $2 per pack and as the sale on ice cream

ended,the price returned to $2 per pint,On the way to the store,how-

ever,Mr,C found $30 lying in the street,He then had $130 to spend on

cigarettes and ice cream,Draw his May budget with a dashed line,With-

out knowing what he purchased,one can determine whether he is better

o than he was in at least one previous month,Which month or months?

He is better off in May than in February.

(c) In fact,Mr,C buys 40 packs of cigarettes and 25 pints of ice cream

in May,Does he satisfy WARP? No.

8.9 (2) In the last chapter,we studied a problem involving food prices

and consumption in Sweden in 1850 and 1890.

(a) Potato consumption was the same in both years,Real income must

have gone up between 1850 and 1890,since the amount of food staples

purchased,as measured by either the Laspeyres or the Paasche quantity

index,rose,The price of potatoes rose less rapidly than the price of either

meat or milk,and at about the same rate as the price of grain flour,So

real income went up and the price of potatoes went down relative to

other goods,From this information,determine whether potatoes were

NAME 107

most likely a normal or an inferior good,Explain your answer.

If potatoes were a normal good,both the

fall in potato price and the rise in income

would increase the demand for potatoes,But

potato consumption did not increase,So

potatoes must be an inferior good.

(b) Can one also tell from these data whether it is likely that pota-

toes were a Gi en good? If potatoes were a

Giffen good,then the fall in the price

of potatoes would decrease demand and the

rise in income would also decrease demand

for potatoes,But potato demand stayed

constant,So potatoes were probably not a

Giffen good.

8.10 (1) Agatha must travel on the Orient Express from Istanbul to

Paris,The distance is 1,500 miles,A traveler can choose to make any

fraction of the journey in a rst-class carriage and travel the rest of the

way in a second-class carriage,The price is 10 cents a mile for a second-

class carriage and 20 cents a mile for a rst-class carriage,Agatha much

prefers rst-class to second-class travel,but because of a misadventure in

an Istanbul bazaar,she has only $200 left with which to buy her tickets.

Luckily,she still has her toothbrush and a suitcase full of cucumber sand-

wiches to eat on the way,Agatha plans to spend her entire $200 on her

tickets for her trip,She will travel rst class as much as she can a ord

to,but she must get all the way to Paris,and $200 is not enough money

to get her all the way to Paris in rst class.

(a) On the graph below,use red ink to show the locus of combinations

of rst- and second-class tickets that Agatha can just a ord to purchase

with her $200,Use blue ink to show the locus of combinations of rst-

and second-class tickets that are su cient to carry her the entire distance

from Istanbul to Paris,Locate the combination of rst- and second-class

miles that Agatha will choose on your graph and label it A.

108 SLUTSKY EQUATION (Ch,8)

0 400 800 1200 1600

400

800

1200

Second-class miles

First-class miles

1600

Red

line

Blue line

Black line

a

Pencil line

b

c

(b) Let m

1

be the number of miles she travels by rst-class coach and m

2

be the number of miles she travels by second-class coach,Write down two

equations that you can solve to nd the number of miles she chooses to

travel by rst-class coach and the number of miles she chooses to travel

by second-class coach.,2m

1

+,1m

2

= 200,m

1

+ m

2

=

1;500.

(c) The number of miles that she travels by second-class coach is

1,000.

(d) Just before she was ready to buy her tickets,the price of second-class

tickets fell to $.05 while the price of rst-class tickets remained at $.20.

On the graph that you drew above,use pencil to show the combinations

of rst-class and second-class tickets that she can a ord with her $200

at these prices,On your graph,locate the combination of rst-class and

second-class tickets that she would now choose,(Remember,she is going

to travel as much rst-class as she can a ord to and still make the 1,500

mile trip on $200.) Label this point B.Howmanymilesdoesshetravel

by second class now? 666.66,(Hint,For an exact solution you

will have to solve two linear equations in two unknowns.) Is second-class

travel a normal good for Agatha? No,Is it a Gi en good for her?

Yes.

NAME 109

8.11 (0) We continue with the adventures of Agatha,from the previous

problem,Just after the price change from $.10 per mile to $.05 per mile

for second-class travel,and just before she had bought any tickets,Agatha

misplaced her handbag,Although she kept most of her money in her sock,

the money she lost was just enough so that at the new prices,she could

exactly a ord the combination of rst- and second-class tickets that she

would have purchased at the old prices,How much money did she lose?

$50,On the graph you started in the previous problem,use black ink

to draw the locus of combinations of rst- and second-class tickets that

she can just a ord after discovering her loss,Label the point that she

chooses with a C,How many miles will she travel by second class now?

1,000.

(a) Finally,poor Agatha nds her handbag again,How many miles will

she travel by second class now (assuming she didn’t buy any tickets before

she found her lost handbag)? 666.66,When the price of second-

class tickets fell from $.10 to $.05,how much of a change in Agatha’s de-

mand for second-class tickets was due to a substitution e ect? None.

How much of a change was due to an income e ect333:33.

110 SLUTSKY EQUATION (Ch,8)

Chapter 9 NAME

Buying and Selling

Introduction,In previous chapters,we studied the behavior of con-

sumers who start out without owning any goods,but who had some money

with which to buy goods,In this chapter,the consumer has an initial en-

dowment,which is the bundle of goods the consumer owns before any

trades are made,A consumer can trade away from his initial endowment

by selling one good and buying the other.

The techniques that you have already learned will serve you well here.

To nd out how much a consumer demands at given prices,you nd his

budget line and then nd a point of tangency between his budget line and

an indi erence curve,To determine a budget line for a consumer who

is trading from an initial endowment and who has no source of income

other than his initial endowment,notice two things,First,the initial

endowment must lie on the consumer’s budget line,This is true because,

no matter what the prices are,the consumer can always a ord his initial

endowment,Second,if the prices are p

1

and p

2

,the slope of the budget

line must be?p

1

=p

2

,This is true,since for every unit of good 1 the

consumer gives up,he can get exactly p

1

=p

2

units of good 2,Therefore

if you know the prices and you know the consumer’s initial endowment,

then you can always write an equation for the consumer’s budget line.

After all,if you know one point on a line and you know its slope,you

can either draw the line or write down its equation,Once you have the

budget equation,you can nd the bundle the consumer chooses,using the

same methods you learned in Chapter 5.

Example,A peasant consumes only rice and sh,He grows some rice and

some sh,but not necessarily in the same proportion in which he wants

to consume them,Suppose that if he makes no trades,he will have 20

units of rice and 5 units of sh,The price of rice is 1 yuan per unit,and

the price of sh is 2 yuan per unit,The value of the peasant’s endowment

is (1 20) + (2 5) = 30,Therefore the peasant can consume any bundle

(R;F) such that (1 R)+(2 F) = 30.

Perhaps the most interesting application of trading from an initial

endowment is the theory of labor supply,To study labor supply,we

consider the behavior of a consumer who is choosing between leisure and

other goods,The only thing that is at all new or \tricky" is nding

the appropriate budget constraint for the problem at hand,To study

labor supply,we think of the consumer as having an initial endowment of

leisure,some of which he may trade away for goods.

In most applications we set the price of \other goods" at 1,The

wage rate is the price of leisure,The role that is played by income in

the ordinary consumer-good model is now played by \full income." A

worker’s full income is the income she would have if she chose to take no

leisure.

112 BUYING AND SELLING (Ch,9)

Example,Sherwin has 18 hours a day which he divides between labor and

leisure,He can work as many hours a day as he wishes for a wage of $5

per hour,He also receives a pension that gives him $10 a day whether he

works or not,The price of other goods is $1 per unit,If Sherwin makes no

trades at all,he will have 18 hours of leisure and 10 units of other goods.

Therefore Sherwin’s initial endowment is 18 hours of leisure a day and

$10 a day for other goods,Let R be the amount of leisure that he has per

day,and let C be the number of dollars he has to spend per day on other

goods,If his wage is $5 an hour,he can a ord to consume bundle (R;C)

if it costs no more per day than the value of his initial endowment,The

value of his initial endowment (his full income) is $10 + ($5 18) = $100

per day,Therefore Sherwin’s budget equation is 5R +C = 100.

9.1 (0) Abishag Appleby owns 20 quinces and 5 kumquats,She has no

income from any other source,but she can buy or sell either quinces or

kumquats at their market prices,The price of kumquats is four times the

price of quinces,There are no other commodities of interest.

(a) How many quinces could she have if she was willing to do without

kumquats? 40,How many kumquats could she have if she was willing

to do without quinces? 10.

010203040

10

20

30

Quinces

Kumquats

40

Red line

Blue line

e

c

Squiggly

line

(b) Draw Abishag’s budget set,using blue ink,and label the endowment

bundle with the letter E,If the price of quinces is 1 and the price of

kumquats is 4,write Abishag’s budget equation,Q +4K =40.

If the price of quinces is 2 and the price of kumquats is 8,write Abishag’s

budget equation,2Q+8K =80,What e ect does doubling both

NAME 113

prices have on the set of commodity bundles that Abishag can a ord?

No effect.

(c) Suppose that Abishag decides to sell 10 quinces,Label her nal

consumption bundle in your graph with the letter C.

(d) Now,after she has sold 10 quinces and owns the bundle labelled C,

suppose that the price of kumquats falls so that kumquats cost the same

as quinces,On the diagram above,draw Abishag’s new budget line,using

red ink.

(e) If Abishag obeys the weak axiom of revealed preference,then there are

some points on her red budget line that we can be sure Abishag will not

choose,On the graph,make a squiggly line over the portion of Abishag’s

red budget line that we can be sure she will not choose.

9.2 (0) Mario has a small garden where he raises eggplant and tomatoes.

He consumes some of these vegetables,and he sells some in the market.

Eggplants and tomatoes are perfect complements for Mario,since the only

recipes he knows use them together in a 1:1 ratio,One week his garden

yielded 30 pounds of eggplant and 10 pounds of tomatoes,At that time

the price of each vegetable was $5 per pound.

(a) What is the monetary value of Mario’s endowment of vegetables?

$200.

(b) On the graph below,use blue ink to draw Mario’s budget line,Mario

ends up consuming 20 pounds of tomatoes and 20 pounds

of eggplant,Draw the indi erence curve through the consumption bundle

that Mario chooses and label this bundle A.

(c) Suppose that before Mario makes any trades,the price of tomatoes

rises to $15 a pound,while the price of eggplant stays at $5 a pound.

What is the value of Mario’s endowment now? $300,Draw his new

budget line,using red ink,He will now choose a consumption bundle

consisting of 15 tomatoes and 15 eggplants.

(d) Suppose that Mario had sold his entire crop at the market for a total

of $200,intending to buy back some tomatoes and eggplant for his own

consumption,Before he had a chance to buy anything back,the price of

tomatoes rose to $15,while the price of eggplant stayed at $5,Draw his

budget line,using pencil or black ink,Mario will now consume 10

pounds of tomatoes and 10 pounds of eggplant.

114 BUYING AND SELLING (Ch,9)

(e) Assuming that the price of tomatoes rose to $15 from $5 before Mario

made any transactions,the change in the demand for tomatoes due to

the substitution e ect was 0,The change in the demand for

tomatoes due to the ordinary income e ect was?10,The change

in the demand for tomatoes due to the endowment income e ect was

+5,The total change in the demand for tomatoes was?5.

010203040

10

20

30

40

Red line

Blue line

a

Black line

Tomatoes

Eggplant

9.3 (0) Lucetta consumes only two goods,A and B,Her only source of

income is gifts of these commodities from her many admirers,She doesn’t

always get these goods in the proportions in which she wants to consume

them in,but she can always buy or sell A at the price p

A

=1andB at

the price p

B

= 2,Lucetta’s utility function is U(a;b)=ab,wherea is the

amount of A she consumes and b istheamountofB she consumes.

(a) Suppose that Lucetta’s admirers give her 100 units of A and 200 units

of B,In the graph below,use red ink to draw her budget line,Label her

initial endowment E.

(b) What are Lucetta’s gross demands for A? 250 units,And for

B? 125 units.

(c) What are Lucetta’s net demands? 150 of A and?75 of

B.

NAME 115

(d) Suppose that before Lucetta has made any trades,the price of good

B falls to 1,and the price of good A stays at 1,Draw Lucetta’s budget

line at these prices on your graph,using blue ink.

(e) Does Lucetta’s consumption of good B rise or fall? It rises.

By how much? 25 units,What happens to Lucetta’s consumption

of good A? It decreases by 100 units.

0 225 300

100

200

300

400

500

600

75

Good A

Good B

Red budget line

Blue budget line

150

e

(f) Suppose that before the price of goodB fell,Lucetta had exchanged all

of her gifts for money,planning to use the money to buy her consumption

bundle later,How much good B will she choose to consume? 250

units,How much good A? 250 units.

(g) Explain why her consumption is di erent depending on whether she

was holding goods or money at the time of the price change,In

the former case,the fall in p

B

makes her

poorer because she is a net seller of good

B,In the latter case,her income doesn’t

116 BUYING AND SELLING (Ch,9)

change.

9.4 (0) Priscilla nds it optimal not to engage in trade at the going

prices and just consumes her endowment,Priscilla has no kinks in her

indi erence curves,and she is endowed with positive amounts of both

goods,Use pencil or black ink to draw a budget line and an indi erence

curve for Priscilla that would be consistent with these facts,Suppose that

the price of good 2 stays the same,but the price of good 1 falls below the

level at which Priscilla made no trade,Use blue ink to show her new bud-

get line,Priscilla satis es the weak axiom of revealed preference,Could

it happen that Priscilla will consume less of good 1 than before? Explain.

No,If p

1

falls,then with the new budget,

she can still afford her old bundle,She

could afford the bundles with less of good

1 than her endowment at the old prices,By

WARP she won’t choose them now.

X2

X1

e

Black

budget

line

Blue

budget

line

9.5 (0) Potatoes are a Gi en good for Paddy,who has a small potato

farm,The price of potatoes fell,but Paddy increased his potato consump-

tion,At rst this astonished the village economist,who thought that a

decrease in the price of a Gi en good was supposed to reduce demand.

But then he remembered that Paddy was a net supplier of potatoes,With

the help of a graph,he was able to explain Paddy’s behavior,In the axes

below,show how this could have happened,Put \potatoes" on the hor-

izontal axis and \all other goods" on the vertical axis,Label the old

equilibrium A and the new equilibrium B.DrawapointC so that the

Slutsky substitution e ect is the movement from A to C and the Slutsky

NAME 117

income e ect is the movement from C to B,On this same graph,you are

also going to have to show that potatoes are a Gi en good,To do this,

draw a budget line showing the e ect of a fall in the price of potatoes if

Paddy didn’t own any potatoes,but only had money income,Label the

new consumption point under these circumstances by D.(Warning:You

probably will need to make a few dry runs on some scratch paper to get

the whole story straight.)

All other goods

Potatoes

b

e

c

a

d

9.6 (0) Recall the travails of Agatha,from the previous chapter,She

had to travel 1,500 miles from Istanbul to Paris,She had only $200 with

which to buy rst-class and second-class tickets on the Orient Express

when the price of rst-class tickets was $.20 and the price of second-class

tickets was $.10,She bought tickets that enabled her to travel all the

way to Paris,with as many miles of rst class as she could a ord,After

she boarded the train,she discovered to her amazement that the price of

second-class tickets had fallen to $.05 while the price of rst-class tickets

remained at $.20,She also discovered that on the train it was possible to

buy or sell rst-class tickets for $.20 a mile and to buy or sell second-class

tickets for $.05 a mile,Agatha had no money left to buy either kind of

ticket,but she did have the tickets that she had already bought.

(a) On the graph below,use pencil to show the combinations of tickets

that she could a ord at the old prices,Use blue ink to show the combi-

nations of tickets that will take her exactly 1,500 miles,Mark the point

that she chooses with the letter A.

118 BUYING AND SELLING (Ch,9)

0 400 800 1200 1600

400

800

1200

Second-class miles

First-class miles

1600

Red line

Blue line

a

Pencil line

(b) Use red ink to draw a line showing all of the combinations of rst-class

and second-class travel that she can a ord when she is on the train,by

trading her endowment of tickets at the new prices that apply on board

the train.

(c) On your graph,show the point that she chooses after nding out

about the price change,Does she choose more,less,or the same amount

of second-class tickets? The same.

9.7 (0) Mr,Cog works in a machine factory,He can work as many

hours per day as he wishes at a wage rate of w.LetC be the number of

dollars he has to spend on consumer goods and let R be the number of

hours of leisure that he chooses.

(a) Suppose that Mr,Cog earns $8 an hour and has 18 hours per day

to devote to labor or leisure,and suppose that he has $16 of nonlabor

income per day,Write an equation for his budget between consumption

and leisure,C+8R = 160,Use blue ink to draw his budget line

in the graph below,His initial endowment is the point where he doesn’t

work,but keeps all of his leisure,Mark this point on the graph below with

the letter A,(When your draw your graph,remember that although Cog

can choose to work and thereby \sell" some of his endowment of leisure,

he cannot \buy leisure" by paying somebody else to loaf for him.) If Mr.

Cog has the utility function U(R;C)=CR,how many hours of leisure

per day will he choose? 10,How many hours per day will he work?

8.

NAME 119

01216204

40

80

120

160

200

240

4

Leisure

Consumption

Black budget line

Red budget line

Blue budget line

8

a

(b) Suppose that Mr,Cog’s wage rate rose to $12 an hour,Use red ink

to draw his new budget line,(He still has $16 a day in nonlabor income.)

If Mr,Cog continued to work exactly as many hours as he did before the

wage increase,how much more money would he have each day to spend

on consumption? $32,But with his new budget line,he chooses to

work 8

1

3

hours,and so his consumption actually increases by

$36.

(c) Suppose that Mr,Cog still receives $8 an hour but that his nonlabor

income rises to $48 per day,Use black ink to draw his budget line,How

many hours does he choose to work? 6.

(d) Suppose that Mr,Cog has a wage of $w perhourandanonlabor

income of $m,As before,assume that he has 18 hours to divide between

labor and leisure,Cog’s budget line has the equation C+wR = m+18w.

Using the same methods you used in the chapter on demand functions,

nd the amount of leisure that Mr,Cog will demand as a function of

wages and of nonlabor income,(Hint,Notice that this is just the same

as nding the demand for R when the price of R is w,the price of C is

120 BUYING AND SELLING (Ch,9)

1,and income is m +18w.) Mr,Cog’s demand function for leisure is

R(w;m)= 9+(m=2w),Mr,Cog’s supply function for labor is

therefore 18?R(w;m)= 9?m=2w.

9.8 (0) Fred has just arrived at college and is trying to gure out how to

supplement the meager checks that he gets from home,\How can anyone

live on $50 a week for spending money?" he asks,But he asks to no

avail,\If you want more money,get a job," say his parents,So Fred

glumly investigates the possibilities,The amount of leisure time that he

has left after allowing for necessary activities like sleeping,brushing teeth,

and studying for economics classes is 50 hours a week,He can work as

many hours per week at a nearby Taco Bell for $5 an hour,Fred’s utility

function for leisure and money to spend on consumption is U(C;L)=CL.

(a) Fred has an endowment that consists of $50 of money to spend on

consumption and 50 hours of leisure,some of which he might \sell"

for money,The money value of Fred’s endowment bundle,including both

his money allowance and the market value of his leisure time is therefore

$300,Fred’s \budget line" for leisure and consumption is like a budget

lineforsomeonewhocanbuythesetwogoodsatapriceof$1perunit

of consumption and a price of $5 per unit of leisure,The only

di erence is that this budget line doesn’t run all the way to the horizontal

axis.

(b) On the graph below,use black ink to show Fred’s budget line,(Hint:

Find the combination of leisure and consumption expenditures that he

could have if he didn’t work at all,Find the combination he would have

if he chose to have no leisure at all,What other points are on your graph?)

On the same graph,use blue ink to sketch the indi erence curves that

give Fred utility levels of 3,000,4,500,and 7,500.

(c) If you maximized Fred’s utility subject to the above budget,how

much consumption would he choose? $150,(Hint,Remember how

to solve for the demand function of someone with a Cobb-Douglas utility

function?)

(d) The amount of leisure that Fred will choose to consume is 30

hours,This means that his optimal labor supply will be 20 hours.

NAME 121

0304506

50

100

150

200

250

300

10

Leisure

Consumption

Black

budget

line

Blue

indifference

curve (3000)

20

Blue indifference curve

(4500)

Blue

indifference

curve

(7500)

9.9 (0) George Johnson earns $5 per hour in his job as a tru e snif-

fer,After allowing time for all of the activities necessary for bodily up-

keep,George has 80 hours per week to allocate between leisure and labor.

Sketch the budget constraints for George resulting from the following

government programs.

(a) There is no government subsidy or taxation of labor income,(Use

blue ink on the graph below.)

020406080

100

200

300

Leisure

Consumption

400

Blue budget line

Red budget line

122 BUYING AND SELLING (Ch,9)

(b) All individuals receive a lump-sum payment of $100 per week from the

government,There is no tax on the rst $100 per week of labor income.

But all labor income above $100 per week is subject to a 50% income tax.

(Use red ink on the graph above.)

(c) If an individual is not working,he receives a payment of $100,If he

works he does not receive the $100,and all wages are subject to a 50%

income tax,(Use blue ink on the graph below.)

020406080

100

200

300

Leisure

Consumption

400

Red budget line

Blue budget line

(d) The same conditions as in Part (c) apply,with the exception that the

rst 20 hours of labor are exempt from the tax,(Use red ink on the graph

above.)

(e) All wages are taxed at 50%,but as an incentive to encourage work,

the government gives a payment of $100 to anyone who works more than

20 hours a week,(Use blue ink on the graph below.)

NAME 123

020406080

100

200

300

Leisure

Consumption

400

Blue budget line

9.10 (0) In the United States,real wage rates in manufacturing have

risen steadily from 1890 to the present,In the period from 1890 to 1930,

the length of the work week was reduced dramatically,But after 1930,

despite continuing growth of real wage rates,the length of the work week

has stayed remarkably constant at about 40 hours per week.

Hourly Wages and Length of Work Week

in U.S,Manufacturing,1890-1983

Sources,Handbook of Labor Statistics,1983 and U.S,Economic History,

by Albert Niemi (p,274),Wages are in 1983 dollars.

Year Wage Hours Worked

1890 1.89 59.0

1909 2.63 51.0

1920 3.11 47.4

1930 3.69 42.1

1940 5.27 38.1

1950 6.86 40.5

1960 8.56 39.7

1970 9.66 39.8

1983 10.74 40.1

124 BUYING AND SELLING (Ch,9)

(a) Use these data to plot a \labor supply curve" on the graph below.

0304506

2

4

6

8

10

12

10

Hourly wage rate (in 1983 dollars)

20

Hours of work per week

(b) At wage rates below $4 an hour,does the workweek get longer or

shorter as the wage rate rises? Shorter.

(c) The data in this table could be consistent with workers choosing var-

ious hours a week to work,given the wage rate,An increase in wages

has both an endowment income e ect and a substitution e ect,The

substitution e ect alone would make for a (longer,shorter) longer

workweek,If leisure is a normal good,the endowment income e ect tends

to make people choose (more,less) more leisure and a (longer,shorter)

shorter workweek,At wage rates below $4 an hour,the (substi-

tution,endowment income) endowment income e ect appears

to dominate,How would you explain what happens at wages above $4

an hour? Substitution and endowment income

effects cancel each other out,so the

work week stays roughly constant.

(d) Between 1890 and 1909,wage rates rose by 39 percent,but

weekly earnings rose by only 20 percent,For this period,the

NAME 125

gain in earnings (overstates,understates) understates the gain

in worker’s wealth,since they chose to take (more,less) more leisure

in 1909 than they took in 1890.

9.11 (0) Professor Mohamed El Hodiri of the University of Kansas,in

a classic tongue-in-cheek article \The Economics of Sleeping," Manifold,

vol,17,1975,o ered the following analysis,\Assume there are 24 hours

in a day,Daily consumption being x and hours of sleep s,the consumer

maximizes a utility function of the form u = x

2

s,wherex = w(24?s),

with w being the wage rate."

(a) In El Hodiri’s model,does the optimal amount of sleeping increase,

decrease,or stay the same as wages increase? Stays the same.

(b) How many hours of sleep per day is best in El Hodiri’s model? 8.

9.12 (0) Wendy and Mac work in fast food restaurants,Wendy gets $4

an hour for the rst 40 hours that she works and $6 an hour for every

hour beyond 40 hours a week,Mac gets $5 an hour no matter how many

hours he works,Each has 80 hours a week to allocate between work and

leisure and neither has any income from sources other than labor,Each

has a utility function U = cr,wherec is consumption and r is leisure.

Each can choose the number of hours to work.

(a) How many hours will Mac choose to work? 40.

(b) Wendy’s budget \line" has a kink in it at the point wherer = 40

and c = 160,Use blue ink for the part of her budget line where she

would be if she does not work overtime,Use red ink for the part where

she would be if she worked overtime.

126 BUYING AND SELLING (Ch,9)

020406080

100

200

300

Leisure

Consumption

400

Blue part of line

Red part of line

(c) The blue line segment that you drew lies on a line with equation

c +4r = 320,The red line that you drew lies on a line with

equation c +6r = 400,(Hint,For the red line,you know one

point on the line and you know its slope.)

(d) If Wendy was paid $4 an hour no matter how many hours she worked,

she would work 40 hours and earn a total of $160 a week.

On your graph,use black ink to draw her indi erence curve through this

point.

(e) Will Wendy choose to work overtime? Yes,What is the best

choice for Wendy from the red budget line? (c;r)= (200;33:3).

How many hours a week will she work? 46.6.

(f) Suppose that the jobs are equally agreeable in all other respects,Since

Wendy and Mac have the same preferences,they will be able to agree

about who has the better job,Who has the better job? Mac,(Hint:

Calculate Wendy’s utility when she makes her best choice,Calculate what

her utility would be if she had Mac’s job and chose the best amount of

time to work.)

NAME 127

9.13 (1) Wally Piper is a plumber,He charges $10 per hour for his work

and he can work as many hours as he likes,Wally has no source of income

other than his labor,He has 168 hours per week to allocate between labor

and leisure,On the graph below,draw Wally’s budget set,showing the

various combinations of weekly leisure and income that Wally can a ord.

0 120 160 200 240

400

800

1200

1600

2000

2400

40

Income

80

Leisure

128

Red

budget

line

(a) Write down Wally’s budget equation,I +10R =1;680.

(b) While self-employed,Wally chose to work 40 hours per week,The

construction rm,Glitz and Drywall,had a rush job to complete,They

o ered Wally $20 an hour and said that he could work as many hours as

he liked,Wally still chose to work only 40 hours per week,On the graph

you drew above,draw in Wally’s new budget line.

(c) Wally has convex preferences and no kinks in his indi erence curves.

On the graph,draw indi erence curves that are consistent with his choice

of working hours when he was self-employed and when he worked for Glitz

and Drywall.

(d) Glitz and Drywall were in a great hurry to complete their project and

wanted Wally to work more than 40 hours,They decided that instead of

paying him $20 per hour,they would pay him only $10 an hour for the

rst 40 hours that he worked per week and $20 an hour for every hour of

128 BUYING AND SELLING (Ch,9)

\overtime" that he worked beyond 40 hours per week,On the graph that

you drew above,use red ink to sketch in Wally’s budget line with this

pay schedule,Draw the indi erence curve through the point that Wally

chooses with this pay schedule,Will Wally work more than 40 hours or

less than 40 hours per week with this pay schedule? More.

9.14 (1) Felicity loves her job,She is paid $10 an hour and can work

as many hours a day as she wishes,She chooses to work only 5 hours

a day,She says the job is so interesting that she is happier working at

this job than she would be if she made the same income without working

at all,A skeptic asks,\If you like the job better than not working at

all,why don’t you work more hours and earn more money?" Felicity,

who is entirely rational,patiently explains that work may be desirable on

average but undesirable on the margin,The skeptic insists that she show

him her indi erence curves and her budget line.

(a) On the axes below,draw a budget line and indi erence curves that are

consistent with Felicity’s behavior and her remarks,Put leisure on the

horizontal axis and income on the vertical axis,(Hint,Where does the

indi erence curve through her actual choice hit the vertical line l = 24?)

Income

Leisure

50

240

24

9.15 (2) Dudley’s utility function is U(C;R)=C?(12?R)

2

,whereR

is the amount of leisure he has per day,He has 16 hours a day to divide

between work and leisure,He has an income of $20 a day from nonlabor

sources,The price of consumption goods is $1 per unit.

(a) If Dudley can work as many hours a day as he likes but gets zero

wages for his labor,how many hours of leisure will he choose? 12.

NAME 129

(b) If Dudley can work as many hours a day as he wishes for a wage rate

of $10 an hour,how many hours will he choose to work? (Hint,Write

down Dudley’s budget constraint,Solve for his labor supply,Remember

that the amount of labor he wishes to supply is 16 minus his demand for

leisure.) 9.

(c) If Dudley’s nonlabor income decreased to $5 a day,how many hours

would he choose to work? 7.

(d) Suppose that Dudley has to pay an income tax of 20 percent on all

of his income,and suppose that his before-tax wage remained at $10 an

hour and his before-tax nonlabor income was $20 per day; how many

hours would he choose to work? 8.

130 BUYING AND SELLING (Ch,9)

Chapter 10 NAME

Intertemporal Choice

Introduction,The theory of consumer saving uses techniques that you

have already learned,In order to focus attention on consumption over

time,we will usually consider examples where there is only one consumer

good,but this good can be consumed in either of two time periods,We

will be using two \tricks." One trick is to treat consumption in period 1

and consumption in period 2 as two distinct commodities,If you make

period-1 consumption the numeraire,then the \price" of period-2 con-

sumption is the amount of period-1 consumption that you have to give

up to get an extra unit of period-2 consumption,This price turns out to

be 1=(1 +r),where r is the interest rate.

The second trick is in the way you treat income in the two di erent

periods,Suppose that a consumer has an income of m

1

in period 1 and

m

2

in period 2 and that there is no inflation,The total amount of period-

1 consumption that this consumer could buy,if he borrowed as much

money as he could possibly repay in period 2,is m

1

+

m

2

1+r

.Asyou

work the exercises and study the text,it should become clear that the

consumer’s budget equation for choosing consumption in the two periods

is always

c

1

+

c

2

1+r

= m

1

+

m

2

1+r

:

This budget constraint looks just like the standard budget constraint that

you studied in previous chapters,where the price of \good 1" is 1,the

price of \good 2" is 1=(1 + r),and \income" is m

1

+

m

2

(1+r)

,Therefore

if you are given a consumer’s utility function,the interest rate,and the

consumer’s income in each period,you can nd his demand for consump-

tion in periods 1 and 2 using the methods you already know,Having

solved for consumption in each period,you can also nd saving,since the

consumer’s saving is just the di erence between his period-1 income and

his period-1 consumption.

Example,A consumer has the utility function U(c

1;c

2

)=c

1

c

2

.Thereis

no inflation,the interest rate is 10%,and the consumer has income 100

in period 1 and 121 in period 2,Then the consumer’s budget constraint

c

1

+c

2

=1:1 = 100 + 121=1:1 = 210,The ratio of the price of good 1 to the

price of good 2 is 1 +r =1:1,The consumer will choose a consumption

bundle so that MU

1

=MU

2

=1:1,But MU

1

= c

2

and MU

2

= c

1

,sothe

consumer must choose a bundle such that c

2

=c

1

=1:1,Take this equation

together with the budget equation to solve for c

1

and c

2

,The solution is

c

1

= 105 and c

2

= 115:50,Since the consumer’s period-1 income is only

100,he must borrow 5 in order to consume 105 in period 1,To pay back

principal and interest in period 2,he must pay 5.50 out of his period-2

income of 121,This leaves him with 115.50 to consume.

You will also be asked to determine the e ects of inflation on con-

132 INTERTEMPORAL CHOICE (Ch,10)

sumer behavior,The key to understanding the e ects of inflation is to

see what happens to the budget constraint.

Example,Suppose that in the previous example,there happened to be

an inflation rate of 6%,and suppose that the price of period-1 goods is

1,Then if you save $1 in period 1 and get it back with 10% interest,

you will get back $1.10 in period 2,But because of the inflation,goods

in period 2 cost 1.06 dollars per unit,Therefore the amount of period-1

consumption that you have to give up to get a unit of period-2 consump-

tion is 1:06=1:10 =,964 units of period-2 consumption,If the consumer’s

money income in each period is unchanged,then his budget equation is

c

1

+,964c

2

= 210,This budget constraint is the same as the budget

constraint would be if there were no inflation and the interest rate were

r,where:964 = 1=(1 + r),The value of r that solves this equation is

known as the real rate of interest,In this case the real rate of interest

is about,038,When the interest rate and inflation rate are both small,

the real rate of interest is closely approximated by the di erence between

the nominal interest rate,(10% in this case) and the inflation rate (6%

in this case),that is,:038,10?:06,As you will see,this is not such a

good approximation if inflation rates and interest rates are large.

10.1 (0) Peregrine Pickle consumes (c

1;c

2

)andearns(m

1;m

2

)inperiods

1 and 2 respectively,Suppose the interest rate is r.

(a) Write down Peregrine’s intertemporal budget constraint in present

value terms,c

1

+

c

2

(1+r)

= m

1

+

m

2

(1+r)

.

(b) If Peregrine does not consume anything in period 1,what is the most

he can consume in period 2? m

1

(1+r)+m

2

,This is the (future

value,present value) of his endowment,Future value.

(c) If Peregrine does not consume anything in period 2,what is the most

he can consume in period 1? m

1

+

m

2

(1+r)

,This is the (future value,

present value) of his endowment,Present value,What is the

slope of Peregrine’s budget line(1 +r).

10.2 (0) Molly has a Cobb-Douglas utility function U(c

1;c

2

)=c

a

1

c

1?a

2

,

where 0 <a<1andwherec

1

and c

2

are her consumptions in periods 1

and 2 respectively,We saw earlier that if utility has the form u(x

1;x

2

)=

x

a

1

x

1?a

2

and the budget constraint is of the \standard" form p

1

x

1

+p

2

x

2

=

m,then the demand functions for the goods are x

1

= am=p

1

and x

2

=

(1?a)m=p

2

.

NAME 133

(a) Suppose that Molly’s income is m

1

in period 1 and m

2

in period 2.

Write down her budget constraint in terms of present values,c

1

+

c

2

=(1 +r)=m

1

+m

2

=(1 +r).

(b) We want to compare this budget constraint to one of the standard

form,In terms of Molly’s budget constraint,what is p

1

1,What

is p

2

1=(1 +r),What is m? m

1

+m

2

=(1 +r).

(c) If a =,2,solve for Molly’s demand functions for consumption in

each period as a function of m

1

,m

2

,andr,Her demand function for

consumption in period 1 is c

1

=,2m

1

+,2m

2

=(1 + r),Her

demand function for consumption in period 2 is c

2

=,8(1+r)m

1

+

:8m

2

.

(d) An increase in the interest rate will decrease her period-1

consumption,It will increase her period-2 consumption and

increase her savings in period 1.

10.3 (0) Nickleby has an income of $2,000 this year,and he expects an

income of $1,100 next year,He can borrow and lend money at an interest

rate of 10%,Consumption goods cost $1 per unit this year and there is

no inflation.

134 INTERTEMPORAL CHOICE (Ch,10)

01234

1

2

3

Consumption this year in 1,000s

Consumption next year in 1,000s

4

e

a

Squiggly

line

Red line

Blue

line

(a) What is the present value of Nickleby’s endowment? $3,000.

What is the future value of his endowment? $3,300,With blue

ink,show the combinations of consumption this year and consumption

next year that he can a ord,Label Nickelby’s endowment with the letter

E.

(b) Suppose that Nickleby has the utility function U(C

1;C

2

)=C

1

C

2

.

Write an expression for Nickleby’s marginal rate of substitution between

consumption this year and consumption next year,(Your answer will be

a function of the variables C

1;C

2

.) MRS =?C

2

=C

1

.

(c) What is the slope of Nickleby’s budget line1:1,Write an

equation that states that the slope of Nickleby’s indi erence curve is equal

to the slope of his budget line when the interest rate is 10%,1:1=

C

2

=C

1

,Also write down Nickleby’s budget equation,C

1

+

C

2

=1:1=3;000.

(d) Solve these two equations,Nickleby will consume 1,500 units

in period 1 and 1,650 units in period 2,Label this point A on your

diagram.

NAME 135

(e) Will he borrow or save in the rst period? Save,How much?

500.

(f) On your graph use red ink to show what Nickleby’s budget line would

be if the interest rate rose to 20%,Knowing that Nickleby chose the

point A at a 10% interest rate,even without knowing his utility function,

you can determine that his new choice cannot be on certain parts of his

new budget line,Draw a squiggly mark over the part of his new budget

line where that choice can not be,(Hint,Close your eyes and think of

WARP.)

(g) Solve for Nickleby’s optimal choice when the interest rate is 20%.

Nickleby will consume 1,458.3 units in period 1 and 1,750

units in period 2.

(h) Will he borrow or save in the rst period? Save,How much?

541.7.

10.4 (0) Decide whether each of the following statements is true or

false,Then explain why your answer is correct,based on the Slutsky

decomposition into income and substitution e ects.

(a) \If both current and future consumption are normal goods,an increase

in the interest rate will necessarily make a saver save more." False.

Substitution effect makes him consume less

in period 1 and save more,For a saver,

income effect works in opposite direction.

Either effect could dominate.

(b) \If both current and future consumption are normal goods,an in-

crease in the interest rate will necessarily make a saver choose more

consumption in the second period." True,The income

and substitution effects both lead to more

consumption in the second period.

10.5 (1) Laertes has an endowment of $20 each period,He can borrow

money at an interest rate of 200%,and he can lend money at a rate of

0%,(Note,If the interest rate is 0%,for every dollar that you save,you

get back $1 in the next period,If the interest rate is 200%,then for every

dollar you borrow,you have to pay back $3 in the next period.)

136 INTERTEMPORAL CHOICE (Ch,10)

(a) Use blue ink to illustrate his budget set in the graph below,(Hint:

The boundary of the budget set is not a single straight line.)

010203040

10

20

30

C1

C2

40

Red line

Blue line

Black line

(b) Laertes could invest in a project that would leave him with m

1

=30

and m

2

= 15,Besides investing in the project,he can still borrow at 200%

interest or lend at 0% interest,Use red ink to draw the new budget set

in the graph above,Would Laertes be better o or worse o by investing

in this project given his possibilities for borrowing or lending? Or can’t

one tell without knowing something about his preferences? Explain.

Better off,If he invests in the project,

he can borrow or lend to get any bundle he

could afford without investing.

(c) Consider an alternative project that would leave Laertes with the

endowment m

1

= 15,m

2

= 30,Again suppose he can borrow and lend

as above,But if he chooses this project,he can’t do the rst project.

Use pencil or black ink to draw the budget set available to Laertes if he

chooses this project,Is Laertes better o or worse o by choosing this

project than if he didn’t choose either project? Or can’t one tell without

knowing more about his preferences? Explain,Can’t tell,He

can afford some things he couldn’t afford

originally,But some things he could afford

before,he can’t afford if he invests in

this project.

10.6 (0) The table below reports the inflation rate and the annual rate

of return on treasury bills in several countries for the years 1984 and 1985.

NAME 137

Inflation Rate and Interest Rate for Selected Countries

% Inflation % Inflation % Interest % Interest

Country Rate,1984 Rate,1985 Rate,1984 Rate,1985

United States 3.6 1.9 9.6 7.5

Israel 304.6 48.1 217.3 210.1

Switzerland 3.1 0.8 3.6 4.1

W,Germany 2.2?0:2 5.3 4.2

Italy 9.2 5.8 15.3 13.9

Argentina 90.0 672.2 NA NA

Japan 0.6 2.0 NA NA

(a) In the table below,use the formula that your textbook gives for the

exact real rate of interest to compute the exact real rates of interest.

(b) What would the nominal rate of return on a bond in Argentina have

to be to give a real rate of return of 5% in 1985? 710:8%,What

would the nominal rate of return on a bond in Japan have to be to give

a real rate of return of 5% in 1985? 7.1%.

(c) Subtracting the inflation rate from the nominal rate of return gives

a good approximation to the real rate for countries with a low rate of

inflation,For the United States in 1984,the approximation gives you

6% while the more exact method suggested by the text gives you

5.79%,But for countries with very high inflation this is a poor

approximation,The approximation gives you?87:3% for Israel

in 1984,while the more exact formula gives you?21:57%,For

Argentina in 1985,the approximation would tell us that a bond yielding

a nominal rate of 677.7% would yield a real interest rate of 5%,This

contrasts with the answer 710.8% that you found above.

138 INTERTEMPORAL CHOICE (Ch,10)

Real Rates of Interest in 1984 and 1985

Country 1984 1985

United States 5.7 5.5

Israel?21:57 109.4

Switzerland 0.5 3.33

W,Germany 3.0 4.4

Italy 5.6 7.6

10.7 (0) We return to the planet Mungo,On Mungo,macroeconomists

and bankers are jolly,clever creatures,and there are two kinds of money,

red money and blue money,Recall that to buy something in Mungo you

have to pay for it twice,once with blue money and once with red money.

Everything has a blue-money price and a red-money price,and nobody

is ever allowed to trade one kind of money for the other,There is a blue-

money bank where you can borrow and lend blue money at a 50% annual

interest rate,There is a red-money bank where you can borrow and lend

red money at a 25% annual interest rate.

A Mungoan named Jane consumes only one commodity,ambrosia,

but it must decide how to allocate its consumption between this year and

next year,Jane’s income this year is 100 blue currency units and no red

currency units,Next year,its income will be 100 red currency units and

no blue currency units,The blue currency price of ambrosia is one b.c.u.

per flagon this year and will be two b.c.u.’s per flagon next year,The red

currency price of ambrosia is one r.c.u,per flagon this year and will be

the same next year.

(a) If Jane spent all of its blue income in the rst period,it would be

enough to pay the blue price for 100 flagons of ambrosia,If Jane

saved all of this year’s blue income at the blue-money bank,it would

have 150 b.c.u.’s next year,This would give it enough blue currency

to pay the blue price for 75 flagons of ambrosia,On the graph

below,draw Jane’s blue budget line,depicting all of those combinations

of current and next period’s consumption that it has enough blue income

to buy.

NAME 139

025 75100

25

50

75

Ambrosia this period

Ambrosia next period

100

50

(b) If Jane planned to spend no red income in the next period and to

borrow as much red currency as it can pay back with interest with next

period’s red income,how much red currency could it borrow? 80.

(c) The (exact) real rate of interest on blue money is?25%,The

real rate of interest on red money is 25%.

(d) On the axes below,draw Jane’s blue budget line and its red budget

line,Shade in all of those combinations of current and future ambrosia

consumption that Jane can a ord given that it has to pay with both

currencies.

140 INTERTEMPORAL CHOICE (Ch,10)

025 75100

25

50

75

Ambrosia this period

Ambrosia next period

100

50

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

Blue

line

Red line

c

(e) It turns out that Jane nds it optimal to operate on its blue budget

line and beneath its red budget line,Find such a point on your graph and

mark it with a C.

(f) On the following graph,show what happens to Jane’s original budget

set if the blue interest rate rises and the red interest rate does not change.

On your graph,shade in the part of the new budget line where Jane’s

new demand could possibly be,(Hint,Apply the principle of revealed

preference,Think about what bundles were available but rejected when

Jane chose to consume at C before the change in blue interest rates.)

NAME 141

025 75100

25

50

75

Ambrosia this period

Ambrosia next period

100

50

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

Blue

line

Shaded region

c

New blue

line

Red

line

10.8 (0) Mr,O,B,Kandle will only live for two periods,In the rst

period he will earn $50,000,In the second period he will retire and live

on his savings,His utility function is U(c

1;c

2

)=c

1

c

2

,wherec

1

is con-

sumption in period 1 and c

2

is consumption in period 2,He can borrow

and lend at the interest rate r =,10.

(a) If the interest rate rises,will his period-1 consumption increase,de-

crease,or stay the same? Stay the same,His demand

for c

1

is,5(m

1

+m

2

=(1 +r)) and m

2

=0.

(b) Would an increase in the interest rate make him consume more or

less in the second period? More,He saves the same

amount,but with higher interest rates,he

gets more back next period.

(c) If Mr,Kandle’s income is zero in period 1,and $ 55,000 in period 2,

would an increase in the interest rate make him consume more,less,or

thesameamountinperiod1? Less.

10.9 (1) Harvey Habit’s utility function is U(c

1;c

2

)=minfc

1;c

2

g,where

c

1

is his consumption of bread in period 1 and c

2

is his consumption of

bread in period 2,The price of bread is $1 per loaf in period 1,The

interest rate is 21%,Harvey earns $2,000 in period 1 and he will earn

$1,100 in period 2.

142 INTERTEMPORAL CHOICE (Ch,10)

(a) Write Harvey’s budget constraint in terms of future value,assuming

no inflation,1:21c

1

+c

2

=3;520.

(b) How much bread does Harvey consume in the rst period and how

much money does he save? (The answer is not necessarily an integer.)

He picks c

1

= c

2

,Substitute into the budget

to find c

1

=3;520=2:21 = 1;592:8,He saves

2;000?3;520=2:21 = 407:2.

(c) Suppose that Harvey’s money income in both periods is the same as

before,the interest rate is still 21%,but there is a 10% inflation rate.

Then in period 2,a loaf of bread will cost $ 1.10,Write down Har-

vey’s budget equation for period-1 and period-2 bread,given this new

information,1:21c

1

+1:1c

2

=3;520.

10.10 (2) In an isolated mountain village,the only crop is corn,Good

harvests alternate with bad harvests,This year the harvest will be 1,000

bushels,Next year it will be 150 bushels,There is no trade with the

outside world,Corn can be stored from one year to the next,but rats

will eat 25% of what is stored in a year,The villagers have Cobb-Douglas

utility functions,U(c

1;c

2

)=c

1

c

2

where c

1

is consumption this year,and

c

2

is consumption next year.

(a) Use red ink to draw a \budget line," showing consumption possibilities

for the village,with this year’s consumption on the horizontal axis and

next year’s consumption on the vertical axis,Put numbers on your graph

to show where the budget line hits the axes.

Next year's consumption

This year's consumption

150

1250

1000

900

1150

1136

1111

Red line

Black line

Blue line

NAME 143

(b) How much corn will the villagers consume this year? 600

bushels,How much will the rats eat? 100 bushels,How

much corn will the villagers consume next year? 450 bushels.

(c) Suppose that a road is built to the village so that now the village is

able to trade with the rest of the world,Now the villagers are able to buy

and sell corn at the world price,which is $1 per bushel,They are also

able to borrow and lend money at an interest rate of 10%,On your graph,

use blue ink to draw the new budget line for the villagers,Solve for the

amount they would now consume in the rst period 568 bushels

and the second period 624 bushels.

(d) Suppose that all is as in the last part of the question except that there

is a transportation cost of $.10 per bushel for every bushel of grain hauled

into or out of the village,On your graph,use black ink or pencil to draw

the budget line for the village under these circumstances.

10.11 (0) The table below records percentage interest rates and inflation

rates for the United States in some recent years,Complete this table.

Inflation and Interest in the United States,1965-1985

Year 1965 1970 1975 1978 1980 1985

CPI,Start of Year 38.3 47.1 66.3 79.2 100.0 130.0

CPI,End of Year 39.4 49.2 69.1 88.1 110.4 133

% Inflation Rate 2.9 4.3 4.2 11.3 10.4 2.3

Nominal Int,Rate 4.0 6.4 5.8 7.2 11.6 7.5

Real Int,Rate 1.1 2.1 1.6?3.7 1.09 5.07

(a) People complained a great deal about the high interest rates in

the late 70s,In fact,interest rates had never reached such heights

in modern times,Explain why such complaints are misleading.

Nominal interest rates were high,but so

was inflation,Real interest rates were

not high,(They were negative in 1978.)

144 INTERTEMPORAL CHOICE (Ch,10)

(b) If you gave up a unit of consumption goods at the beginning of 1985

and saved your money at interest,you could use the proceeds of your

saving to buy 1.05 units of consumption goods at the beginning of

1986,If you gave up a unit of consumption goods at the beginning of

1978 and saved your money at interest,you would be able to use the

proceeds of your saving to buy,96 units of consumption goods at the

beginning of 1979.

10.12 (1) Marsha Mellow doesn’t care whether she consumes in period

1 or in period 2,Her utility function is simply U(c

1;c

2

)=c

1

+ c

2

.Her

initial endowment is $20 in period 1 and $40 in period 2,In an antique

shop,she discovers a cookie jar that is for sale for $12 in period 1 and that

she is certain she can sell for $20 in period 2,She derives no consumption

bene ts from the cookie jar,and it costs her nothing to store it for one

period.

(a) On the graph below,label her initial endowment,E,and use blue ink

to draw the budget line showing combinations of period-1 and period-2

consumption that she can a ord if she doesn’t buy the cookie jar,On the

same graph,label the consumption bundle,A,that she would have if she

did not borrow or lend any money but bought the cookie jar in period 1,

sold it in period 2,and used the proceeds to buy period-2 consumption.

If she cannot borrow or lend,should Marsha invest in the cookie jar?

Yes.

(b) Suppose that Marsha can borrow and lend at an interest rate of 50%.

On the graph where you labelled her initial endowment,draw the budget

line showing all of the bundles she can a ord if she invests in the cookie

jar and borrows or lends at the interest rate of 50%,On the same graph

use red ink to draw one or two of Marsha’s indi erence curves.

020406080

20

40

60

Period-1 consumption

Period-2 consumption

80

,

e

a

Blue

line

Red

curves

NAME 145

(c) Suppose that instead of consumption in the two periods being per-

fect substitutes,they are perfect complements,so that Marsha’s utility

function is minfc

1;c

2

g,If she cannot borrow or lend,should she buy the

cookie jar? No,If she can borrow and lend at an interest rate of 50%,

should she invest in the cookie jar? Yes,If she can borrow or lend as

much at an interest rate of 100%,should she invest in the cookie jar?

No.

146 INTERTEMPORAL CHOICE (Ch,10)

Chapter 11 NAME

Asset Markets

Introduction,The fundamental equilibrium condition for asset markets

is that in equilibrium the rate of return on all assets must be the same.

Thus if you know the rate of interest and the cash flow generated by an

asset,you can predict what its market equilibrium price will be,This

condition has many interesting implications for the pricing of durable

assets,Here you will explore several of these implications.

Example,A drug manufacturing rm owns the patent for a new medicine.

The patent will expire on January 1,1996,at which time anyone can pro-

duce the drug,Whoever owns the patent will make a pro t of $1,000,000

per year until the patent expires,For simplicity,let us suppose that prof-

its for any year are all collected on December 31,The interest rate is

5%,Let us gure out what the selling price of the patent rights will be

on January 1,1993,On January 1,1993,potential buyers realize that

owning the patent will give them $1,000,000 every year starting 1 year

from now and continuing for 3 years,The present value of this cash flow

is

$

1;000;000

(1:05)

+

1;000;000

(1:05)

2

+

1;000;000

(1:05)

3

$2;723;248:

Nobody would pay more than this amount for the patent since if you put

$2,723,248 at 5% interest,you could collect $1,000,000 a year from the

bank for 3 years,starting 1 year from now,The patent wouldn’t sell for

less than $2,723,248,since if it sold for less,one would get a higher rate

of return by investing in this patent than one could get from investing in

anything else,What will the price of the patent be on January 1,1994?

At that time,the patent is equivalent to a cash flow of $1,000,000 in 1

year and another $1,000,000 in 2 years,The present value of this flow,

viewed from the standpoint of January 1,1994,will be

$

1;000;000

(1:05)

+

1;000;000

(1:05)

2

$1;859;310:

A slightly more di cult problem is one where the cash flow from an

asset depends on how the asset is used,To nd the price of such an asset,

one must ask what will be the present value of the cash flow that the asset

yields if it is managed in such a way as to maximize its present value.

Example,People will be willing to pay $15 a bottle to drink a certain wine

this year,Next year they would be willing to pay $25,and the year after

that they would be willing to pay $26,After that,it starts to deteriorate

and the amount people are willing to pay to drink it falls,The interest

rate is 5%,We can determine not only what the wine will sell for but

also when it will be drunk,If the wine is drunk in the rst year,it would

have to sell for $15,But no rational investor is going to sell the wine for

148 ASSET MARKETS (Ch,11)

$15 in the rst year,because it will sell for $25 one year later,This is a

66:66% rate of return,which is better than the rate of interest,When the

interest rate is 5%,investors are willing to pay at least $25=1:05 = $23:81

for the wine,So investors must outbid drinkers,and none will be drunk

this year,Will investors want to hold onto the wine for 2 years? In 2

years,the wine will be worth $26,so the present value of buying the wine

and storing it for 2 years is $26=(1:05)

2

= $23:58,This is less than the

present value of holding the wine for 1 year and selling it for $25,So,we

conclude that the wine will be drunk after 1 year,Its current selling price

will be $23:81,and 1 year from now,it will sell for $25.

11.0 Warm Up Exercise,Here are a few problems on present val-

ues,In all of the following examples,assume that you can both borrow

and lend at an annual interest rate of r andthattheinterestratewill

remain the same forever.

(a) You would be indi erent between getting $1 now and 1+r dollars,

one year from now,because if you put the dollar in the bank,then one

year from now you could get back 1+r dollars from the bank.

(b) You would be indi erent between getting 1 dollar(s) one

year from now and getting $1=(1 +r) dollars now,because 1=(1 +r)

deposited in the bank right now would enable you to withdraw principal

and interest worth $1.

(c) For any X>0,you would be indi erent between getting X=(1 +

r) dollars right now and $X one year from now,The present value of

$X received one year from now is X=(1 +r) dollars.

(d) The present value of an obligation to pay $X one year from now is

X=(1 +r) dollars.

(e) The present value of $X,to be received 2 years from now,is

X=(1 +r)

2

dollars.

(f) The present value of an asset that pays X

t

dollars t years from now

is X

t

=(1 +r)

t

dollars.

NAME 149

(g) The present value of an asset that pays $X

1

one year from now,$X

2

in

two years,and $X

10

ten years from now is X

1

=(1+r)+X

2

=(1+

r)

2

+X

10

=(1 +r)

10

dollars.

(h) Thepresentvalueofanassetthatpaysaconstantamount,$X per

year forever can be computed in two di erent ways,One way is to gure

out the amount of money you need in the bank so that the bank would

give you $X per year,forever,without ever exhausting your principal.

The annual interest received on a bank account of X=r dollars will

be $X,Therefore having X=r dollars right now is just as good as

getting $X a year forever.

(i) Another way to calculate the present value of $X a year forever is to

evaluate the in nite series

P

1

i=1

X=(1+r)

i

,This series is known

as a geometric series,Whenever r>0,this sum is well de ned

and is equal to X=r.

(j) If the interest rate is 10%,the present value of receiving $1,000 one

year from now will be,to the nearest dollar,$909,The present

value of receiving $1,000 a year forever,will be,to the nearest dollar,

$10,000.

(k) If the interest rate is 10%,what is the present value of an asset that

requires you to pay out $550 one year from now and will pay you back

$1,210 two years from now550=(1:1) + 1;210=(1:1)

2

=

500 dollars.

11.1 (0) An area of land has been planted with Christmas trees,On

December 1,ten years from now,the trees will be ready for harvest,At

that time,the standing Christmas trees can be sold for $1,000 per acre.

The land,after the trees have been removed,will be worth $200 per acre.

There are no taxes or operating expenses,but also no revenue from this

land until the trees are harvested,The interest rate is 10%.

(a) What can we expect the market price of the land to be?

$1;200=(1:1)

10

$463 per acre.

150 ASSET MARKETS (Ch,11)

(b) Suppose that the Christmas trees do not have to be sold after 10

years,but could be sold in any year,Their value if they are cut before

they are 10 years old is zero,After the trees are 10 years old,an acre of

trees is worth $1,000 and its value will increase by $100 per year for the

next 20 years,After the trees are cut,the land on which the trees stood

can always be sold for $200 an acre,When should the trees be cut to

maximize the present value of the payments received for trees and land?

After 10 years,What will be the market price of an acre of

land? Still $463.

11.2 (0) Publicity agents for the Detroit Felines announce the signing

of a phenomenal new quarterback,Archie Parabola,They say that the

contract is worth $1,000,000 and will be paid in 20 installments of $50,000

per year starting one year from now and with one new installment each

year for next 20 years,The contract contains a clause that guarantees he

will get all of the money even if he is injured and cannot play a single game.

Sports writers declare that Archie has become an \instant millionaire."

(a) Archie’s brother,Fenwick,who majored in economics,explains to

Archie that he is not a millionaire,In fact,his contract is worth less than

half a million dollars,Explain in words why this is so.

The present value of $50,000 a year for 20

years is less than $1,000,000,since the

present value of a dollar received in the

future is less than $1.

Archie’s college course on \Sports Management" didn’t cover present

values,So his brother tried to reason out the calculation for him,Here

is how it goes:

(b) Suppose that the interest rate is 10% and is expected to remain at

10% forever,How much would it cost the team to buy Archie a perpetuity

that would pay him and his heirs $1 per year forever,startingin1year?

$10.

(c) How much would it cost to buy a perpetuity that paid $50,000 a year

forever,starting in one year? $500,000.

In the last part,you found the present value of Archie’s contract

if he were going to get $50,000 a year forever,But Archie is not going

to get $50;000 a year forever,The payments stop after 20 years,The

present value of Archie’s actual contract is the same as the present value

of a contract that pays him $50,000 a year forever,but makes him pay

back $50,000 each year,forever,starting 21 years from now,Therefore

you can nd the present value of Archie’s contract by subtracting the

NAME 151

present value of $50,000 a year forever,starting 21 years from now from

the present value of $50,000 a year forever.

(d) If the interest rate is and will remain at 10%,a stream of payments

of $50,000 a year,starting 21 years from now has the same present value

as a lump sum of $ 500,000 to be received all at once,exactly 20

years from now.

(e) If the interest rate is and will remain at 10%,what is the present value

of $50,000 per year forever,starting 21 years from now? $75,000.

(Hint,The present value of $1 to be paid in 20 years is 1=(1+r)

20

=,15.)

(f) Now calculate the present value of Archie’s contract,8:50

50;000 = $425;000.

11.3 (0) Professor Thesis is puzzling over the formula for the present

value of a stream of payments of $1 a year,starting 1 year from now and

continuing forever,He knows that the value of this stream is expressed

by the in nite series

S =

1

1+r

+

1

(1 +r)

2

+

1

(1 +r)

3

+:::;

but he can’t remember the simpli ed formula for this sum,All he knows

is that if the rst payment were to arrive today,rather than a year from

now,the present value of the sum would be $1 higher,So he knows that

S +1=1+

1

(1 +r)

+

1

(1 +r)

2

+

1

(1 +r)

3

+::::

Professor Antithesis su ers from a similar memory lapse,He can’t

remember the formula for S either,But,he knows that the present value

of $1 a year forever,starting right now has to be 1 + r times as large as

the present value of $1 a year,starting a year from now,(This is true

because if you advance any income stream by a year,you multiply its

present value by 1+r.) That is,

1+

1

(1 +r)

+

1

(1 +r)

2

+

1

(1 +r)

3

+:::=(1+r)S:

(a) If Professor Thesis and Professor Antithesis put their knowledge to-

gether,they can express a simple equation involving only the variable S.

This equation is S +1= (1 +r)S,Solving this equation,they nd

that S = 1=r.

152 ASSET MARKETS (Ch,11)

(b) The two professors have also forgotten the formula for the present

value of a stream of $1 per year starting next year and continuing for K

years,They agree to call this number S(K) and they see that

S(K)=

1

(1 +r)

+

1

(1 +r)

2

+:::+

1

(1 +r)

K

:

Professor Thesis notices that if each of the payments came 1 year earlier,

the present value of the resulting stream of payments would be

1+

1

(1 +r)

+

1

(1 +r)

2

+:::+

1

(1 +r)

K?1

= S(K)+1?

1

(1 +r)

K

:

Professor Antithesis points out that speeding up any stream of payments

by a year is also equivalent to multiplying its present value by (1 + r).

Putting their two observations together,the two professors noticed an

equation that could be solved for S(K),This equation is S(K)+1?

1

(1+r)

K

= S(K)(1 +r),Solving this equation for S(K),they nd

that the formula for S(K)is S(K)=(1?

1

(1+r)

K

)=r.

Calculus 11.4 (0) You are the business manager of P,Bunyan Forests,Inc.,and

are trying to decide when you should cut your trees,The market value of

the lumber that you will get if you let your trees reach the age of t years

is given by the function W(t)=e

:20t?:001t

2

,Mr,Bunyan can earn an

interest rate of 5% per year on money in the bank.

The rate of growth of the market value of the trees will be greater

than 5% until the trees reach 75 years of age,(Hint,It follows

from elementary calculus that if F(t)=e

g(t)

,thenF

0

(t)=F(t)=g

0

(t).)

(a) If he is only interested in the trees as an investment,how old should

Mr,Bunyan let the trees get? 75 years.

(b) At what age do the trees have the greatest market value? 100

years.

11.5 (0) You expect the price of a certain painting to rise by 8% per

year forever,The market interest rate for borrowing and lending is 10%.

Assume there are no brokerage costs in purchasing or selling.

(a) If you pay a price of $x for the painting now and sell it in a year,how

much has it cost you to hold the painting rather than to have loaned the

$x at the market interest rate? It has cost,02x.

NAME 153

(b) You would be willing to pay $100 a year to have the painting on your

walls,Write an equation that you can solve for the price x at which you

would be just willing to buy the painting,02x = 100.

(c) How much should you be willing to pay to buy the painting?

$5,000.

11.6 (2) Ashley is thinking of buying a truckload of wine for investment

purposes,He can borrow and lend as much as he likes at an annual

interest rate of 10%,He is looking at three kinds of wine,To keep our

calculations simple,let us assume that handling and storage costs are

negligible.

Wine drinkers would pay exactly $175 a case to drink Wine A today.

But if Wine A is allowed to mature for one year,it will improve,In fact

wine drinkers will be willing to pay $220 a case to drink this wine one

year from today,After that,the wine gradually deteriorates and becomes

less valuable every year.

From now until one year from now,Wine B is indistinguishable

from Wine A,But instead of deteriorating after one year,Wine B will

improve,In fact the amount that wine drinkers would be willing to pay

to drink Wine B will be $220 a case in one year and will rise by $10 per

case per year for the next 30 years.

Wine drinkers would be willing to pay $100 per case to drink Wine

C right now,But one year from now,they will be willing to pay $250

per case to drink it and the amount they will be willing to pay to drink

it will rise by $50 per case per year for the next 20 years.

(a) What is the most Ashley would be willing to pay per case for Wine

A? $200.

(b) What is the most Ashley would be willing to pay per case for Wine

B? $200,(Hint,When will Wine B be drunk?)

(c) How old will Wine C be when it rst becomes worthwhile for investors

to sell o their holdings and for drinkers to drink it? 6 years.

(Hint,When does the rate of return on holding wine get to 10%?)

(d) What will the price of Wine C beatthetimeitis rstdrunk?

$500 per case.

154 ASSET MARKETS (Ch,11)

(e) What is the most that Ashley would be willing to pay today for a case

of Wine C? (Hint,What is the present value of his investment if he sells

it to a drinker at the optimal time?) Express your answer in exponential

notation without calculating it out,$500=1:1

6

.

11.7 (0) Fisher Brown is taxed at 40% on his income from ordinary

bonds,Ordinary bonds pay 10% interest,Interest on municipal bonds is

not taxed at all.

(a) If the interest rate on municipal bonds is 7%,should he buy municipal

bonds or ordinary bonds? Brown should buy municipal

bonds.

(b) Hunter Black makes less money than Fisher Brown and is taxed at

only 25% on his income from ordinary bonds,Which kind of bonds should

he buy? Black should buy ordinary bonds.

(c) If Fisher has $1,000,000 in bonds and Hunter has $10,000 in bonds,

how much tax does Fisher pay on his interest from bonds? 0.

How much tax does Hunter pay on his interest from bonds? $250.

(d) The government is considering a new tax plan under which no interest

income will be taxed,If the interest rates on the two types of bonds do

not change,and Fisher and Hunter are allowed to adjust their portfolios,

how much will Fisher’s after-tax income be increased? $30,000.

How much will Hunter’s after-tax income be increased? $250.

(e) What would the change in the tax law do to the demand for municipal

bonds if the interest rates did not change? It would reduce

it to zero.

(f) What interest rate will new issues of municipal bonds have to pay

in order to attract purchasers? They will have to pay

10%.

NAME 155

(g) What do you think will happen to the market price of the old mu-

nicipal bonds,which had a 7% yield originally? The price of

the old bonds will fall until their yield

equals 10%.

11.8 (0) In the text we discussed the market for oil assuming zero

production costs,but now suppose that it is costly to get the oil out of

the ground,Suppose that it costs $5 dollars per barrel to extract oil from

the ground,Let the price in period t be denoted by p

t

and let r be the

interest rate.

(a) If a rm extracts a barrel of oil in period t,howmuchpro tdoesit

make in period t? p

t

5.

(b) If a rm extracts a barrel of oil in period t+1,how muchpro tdoes

it make in period t+1? p

t+1

5.

(c) What is the present value of the pro ts from extracting a barrel of oil

in period t+1? (p

t+1

5)=(1+r)

t+1

,What is the present value of

pro t from extracting a barrel of oil in periodt? (p

t

5)=(1+r)

t

.

(d) If the rm is willing to supply oil in each of the two periods,what

must be true about the relation between the present value of pro ts

from sale of a barrel of oil in the two periods? The present

values must be equal,Express this relation as an equation.

p

t+1

5

(1+r)

t+1

=

p

t

5

(1+r)

t

.

(e) Solve the equation in the above part for p

t+1

as a function of p

t

and

r,p

t+1

=(1+r)p

t

5r.

(f) Is the percentage rate of price increase between periods larger or

smaller than the interest rate? The percent change in

price is smaller.

11.9 (0) Dr,No owns a bond,serial number 007,issued by the James

Company,The bond pays $200 for each of the next three years,at which

time the bond is retired and pays its face value of $2,000.

156 ASSET MARKETS (Ch,11)

(a) How much is the James bond 007 worth to Dr,No at an interest rate

of 10%? 200=1:1+200=1:1

2

+200=1:1

3

+2;000=1:1

3

=

2;000.

(b) How valuable is James bond 007 at an interest rate of 5%?

200=1:05 + 200=1:05

2

+ 200=1:05

3

+2;000=1:05

3

=

2;272:32.

(c) Ms,Yes o ers Dr,No $2,200 for the James bond 007,Should Dr,No

say yes or no to Ms,Yes if the interest rate is 10%? Yes,What if

the interest rate is 5%? No.

(d) In order to destroy the world,Dr,No hires Professor Know to develop

a nasty zap beam,In order to lure Professor Know from his university

position,Dr,No will have to pay the professor $200 a year,The nasty

zap beam will take three years to develop,at the end of which it can be

built for $2,000,If the interest rate is 5%,how much money will Dr,No

need today to nance this dastardly program? $2,272.32,

which is the present value calculated in

the first part of the problem,If the interest rate

were 10%,would the world be in more or less danger from Dr,No?

More danger,since the dastardly plan is

now cheaper.

11.10 (0) Chillingsworth owns a large,poorly insulated home,His

annual fuel bill for home heating averages $300 per year,An insulation

contractor suggests to him the following options.

Plan A,Insulate just the attic,If he does this,he will permanently

reduce his fuel consumption by 15%,Total cost of insulating the attic is

$300.

Plan B,Insulate the attic and the walls,If he does this,he will perma-

nently reduce his fuel consumption by 20%,Total cost of insulating the

attic and the walls is $500.

Plan C,Insulate the attic and the walls,and install a solar heating unit.

If he does this,he will permanently reduce his fuel costs to zero,Total cost

of this option is $7,000 for the solar heater and $500 for the insulating.

NAME 157

(a) Assume for simplicity of calculations that the house and the insulation

will last forever,Calculate the present value of the dollars saved on fuel

from each of the three options if the interest rate is 10%,The present

values are,Plan A? $450,Plan B? $600,Plan C? $3,000.

(b) Each plan requires an expenditure of money to undertake,The di er-

ence between the present value and the present cost of each plan is,Plan

A? 450?300 = 150,Plan B? 600?500 = 100,Plan

C? 3;000?7;500 =?4;500.

(c) If the price of fuel is expected to remain constant,which option should

he choose if he can borrow and lend at an annual interest rate of 10%?

A.

(d) Which option should he choose if he can borrow and lend at an annual

rate of 5%? B.

(e) Suppose that the government o ers to pay half of the cost of any

insulation or solar heating device,Which option would he now choose at

interest rates 10%? B,5%? C.

(f) Suppose that there is no government subsidy but that fuel prices are

expected to rise by 5% per year,What is the present value of fuel savings

from each of the three proposals if interest rates are 10%? (Hint,If

a stream of income is growing at x% and being discounted at y%,its

present value should be the same as that of a constant stream of income

discounted at (y?x)%.) Plan A? $900,Plan B? $1,200.

Plan C? $6,000,Which proposal should Chillingsworth choose if

interest rates are 10%? B,5%? C.

11.11 (1) Have you ever wondered if a college education is nancially

worthwhile? The U.S,Census Bureau collects data on income and educa-

tion that throws some light on this question,A recent census publication

(Current Population Reports,Series P-70,No,11) reports the average an-

nual wage income in 1984 of persons aged 35{44 by the level of schooling

achieved,The average wage income of high school graduates was $13,000

per year,The average wage income of persons with bachelor’s degrees

was $24,000 per year,The average wage income of persons with master’s

degrees was $28,000 per year,The average wage income of persons with

Ph.D.’s was $40,000 per year,These income di erences probably over-

state the return to education itself,because it is likely that those people

who get more education tend to be more able than those who get less.

158 ASSET MARKETS (Ch,11)

Some of the income di erence is,therefore,a return to ability rather than

to education,But just to get a rough idea of returns to education,let us

see what would be the return if the reported wage di erences are all due

to education.

(a) Suppose that you have just graduated from high school at age 18,You

want to estimate the present value of your lifetime earnings if you do not

go to college but take a job immediately,To do this,you have to make

some assumptions,Assume that you would work for 47 years,until you

are 65 and then retire,Assume also that you would make $13,000 a year

for the rest of your life,(If you were going to do this more carefully,you

would want to take into account that people’s wages vary with their age,

but let’s keep things simple for this problem.) Assume that the interest

rate is 5%,Find the present value of your lifetime earnings,(Hint,First

nd out the present value of $13,000 a year forever,Subtract from this

the present value of $13,000 a year forever,starting 47 years from now.)

$233,753.

(b) Again,supposing you have just graduated from high school at age 18,

and you want to estimate the present value of your life time earnings if

you go to college for 4 years and do not earn any wages until you graduate

from college,Assume that after graduating from college,you would work

for 43 years at $24,000 per year,What would be the present value of your

lifetime earnings? $346,441.

(c) Now calculate the present value of your lifetime earnings if you get a

master’s degree,Assume that if you get a master’s,you have no earnings

for 6 years and then you work for 41 years at $28,000 per year,What

would be the present value of your lifetime income? $361,349.

(d) Finally calculate the present value of your lifetime earnings if you get

a Ph.D,Assume that if you get a Ph.D.,you will have no earnings for 8

years and then you work for 39 years at $40,000 per year,What would

be the present value of your lifetime income? $460,712.

(e) Consider the case of someone who married right after nishing high

school and stopped her education at that point,Suppose that she is now

45 years old,Her children are nearly adults,and she is thinking about

going back to work or going to college,Assuming she would earn the

average wage for her educational level and would retire at age 65,what

would be the present value of her lifetime earnings if she does not go to

college? $162,000.

NAME 159

(f) What would be the present value of her lifetime earnings if she goes to

college for 4 years and then takes a job until she is 65? $213,990.

(g) If college tuition is $5,000 per year,is it nancially worthwhile

for her to go to college? Explain,Yes,the gain in

the present value of her income exceeds the

present value of tuition.

11.12 (0) As you may have noticed,economics is a di cult major,Are

their any rewards for all this e ort? The U.S,census publication discussed

in the last problem suggests that there might be,There are tables report-

ing wage income by the eld in which one gets a degree,For bachelor’s

degrees,the most lucrative majors are economics and engineering,The

average wage incomes for economists are about $28,000 per year and for

engineers are about $27,000,Psychology majors average about $15,000 a

year and English majors about $14,000 per year.

(a) Can you think of any explanation for these di erences? Some

might say that economics,like accounting

or mortuary science,is so boring the

pay has to be high to get you to do

it,Others would suggest that the ability

to do well in economics is scarce and

is valued by the marketplace,Perhaps

the English majors and psychology majors

include disproportionately many persons

who are not full-time participants in the

labor force,No doubt there are several

other good partial explanations.

(b) The same table shows that the average person with an advanced degree

in business earns $38,000 per year and the average person with a degree

in medicine earns $45,000 per year,Suppose that an advanced degree

in business takes 2 years after one spends 4 years getting a bachelor’s

degree and that a medical degree takes 4 years after getting a bachelor’s

160 ASSET MARKETS (Ch,11)

degree,Suppose that you are 22 years old and have just nished college,If

r =,05,nd the present value of lifetime earnings for a graduating senior

who will get an advanced degree in business and earn the average wage

rate for someone with this degree until retiring at 65,$596,000.

Make a similar calculation for medicine,$630,000.

11.13 (0) On the planet Stinko,the principal industry is turnip growing.

For centuries the turnip elds have been fertilized by guano which was

deposited by the now-extinct giant scissor-billed kiki-bird,It costs $5 per

ton to mine kiki-bird guano and deliver it to the elds,Unfortunately,the

country’s stock of kiki-bird guano is about to be exhausted,Fortunately

the scientists on Stinko have devised a way of synthesizing kiki-guano from

political science textbooks and swamp water,This method of production

makes it possible to produce a product indistinguishable from kiki-guano

and to deliver it to the turnip elds at a cost of $30 per ton,The interest

rate on Stinko is 10%,There are perfectly competitive markets for all

commodities.

(a) Given the current price and the demand function for kiki-guano,the

last of the deposits on Stinko will be exhausted exactly one year from

now,Next year,the price of kiki-guano delivered to the elds will have

to be $30,so that the synthetic kiki-guano industry will just break even.

The owners of the guano deposits know that next year,they would get a

net return of $25 a ton for any guano they have left to sell,In equilibrium,

what must be the current price of kiki-guano delivered to the turnip elds?

The price of guano delivered to the field

must be the $5 + the present value of $25.

This is 5+25=1:1=27:73,(Hint,In equilibrium,sellers

must be indi erent between selling their kiki-guano right now or at any

other time before the total supply is exhausted,But we know that they

must be willing to sell it right up until the day,one year from now,when

the supply will be exhausted and the price will be $30,the cost of synthetic

guano.)

(b) Suppose that everything is as we have said previously except that the

deposits of kiki-guano will be exhausted 10 years from now,What must

be the current price of kiki-guano? (Hint,1:1

10

=2:59.)

5+25=(1:1)

10

=14:65.

Chapter 12 NAME

Uncertainty

Introduction,In Chapter 11,you learned some tricks that allow you to

use techniques you already know for studying intertemporal choice,Here

you will learn some similar tricks,so that you can use the same methods

to study risk taking,insurance,and gambling.

One of these new tricks is similar to the trick of treating commodi-

ties at di erent dates as di erent commodities,This time,we invent

new commodities,which we call contingent commodities.Ifeitheroftwo

events A or B could happen,then we de ne one contingent commodity

as consumption if A happens and another contingent commodity as con-

sumption if B happens,The second trick is to nd a budget constraint

that correctly speci es the set of contingent commodity bundles that a

consumer can a ord.

This chapter presents one other new idea,and that is the notion

of von Neumann-Morgenstern utility,A consumer’s willingness to take

various gambles and his willingness to buy insurance will be determined

by how he feels about various combinations of contingent commodities.

Often it is reasonable to assume that these preferences can be expressed

by a utility function that takes the special form known as von Neumann-

Morgenstern utility,The assumption that utility takes this form is called

the expected utility hypothesis,If there are two events,1 and 2 with

probabilities

1

and

2

,and if the contingent consumptions are c

1

and

c

2

,then the von Neumann-Morgenstern utility function has the special

functional form,U(c

1;c

2

)=

1

u(c

1

)+

2

u(c

2

),The consumer’s behavior

is determined by maximizing this utility function subject to his budget

constraint.

Example,You are thinking of betting on whether the Cincinnati Reds

will make it to the World Series this year,A local gambler will bet with

you at odds of 10 to 1 against the Reds,You think the probability that

the Reds will make it to the World Series is =,2,If you don’t bet,

you are certain to have $1,000 to spend on consumption goods,Your

behavior satis es the expected utility hypothesis and your von Neumann-

Morgenstern utility function is

1

p

c

1

+

2

p

c

2

.

The contingent commodities are dollars if the Reds make the World

Series and dollars if the Reds don’t make the World Series.Letc

W

be

your consumption contingent on the Reds making the World Series and

c

NW

be your consumption contingent on their not making the Series.

Betting on the Reds at odds of 10 to 1 means that if you bet $x on the

Reds,then if the Reds make it to the Series,you make a net gain of $10x,

but if they don’t,you have a net loss of $x,Since you had $1,000 before

betting,if you bet $x on the Reds and they made it to the Series,you

would have c

W

=1;000 + 10x to spend on consumption,If you bet $x

on the Reds and they didn’t make it to the Series,you would lose $x,

162 UNCERTAINTY (Ch,12)

and you would have c

NW

=1;000?x,By increasing the amount $x that

you bet,you can make c

W

larger and c

NW

smaller,(You could also bet

against the Reds at the same odds,If you bet $x against the Reds and

they fail to make it to the Series,you make a net gain of,1x and if they

make it to the Series,you lose $x,If you work through the rest of this

discussion for the case where you bet against the Reds,you will see that

the same equations apply,with x being a negative number.) We can use

the above two equations to solve for a budget equation,From the second

equation,we have x =1;000?c

NW

,Substitute this expression for x into

the rst equation and rearrange terms to nd c

W

+10c

NW

=11;000,or

equivalently,:1c

W

+ c

NW

=1;100,(The same budget equation can be

written in many equivalent ways by multiplying both sides by a positive

constant.)

Then you will choose your contingent consumption bundle (c

W;c

NW

)

to maximize U(c

W;c

NW

)=:2

p

c

W

+,8

p

c

NW

subject to the budget

constraint,:1c

W

+c

NW

=1;100,Using techniques that are now familiar,

you can solve this consumer problem,From the budget constraint,you

see that consumption contingent on the Reds making the World Series

costs 1=10 as much as consumption contingent on their not making it,If

you set the marginal rate of substitution between c

W

and c

NW

equal to

the price ratio and simplify the resulting expression,you will nd that

c

NW

=,16c

W

,This equation,together with the budget equation implies

that c

W

=$4;230:77 and c

NW

= $676:92,You achieve this bundle by

betting $323:08 on the Reds,If the Reds make it to the Series,you will

have $1;000 + 10 323:08 = $4;230:80,If not,you will have $676:92.

(We rounded the solutions to the nearest penny.)

12.1 (0) In the next few weeks,Congress is going to decide whether

or not to develop an expensive new weapons system,If the system is

approved,it will be very pro table for the defense contractor,General

Statics,Indeed,if the new system is approved,the value of stock in

General Statics will rise from $10 per share to $15 a share,and if the

project is not approved,the value of the stock will fall to $5 a share,In

his capacity as a messenger for Congressman Kickback,Buzz Condor has

discovered that the weapons system is much more likely to be approved

than is generally thought,On the basis of what he knows,Condor has

decided that the probability that the system will be approved is 3/4 and

the probability that it will not be approved is 1/4,Let c

A

be Condor’s

consumption if the system is approved and c

NA

be his consumption if

the system is not approved,Condor’s von Neumann-Morgenstern utility

function is U(c

A;c

NA

)=:75 lnc

A

+,25 lnc

NA

,Condor’s total wealth is

$50,000,all of which is invested in perfectly safe assets,Condor is about

to buy stock in General Statics.

(a) If Condor buys x shares of stock,and if the weapons system is ap-

proved,he will make a pro t of $5 per share,Thus the amount he can

consume,contingent on the system being approved,is c

A

= $50;000+5x.

If Condor buys x shares of stock,and if the weapons system is not ap-

proved,then he will make a loss of $ 5 per share,Thus the

NAME 163

amount he can consume,contingent on the system not being approved,is

c

NA

= 50;000?5x.

(b) You can solve for Condor’s budget constraint on contingent commod-

ity bundles (c

A;c

NA

) by eliminating x from these two equations,His bud-

get constraint can be written as,5 c

A

+,5 c

NA

=50;000.

(c) Buzz Condor has no moral qualms about trading on inside informa-

tion,nor does he have any concern that he will be caught and punished.

To decide how much stock to buy,he simply maximizes his von Neumann-

Morgenstern utility function subject to his budget,If he sets his marginal

rate of substitution between the two contingent commodities equal to

their relative prices and simpli es the equation,he nds that c

A

=c

NA

=

3,(Reminder,Where a is any constant,the derivative of alnx

with respect to x is a=x.)

(d) Condor nds that his optimal contingent commodity bundle is

(c

A;c

NA

)= (75,000,25,000),To acquire this contingent com-

modity bundle,he must buy 5,000 shares of stock in General Statics.

12.2 (0) Willy owns a small chocolate factory,located close to a river

that occasionally floods in the spring,with disastrous consequences,Next

summer,Willy plans to sell the factory and retire,The only income he

will have is the proceeds of the sale of his factory,If there is no flood,

the factory will be worth $500,000,If there is a flood,then what is left

of the factory will be worth only $50,000,Willy can buy flood insurance

at a cost of $.10 for each $1 worth of coverage,Willy thinks that the

probability that there will be a flood this spring is 1/10,Let c

F

denote the

contingent commodity dollars if there is a flood and c

NF

denote dollars

if there is no flood,Willy’s von Neumann-Morgenstern utility function is

U(c

F;c

NF

)=:1

p

c

F

+:9

p

c

NF

.

(a) If he buys no insurance,then in each contingency,Willy’s consumption

will equal the value of his factory,so Willy’s contingent commodity bundle

will be (c

F;c

NF

)= (50;000;500;000).

(b) To buy insurance that pays him $x in case of a flood,Willy must

pay an insurance premium of,1x,(The insurance premium must be

paid whether or not there is a flood.) If Willy insures for $x,thenif

there is a flood,he gets $x in insurance bene ts,Suppose that Willy has

contracted for insurance that pays him $x in the event of a flood,Then

after paying his insurance premium,he will be able to consume c

F

=

50;000 +:9x,If Willy has this amount of insurance and there is

no flood,then he will be able to consume c

NF

= 500;000?:1x.

164 UNCERTAINTY (Ch,12)

(c) You can eliminate x from the two equations for c

F

and c

NF

that

you found above,This gives you a budget equation for Willy,Of course

there are many equivalent ways of writing the same budget equation,

since multiplying both sides of a budget equation by a positive constant

yields an equivalent budget equation,The form of the budget equation in

which the \price" of c

NF

is 1 can be written as,9c

NF

+,1 c

F

=

455,000.

(d) Willy’s marginal rate of substitution between the two contingent com-

modities,dollars if there is no flood and dollars if there is a flood,is

MRS(c

NF;c

F

)=?

:9

p

c

F

:1

p

c

NF

,To nd his optimal bundle of contingent

commodities,you must set this marginal rate of substitution equal to

the number =?9,Solving this equation,you nd that Willy will

choose to consume the two contingent commodities in the ratio

c

NF

=c

F

=1.

(e) Since you know the ratio in which he will consume c

NF

and c

F

,and

you know his budget equation,you can solve for his optimal consumption

bundle,which is (c

NF;c

F

)= (455;000; 455;000),Willy will

buy an insurance policy that will pay him $450,000 if there is a

flood,The amount of insurance premium that he will have to pay is

$45,000.

12.3 (0) Clarence Bunsen is an expected utility maximizer,His pref-

erences among contingent commodity bundles are represented by the ex-

pected utility function

u(c

1;c

2;

1;

2

)=

1

p

c

1

+

2

p

c

2

:

Clarence’s friend,Hjalmer Ingqvist,has o ered to bet him $1,000 on the

outcome of the toss of a coin,That is,if the coin comes up heads,Clarence

must pay Hjalmer $1,000 and if the coin comes up tails,Hjalmer must

pay Clarence $1,000,The coin is a fair coin,so that the probability of

heads and the probability of tails are both 1/2,If he doesn’t accept the

bet,Clarence will have $10,000 with certainty,In the privacy of his car

dealership o ce over at Bunsen Motors,Clarence is making his decision.

(Clarence uses the pocket calculator that his son,Elmer,gave him last

Christmas,You will nd that it will be helpful for you to use a calculator

too.) Let Event 1 be \coin comes up heads" and let Event 2 be \coin

comes up tails."

NAME 165

(a) If Clarence accepts the bet,then in Event 1,he will have 9,000

dollars and in Event 2,he will have 11,000 dollars.

(b) Since the probability of each event is 1/2,Clarence’s expected utility

for a gamble in which he gets c

1

in Event 1 and c

2

in Event 2 can be

described by the formula

1

2

p

c

1

+

1

2

p

c

2

,Therefore Clarence’s

expected utility if he accepts the bet with Hjalmer will be 99.8746.

(Use that calculator.)

(c) If Clarence decides not to bet,then in Event 1,he will have

10,000 dollars and in Event 2,he will have 10,000 dollars.

Therefore if he doesn’t bet,his expected utility will be 100.

(d) Having calculated his expected utility if he bets and if he does not bet,

Clarence determines which is higher and makes his decision accordingly.

Does Clarence take the bet? No.

12.4 (0) It is a slow day at Bunsen Motors,so since he has his calcu-

lator warmed up,Clarence Bunsen (whose preferences toward risk were

described in the last problem) decides to study his expected utility func-

tion more closely.

(a) Clarence rst thinks about really big gambles,What if he bet his

entire $10,000 on the toss of a coin,where he loses if heads and wins if

tails? Then if the coin came up heads,he would have 0 dollars and if it

came up tails,he would have $20,000,His expected utility if he took the

bet would be 70.71,while his expected utility if he didn’t take the

bet would be 100,Thereforeheconcludesthathewouldnottake

such a bet.

(b) Clarence then thinks,\Well,of course,I wouldn’t want to take a

chance on losing all of my money on just an ordinary bet,But,what

if somebody o ered me a really good deal,Suppose I had a chance to

bet where if a fair coin came up heads,I lost my $10,000,but if it came

up tails,I would win $50,000,Would I take the bet? If I took the bet,

my expected utility would be 122.5,If I didn’t take the bet,my

expected utility would be 100,Therefore I should take the bet."

166 UNCERTAINTY (Ch,12)

(c) Clarence later asks himself,\If I make a bet where I lose my $10,000

if the coin comes up heads,what is the smallest amount that I would have

to win in the event of tails in order to make the bet a good one for me

to take?" After some trial and error,Clarence found the answer,You,

too,might want to nd the answer by trial and error,but it is easier to

nd the answer by solving an equation,On the left side of your equation,

you would write down Clarence’s utility if he doesn’t bet,On the right

side of the equation,you write down an expression for Clarence’s utility

if he makes a bet such that he is left with zero consumption in Event 1

and x in Event 2,Solve this equation for x,The answer to Clarence’s

question is where x =10;000,The equation that you should write is

100 =

1

2

p

x,The solution is x = 40;000.

(d) Your answer to the last part gives you two points on Clarence’s in-

di erence curve between the contingent commodities,money in Event 1

and money in Event 2,(Poor Clarence has never heard of indi erence

curves or contingent commodities,so you will have to work this part for

him,while he heads over to the Chatterbox Cafe for morning co ee.) One

of these points is where money in both events is $10,000,On the graph

below,label this point A,The other is where money in Event 1 is zero

and money in Event 2 is 40,000,On the graph below,label this

point B.

010203040

10

20

30

Money in Event 1 (x 1,000)

Money in Event 2 (x 1,000)

40

a

b

c

d

(e) You can quickly nd a third point on this indi erence curve,The

coin is a fair coin,and Clarence cares whether heads or tails turn up only

because that determines his prize,Therefore Clarence will be indi erent

between two gambles that are the same except that the assignment of

prizes to outcomes are reversed,In this example,Clarence will be indif-

ferent between point B on the graph and a point in which he gets zero if

Event 2 happens and 40,000 if Event 1 happens,Find this point

on the Figure above and label it C.

NAME 167

(f) Another gamble that is on the same indi erence curve for Clarence

as not gambling at all is the gamble where he loses $5,000 if heads turn

up and where he wins 6,715.73 dollars if tails turn up,(Hint,To

solve this problem,put the utility of not betting on the left side of an

equation and on the right side of the equation,put the utility of having

$10;000?$5;000 in Event 1 and $10;000 +x in Event 2,Then solve the

resulting equation for x.) On the axes above,plot this point and label it

D,Now sketch in the entire indi erence curve through the points that

you have labeled.

12.5 (0) Hjalmer Ingqvist’s son-in-law,Earl,has not worked out very

well,It turns out that Earl likes to gamble,His preferences over contin-

gent commodity bundles are represented by the expected utility function

u(c

1;c

2;

1;

2

)=

1

c

2

1

+

2

c

2

2

:

(a) Just the other day,some of the boys were down at Skoog’s tavern

when Earl stopped in,They got to talking about just how bad a bet they

could get him to take,At the time,Earl had $100,Kenny Olson shu ed

a deck of cards and o ered to bet Earl $20 that Earl would not cut a spade

from the deck,Assuming that Earl believed that Kenny wouldn’t cheat,

the probability that Earl would win the bet was 1/4 and the probability

that Earl would lose the bet was 3/4,If he won the bet,Earl would

have 120 dollars and if he lost the bet,he would have 80

dollars,Earl’s expected utility if he took the bet would be 8,400,

and his expected utility if he did not take the bet would be 10,000.

Therefore he refused the bet.

(b) Just when they started to think Earl might have changed his ways,

Kenny o ered to make the same bet with Earl except that they would

bet $100 instead of $20,What is Earl’s expected utility if he takes that

bet? 10,000,Would Earl be willing to take this bet? He is

just indifferent about taking it or not.

(c) Let Event 1 be the event that a card drawn from a fair deck of cards is

a spade,Let Event 2 be the event that the card is not a spade,Earl’s pref-

erences between income contingent on Event 1,c

1

,and income contingent

on Event 2,c

2

,can be represented by the equation u =

1

4

c

2

1

+

3

4

c

2

2

.

Use blue ink on the graph below to sketch Earl’s indi erence curve passing

through the point (100;100).

168 UNCERTAINTY (Ch,12)

0 50 100 150 200

50

100

150

Money in Event 1

Money in Event 2

200

Blue curve

Red curves

(d) On the same graph,let us draw Hjalmer’s son-in-law Earl’s indif-

ference curves between contingent commodities where the probabilities

are di erent,Suppose that a card is drawn from a fair deck of cards.

Let Event 1 be the event that the card is black,Let event 2 be the event

that the card drawn is red,Suppose each event has probability 1/2,Then

Earl’s preferences between income contingent on Event 1 and income con-

tingent on Event 2 are represented by the formula u =

1

2

c

2

1

+

1

2

c

2

2

.

On the graph,use red ink to show two of Earl’s indi erence curves,in-

cluding the one that passes through (100;100).

12.6 (1) Sidewalk Sam makes his living selling sunglasses at the board-

walk in Atlantic City,If the sun shines Sam makes $30,and if it rains

Sam only makes $10,For simplicity,we will suppose that there are only

two kinds of days,sunny ones and rainy ones.

(a) One of the casinos in Atlantic City has a new gimmick,It is accepting

bets on whether it will be sunny or rainy the next day,The casino sells

dated \rain coupons" for $1 each,If it rains the next day,the casino will

give you $2 for every rain coupon you bought on the previous day,If it

doesn’t rain,your rain coupon is worthless,In the graph below,mark

Sam’s \endowment" of contingent consumption if he makes no bets with

the casino,and label it E.

NAME 169

010203040

10

20

30

Cs

Cr

40

e

a

Blue

line

Red

line

(b) On the same graph,mark the combination of consumption contingent

on rain and consumption contingent on sun that he could achieve by

buying 10 rain coupons from the casino,Label it A.

(c) On the same graph,use blue ink to draw the budget line representing

all of the other patterns of consumption that Sam can achieve by buying

rain coupons,(Assume that he can buy fractional coupons,but not neg-

ative amounts of them.) What is the slope of Sam’s budget line at points

above and to the left of his initial endowment? The slope is

1.

(d) Suppose that the casino also sells sunshine coupons,These tickets

also cost $1,With these tickets,the casino gives you $2 if it doesn’t rain

and nothing if it does,On the graph above,use red ink to sketch in the

budget line of contingent consumption bundles that Sam can achieve by

buying sunshine tickets.

(e) If the price of a dollar’s worth of consumption when it rains is set equal

to 1,what is the price of a dollar’s worth of consumption if it shines?

The price is 1.

12.7 (0) Sidewalk Sam,from the previous problem,has the utility func-

tion for consumption in the two states of nature

u(c

s;c

r; )=c

1?

s

c

r;

where c

s

is the dollar value of his consumption if it shines,c

r

is the dollar

value of his consumption if it rains,and is the probability that it will

rain,The probability that it will rain is =,5.

170 UNCERTAINTY (Ch,12)

(a) How many units of consumption is it optimal for Sam to consume

conditional on rain? 20 units.

(b) How many rain coupons is it optimal for Sam to buy? 10.

12.8 (0) Sidewalk Sam’s brother Morgan von Neumanstern is an ex-

pected utility maximizer,His von Neumann-Morgenstern utility function

for wealth is u(c)=lnc,Sam’s brother also sells sunglasses on another

beach in Atlantic City and makes exactly the same income as Sam does.

He can make exactly the same deal with the casino as Sam can.

(a) If Morgan believes that there is a 50% chance of rain and a 50% chance

of sun every day,what would his expected utility of consuming (c

s;c

r

)

be? u =

1

2

lnc

s

+

1

2

lnc

r

.

(b) How does Morgan’s utility function compare to Sam’s? Is one a

monotonic transformation of the other? Morgan’s utility

function is just the natural log of Sam’s,

so the answer is yes.

(c) What will Morgan’s optimal pattern of consumption be? Answer:

Morgan will consume 20 on the sunny days and 20 on

the rainy days,How does this compare to Sam’s consumption? This

is the same as Sam’s consumption.

12.9 (0) Billy John Pigskin of Mule Shoe,Texas,has a von Neumann-

Morgenstern utility function of the form u(c)=

p

c:Billy John also weighs

about 300 pounds and can outrun jackrabbits and pizza delivery trucks.

Billy John is beginning his senior year of college football,If he is not

seriously injured,he will receive a $1,000,000 contract for playing pro-

fessional football,If an injury ends his football career,he will receive a

$10,000 contract as a refuse removal facilitator in his home town,There

is a 10% chance that Billy John will be injured badly enough to end his

career.

(a) What is Billy John’s expected utility? We calculate

:1

p

10;000 +:9

p

1;000;000 = 910.

NAME 171

(b) If Billy John pays $p for an insurance policy that would give him

$1,000,000 if he su ered a career-ending injury while in college,then he

would be sure to have an income of $1;000;000?p no matter what hap-

pened to him,Write an equation that can be solved to nd the largest

price that Billy John would be willing to pay for such an insurance policy.

The equation is 910 =

q

1;000;000?p.

(c) Solve this equation for p,p = 171;900.

12.10 (1) You have $200 and are thinking about betting on the Big

Game next Saturday,Your team,the Golden Boars,are scheduled to

play their traditional rivals the Robber Barons,It appears that the going

odds are 2 to 1 against the Golden Boars,That is to say if you want

to bet $10 on the Boars,you can nd someone who will agree to pay

you $20 if the Boars win in return for your promise to pay him $10 if

the Robber Barons win,Similarly if you want to bet $10 on the Robber

Barons,you can nd someone who will pay you $10 if the Robber Barons

win,in return for your promise to pay him $20 if the Robber Barons lose.

Suppose that you are able to make as large a bet as you like,either on

the Boars or on the Robber Barons so long as your gambling losses do

not exceed $200,(To avoid tedium,let us ignore the possibility of ties.)

(a) If you do not bet at all,you will have $200 whether or not the Boars

win,If you bet $50 on the Boars,then after all gambling obligations are

settled,you will have a total of 300 dollars if the Boars win and

150 dollars if they lose,On the graph below,use blue ink to draw a

line that represents all of the combinations of \money if the Boars win"

and \money if the Robber Barons win" that you could have by betting

from your initial $200 at these odds.

172 UNCERTAINTY (Ch,12)

0 100 200 300 400

100

200

300

Money if the Boars win

Money if the Boars lose

400

e

c

d

Red line

Blue line

(b) Label the point on this graph where you would be if you did not bet

at all with an E.

(c) After careful thought you decide to bet $50 on the Boars,Label the

point you have chosen on the graph with a C,Suppose that after you have

made this bet,it is announced that the star Robber Baron quarterback

su ered a sprained thumb during a tough economics midterm examination

and will miss the game,The market odds shift from 2 to 1 against the

Boars to \even money" or 1 to 1,That is,you can now bet on either

team and the amount you would win if you bet on the winning team is

the same as the amount that you would lose if you bet on the losing team.

You cannot cancel your original bet,but you can make new bets at the

new odds,Suppose that you keep your rst bet,but you now also bet

$50 on the Robber Barons at the new odds,If the Boars win,then after

you collect your winnings from one bet and your losses from the other,

how much money will you have left? $250,If the Robber Barons

win,how much money will you have left after collecting your winnings

and paying o your losses? $200.

(d) Use red ink to draw a line on the diagram you made above,showing

the combinations of \money if the Boars win" and \money if the Robber

Barons win" that you could arrange for yourself by adding possible bets

at the new odds to the bet you made before the news of the quarterback’s

misfortune,On this graph,label the point D that you reached by making

the two bets discussed above.

12.11 (2) The certainty equivalent of a lottery is the amount of money

you would have to be given with certainty to be just as well-o with that

lottery,Suppose that your von Neumann-Morgenstern utility function

NAME 173

over lotteries that give you an amount x if Event 1 happens and y if

Event 1 does not happen is U(x;y; )=

p

x+(1? )

p

y,where is the

probability that Event 1 happens and 1? is the probability that Event

1 does not happen.

(a) If =,5,calculate the utility of a lottery that gives you $10,000

if Event 1 happens and $100 if Event 1 does not happen,55 =

:5 100 +:5 10:

(b) If you were sure to receive $4,900,what would your utility be? 70.

(Hint,If you receive $4,900 with certainty,then you receive $4,900 in

both events.)

(c) Given this utility function and =,5,write a general formula for the

certainty equivalent of a lottery that gives you $x if Event 1 happens and

$y if Event 1 does not happen,(:5x

1=2

+:5y

1=2

)

2

.

(d) Calculate the certainty equivalent of receiving $10,000 if Event 1 hap-

pens and $100 if Event 1 does not happen,$3,025.

12.12 (0) Dan Partridge is a risk averter who tries to maximize the

expected value of

p

c,wherec is his wealth,Dan has $50,000 in safe

assets and he also owns a house that is located in an area where there

are lots of forest res,If his house burns down,the remains of his house

and the lot it is built on would be worth only $40,000,giving him a total

wealth of $90,000,If his home doesn’t burn,it will be worth $200,000

and his total wealth will be $250,000,The probability that his home will

burn down is,01.

(a) Calculate his expected utility if he doesn’t buy re insurance.

$498.

(b) Calculate the certainty equivalent of the lottery he faces if he doesn’t

buy re insurance,$248,004.

(c) Suppose that he can buy insurance at a price of $1 per $100 of in-

surance,For example if he buys $100,000 worth of insurance,he will pay

$1,000 to the company no matter what happens,but if his house burns,

he will also receive $100,000 from the company,If Dan buys $160,000

worth of insurance,he will be fully insured in the sense that no matter

what happens his after-tax wealth will be $248,400.

174 UNCERTAINTY (Ch,12)

(d) Therefore if he buys full insurance,the certainty equivalent of his

wealth is $248,400,and his expected utility is

p

248;800.

12.13 (0) Portia has been waiting a long time for her ship to come in

and has concluded that there is a 25% chance that it will arrive today,If

it does come in today,she will receive $1,600,If it does not come in today,

it will never come and her wealth will be zero,Portia has a von Neumann-

Morgenstern utility such that she wants to maximize the expected value

of

p

c,wherec is total income,What is the minimum price at which she

will sell the rights to her ship? $100.

Chapter 13 NAME

Risky Assets

Introduction,Here you will solve the problems of consumers who wish

to divide their wealth optimally between a risky asset and a safe asset.

The expected rate of return on a portfolio is just a weighted average of

the rate of return on the safe asset and the expected rate of return on

the risky asset,where the weights are the fractions of the consumer’s

wealth held in each,The standard deviation of the portfolio return is

just the standard deviation of the return on the risky asset times the

fraction of the consumer’s wealth held in the risky asset,Sometimes

you will look at the problem of a consumer who has preferences over

the expected return and the risk of her portfolio and who faces a budget

constraint,Since a consumer can always put all of her wealth in the

safe asset,one point on this budget constraint will be the combination

of the safe rate of return and no risk (zero standard deviation),Now

as the consumer puts x percent of her wealth into the risky asset,she

gains on that amount the di erence between the expected rate of return

for the risky asset and the rate of return on the safe asset,But she also

absorbs some risk,So the slope of the budget line will be the di erence

between the two returns divided by the standard deviation of the portfolio

that has x percent of the consumer’s wealth invested in the risky asset.

You can then apply the usual indi erence curve{budget line analysis to

nd the consumer’s optimal choice of risk and expected return given her

preferences,(Remember that if the standard deviation is plotted on the

horizontal axis and if less risk is preferred to more,the better bundles will

lie to the northwest.) You will also be asked to apply the result from the

Capital Asset Pricing Model that the expected rate of return on any asset

is equal to the sum of the risk-free rate of return plus the risk adjustment.

Remember too that the expected rate of return on an asset is its expected

change in price divided by its current price.

13.1 (3) Ms,Lynch has a choice of two assets,The rst is a risk-free

assetthato ersarateofreturnofr

f

,and the second is a risky asset (a

china shop that caters to large mammals) that has an expected rate of

return of r

m

and a standard deviation of

m

.

(a) If x is the percent of wealth Ms,Lynch invests in the risky asset,

what is the equation for the expected rate of return on the portfolio?

r

x

= xr

m

+(1?x)r

f

,What is the equation for the standard

deviation of the portfolio?

x

= x

m

.

176 RISKY ASSETS (Ch,13)

(b) By solving the second equation above for x and substituting the result

into the rst equation,derive an expression for the rate of return on the

portfolio in terms of the portfolio’s riskiness,r

x

=

r

m

r

f

m

x

+r

f

.

(c) Suppose that Ms,Lynch can borrow money at the interest rate r

f

and invest it in the risky asset,If r

m

= 20,r

f

= 10,and

m

= 10,what

will be Ms,Lynch’s expected return if she borrows an amount equal to

100% of her initial wealth and invests it in the risky asset? (Hint,This

is just like investing 200% of her wealth in the risky asset.) Apply

the formula r

x

= xr

m

+(1?x)r

f

with x =2 to

get r

x

=2 20?1 10 = 30.

(d) Suppose that Ms,Lynch can borrow or lend at the risk-free rate,If

r

f

is 10%,r

m

is 20%,and

m

is 10%,what is the formula for the \budget

line" Ms,Lynch faces? r

x

=

x

+10,Plot this budget line in the

graph below.

010203040

10

20

30

Standard deviation

Expected return

40

Budget line

U=0

U=5

U=10

(e) Which of the following risky assets would Ms,Lynch prefer to her

present risky asset,assuming she can only invest in one risky asset at a

time and that she can invest a fraction of her wealth in whichever risky

asset she chooses? Write the word \better," \worse," or \same" after

each of the assets.

Asset A with r

a

=17% and

a

=5%,Better.

Asset B with r

b

=30% and

b

= 25%,Worse.

NAME 177

Asset C with r

c

=11% and

c

=1%,Same.

Asset D with r

d

=25% and

d

= 14%,Better.

(f) Suppose Ms,Lynch’s utility function has the formu(r

x;

x

)=r

x

2

x

.

How much of her portfolio will she invest in the original risky asset?

(You might want to graph a few of Ms,Lynch’s indi erence curves be-

fore answering; e.g.,graph the combinations of r

x

and

x

that imply

u(r

x;

x

)=0;1;:::),She will not invest anything

in the risky asset.

13.2 (3) Fenner Smith is contemplating dividing his portfolio between

two assets,a risky asset that has an expected return of 30% and a standard

deviation of 10%,and a safe asset that has an expected return of 10%

and a standard deviation of 0%.

(a) If Mr,Smith invests x percent of his wealth in the risky asset,what

will be his expected return? r

x

=30x+ 10(1?x).

(b) If Mr,Smith invests x percent of his wealth in the risky asset,what

will be the standard deviation of his wealth?

x

=10x.

(c) Solve the above two equations for the expected return on Mr,Smith’s

wealth as a function of the standard deviation he accepts,The

budget line is r

x

=2

x

+10.

(d) Plot this \budget line" on the graph below.

0 5 10 15 20

10

20

30

Standard deviation

Expected return

40

Budget line

Indifference

curves

Optimal choice

178 RISKY ASSETS (Ch,13)

(e) If Mr,Smith’s utility function is u(r

x;

x

)=minfr

x;30?2

x

g,then

Mr,Smith’s optimal value of r

x

is 20,and his optimal value of

x

is 5,(Hint,You will need to solve two equations in two unknowns.

One of the equations is the budget constraint.)

(f) Plot Mr,Smith’s optimal choice and an indi erence curve through it

in the graph.

(g) What fraction of his wealth should Mr,Smith invest in the risky asset?

Using the answer to Part (a),we find an x

that solves 20 = r

x

=30x + 10(1?x),The

answer is x =,5.

13.3 (2) Assuming that the Capital Asset Pricing Model is valid,com-

plete the following table,In this table p

0

is the current price of asset i

and Ep

1

is the expected price of asset i in the next period.

r

f

r

m

r

i

i

p

0

Ep

1

10 20 10 0 100 110

10 20 25 1.5 100 125

10 15 20 2 200 240

0 30 20 2=3 40 48

10 22 10 0 80 88

13.4 (2) Farmer Alf Alpha has a pasture located on a sandy hill,The

return to him from this pasture is a random variable depending on how

much rain there is,In rainy years the yield is good; in dry years the yield

is poor,The market value of this pasture is $5,000,The expected return

from this pasture is $500 with a standard deviation of $100,Every inch

of rain above average means an extra $100 in pro t and every inch of rain

below average means another $100 less pro t than average,Farmer Alf

has another $5,000 that he wants to invest in a second pasture,There are

two possible pastures that he can buy.

(a) One is located on low land that never floods,This pasture yields

an expected return of $500 per year no matter what the weather is like.

What is Alf Alpha’s expected rate of return on his total investment if he

buys this pasture for his second pasture? 10%,What is the standard

deviation of his rate of return in this case? 10%.

NAME 179

(b) Another pasture that he could buy is located on the very edge of the

river,This gives very good yields in dry years but in wet years it floods.

This pasture also costs $5,000,The expected return from this pasture is

$500 and the standard deviation is $100,Every inch of rain below average

means an extra $100 in pro t and every inch of rain above average means

another $100 less pro t than average,If Alf buys this pasture and keeps

his original pasture on the sandy hill,what is his expected rate of return

on his total investment? 10%,What is the standard deviation of the

rate of return on his total investment in this case? 0%.

(c) If Alf is a risk averter,which of these two pastures should he buy

and why? He should choose the second pasture

since it has the same expected return and

lower risk.

180 RISKY ASSETS (Ch,13)

Chapter 14 NAME

Consumer’s Surplus

Introduction,In this chapter you will study ways to measure a con-

sumer’s valuation of a good given the consumer’s demand curve for it.

The basic logic is as follows,The height of the demand curve measures

how much the consumer is willing to pay for the last unit of the good

purchased|the willingness to pay for the marginal unit,Therefore the

sum of the willingnesses-to-pay for each unit gives us the total willingness

to pay for the consumption of the good.

In geometric terms,the total willingness to pay to consume some

amount of the good is just the area under the demand curve up to that

amount,This area is called gross consumer’s surplus or total bene t

of the consumption of the good,If the consumer has to pay some amount

in order to purchase the good,then we must subtract this expenditure in

order to calculate the (net) consumer’s surplus.

When the utility function takes the quasilinear form,u(x)+m,the

area under the demand curve measures u(x),and the area under the

demand curve minus the expenditure on the other good measures u(x)+

m,Thus in this case,consumer’s surplus serves as an exact measure of

utility,and the change in consumer’s surplus is a monetary measure of a

change in utility.

If the utility function has a di erent form,consumer’s surplus will not

be an exact measure of utility,but it will often be a good approximation.

However,if we want more exact measures,we can use the ideas of the

compensating variation and the equivalent variation.

Recall that the compensating variation is the amount of extra income

that the consumer would need at the new prices to be as well o as she

was facing the old prices; the equivalent variation is the amount of money

that it would be necessary to take away from the consumer at the old

prices to make her as well o as she would be,facing the new prices.

Although di erent in general,the change in consumer’s surplus and the

compensating and equivalent variations will be the same if preferences are

quasilinear.

In this chapter you will practice:

Calculating consumer’s surplus and the change in consumer’s surplus

Calculating compensating and equivalent variations

Example,Suppose that the inverse demand curve is given by P(q)=

100?10q and that the consumer currently has 5 units of the good,How

much money would you have to pay him to compensate him for reducing

his consumption of the good to zero?

Answer,The inverse demand curve has a height of 100 when q =0

and a height of 50 when q = 5,The area under the demand curve is a

trapezoid with a base of 5 and heights of 100 and 50,We can calculate

182 CONSUMER’S SURPLUS (Ch,14)

the area of this trapezoid by applying the formula

Area of a trapezoid = base

1

2

(height

1

+height

2

):

In this case we have A =5

1

2

(100 + 50) = $375.

Example,Suppose now that the consumer is purchasing the 5 units at a

price of $50 per unit,If you require him to reduce his purchases to zero,

how much money would be necessary to compensate him?

In this case,we saw above that his gross bene ts decline by $375.

On the other hand,he has to spend 5 50 = $250 less,The decline in

net surplus is therefore $125.

Example,Suppose that a consumer has a utility function u(x

1;x

2

)=

x

1

+ x

2

,Initially the consumer faces prices (1;2) and has income 10.

If the prices change to (4;2),calculate the compensating and equivalent

variations.

Answer,Since the two goods are perfect substitutes,the consumer

will initially consume the bundle (10;0) and get a utility of 10,After the

prices change,she will consume the bundle (0;5) and get a utility of 5.

After the price change she would need $20 to get a utility of 10; therefore

the compensating variation is 20?10 = 10,Before the price change,she

would need an income of 5 to get a utility of 5,Therefore the equivalent

variation is 10?5=5.

14.1 (0) Sir Plus consumes mead,and his demand function for tankards

of mead is given by D(p) = 100?p,wherep is the price of mead in

shillings.

(a) If the price of mead is 50 shillings per tankard,how many tankards of

mead will he consume? 50.

(b) How much gross consumer’s surplus does he get from this consump-

tion? 3,750.

(c) How much money does he spend on mead? 2,500.

(d) What is his net consumer’s surplus from mead consumption?

1,250.

14.2 (0) Here is the table of reservation prices for apartments taken

from Chapter 1:

Person=ABCDEFGH

Price = 40 25 30 35 10 18 15 5

NAME 183

(a) If the equilibrium rent for an apartment turns out to be $20,which

consumers will get apartments? A,B,C,D.

(b) If the equilibrium rent for an apartment turns out to be $20,what

is the consumer’s (net) surplus generated in this market for person A?

20,For person B? 5.

(c) If the equilibrium rent is $20,what is the total net consumers’ surplus

generated in the market? 50.

(d) If the equilibrium rent is $20,what is the total gross consumers’

surplus in the market? 130.

(e) If the rent declines to $19,how much does the gross surplus increase?

0.

(f) If the rent declines to $19,how much does the net surplus increase?

4.

Calculus 14.3 (0) Quasimodo consumes earplugs and other things,His utility

function for earplugs x and money to spend on other goods y is given by

u(x;y) = 100x?

x

2

2

+y:

(a) What kind of utility function does Quasimodo have? Quasilinear.

(b) What is his inverse demand curve for earplugs? p = 100?x.

(c) If the price of earplugs is $50,how many earplugs will he consume?

50.

(d) If the price of earplugs is $80,how many earplugs will he consume?

20.

(e) Suppose that Quasimodo has $4,000 in total to spend a month,What

is his total utility for earplugs and money to spend on other things if the

price of earplugs is $50? $5,250.

184 CONSUMER’S SURPLUS (Ch,14)

(f) What is his total utility for earplugs and other things if the price of

earplugs is $80? $4,200.

(g) Utility decreases by 1,050 when the price changes from $50 to

$80.

(h) What is the change in (net) consumer’s surplus when the price changes

from $50 to $80? 1,050.

14.4 (2) In the graph below,you see a representation of Sarah Gamp’s

indi erence curves between cucumbers and other goods,Suppose that

the reference price of cucumbers and the reference price of \other goods"

are both 1.

Cucumbers

Other goods

0

40

30

20

10

10 20 30 40

B

A

(a) What is the minimum amount of money that Sarah would need in

order to purchase a bundle that is indi erent to A? 20.

(b) What is the minimum amount of money that Sarah would need in

order to purchase a bundle that is indi erent to B? 30.

(c) Suppose that the reference price for cucumbers is 2 and the reference

price for other goods is 1,How much money does she need in order to

purchase a bundle that is indi erent to bundle A? 30.

(d) What is the minimum amount of money that Sarah would need to

purchase a bundle that is indi erent to B using these new prices? 40.

NAME 185

(e) No matter what prices Sarah faces,the amount of money she needs

to purchase a bundle indi erent to A must be (higher,lower) than the

amount she needs to purchase a bundle indi erent to B,lower.

14.5 (2) Bernice’s preferences can be represented by u(x;y)=minfx;yg,

where x is pairs of earrings and y is dollars to spend on other things,She

faces prices (p

x;p

y

)=(2;1) and her income is 12.

(a) Draw in pencil on the graph below some of Bernice’s indi erence

curves and her budget constraint,Her optimal bundle is 4 pairs

of earrings and 4 dollars to spend on other things.

0481216

4

8

12

Pairs of earrings

Dollars for other things

16

Black line

Pencil lines

Red

line

Blue lines

(b) The price of a pair of earrings rises to $3 and Bernice’s income stays

the same,Using blue ink,draw her new budget constraint on the graph

above,Her new optimal bundle is 3 pairs of earrings and

3 dollars to spend on other things.

(c) What bundle would Bernice choose if she faced the original prices and

had just enough income to reach the new indi erence curve? (3;3).

Draw with red ink the budget line that passes through this bundle at

the original prices,How much income would Bernice need at the original

prices to have this (red) budget line? $9.

186 CONSUMER’S SURPLUS (Ch,14)

(d) The maximum amount that Bernice would pay to avoid the price

increase is $3,This is the (compensating,equivalent) variation in

income,Equivalent.

(e) What bundle would Bernice choose if she faced the new prices and had

just enough income to reach her original indi erence curve? (4;4).

Draw with black ink the budget line that passes through this bundle at

the new prices,How much income would Bernice have with this budget?

$16.

(f) In order to be as well-o as she was with her original bundle,Bernice’s

original income would have to rise by $4,This is the (compensating,

equivalent) variation in income,Compensating.

Calculus 14.6 (0) Ulrich likes video games and sausages,In fact,his preferences

can be represented by u(x;y)=ln(x +1)+y where x is the number of

video games he plays and y is the number of dollars that he spends on

sausages,Let p

x

be the price of a video game and m be his income.

(a) Write an expression that says that Ulrich’s marginal rate of substi-

tution equals the price ratio,( Hint,Remember Donald Fribble from

Chapter 6?) 1=(x+1)=p

x

.

(b) Since Ulrich has quasilinear preferences,you can solve this

equation alone to get his demand function for video games,which is

x =1=p

x

1,His demand function for the dollars to spend on

sausages is y = m?1+p.

(c) Video games cost $:25 and Ulrich’s income is $10,Then Ulrich de-

mands 3 video games and 9.25 dollars’ worth of sausages.

His utility from this bundle is 10.64,(Round o to two decimal

places.)

(d) If we took away all of Ulrich’s video games,how much money would

he need to have to spend on sausages to be just as well-o as before?

$10.64.

NAME 187

(e) Now an amusement tax of $.25 is put on video games and is passed

on in full to consumers,With the tax in place,Ulrich demands 1

video game and 9.5 dollars’ worth of sausages,His utility from this

bundle is 10.19,(Round o to two decimal places.)

(f) Now if we took away all of Ulrich’s video games,how much money

would he have to have to spend on sausages to be just as well-o as with

the bundle he purchased after the tax was in place? $10.19.

(g) What is the change in Ulrich’s consumer surplus due to the tax?

:45 How much money did the government collect from Ulrich by

means of the tax? $.25.

Calculus 14.7 (1) Lolita,an intelligent and charming Holstein cow,consumes

only two goods,cow feed (made of ground corn and oats) and hay,Her

preferences are represented by the utility function U(x;y)=x?x

2

=2+y,

where x is her consumption of cow feed and y is her consumption of hay.

Lolita has been instructed in the mysteries of budgets and optimization

and always maximizes her utility subject to her budget constraint,Lolita

has an income of $m that she is allowed to spend as she wishes on cow

feed and hay,The price of hay is always $1,and the price of cow feed will

be denoted by p,where0<p 1.

(a) Write Lolita’s inverse demand function for cow feed,(Hint,Lolita’s

utility function is quasilinear,When y is the numeraire and the price of

x is p,the inverse demand function for someone with quasilinear utility

f(x)+y is found by simply setting p = f

0

(x).) p =1?x.

(b) If the price of cow feed is p and her income is m,howmuchhaydoes

Lolita choose? (Hint,The money that she doesn’t spend on feed is used

to buy hay.) m?p(1?p).

(c) Plug these numbers into her utility function to nd out the utility level

that she enjoys at this price and this income,u = m+(1?p)

2

=2.

(d) Suppose that Lolita’s daily income is $3 and that the price of feed is

$:50,What bundle does she buy? (1=2;11=4),What bundle would

she buy if the price of cow feed rose to $1? (0;3).

188 CONSUMER’S SURPLUS (Ch,14)

(e) How much money would Lolita be willing to pay to avoid having the

price of cow feed rise to $1? 1=8,This amount is known as the

equivalent variation.

(f) Suppose that the price of cow feed rose to $1,How much extra money

would you have to pay Lolita to make her as well-o as she was at the

old prices? 1=8,This amount is known as the compensating

variation,Which is bigger,the compensating or the equivalent variation,

or are they the same? Same.

(g) At the price $.50 and income $3,how much (net) consumer’s surplus

is Lolita getting? 1=8.

14.8 (2) F,Flintstone has quasilinear preferences and his inverse demand

function for Brontosaurus Burgers is P(b)=30?2b,Mr,Flintstone is

currently consuming 10 burgers at a price of 10 dollars.

(a) How much money would he be willing to pay to have this amount

rather than no burgers at all? $200,What is his level of (net)

consumer’s surplus? $100.

(b) The town of Bedrock,the only supplier of Brontosaurus Burgers,

decides to raise the price from $10 a burger to $14 a burger,What

is Mr,Flintstone’s change in consumer’s surplus? At price

$10,consumer’s surplus is $100,At $14,

he demands 8 burgers,for net consumer’s

surplus of

1

2

(16 8) = 64,The change in

consumer’s surplus is?$36.

14.9 (1) Karl Kapitalist is willing to produce p=2?20 chairs at every

price,p>40,At prices below 40,he will produce nothing,If the price

of chairs is $100,Karl will produce 30 chairs,At this price,how

much is his producer’s surplus?

1

2

(60 30) = 900.

14.10 (2) Ms,Q,Moto loves to ring the church bells for up to 10

hours a day,Where m is expenditure on other goods,and x is hours of

bell ringing,her utility is u(m;x)=m +3x for x 10,If x>10,she

develops painful blisters and is worse o than if she didn’t ring the bells.

NAME 189

Her income is equal to $100 and the sexton allows her to ring the bell for

10 hours.

(a) Due to complaints from the villagers,the sexton has decided to restrict

Ms,Moto to 5 hours of bell ringing per day,This is bad news for Ms.

Moto,In fact she regards it as just as bad as losing $15 dollars of

income.

(b) The sexton relents and o ers to let her ring the bells as much as she

likes so long as she pays $2 per hour for the privilege,How much ringing

does she do now? 10 hours,This tax on her activities is as bad

as a loss of how much income? $20.

(c) The villagers continue to complain,The sexton raises the price of

bell ringing to $4 an hour,How much ringing does she do now? 0

hours,This tax,as compared to the situation in which she could

ring the bells for free,is as bad as a loss of how much income? $30.

190 CONSUMER’S SURPLUS (Ch,14)

Chapter 15 NAME

Market Demand

Introduction,Some problems in this chapter will ask you to construct

the market demand curve from individual demand curves,The market

demand at any given price is simply the sum of the individual demands at

that price,The key thing to remember in going from individual demands

to the market demand is to add quantities,Graphically,you sum the

individual demands horizontally to get the market demand,The market

demand curve will have a kink in it whenever the market price is high

enough that some individual demand becomes zero.

Sometimes you will need to nd a consumer’s reservation price for

a good,Recall that the reservation price is the price that makes the

consumer indi erent between having the good at that price and not hav-

ing the good at all,Mathematically,the reservation price p

satis es

u(0;m)=u(1;m?p

),where m is income and the quantity of the other

good is measured in dollars.

Finally,some of the problems ask you to calculate price and/or in-

come elasticities of demand,These problems are especially easy if you

know a little calculus,If the demand function is D(p),and you want to

calculate the price elasticity of demand when the price is p,you only need

to calculate dD(p)=dp and multiply it by p=q.

15.0 Warm Up Exercise,(Calculating elasticities.) Here are

some drills on price elasticities,For each demand function,nd an ex-

pression for the price elasticity of demand,The answer will typically be

a function of the price,p,As an example,consider the linear demand

curve,D(p)=30?6p.ThendD(p)=dp =?6andp=q = p=(30?6p),so

the price elasticity of demand is?6p=(30?6p).

(a) D(p)=60?p,?p=(60?p).

(b) D(p)=a?bp,?bp=(a?bp).

(c) D(p)=40p

2

,?2.

(d) D(p)=Ap

b

,?b.

(e) D(p)=(p+3)

2

,?2p=(p+3).

192 MARKET DEMAND (Ch,15)

(f) D(p)=(p+a)

b

,?bp=(p+a).

15.1 (0) In Gas Pump,South Dakota,there are two kinds of consumers,

Buick owners and Dodge owners,Every Buick owner has a demand func-

tion for gasoline D

B

(p)=20?5p for p 4andD

B

(p)=0ifp>4.

Every Dodge owner has a demand function D

D

(p)=15?3p for p 5

and D

D

(p)=0forp>5,(Quantities are measured in gallons per week

and price is measured in dollars.) Suppose that Gas Pump has 150 con-

sumers,100 Buick owners,and 50 Dodge owners.

(a) If the price is $3,what is the total amount demanded by each indi-

vidual Buick Owner? 5,And by each individual Dodge owner?

6.

(b) What is the total amount demanded by all Buick owners? 500.

What is the total amount demanded by all Dodge owners? 300.

(c) What is the total amount demanded by all consumers in Gas Pump

at a price of 3? 800.

(d) On the graph below,use blue ink to draw the demand curve repre-

senting the total demand by Buick owners,Use black ink to draw the

demand curve representing total demand by Dodge owners,Use red ink

to draw the market demand curve for the whole town.

(e) At what prices does the market demand curve have kinks? At

p =4 and p =5.

(f) When the price of gasoline is $1 per gallon,how much does weekly

demand fall when price rises by 10 cents? 65 gallons.

(g) When the price of gasoline is $4.50 per gallon,how much does weekly

demand fall when price rises by 10 cents? 15 gallons.

(h) When the price of gasoline is $10 per gallon,how much does weekly

demand fall when price rises by 10 cents? Remains at zero.

NAME 193

0 1500 2000 2500 3000

1

2

3

4

5

6

500

Dollars per gallon

1000

Gallons per week

Blue line

Black

line

Red line

15.2 (0) For each of the following demand curves,compute the inverse

demand curve.

(a) D(p)=maxf10?2p;0g,p(q)=5?q=2 if q<10.

(b) D(p) = 100=

p

p,p(q)=10;000=q

2

.

(c) lnD(p)=10?4p,p(q)=(10?lnq)=4.

(d) lnD(p)=ln20?2lnp,p(q)=

q

20=q.

15.3 (0) The demand function of dog breeders for electric dog polishers

is q

b

=maxf200?p;0g,and the demand function of pet owners for electric

dog polishers is q

o

=maxf90?4p;0g.

(a) At price p,what is the price elasticity of dog breeders’ demand for

electric dog polishersp=(200?p),What is the price elasticity

of pet owners’ demand4p=(90?4p).

194 MARKET DEMAND (Ch,15)

(b) At what price is the dog breeders’ elasticity equal to?1? $100.

At what price is the pet owners’ elasticity equal to?1? $11.25.

(c) On the graph below,draw the dog breeders’ demand curve in blue

ink,the pet owners’ demand curve in red ink,and the market demand

curve in pencil.

(d) Find a nonzero price at which there is positive total demand for dog

polishers and at which there is a kink in the demand curve,$22.50.

What is the market demand function for prices below the kink? 290?

5p,What is the market demand function for prices above the kink?

200?p.

(e) Where on the market demand curve is the price elasticity equal to

1? $100,At what price will the revenue from the sale of electric

dog polishers be maximized? $100,If the goal of the sellers is to

maximize revenue,will electric dog polishers be sold to breeders only,to

pet owners only,or to both? Breeders only.

NAME 195

0 150 200 250 300

50

100

150

200

250

300

50

Price

100

Quantity

Blue line

Red

line

Pencil line

22.5

90 290

Calculus 15.4 (0) The demand for kitty litter,in pounds,is lnD(p)=1;000?

p+lnm,wherep is the price of kitty litter and m is income.

(a) What is the price elasticity of demand for kitty litter when p =2and

m = 5002,When p =3andm = 5003,When p =4and

m =1;5004.

(b) What is the income elasticity of demand for kitty litter when p =2

and m = 500? 1,When p =2andm =1;000? 1.

When p =3andm =1;500? 1.

196 MARKET DEMAND (Ch,15)

(c) What is the price elasticity of demand when price is p and income is

mp,The income elasticity of demand? 1.

Calculus 15.5 (0) The demand function for drangles is q(p)=(p +1)

2

.

(a) What is the price elasticity of demand at price p2p=(p+1).

(b) At what price is the price elasticity of demand for drangles equal to

1? When the price equals 1.

(c) Write an expression for total revenue from the sale of drangles as

a function of their price,R(p)=pq = p=(p +1)

2

,Use

calculus to nd the revenue-maximizing price,Don’t forget to check the

second-order condition,Differentiating and solving

gives p =1.

(d) Suppose that the demand function for drangles takes the more general

form q(p)=(p+a)

b

where a>0andb>1,Calculate an expression for

the price elasticity of demand at price p,?bp=(p+a),At what

price is the price elasticity of demand equal to?1? p = a=(b?1).

15.6 (0) Ken’s utility function is u

K

(x

1;x

2

)=x

1

+ x

2

and Barbie’s

utility function is u

B

(x

1;x

2

)=(x

1

+1)(x

2

+ 1),A person can buy 1

unit of good 1 or 0 units of good 1,It is impossible for anybody to buy

fractional units or to buy more than 1 unit,Either person can buy any

quantity of good 2 that he or she can a ord at a price of $1 per unit.

(a) Where m is Barbie’s wealth and p

1

is the price of good 1,write an

equation that can be solved to nd Barbie’s reservation price for good 1.

(m?p

1

+1)2=m +1,What is Barbie’s reservation price

for good 1? p =(m+1)=2,What is Ken’s reservation price for

good 1? $1.

(b) If Ken and Barbie each have a wealth of 3,plot the market demand

curve for good 1.

NAME 197

01234

1

2

3

4

Price

Quantity

15.7 (0) The demand function for yo-yos is D(p;M)=4?2p +

1

100

M,

where p is the price of yo-yos and M is income,If M is 100 and p is 1,

(a) What is the income elasticity of demand for yo-yos? 1=3.

(b) What is the price elasticity of demand for yo-yos2=3.

15.8 (0) If the demand function for zarfs is P =10?Q,

(a) At what price will total revenue realized from their sale be at a max-

imum? P =5.

(b) How many zarfs will be sold at that price? Q =5.

15.9 (0) The demand function for football tickets for a typical game at a

large midwestern university is D(p) = 200;000?10;000p,The university

has a clever and avaricious athletic director who sets his ticket prices so

as to maximize revenue,The university’s football stadium holds 100,000

spectators.

(a) Write down the inverse demand function,p(q)=20?

q=10;000.

198 MARKET DEMAND (Ch,15)

(b) Write expressions for total revenue R(q)=20q?q

2

=10;000

and marginal revenue MR =20?q=5;000 as a function of the

number of tickets sold.

(c) On the graph below,use blue ink to draw the inverse demand function

and use red ink to draw the marginal revenue function,On your graph,

also draw a vertical blue line representing the capacity of the stadium.

0 20 40 60 80 100 120 140 160

5

10

15

20

25

30

Price

Quantity x 1000

Red line

Red line

Black line

Blue line

Stadium capacity

(d) What price will generate the maximum revenue? $10,What

quantity will be sold at this price? 100,000.

(e) At this quantity,what is marginal revenue? 0,At this quantity,

what is the price elasticity of demand1,Will the stadium be full?

Yes.

(f) A series of winning seasons caused the demand curve for football

tickets to shift upward,The new demand function is q(p) = 300;000?

10;000p,What is the new inverse demand function? p(q)=30?

q=10;000.

NAME 199

(g) Write an expression for marginal revenue as a function of output.

MR(q)= 30?q=5;000,Use red ink to draw the new demand

function and use black ink to draw the new marginal revenue function.

(h) Ignoring stadium capacity,what price would generate maximum

revenue? $15,What quantity would be sold at this price?

150,000.

(i) As you noticed above,the quantity that would maximize total revenue

given the new higher demand curve is greater than the capacity of the

stadium,Clever though the athletic director is,he cannot sell seats he

hasn’t got,He notices that his marginal revenue is positive for any number

of seats that he sells up to the capacity of the stadium,Therefore,in order

to maximize his revenue,he should sell 100,000 tickets at a price

of $20.

(j) When he does this,his marginal revenue from selling an extra seat

is 10,The elasticity of demand for tickets at this price quantity

combination is =?2.

15.10 (0) The athletic director discussed in the last problem is consid-

ering the extra revenue he would gain from three proposals to expand the

size of the football stadium,Recall that the demand function he is now

facing is given by q(p) = 300;000?10;000p.

(a) How much could the athletic director increase the total revenue per

game from ticket sales if he added 1,000 new seats to the stadium’s capac-

ity and adjusted the ticket price to maximize his revenue? 9,900.

(b) How much could he increase the revenue per game by adding 50,000

new seats? $250,000,60,000 new seats? (Hint,The athletic

director still wants to maximize revenue.) $250,000.

(c) A zealous alumnus o ers to build as large a stadium as the athletic

director would like and donate it to the university,There is only one hitch.

The athletic director must price his tickets so as to keep the stadium full.

If the athletic director wants to maximize his revenue from ticket sales,

how large a stadium should he choose? 150,000 seats.

200 MARKET DEMAND (Ch,15)

Chapter 16 NAME

Equilibrium

Introduction,Supply and demand problems are bread and butter for

economists,In the problems below,you will typically want to solve for

equilibrium prices and quantities by writing an equation that sets supply

equal to demand,Where the price received by suppliers is the same as the

price paid by demanders,one writes supply and demand as functions of

the same price variable,p,and solves for the price that equalizes supply

and demand,But if,as happens with taxes and subsidies,suppliers face

di erent prices from demanders,it is a good idea to denote these two

prices by separate variables,p

s

and p

d

,Then one can solve for equilibrium

by solving a system of two equations in the two unknowns p

s

and p

d

.The

two equations are the equation that sets supply equal to demand and

the equation that relates the price paid by demanders to the net price

received by suppliers.

Example,The demand function for commodity x is q =1;000?10p

d

,

where p

d

is the price paid by consumers,The supply function for x is

q = 100 + 20p

s

,wherep

s

is the price received by suppliers,For each unit

sold,the government collects a tax equal to half of the price paid by con-

sumers,Let us nd the equilibrium prices and quantities,In equilibrium,

supply must equal demand,so that 1;000?10p

d

= 100 + 20p

s

,Since the

government collects a tax equal to half of the price paid by consumers,

it must be that the sellers only get half of the price paid by consumers,

so it must be that p

s

= p

d

=2,Now we have two equations in the two

unknowns,p

s

and p

d

,Substitute the expression p

d

=2forp

s

in the rst

equation,and you have 1;000?10p

d

= 100 + 10p

d

,Solve this equation

to nd p

d

= 45,Then p

s

=22:5andq = 550.

16.1 (0) The demand for yak butter is given by 120?4p

d

and the

supply is 2p

s

30,where p

d

is the price paid by demanders and p

s

is

the price received by suppliers,measured in dollars per hundred pounds.

Quantities demanded and supplied are measured in hundred-pound units.

(a) On the axes below,draw the demand curve (with blue ink) and the

supply curve (with red ink) for yak butter.

202 EQUILIBRIUM (Ch,16)

0 40 60 80 100

Yak butter

20

40

60

80

Price

20 120

Blue line

Red line

p1

q1q2

p2

(b) Write down the equation that you would solve to nd the equilibrium

price,Solve 120?4p =2p?30.

(c) What is the equilibrium price of yak butter? $25,What is the

equilibrium quantity? 20,Locate the equilibrium price and quantity

on the graph,and label them p

1

and q

1

.

(d) A terrible drought strikes the central Ohio steppes,traditional home-

land of the yaks,The supply schedule shifts to 2p

s

60,The demand

schedule remains as before,Draw the new supply schedule,Write down

the equation that you would solve to nd the new equilibrium price of

yak butter,120?4p =2p?60.

(e) The new equilibrium price is 30 and the quantity is 0.

Locate the new equilibrium price and quantity on the graph and label

them p

2

and q

2

.

(f) The government decides to relieve stricken yak butter consumers and

producers by paying a subsidy of $5 per hundred pounds of yak butter

to producers,If p

d

is the price paid by demanders for yak butter,what

is the total amount received by producers for each unit they produce?

p

d

+5,When the price paid by consumers is p

d

,how much yak butter

is produced? 2p

d

50.

NAME 203

(g) Write down an equation that can be solved for the equilibrium price

paid by consumers,given the subsidy program,2p

d

50 =

120? 4p

d

,What are the equilibrium price paid by consumers

and the equilibrium quantity of yak butter now? p

d

= 170=6,

q = 170=3?50 = 20=3.

(h) Suppose the government had paid the subsidy to consumers rather

than producers,What would be the equilibrium net price paid by con-

sumers? 170=6,The equilibrium quantity would be 20=3.

16.2 (0) Here are the supply and demand equations for throstles,where

p is the price in dollars:

D(p)=40?p

S(p)=10+p:

On the axes below,draw the demand and supply curves for throstles,

using blue ink.

010203040

10

20

30

40

Price

Throstles

Demand

Supply

Deadweight

loss

(a) The equilibrium price of throstles is 15 and the equilibrium

quantity is 25.

(b) Suppose that the government decides to restrict the industry to selling

only 20 throstles,At what price would 20 throstles be demanded? 20.

How many throstles would suppliers supply at that price? 30,At what

price would the suppliers supply only 20 units? $10.

204 EQUILIBRIUM (Ch,16)

(c) The government wants to make sure that only 20 throstles are bought,

but it doesn’t want the rms in the industry to receive more than the

minimum price that it would take to have them supply 20 throstles,One

way to do this is for the government to issue 20 ration coupons,Then

in order to buy a throstle,a consumer would need to present a ration

coupon along with the necessary amount of money to pay for the good.

If the ration coupons were freely bought and sold on the open market,

what would be the equilibrium price of these coupons? $10.

(d) On the graph above,shade in the area that represents the deadweight

loss from restricting the supply of throstles to 20,How much is this ex-

pressed in dollars? (Hint,What is the formula for the area of a triangle?)

$25.

16.3 (0) The demand curve for ski lessons is given by D(p

D

) = 100?2p

D

and the supply curve is given by S(p

S

)=3p

S

.

(a) What is the equilibrium price? $20,What is the equilibrium

quantity? 60.

(b) A tax of $10 per ski lesson is imposed on consumers,Write an equation

that relates the price paid by demanders to the price received by suppliers.

p

D

= p

S

+10,Write an equation that states that supply equals

demand,100?2p

D

=3p

S

.

(c) Solve these two equations for the two unknowns p

S

and p

D

.With

the $10 tax,the equilibrium price p

D

paid by consumers would be $26

per lesson,The total number of lessons given would be 48.

(d) A senator from a mountainous state suggests that although ski lesson

consumers are rich and deserve to be taxed,ski instructors are poor and

deserve a subsidy,He proposes a $6 subsidy on production while main-

taining the $10 tax on consumption of ski lessons,Would this policy have

any di erent e ects for suppliers or for demanders than a tax of $4 per

lesson? No.

16.4 (0) The demand curve for salted cod sh is D(P) = 200?5P and

the supply curve S(P)=5P.

NAME 205

(a) On the graph below,use blue ink to draw the demand curve and the

supply curve,The equilibrium market price is $20 and the equilibrium

quantity sold is 100.

0 50 100 150 200

10

20

30

40

Price

Quantity of codfish

Demand

Blue Supply

Deadweight

loss

Red

supply

(b) A quantity tax of $2 per unit sold is placed on salted cod sh,Use red

ink to draw the new supply curve,where the price on the vertical axis

remains the price per unit paid by demanders,The new equilibrium price

paid by the demanders will be $21 and the new price received by the

suppliers will be $19,The equilibrium quantity sold will be 95.

(c) The deadweight loss due to this tax will be 5=2 5=2,On

your graph,shade in the area that represents the deadweight loss.

16.5 (0) The demand function for merino ewes is D(P) = 100=P,and

the supply function is S(P)=P.

(a) What is the equilibrium price? $10.

206 EQUILIBRIUM (Ch,16)

(b) What is the equilibrium quantity? 10.

(c) An ad valorem tax of 300% is imposed on merino ewes so that the

price paid by demanders is four times the price received by suppliers.

What is the equilibrium price paid by the demanders for merino ewes

now? $20,What is the equilibrium price received by the suppliers

for merino ewes? $5,What is the equilibrium quantity? 5.

16.6 (0) Schrecklich and LaMerde are two justi ably obscure nineteenth-

century impressionist painters,The world’s total stock of paintings by

Schrecklich is 100,and the world’s stock of paintings by LaMerde is 150.

The two painters are regarded by connoisseurs as being very similar in

style,Therefore the demand for either painter’s work depends both on its

own price and the price of the other painter’s work,The demand function

for Schrecklichs is D

S

(P) = 200?4P

S

2P

L

,and the demand function for

LaMerdes is D

L

(P) = 200?3P

L

P

S

,whereP

S

and P

L

are respectively

the price in dollars of a Schrecklich painting and a LaMerde painting.

(a) Write down two simultaneous equations that state the equilibrium

condition that the demand for each painter’s work equals supply.

The equations are 200?4P

S

2P

L

= 100 and

200?3P

L

P

S

= 150.

(b) Solving these two equations,one nds that the equilibrium price of

Schrecklichs is 20 and the equilibrium price of LaMerdes is 10.

(c) On the diagram below,draw a line that represents all combinations of

prices for Schrecklichs and LaMerdes such that the supply of Schrecklichs

equals the demand for Schrecklichs,Draw a second line that represents

those price combinations at which the demand for LaMerdes equals the

supply of LaMerdes,Label the unique price combination at which both

markets clear with the letter E.

NAME 207

010203040

10

20

30

40

Pl

Ps

e

Schrecklich

La Mendes

Red line

e'

(d) A re in a bowling alley in Hamtramck,Michigan,destroyed one of

the world’s largest collections of works by Schrecklich,The re destroyed

a total of 10 Schrecklichs,After the re,the equilibrium price of Schreck-

lichs was 23 and the equilibrium price of LaMerdes was 9.

(e) On the diagram you drew above,use red ink to draw a line that shows

the locus of price combinations at which the demand for Schrecklichs

equals the supply of Schrecklichs after the re,On your diagram,label

the new equilibrium combination of prices E

0

.

16.7 (0) The price elasticity of demand for oatmeal is constant and

equal to?1,When the price of oatmeal is $10 per unit,the total amount

demanded is 6,000 units.

(a) Write an equation for the demand function,q =60;000=p.

Graph this demand function below with blue ink,(Hint,If the demand

curve has a constant price elasticity equal to,thenD(p)=ap

for some

constant a,You have to use the data of the problem to solve for the

constants a and that apply in this particular case.)

208 EQUILIBRIUM (Ch,16)

046810

Quantity (thousands)

5

10

15

20

Price

2 12

e

Red lines

Blue lines

(b) If the supply is perfectly inelastic at 5,000 units,what is the equilib-

rium price? $12,Show the supply curve on your graph and label the

equilibrium with an E.

(c) Suppose that the demand curve shifts outward by 10%,Write down

the new equation for the demand function,q =66;000=p,Sup-

pose that the supply curve remains vertical but shifts to the right by 5%.

Solve for the new equilibrium price 12:51 and quantity 5;250.

(d) By what percentage approximately did the equilibrium price rise?

It rose by about 5 percent,Use red ink to draw the

new demand curve and the new supply curve on your graph.

(e) Suppose that in the above problem the demand curve shifts outward

by x% and the supply curve shifts right by y%,By approximately what

percentage will the equilibrium price rise? By about (x?y)

percent.

16.8 (0) An economic historian* reports that econometric studies in-

dicate for the pre{Civil War period,1820{1860,the price elasticity of

demand for cotton from the American South was approximately?1,Due

to the rapid expansion of the British textile industry,the demand curve

for American cotton is estimated to have shifted outward by about 5%

per year during this entire period.

* Gavin Wright,The Political Economy of the Cotton South,W.W.

Norton,1978.

NAME 209

(a) If during this period,cotton production in the United States grew by

3% per year,what (approximately) must be the rate of change of the price

of cotton during this period? It would rise by about 2%

a year.

(b) Assuming a constant price elasticity of?1,and assuming that when

the price is $20,the quantity is also 20,graph the demand curve for

cotton,What is the total revenue when the price is $20? 400,What

is the total revenue when the price is $10? 400.

010203040

10

20

30

40

Price of cotton

Quantity of cotton

(c) If the change in the quantity of cotton supplied by the United States is

to be interpreted as a movement along an upward-sloping long-run supply

curve,what would the elasticity of supply have to be? (Hint,From 1820

to 1860 quantity rose by about 3% per year and price rose by 2 %

per year,[See your earlier answer.] If the quantity change is a movement

along the long-run supply curve,then the long-run price elasticity must

be what?) 1.5 %.

(d) The American Civil War,beginning in 1861,had a devastating e ect

on cotton production in the South,Production fell by about 50% and

remained at that level throughout the war,What would you predict

would be the e ect on the price of cotton? It would double

if demand didn’t change.

210 EQUILIBRIUM (Ch,16)

(e) What would be the e ect on total revenue of cotton farmers in the

South? Since the demand has elasticity of

1,the revenue would stay the same.

(f) The expansion of the British textile industry ended in the 1860s,

and for the remainder of the nineteenth century,the demand curve for

American cotton remained approximately unchanged,By about 1900,

the South approximately regained its prewar output level,What do you

think happened to cotton prices then? They would recover

to their old levels.

16.9 (0) The number of bottles of chardonnay demanded per year is

$1;000;000?60;000P,whereP is the price per bottle (in U.S,dollars).

The number of bottles supplied is 40;000P.

(a) What is the equilibrium price? $10,What is the equilibrium

quantity? 400;000.

(b) Suppose that the government introduces a new tax such that the wine

maker must pay a tax of $5 per bottle for every bottle that he produces.

What is the new equilibrium price paid by consumers? $12,What is

the new price received by suppliers? $7,What is the new equilibrium

quantity? 280,000.

16.10 (0) The inverse demand function for bananas is P

d

=18?3Q

d

and the inverse supply function is P

s

=6+Q

s

,where prices are measured

in cents.

(a) If there are no taxes or subsidies,what is the equilibrium quantity?

3,What is the equilibrium market price? 9 cents.

(b) If a subsidy of 2 cents per pound is paid to banana growers,then

in equilibrium it still must be that the quantity demanded equals the

quantity supplied,but now the price received by sellers is 2 cents higher

than the price paid by consumers,What is the new equilibrium quantity?

3.5,What is the new equilibrium price received by suppliers? 9.5

cents,What is the new equilibrium price paid by demanders? 7.5

cents.

NAME 211

(c) Express the change in price as a percentage of the original price.

-16.66%,If the cross-elasticity of demand between bananas and

apples is +.5,what will happen to the quantity of apples demanded as a

consequence of the banana subsidy,if the price of apples stays constant?

(State your answer in terms of percentage change.) -8.33%.

16.11 (1) King Kanuta rules a small tropical island,Nutting Atoll,

whose primary crop is coconuts,If the price of coconuts is P,thenKing

Kanuta’s subjects will demand D(P)=1;200?100P coconuts per week

for their own use,The number of coconuts that will be supplied per week

by the island’s coconut growers is S(p) = 100P.

(a) The equilibrium price of coconuts will be 6 and the equilib-

rium quantity supplied will be 600.

(b) One day,King Kanuta decided to tax his subjects in order to collect

coconuts for the Royal Larder,The king required that every subject

who consumed a coconut would have to pay a coconut to the king as a

tax,Thus,if a subject wanted 5 coconuts for himself,he would have

to purchase 10 coconuts and give 5 to the king,When the price that

is received by the sellers is p

S

,how much does it cost one of the king’s

subjects to get an extra coconut for himself? 2p

S

.

(c) When the price paid to suppliers is p

S

,how many coconuts will the

king’s subjects demand for their own consumption? (Hint,Express p

D

in terms of p

S

and substitute into the demand function.) Since

p

D

=2p

S

,they consume 1;200?200p

S

.

(d) Since the king consumes a coconut for every coconut consumed by

the subjects,the total amount demanded by the king and his subjects is

twice the amount demanded by the subjects,Therefore,when the price

received by suppliers is p

S

,the total number of coconuts demanded per

week by Kanuta and his subjects is 2;400?400p

S

.

(e) Solve for the equilibrium value of p

S

24/5,the equilibrium total

number of coconuts produced 480,and the equilibrium total number

of coconuts consumed by Kanuta’s subjects,240.

212 EQUILIBRIUM (Ch,16)

(f) King Kanuta’s subjects resented paying the extra coconuts to the

king,and whispers of revolution spread through the palace,Worried by

the hostile atmosphere,the king changed the coconut tax,Now,the

shopkeepers who sold the coconuts would be responsible for paying the

tax,For every coconut sold to a consumer,the shopkeeper would have to

pay one coconut to the king,This plan resulted in 480=2 = 240

coconuts being sold to the consumers,The shopkeepers got 24=5 per

coconut after paying their tax to the king,and the consumers paid a price

of 48=5 per coconut.

Chapter 17 NAME

Auctions

Introduction,An auction is described by a set of rules,The rules

specify bidding procedures for participants and the way in which the

array of bids made determines who gets the object being sold and how

much each bidder pays,Those who are trying to sell an object by auction

typically do not know the willingness to pay of potential buyers but have

some probabilistic expectations,Sellers are interested in nding rules that

maximize their expected revenue from selling the object.

Social planners are often interested not only in the revenue generated

from an auction method,but also in its e ciency,In the absence of

externalities,an auction for a single object will be e cient only if the

object is sold to the buyer who values it most highly.

17.1 (1) At Toivo’s auction house in Ishpemming,Michigan,a beautiful

stu ed moosehead is being sold by auction,There are 5 bidders in atten-

dance,Aino,Erkki,Hannu,Juha,and Matti,The moosehead is worth

$100 to Aino,$20 to Erkki,and $5 to each of the others,The bidders do

not collude and they don’t know each others’ valuations.

(a) If the auctioneer sells it in an English auction,who would get the

moosehead and approximately how much would the buyer pay? Aino

would get it for $20.

(b) If the auctioneer sells it in a sealed-bid,second-price auction and if

no bidder knows the others’ values for the moosehead,how much should

Aino bid in order to maximize his expected gain? $100 How much

should Erkki bid? $20 How much would each of the others bid?

$5 Who would get the moosehead and how much would he pay?

Aino would get it for $20.

17.2 (2) Charlie Plopp sells used construction equipment in a quiet

Oklahoma town,He has run short of cash and needs to raise money

quickly by selling an old bulldozer,If he doesn’t sell his bulldozer to a

customer today,he will have to sell it to a wholesaler for $1,000.

Two kinds of people are interested in buying bulldozers,These are

professional bulldozer operators and people who use bulldozers only for

recreational purposes on weekends,Charlie knows that a professional

bulldozer operator would be willing to pay $6,000 for his bulldozer but no

214 AUCTIONS (Ch,17)

more,while a weekend recreational user would be willing to pay $4;500

but no more,Charlie puts a sign in his window,\Bulldozer Sale Today."

Charlie is disappointed to discover that only two potential buyers

have come to his auction,These two buyers evidently don’t know each

other,Charlie believes that the probability that either is a professional

bulldozer operator is independent of the other’s type and he believes that

each of them has a probability of 1/2 of being a professional bulldozer

operator and a probability of 1/2 of being a recreational user.

Charlie considers the following three ways of selling the bulldozer:

Method 1,Post a price of $6,000 and if nobody takes the bulldozer

at that price,sell it to the wholesaler.

Method 2,Post a price equal to a recreational bulldozer user’s buyer

value and sell it to anyone who o ers that price.

Method 3,Run a sealed-bid auction and sell the bulldozer to the

high bidder at the second highest bid (if there is a tie,choose one of

the high bidders at random and sell the bulldozer to this bidder at

the price bid by both bidders.)

(a) What is the probability that both potential buyers are professional

bulldozer operators? 1/4,What is the probability that both are recre-

ational bulldozer users? 1/4,What is the probability that one of them

is of each type? 1/2.

(b) If Charlie sells by method 1,what is the probability that he will be

able to sell the bulldozer to one of the two buyers? 3/4,What is

the probability that he will have to sell the bulldozer to the wholesaler?

1/4,What is his expected revenue? $(3=4) $6;000 +

(1=4) $1;000 = $4;750:

(c) If Charlie sells by method 2,how much will he receive for his bulldozer?

$4,500.

(d) Suppose that Charlie sells by method 3 and that both potential

buyers bid rationally,If both bidders are professional bulldozer oper-

ators,how much will each bid? $6,000,How much will Char-

lie receive for his bulldozer? $6,000,If one bidder is a profes-

sional bulldozer operator and one is a recreational user,what bids will

Charlie receive? Professional bids $6,000.

Recreational user bids $4,500,Who will get

NAME 215

the bulldozer? The professional,How much money will

Charlie get for his bulldozer? $4,500,If both bidders are recre-

ational bulldozer users,how much will each bid? $4,500,How

much will Charlie receive for his bulldozer? $4,500,What will be

Charlie’s expected revenue from selling the bulldozer by method 3?

$(1=4) $6;000 + (3=4) $4;500 = $4;875.

(e) Which of the three methods will give Charlie the highest expected

revenue? Method 3.

17.3 (2) We revisit our nancially a icted friend,Charlie Plopp,This

time we will look at a slightly generalized version of the same problem,All

else is as before,but the willingness to pay of recreational bulldozers is an

amount C<$6;000 which is known to Charlie,In the previous problem

we dealt with the special case where C =$4;500,Now we want to explore

the way in which the sales method that gives Charlie the highest expected

revenue depends on the size of C.

(a) What will Charlie’s expected revenue be if he posts a price equal to

the reservation price of professional bulldozer operators? $(3=4)

$6;000 + (1=4) $1;000 = $4;750:

(b) If Charlie posts a price equal to the reservation price C of recreational

bulldozer operators,what is his expected revenue? $C.

(c) If Charlie sells his bulldozer by method 3,the second-price sealed-bid

auction,what is his expected revenue? (The answer is a function of C.)

$(1=4) $6;000 + (3=4) $C =$1;500 + (3=4)C.

(d) Show that selling by method 3 will give Charlie a higher expected pay-

o than selling by method 2 if C<$6;000,With method 3,

each bidder will bid his true valuation.

If both bidders have valuations of

$6,000,he will get $6,000,Otherwise,he

will get $C,His expected payoff is then

216 AUCTIONS (Ch,17)

$1;500+(3=4)C,With method 2 he gets $C.

But $1;500 + (3=4)C>Cwhenever C<$6;000.

(e) For what values of C is Charlie better o selling by method 2 than by

method 1? C>$(3=4)6;000 + (1=4)1;000 = 4;759.

(f) For what values of C is Charlie better o selling by method 1 than by

method 3? This happens when 4;750 > 1;500+

3

4C

,

which is the case whenever C<4;333:33.

17.4 (3) Yet again we tread the dusty streets of Charlie Plopp’s home

town,Everything is as in the previous problem,Professional bulldozer

operators are willing to pay $6,000 for a bulldozer and recreational users

are willing to pay C,Charlie is just about to sell his bulldozer when a

third potential buyer appears,Charlie believes that this buyer,like the

other two,is equally likely to be a professional bulldozer operator as a

recreational bulldozer operator and that this probability is independent

of the types of the other two.

(a) With three buyers,Charlie’s expected revenue from using method 1

is 5375,his expected revenue from using method 2 is C,and

his expected revenue from using method 3 is $3;000 + (C=2).

(b) At which values of C would method 1 give Charlie a higher expected

revenue than either of the other two methods of selling proposed above?

C<$4;750.

(c) At which values of C (if any) would method 2 give Charlie a higher

expected revenue than either of the other two methods of selling proposed

above? None.

(d) At which values of C would method 3 give Charlie a higher expected

revenue than either of the other two methods of selling proposed above?

C>$4;750

17.5 (2) General Scooters has decided to replace its old assembly line

with a new one that makes extensive use of robots,There are two con-

tractors who would be able to build the new assembly line,General

Scooters’s industrial spies and engineers have done some exploratory re-

search of their own on the costs of building the new assembly line for each

NAME 217

of the two contractors,They have discovered that for each rm,this cost

will take one of three possible values H,M,andL,whereH>M>L.

Unfortunately,General Scooters has not been able to determine whether

the costs of either of the rms are H,M,orL,The best information that

General Scooters’s investigators have been able to give it is that for each

contractor the probability is 1/3 that the cost is H,1/3 that the cost is

M,and 1/3 that the cost is L and that the probability distribution of

costs is independent between the two contractors,Each contractor knows

its own costs but thinks that the other’s costs are equally likely to be

H,M,orL,General Scooters is con dent that the contractors will not

collude.

(a) Accountants at General Scooters suggested that General Scoooters

accept sealed bids from the two contractors for constructing the assembly

line and that it announce that it will award the contract to the low bidder

but will pay the low bidder the amount bid by the other contractor,(If

there is a tie for low bidder,one of the bidders will be selected at random

to get the contract.) If this is done,what bidding strategy should each

of the contractors use (assuming that they cannot collude) in order to

maximize their expected pro ts? Each would bid his

true valuation.

(b) Suppose that General Scooters uses the bidding mechanism suggested

by the accountants,What is the probability that it will have to pay H

to get the job done? 5/9 What is the probability that it will have

to pay M? 1/3 What is the probability that it will have to pay L?

1/9 Write an expression in terms of the variables H,M,andL for the

expected cost of the project to General Scooters,H

5

9

+M

3

9

+L

1

9

(c) When the distinguished-looking,silver-haired chairman of General

Scooters was told of the accountants’ suggested bidding scheme,he was

outraged,\What a stupid bidding system! Any fool can see that it is

more pro table for us to pay the lower of the two bids,Why on earth

would you ever want to pay the higher bid rather than the lower one?"

he roared.

A timid-looking accountant summoned up his courage and answered

the chairman’s question,What answer would you suggest that he

make? The amount that contractors will bid

depends on the rules of the auction,If

you contract to the low bidder at the low

218 AUCTIONS (Ch,17)

bidder’s bid,then all bidders will bid

a higher amount than they would if you

contract at the second lowest bid.

(d) The chairman ignored the accountants and proposed the following

plan,\Let us award the contract by means of sealed bids,but let us do it

wisely,Since we know that the contractors’ costs are either H,M,orL,

we will accept only bids of H,M,orL,and we will award the contract

to the low bidder at the price he himself bids,(If there is a tie,we will

randomly select one of the bidders and award it to him at his bid.)"

If the chairman’s scheme is adopted,would it ever be worthwhile for

a contractor with costs of L to bid L? No,If he bids L,

he is sure to make zero profits whether

or not he gets the contract,If he bids

higher than L there is a chance that he

might get the contract and make a profit.

(e) Suppose that the chairman’s bidding scheme is adopted and that both

contractors use the strategy of padding their bids in the following way,A

contractor will bid M if her costs are L,and she will bid H if her costs are

H or M,If contractors use this strategy,what is the expected cost of the

project to General Scooters? H

2

3

+ M

1

3

Which of the two schemes

will result in a lower expected cost for General Scooters,the accountants’

scheme or the chairman’s scheme?* The accountants’

scheme.

(f) We have not yet demonstrated that the bid-padding strategies pro-

posed above are equilibrium strategies for bidders,Here we will show that

this is the case for some (but not all) values of H,M,andL,Suppose

that you are one of the two contractors,You believe that the other con-

tractor is equally likely to have costs of H,M,orL andthathewillbid

* The chairman’s scheme might not have worked out so badly for Gen-

eral Scooters if he had not insisted that the only acceptable bids are H,

M,andL,If bidders had been allowed to bid any number between L

and H,then the only equilibrium in bidding strategies would involve the

use of mixed strategies,and if the contractors used these strategies,the

expected cost of the project to General Scooters would be the same as it

is with the second-bidder auction proposed by the accountants.

NAME 219

H when his costs are M or H and he will bid M when his costs are L.

Obviously if your costs are H,you can do no better than to bid H.If

your costs are M,your expected pro ts will be positive if you bid H and

negative or zero if you bid L or M,What if your costs are L? For what

values of H,M,andL will the best strategy available to you be to bid

H? 5M?4H>L

17.6 (3) Late in the day at an antique rug auction there are only two

bidders left,April and Bart,The last rug is brought out and each bidder

takes a look at it,The seller says that she will accept sealed bids from

each bidder and will sell the rug to the highest bidder at the highest

bidder’s bid.

Each bidder believes that the other is equally likely to value the

rug at any amount between 0 and $1,000,Therefore for any number X

between 0 and 1,000,each bidder believes that the probability that the

other bidder values the rug at less than X is X=1;000,The rug is actually

worth $800 to April,If she gets the rug,her pro t will be the di erence

between $800 and what she pays for it,and if she doesn’t get the rug,

her pro t will be zero,She wants to make her bid in such a way as to

maximize her expected pro t.

(a) Suppose that April thinks that Bart will bid exactly what the rug is

worth to him,If she bids $700 for the rug,what is the probability that

she will get the rug? 7/10,If she gets the rug for $700,what is her

pro t? $100,What is her expected pro t if she bids $700? $70.

(b) Suppose that Bart will pay exactly what the rug is worth to him.

If April bids $600 for the rug,what is the probability that she will get

the rug? 6/10,What is her pro t if she gets the rug for $600?

$200,What is her expected pro t if she bids $600? $120.

(c) Again suppose that Bart will bid exactly what the rug is worth to

him,If April bids $x for the rug (where x is a number between 0 and

1,000) what is the probability that she will get the rug? x=1;000

What is her pro t if she gets the rug? $800-x Write a formula for

her expected pro t if she bids $x,$(800?x)(x=1;000),Find

the bid x that maximizes her expected pro t,(Hint,Take a derivative.)

x = 400.

220 AUCTIONS (Ch,17)

(d) Now let us go a little further toward nding a general answer,Suppose

that the value of the rug to April is $V and she believes that Bart will

bid exactly what the rug is worth to him,Write a formula that expresses

her expected pro t in terms of the variables V and x if she bids $x.

$(V?x)(x=1;000) Now calculate the bid $x that will maximize

her expected pro t,(Same hint,Take a derivative.) x = V=2.

17.7 (3) If you did the previous problem correctly,you found that if

April believes that Bart will bid exactly as much as the rug is worth to

him,then she will bid only half as much as the rug is worth to her,If

this is the case,it doesn’t seem reasonable for April to believe that Bart

will bid his full value,Let’s see what would the best thing for April to do

if she believed that Bart would bid only half as much as the rug is worth

to him.

(a) If Bart always bids half of what the rug is worth to him,what is

the highest amount that Bart would ever bid? $500,Why would it

never pay for April to bid more than $500.01? She can get it

for sure by bidding just over $500,since

Bart will never bid more than $500.

(b) Suppose that the the rug is worth $800 to April and she bids $300 for

it,April will only get the rug if the value of the rug to Bart is less than

$600 What is the probability that she will get the rug if she bids $300

for it? 6/10,What is her pro t if she bids $300 and gets the rug?

$500,What is her expected pro t if she bids $300? $300.

(c) Suppose that the rug is worth $800 to April,What is the probability

that she will get it if she bids $x where $x<$500? 2x=1;000,Write

a formula for her expected pro t as a function of her bid $x when the rug

is worth $800 to her,$(800?x)2x=1;000,What bid maximizes

her expected pro t in this case? $400.

NAME 221

(d) Suppose that April values the rug at $V and she believes that Bart

will bid half of his true value,Show that the best thing for April is to

bidhalfofherowntruevalue,Maximize (V?x)2x=1;000:

The derivative with respect to x is 0

when x = V=2.

(e) Suppose that April believes that Bart will bid half of his actual value

and Bart believes that April will bid half of her actual value,Suppose also

that they both act to maximize their expected pro t given these beliefs.

Will these beliefs be self-con rming in the sense that given these beliefs,

each will take the action that the other expects? Yes.

17.8 (2) Rod’s Auction House in Bent Crankshaft,Oregon,holds sealed-

bid used-car auctions every Tuesday,Each used car is sold to the highest

bidder at the second-highest bidder’s bid,On average,half of the cars

that are sold at Rod’s Auction House are lemons and half are good used

cars,A good used car is worth $1,000 to any buyer and a lemon is worth

only $100 to any buyer,Buyers are allowed to look over the used cars for

a few minutes before they are auctioned,Almost all of the buyers who

attend the auctions can do no better than random choice at picking good

cars from among the lemons,The only exception is Al Crankcase,Al can

sometimes,but not always,detect a lemon by licking the oil o of the

dipstick,A good car will never fail Al’s taste test,but 1/3 of the lemons

fail his test,Al attends every auction,licks every dipstick,and taking

into account the results of his taste test,bids his expected value for every

car.

(a) This auction environment is an example of a (common,private)

common value auction.

(b) If a car passes Al’s taste test,what is the probability that it is a good

used car? 3/4

(c) If a car fails Al’s taste test,what is the probability that it is a good

used car? 0

(d) How much will Al bid for a car that passes his taste test? 3=5

1;000 + 2=5 100 = $640 How much will he bid for a car

that fails his taste test? $100

222 AUCTIONS (Ch,17)

(e) Suppose that for each car,a naive bidder at Rod’s Auction House bid

his expected value for a randomly selected car from among those available.

How much would he bid? $550

(f) Given that Al bids his expected value for every used car and the naive

bidders bid the expected value of a randomly selected car,will a naive

bidder ever get a car that passed Al’s taste test? No

(g) What is the expected value of cars that naive bidders get if they always

bid their expected values for a randomly selected car? $100 Will naive

bidders make money,lose money,or break even if they follow this policy?

Lose money.

(h) If the bidders other than Al bid their expected value for a car,given

that it has failed Al’s taste test,how much will they bid? $100

(i) If bidders other than Al bid their expected values for cars that fail Al’s

taste test,and Al bids his expected value for all cars,given the results

of the test,who will get the good cars and at what price? (Recall that

cars are sold to the highest bidder at the second-highest bid.) Al

will get all of the good cars and he will

pay $100 for them.

(j) What will Al’s expected pro t be on a car that passes his test?

$540

17.9 (3) Steve and Leroy buy antique paintings at an art gallery in

Fresno,California,Eighty percent of the paintings that are sold at the

gallery are fakes,and the rest are genuine,After a painting is purchased,

it will be carefully analyzed,and then everybody will know for certain

whether it is genuine or a fake,A genuine antique is worth $1,000,A

fake is worthless,Before they place their bids,buyers are allowed to

inspect the paintings briefly and then must place their bids,Because

they are allowed only a brief inspection,Steve and Leroy each try to

guess whether the paintings are fakes by smelling them,Steve nds that

if a painting fails his sni test,then it is certainly a fake,However,he

cannot detect all fakes,In fact the probability that a fake passes Steve’s

sni test is 1/2,Leroy detects fakes in the same way as Steve,Half of

the fakes fail his sni test and half of them pass his sni test,Genuine

paintings are sure to pass Leroy’s sni test,For any fake,the probability

that Steve recognizes it as a fake is independent of the probability that

Leroy recognizes it as a fake.

NAME 223

The auction house posts a price for each painting,Potential buyers

can submit a written o er to buy at the posted price on the day of the

sale,If more than one person o ers to buy the painting,the auction house

will select one of them at random and sell to that person at the posted

price.

(a) One day,as the auction house is about to close,Steve arrives and

discovers that neither Leroy nor any other bidders have appeared,He

sni s a painting,and it passes his test,Given that it has passed his test,

what is the probability that it is a good painting? (Hint,Since fakes are

much more common than good paintings,the number of fakes that pass

Steve’s test will exceed the number of genuine antiques that pass his test.)

1/3 Steve realizes that he can buy the painting for the posted price

if he wants it,What is the highest posted price at which he would be

willing to buy the painting? $333.33.

(b) On another day,Steve and Leroy see each other at the auction,sni ng

all of the paintings,No other customers have appeared at the auction

house,In deciding how much to bid for a painting that passes his sni

test,Steve considers the following,If a painting is selected at random and

sni ed by both Steve and Leroy,there are ve possible outcomes,Fill in

the blanks for the probability of each.

A,Genuine and passes both dealers’ tests,Probability,.2

B,Fake and passes both dealers’ tests,Probability,.2

C,Fake and passes Steve’s test but fails Leroy’s,Probability,.2

D,Fake and passes Leroy’s test but fails Steve’s,Probability:

.2

E,Fake and fails both dealers’ tests,Probability,.2

(c) On the day when Steve and Leroy are the only customers,the auction

house sets a reserve price of $300,Suppose that Steve believes that Leroy

will o er to buy any painting that passes his sni test,Recall that if Steve

and Leroy both bid on a painting,the probability that Steve gets it is only

1/2,If Steve decides to bid on every painting that passes his own sni

test,what is the probability that a randomly selected painting is genuine

and that Steve is able to buy it?,1 What is the probability that

a randomly selected painting is a fake and that Steve will bid on it and

get it?,3 If Steve o ers to pay $300 for every painting that

passes his sni test,will his expected pro t be positive or negative?

negative Suppose that Steve knows that Leroy is willing to pay the

224 AUCTIONS (Ch,17)

reserve price for any painting that passes Leroy’s sni test,What is the

highest reserve price that Steve should be willing to pay for a painting

that passes his own sni test? $250

17.10 (2) Every day the Repo nance company holds a sealed-bid,

second-price auction in which it sells a repossessed automobile,There are

only three bidders who bid on these cars,Arnie,Barney,and Carny,Each

of these bidders is a used-car dealer whose willingness to pay for another

used car fluctuates randomly from day to day in response to variation in

demand at his car lot,The value of one of these used cars to any dealer,

on any given day is a random variable which takes a high value $H with

probability 1=2andalowvalue$L with probability 1=2,The value that

each dealer places on a car on a given day is independent of the values

placed by the other dealers.

Each day the used-car dealers submit written bids for the used car

being auctioned,The Repo nance company will sell the car to the dealer

with the highest bid at the price bid by the second-highest bidder,If there

is a tie for the highest bid,then the second-highest bid is equal to the

highest bid and so that day’s car will be sold to a randomly selected top

bidder at the price bid by all top bidders.

(a) How much should a dealer bid for a used car on a day when he places

a value of $H on a used car? $H How much should a dealer bid for a

used car on a day when he places a value of $L on a used car? $L

(b) If the dealers do not collude,how much will Repo get for a used car

on days when two or three dealers value the car at $H? $H How much

will Repo get for a used car on days when fewer than two dealers value

the car at $H? $L

(c) On any given day,what is the probability that Repo receives $H for

that day’s used car? 1/2 What is the probability that Repo receives

$L for that day’s used car? 1/2 What is Repo’s expected revenue from

the sale? $(H +L)=2

(d) If there is no collusion and every dealer bids his actual valuation for

every used car,what is the probability on any given day that Arnie gets

a car for a lower price than the value he places on it? (Hint,This will

happen only if the car is worth $H to Arnie and $L to the other dealers.)

1/8 Suppose that we measure a car dealer’s pro t by the di erence

NAME 225

between what a car is worth to him and what he pays for it,On a

randomly selected day,what is Arnie’s expected pro t? (H?L)=8

(e) The expected total pro t of all participants in the market is the sum

of the expected pro ts of the three car dealers and the expected revenue

realized by Repo,Used cars are sold by a sealed-bid,second-price auction

and the dealers do not collude,What is the sum of the expected pro ts of

all participants in the market? (3(H?L)=8)+((H+L)=2) =

(7=8)H +(1=8)L

17.11 (3) This problem (and the two that follow) concerns collusion

among bidders in sealed-bid auctions,Many writers have found evidence

that collusive bidding occurs,The common name for a group that prac-

tices collusive bidding is a \bidding ring."*

Arnie,Barney,and Carny of the previous problem happened to meet

at a church social and got to talking about the high prices they were

paying for used cars and the low pro ts they were making,Carny com-

plained,\About half the time the used cars go for $H,and when that

happens,none of us makes any money." Arnie got a thoughtful look and

then whispered,\Why don’t we agree to always bid $L in Repo’s used-car

auctions?" Barney said,\I’m not so sure that’s a good idea,If we all bid

$L,then we will save some money,but the trouble is,when we all bid the

same,we are just as likely to get the car if we have a low value as we are

to get it if we have a high value,When we bid what we think its worth,

then it always goes to one of the people who value it most."

(a) If Arnie,Barney,and Carny agree to always bid $L,thenonanygiven

day,what is the probability that Barney gets the car for $L when it is

actually worth $H to him? 1/6 What is Barney’s expected pro t per

day? $(H?L)=6

(b) Do the three dealers make higher expected pro ts with this collusive

agreement than they would if they did not collude? Explain,Yes.

(H?L)=6 > (H?L)=8

* Our discussion draws extensively on a paper,\Collusive Bidder Be-

havior at Single-Object,Second-price,and English Auctions" by Daniel

Graham and Robert Marshall in the Journal of Political Economy,1987.

226 AUCTIONS (Ch,17)

(c) Calculate the expected total pro ts of all participants in the market

(including Repo as well as the three dealers) in the case where the dealers

collude,3(H?L)=6+L =(H+L)=2 Are these expected total

pro ts larger or smaller than they are when the dealers do not collude?

Smaller

(d) The cars are said to be allocated e ciently if a car never winds up

in the hands of a dealer who values it less than some other dealer values

it,With a sealed-bid,second-price auction,if there is no collusion,are

the cars allocated e ciently? Yes,If the dealers collude as in this

problem,are the cars allocated e ciently? No.

17.12 (2) Arnie,Barney,and Carny happily practiced the strategy of

\always bid low" for several weeks,until one day Arnie had another idea.

Arnie proposed to the others,\When we all bid $L,it sometimes happens

that the one who gets the week’s car values it at only $L although it is

worth $H to somebody else,I’ve thought of a scheme that will increase

pro ts for all of us." Here is Arnie’s scheme,Every day,before Repo holds

its auction,Arnie,Barney and Carny will hold a sealed-bid,second-price

preauction auction among themselves in which they bid for the right to

be the only high bidder in that day’s auction,The dealer who wins this

preauction bidding can bid anything he likes,while the other two bidders

must bid $L,A preauction auction like this is known is a \knockout."

The revenue that is collected from the \knockout" auction is divided

equally among Arnie,Barney,and Carny,For this problem,assume that

in the knockout auction,each bidder bids his actual value of winning the

knockout auction.*

(a) If the winner of the knockout auction values the day’s used car at

$H,then he knows that he can bid $H for this car in Repo’s second-price

sealed-bid auction and he will get it for a price of $L,Therefore the value

of winning the knockout auction to someone who values a used car at $H

must be $H?L,The value of winning the knockout auction to

someone who values a used car at $L is 0

* It is not necessarily the case that this is the best strategy in the

knockout auction,since one’s bids a ect the revenue redistributed from

the auction as well as who gets the right to bid,Graham and Marshall

present a variation on this mechanism that ensures \honest" bidding in

the knockout auction.

NAME 227

(b) On a day when one dealer values the used car at $H and the other

two value it at $L,the dealer with value $H will bid $H?L in

the knockout auction and the other two dealers will bid 0 In this

case,in the knockout auction,the dealer pays 0 for the right to

be the only high bidder in Repo’s auction,In this case,the day’s used

car will go to the only dealer with value $H and he pays Repo $L for

it,On this day,the dealer with the high buyer value makes a total pro t

of $H?L

(c) We continue to assume that in the knockout auction,dealers bid

their actual values of winning the knockout,On days when two or more

buyers value the used car at $H,the winner of the knockout auction pays

H?L for the right to be the only high bidder in Repo’s auction.

(d) If Arnie’s scheme is adopted,what is the expected total pro t of each

of the three car dealers? (Remember to include each dealer’s share of the

revenue from the knockout auction.) 7( H-L)/8

17.13 (2) After the passage of several weeks during which Repo never got

more than one high bid for a car,the Repo folks guessed that something

was amiss,Some members of the board of directors proposed hiring a hit

man to punish Arnie,Barney,and Carny,but cooler heads prevailed and

they decided instead to hire an economist who had studied Intermediate

Microeconomics,The economist suggested,\Why don’t you set a reserve

price $R which is just a little bit lower than $H (but of course much

larger than $L)? If you get at least one bid of $R,sellitfor$R to one

of these bidders,and if you don’t get a bid as large as your $R,then just

dump that day’s car into the river,(Sadly,the environmental protection

authorities in Repo’s hometown are less than vigilant.) \But what a

waste," said a Repo o cial,\Just do the math," replied the economist.

(a) The economist continued,\If Repo sticks to its guns and refuses to

sell at any price below $R,then even if Arnie,Barney,and Carny collude,

the best they can do is for each to bid $R when they value a car at $H

and to bid nothing when they value it at $L." If they follow this strategy,

the probability that Repo can sell a given car for $R is 7/8,soRepo’s

expected pro t will be $(7=8)R.

(b) Setting a reserve price that is just slightly below $H and destroying

cars for which it gets no bid will be more pro table for Repo than setting

no reservation price if the ratio H=L is greater than 7/8 and less

pro table if H=L is less than 7/8

228 AUCTIONS (Ch,17)

Chapter 18 NAME

Technology

Introduction,In this chapter you work with production functions,re-

lating output of a rm to the inputs it uses,This theory will look familiar

to you,because it closely parallels the theory of utility functions,In utility

theory,an indi erence curve is a locus of commodity bundles,all of which

give a consumer the same utility,In production theory,an isoquant is a lo-

cus of input combinations,all of which give the same output,In consumer

theory,you found that the slope of an indi erence curve at the bundle

(x

1;x

2

) is the ratio of marginal utilities,MU

1

(x

1;x

2

)=MU

2

(x

1;x

2

),In

production theory,the slope of an isoquant at the input combination

(x

1;x

2

) is the ratio of the marginal products,MP

1

(x

1;x

2

)=MP

2

(x

1;x

2

).

Most of the functions that we gave as examples of utility functions can

also be used as examples of production functions.

There is one important di erence between production functions and

utility functions,Remember that utility functions were only \unique up to

monotonic transformations." In contrast,two di erent production func-

tions that are monotonic transformations of each other describe di erent

technologies.

Example,If the utility function U(x

1;x

2

)=x

1

+x

2

represents a person’s

preferences,then so would the utility function U

(x

1;x

2

)=(x

1

+ x

2

)

2

.

A person who had the utility function U

(x

1;x

2

)wouldhavethesame

indi erence curves as a person with the utility function U(x

1;x

2

)and

would make the same choices from every budget,But suppose that one

rm has the production function f(x

1;x

2

)=x

1

+x

2

,and another has the

production function f

(x

1;x

2

)=(x

1

+x

2

)

2

.Itistruethatthetwo rms

will have the same isoquants,but they certainly do not have the same

technology,If both rms have the input combination (x

1;x

2

)=(1;1),

then the rst rm will have an output of 2 and the second rm will have

an output of 4.

Now we investigate \returns to scale." Here we are concerned with

the change in output if the amount of every input is multiplied by a

number t>1,If multiplying inputs by t multiplies output by more than

t,then there are increasing returns to scale,If output is multiplied by

exactly t,there are constant returns to scale,If output is multiplied by

less than t,then there are decreasing returns to scale.

Example,Consider the production function f(x

1;x

2

)=x

1=2

1

x

3=4

2

.Ifwe

multiply the amount of each input by t,then output will be f(tx

1;tx

2

)=

(tx

1

)

1=2

(tx

2

)

3=4

.Tocomparef(tx

1;tx

2

)tof(x

1;x

2

),factor out the

expressions involving t from the last equation,You get f(tx

1;tx

2

)=

t

5=4

x

1=2

1

x

3=4

2

= t

5=4

f(x

1;x

2

),Therefore when you multiply the amounts

of all inputs by t,you multiply the amount of output by t

5=4

,This means

there are increasing returns to scale.

230 TECHNOLOGY (Ch,18)

Example,Let the production function be f(x

1;x

2

)=minfx

1;x

2

g.Then

f(tx

1;tx

2

)=minftx

1;tx

2

g=mintfx

1;x

2

g= tminfx

1;x

2

g= tf(x

1;x

2

):

Therefore when all inputs are multiplied by t,output is also multiplied by

t,It follows that this production function has constant returns to scale.

You will also be asked to determine whether the marginal product

of each single factor of production increases or decreases as you increase

the amount of that factor without changing the amount of other factors.

Those of you who know calculus will recognize that the marginal product

of a factor is the rst derivative of output with respect to the amount

of that factor,Therefore the marginal product of a factor will decrease,

increase,or stay constant as the amount of the factor increases depending

on whether the second derivative of the production function with respect

to the amount of that factor is negative,positive,or zero.

Example,Consider the production function f(x

1;x

2

)=x

1=2

1

x

3=4

2

.The

marginal product of factor 1 is

1

2

x

1=2

1

x

3=4

2

,This is a decreasing function

of x

1

,as you can verify by taking the derivative of the marginal product

with respect to x

1

,Similarly,you can show that the marginal product of

x

2

decreases as x

2

increases.

18.0 Warm Up Exercise,The rst part of this exercise is to cal-

culate marginal products and technical rates of substitution for several

frequently encountered production functions,As an example,consider

the production function f(x

1;x

2

)=2x

1

+

p

x

2

,The marginal product of

x

1

is the derivative of f(x

1;x

2

) with respect to x

1

,holding x

2

xed,This

is just 2,The marginal product of x

2

is the derivative of f(x

1;x

2

)with

respect to x

2

,holding x

1

xed,which in this case is

1

2

p

x

2

.TheTRS is

MP

1

=MP

2

=?4

p

x

2

,Those of you who do not know calculus should

ll in this table from the answers in the back,The table will be a useful

reference for later problems.

NAME 231

Marginal Products and Technical Rates of Substitution

f(x

1;x

2

) MP

1

(x

1;x

2

) MP

2

(x

1;x

2

) TRS(x

1;x

2

)

x

1

+2x

2

1 2?1=2

ax

1

+bx

2

a b?a=b

50x

1

x

2

50x

2

50x

1

x

2

x

1

x

1=4

1

x

3=4

2

1

4

x

3=4

1

x

3=4

2

3

4

x

1=4

1

x

1=4

2

x

2

3x

1

Cx

a

1

x

b

2

Cax

a?1

1

x

b

2

Cbx

a

1

x

b?1

2

ax

2

bx

1

(x

1

+2)(x

2

+1) x

2

+1 x

1

+2?

x

2

+1

x

1

+2

(x

1

+a)(x

2

+b) x

2

+b x

1

+a?

x

2

+b

x

1

+a

ax

1

+b

p

x

2

a

b

2

p

x

2

2a

p

x

2

b

x

a

1

+x

a

2

ax

a?1

1

ax

a?1

2

x

1

x

2

a?1

(x

a

1

+x

a

2

)

b

bax

a?1

1

(x

a

1

+x

a

2

)

b?1

bax

a?1

2

(x

a

1

+x

a

2

)

b?1

x

1

x

2

a?1

232 TECHNOLOGY (Ch,18)

Returns to Scale and Changes in Marginal Products

For each production function in the table below,put an I,C,orD in

the rst column if the production function has increasing,constant,or

decreasing returns to scale,Put an I,C,orD in the second (third)

column,depending on whether the marginal product of factor 1 (factor

2) is increasing,constant,or decreasing,as the amount of that factor

alone is varied.

f(x

1;x

2

) Scale MP

1

MP

2

x

1

+2x

2

C C C

p

x

1

+2x

2

D D D

:2x

1

x

2

2

I C I

x

1=4

1

x

3=4

2

C D D

x

1

+

p

x

2

D C D

(x

1

+1)

:5

(x

2

)

:5

D D D

x

1=3

1

+x

1=3

2

3

C D D

18.1 (0) Prunella raises peaches,Where L is the number of units of

labor she uses and T is the number of units of land she uses,her output

is f(L;T)=L

1

2

T

1

2

bushels of peaches.

(a) On the graph below,plot some input combinations that give her an

output of 4 bushels,Sketch a production isoquant that runs through these

points,The points on the isoquant that gives her an output of 4 bushels

all satisfy the equation T = 16=L.

NAME 233

0481216

2

4

6

L

T

8

(b) This production function exhibits (constant,increasing,decreasing)

returns to scale,Constant returns to scale.

(c) In the short run,Prunella cannot vary the amount of land she uses.

On the graph below,use blue ink to draw a curve showing Prunella’s

output as a function of labor input if she has 1 unit of land,Locate the

points on your graph at which the amount of labor is 0,1,4,9,and

16 and label them,The slope of this curve is known as the marginal

product of labor,Is this curve getting steeper or flatter

as the amount of labor increase? Flatter.

0481216

2

4

6

Labour

Output

8

Blue line

Red line

Red MPL line

234 TECHNOLOGY (Ch,18)

(d) Assuming she has 1 unit of land,how much extra output does she

get from adding an extra unit of labor when she previously used 1 unit of

labor?

p

2?1,41,4 units of labor?

p

5?2,24,If

you know calculus,compute the marginal product of labor at the input

combination (1;1) and compare it with the result from the unit increase

in labor output found above,Derivative is 1=2

p

L,so

the MP is,5 when L =1 and,25 when L =4.

(e) In the long run,Prunella can change her input of land as well as

of labor,Suppose that she increases the size of her orchard to 4 units

of land,Use red ink to draw a new curve on the graph above showing

output as a function of labor input,Also use red ink to draw a curve

showing marginal product of labor as a function of labor input when the

amount of land is xed at 4.

18.2 (0) Supposex

1

and x

2

areusedin xedproportionsandf(x

1;x

2

)=

minfx

1;x

2

g.

(a) Suppose that x

1

<x

2

,The marginal product for x

1

is 1

and (increases,remains constant,decreases) remains constant

for small increases in x

1

.Forx

2

the marginal product is 0,

and (increases,remains constant,decreases) remains constant

for small increases in x

2

,The technical rate of substitution between x

2

and x

1

is infinity,This technology demonstrates (increasing,

constant,decreasing) constant returns to scale.

(b) Suppose that f(x

1;x

2

)=minfx

1;x

2

g and x

1

= x

2

= 20,What is

the marginal product of a small increase in x

1

0,What is the

marginal product of a small increase in x

2

0,The marginal

product of x

1

will (increase,decrease,stay constant) increase if

the amount of x

2

is increased by a little bit.

Calculus 18.3 (0) Suppose the production function is Cobb-Douglas and

f(x

1;x

2

)=x

1=2

1

x

3=2

2

.

(a) Write an expression for the marginal product of x

1

at the point

(x

1;x

2

).

1

2

x

1=2

1

x

3=2

2

.

NAME 235

(b) The marginal product of x

1

(increases,decreases,remains constant)

decreases for small increases in x

1

,holding x

2

xed.

(c) The marginal product of factor 2 is 3=2x

1=2

1

x

1=2

2

,and it (in-

creases,remains constant,decreases) increases for small increases

in x

2

.

(d) An increase in the amount of x

2

(increases,leaves unchanged,de-

creases) increases the marginal product of x

1

.

(e) The technical rate of substitution between x

2

and x

1

is?x

2

=3x

1

.

(f) Does this technology have diminishing technical rate of substitution?

Yes.

(g) This technology demonstrates (increasing,constant,decreasing)

increasing returns to scale.

18.4 (0) The production function for fragles is f(K;L)=L=2+

p

K,

where L is the amount of labor used and K the amount of capital used.

(a) There are (constant,increasing,decreasing) decreasing re-

turns to scale,The marginal product of labor is constant (con-

stant,increasing,decreasing).

(b) In the short run,capital is xed at 4 units,Labor is variable,On the

graph below,use blue ink to draw output as a function of labor input in

the short run,Use red ink to draw the marginal product of labor as a

function of labor input in the short run,The average product of labor is

de ned as total output divided by the amount of labor input,Use black

ink to draw the average product of labor as a function of labor input in

the short run.

236 TECHNOLOGY (Ch,18)

0481216

2

4

6

Labour

Fragles

8

Black line

Blue

line

Red line

18.5 (0) General Monsters Corporation has two plants for producing

juggernauts,one in Flint and one in Inkster,The Flint plant produces

according to f

F

(x

1;x

2

)=minfx

1;2x

2

g and the Inkster plant produces

according to f

I

(x

1;x

2

)=minf2x

1;x

2

g,wherex

1

and x

2

are the inputs.

(a) On the graph below,use blue ink to draw the isoquant for 40 jugger-

nauts at the Flint plant,Use red ink to draw the isoquant for producing

40 juggernauts at the Inkster plant.

NAME 237

020406080

20

40

60

X2

80

,

a

b

c

Blue isoquant

Red

isoquant

Black isoquant

X1

(b) Suppose that the rm wishes to produce 20 juggernauts at each plant.

How much of each input will the rm need to produce 20 juggernauts

at the Flint plant? x

1

=20;x

2

=10,How much of each

input will the rm need to produce 20 juggernauts at the Inkster plant?

x

1

=10;x

2

=20,Label with an a on the graph,the point

representing the total amount of each of the two inputs that the rm

needs to produce a total of 40 juggernauts,20 at the Flint plant and 20

at the Inkster plant.

(c) Label with a b on your graph the point that shows how much of each

of the two inputs is needed in toto if the rm is to produce 10 juggernauts

in the Flint plant and 30 juggernauts in the Inkster plant,Label with a

c the point that shows how much of each of the two inputs that the rm

needs in toto if it is to produce 30 juggernauts in the Flint plant and

10 juggernauts in the Inkster plant,Use a black pen to draw the rm’s

isoquant for producing 40 units of output if it can split production in any

manner between the two plants,Is the technology available to this rm

convex? Yes.

18.6 (0) You manage a crew of 160 workers who could be assigned to

make either of two products,Product A requires 2 workers per unit of

output,Product B requires 4 workers per unit of output.

(a) Write an equation to express the combinations of products A and

B that could be produced using exactly 160 workers,2A +4B =

160,On the diagram below,use blue ink to shade in the area depicting

238 TECHNOLOGY (Ch,18)

the combinations of A and B that could be produced with 160 workers.

(Assume that it is also possible for some workers to do nothing at all.)

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

020406080

20

40

60

B

80

,

A

Red

shading

Blue

shading

Black

shading

a

(b) Suppose now that every unit of product A that is produced requires

the use of 4 shovels as well as 2 workers and that every unit of product B

produced requires 2 shovels and 4 workers,On the graph you have just

drawn,use red ink to shade in the area depicting combinations of A and

B that could be produced with 180 shovels if there were no worries about

the labor supply,Write down an equation for the set of combinations of

A and B that require exactly 180 shovels,4A+2B = 180.

(c) On the same diagram,use black ink to shade the area that repre-

sents possible output combinations when one takes into account both the

limited supply of labor and the limited supply of shovels.

(d) On your diagram locate the feasible combination of inputs that use up

all of the labor and all of the shovels,If you didn’t have the graph,what

equations would you solve to determine this point? 2A +4B =

160 and 4A+2B = 180.

(e) If you have 160 workers and 180 shovels,what is the largest amount of

product A that you could produce? 45 units,If you produce this

amount,you will not use your entire supply of one of the inputs,Which

one? Workers,How many will be left unused? 70.

18.7 (0) A rm has the production function f(x;y)=minf2x;x + yg.

On the graph below,use red ink to sketch a couple of production isoquants

for this rm,A second rm has the production function f(x;y)=x +

minfx;yg,Do either or both of these rms have constant returns to scale?

NAME 239

Both do,On the same graph,use black ink to draw a couple of

isoquants for the second rm.

010203040

10

20

30

40

y

x

Black

isoquants

Red

isoquants

18.8 (0) Suppose the production function has the form

f(x

1;x

2;x

3

)=Ax

a

1

x

b

2

x

c

3;

where a+b+c>1.Provethatthereareincreasingreturnstoscale.

For any t>1,f(tx

1;tx

2;tx

3

)=A(tx

1

)

a

(tx

2

)

b

(tx

3

)

c

=

t

a+b+c

f(x

1;x

2;x

3

) >tf(x

1;x

2;x

3

).

18.9 (0) Suppose that the production function is f(x

1;x

2

)=Cx

a

1

x

b

2

,

where a,b,andC are positive constants.

(a) For what positive values of a,b,andC are there decreasing returns

to scale? All C>0 and a + b<1,constant returns to

scale? All C>0 and a + b =1,increasing returns to

scale? All C>0 and a+b>1.

(b) For what positive values of a,b,andC is there decreasing marginal

product for factor 1? All C>0 and b>0 and a<1.

240 TECHNOLOGY (Ch,18)

(c) For what positive values of a,b,andC is there diminishing technical

rate of substitution? For all positive values.

18.10 (0) Suppose that the production function is f(x

1;x

2

)=

(x

a

1

+x

a

2

)

b

,wherea and b are positive constants.

(a) For what positive values of a and b are there decreasing returns to

scale? ab< 1,Constant returns to scale? ab =1,Increasing

returns to scale? ab> 1.

18.11 (0) Suppose that a rm has the production function f(x

1;x

2

)=

p

x

1

+x

2

2

.

(a) The marginal product of factor 1 (increases,decreases,stays constant)

decreases as the amount of factor 1 increases,The marginal

product of factor 2 (increases,decreases,stays constant) increases

as the amount of factor 2 increases.

(b) This production function does not satisfy the de nition of increasing

returns to scale,constant returns to scale,or decreasing returns to scale.

How can this be? Returns to scale are different

depending on the ratio in which the factors

are used,Find a combination of inputs such that doubling the

amount of both inputs will more than double the amount of output.

x

1

=1,x

2

=4,for example,Find a combination of

inputs such that doubling the amount of both inputs will less than double

output,x

1

=4,x

2

=0,for example.

Chapter 19 NAME

Profit Maximization

Introduction,A rm in a competitive industry cannot charge more than

the market price for its output,If it also must compete for its inputs,then

it has to pay the market price for inputs as well,Suppose that a pro t-

maximizing competitive rm can vary the amount of only one factor and

that the marginal product of this factor decreases as its quantity increases.

Then the rm will maximize its pro ts by hiring enough of the variable

factor so that the value of its marginal product is equal to the wage,Even

if a rm uses several factors,only some of them may be variable in the

short run.

Example,A rm has the production function f(x

1;x

2

)=x

1=2

1

x

1=2

2

,Sup-

pose that this rm is using 16 units of factor 2 and is unable to vary this

quantity in the short run,In the short run,the only thing that is left for

the rm to choose is the amount of factor 1,Let the price of the rm’s

output be p,and let the price it pays per unit of factor 1 be w

1

.We

want to nd the amount of x

1

that the rm will use and the amount of

output it will produce,Since the amount of factor 2 used in the short run

must be 16,we have output equal to f(x

1;16) = 4x

1=2

1

.Themarginal

product of x

1

is calculated by taking the derivative of output with respect

to x

1

,This marginal product is equal to 2x

1=2

1

,Setting the value of the

marginal product of factor 1 equal to its wage,we have p2x

1=2

1

= w

1

.

Now we can solve this for x

1

,We nd x

1

=(2p=w

1

)

2

,Plugging this

into the production function,we see that the rm will choose to produce

4x

1=2

1

=8p=w

1

units of output.

In the long run,a rm is able to vary all of its inputs,Consider

the case of a competitive rm that uses two inputs,Then if the rm is

maximizing its pro ts,it must be that the value of the marginal product

of each of the two factors is equal to its wage,This gives two equations in

the two unknown factor quantities,If there are decreasing returns to scale,

these two equations are enough to determine the two factor quantities,If

there are constant returns to scale,it turns out that these two equations

are only su cient to determine the ratio in which the factors are used.

In the problems on the weak axiom of pro t maximization,you are

asked to determine whether the observed behavior of rms is consistent

with pro t-maximizing behavior,To do this you will need to plot some of

the rm’s isopro t lines,An isopro t line relates all of the input-output

combinations that yield the same amount of pro t for some given input

and output prices,To get the equation for an isopro t line,just write

down an equation for the rm’s pro ts at the given input and output

prices,Then solve it for the amount of output produced as a function

of the amount of the input chosen,Graphically,you know that a rm’s

behavior is consistent with pro t maximization if its input-output choice

242 PROFIT MAXIMIZATION (Ch,19)

in each period lies below the isopro t lines of the other periods.

19.1 (0) The short-run production function of a competitive rm is

given by f(L)=6L

2=3

,whereL istheamountoflaborituses,(For

those who do not know calculus|if total output is aL

b

,wherea and b

are constants,and where L is the amount of some factor of production,

then the marginal product of L is given by the formula abL

b?1

.) The cost

per unit of labor is w = 6 and the price per unit of output is p =3.

(a) Plot a few points on the graph of this rm’s production function and

sketch the graph of the production function,using blue ink,Use black

ink to draw the isopro t line that passes through the point (0;12),the

isopro t line that passes through (0;8),and the isopro t line that passes

through the point (0;4),What is the slope of each of the isopro t lines?

They all have slope 2,How many points on the isopro t

line through (0;12) consist of input-output points that are actually pos-

sible? None,Make a squiggly line over the part of the isopro t line

through (0;4) that consists of outputs that are actually possible.

(b) How many units of labor will the rm hire? 8,How much

output will it produce? 24,If the rm has no other costs,how much

will its total pro ts be? 24.

0 8 12 16 20

Labour input

12

24

36

48

Output

424

8

4

Black lines

Blue curve

Squiggly line

13.3

Red line

NAME 243

(c) Suppose that the wage of labor falls to 4,and the price of output

remains at p,On the graph,use red ink to draw the new isopro t line

for the rm that passes through its old choice of input and output,Will

the rm increase its output at the new price? Yes,Explain why,

referring to your diagram,As the diagram shows,the

firm can reach a higher isoprofit line by

increasing output.

Calculus 19.2 (0) A Los Angeles rm uses a single input to produce a recreational

commodity according to a production function f(x)=4

p

x,wherex is

the number of units of input,The commodity sells for $100 per unit,The

input costs $50 per unit.

(a) Write down a function that states the rm’s pro t as a function of

the amount of input,= 400

p

x?50x.

(b) What is the pro t-maximizing amount of input? 16,of output?

16,How much pro ts does it make when it maximizes pro ts?

$800.

(c) Suppose that the rm is taxed $20 per unit of its output and the price

of its input is subsidized by $10,What is its new input level? 16.

What is its new output level? 16,How much pro t does it make now?

$640,(Hint,A good way to solve this is to write an expression for the

rm’s pro t as a function of its input and solve for the pro t-maximizing

amount of input.)

(d) Suppose that instead of these taxes and subsidies,the rm is taxed

at 50% of its pro ts,Write down its after-tax pro ts as a function of the

amount of input,=,50 (400

p

x?50x),What is the

pro t-maximizing amount of output? 16,How much pro t does it

make after taxes? $400.

19.3 (0) Brother Jed takes heathens and reforms them into righteous

individuals,There are two inputs needed in this process,heathens (who

are widely available) and preaching,The production function has the

following form,r

p

=minfh;pg,wherer

p

is the number of righteous

244 PROFIT MAXIMIZATION (Ch,19)

persons produced,h is the number of heathens who attend Jed’s sermons,

and p is the number of hours of preaching,For every person converted,

Jed receives a payment of s from the grateful convert,Sad to say,heathens

do not flock to Jed’s sermons of their own accord,Jed must o er heathens

apaymentofw to attract them to his sermons,Suppose the amount of

preaching is xed at p and that Jed is a pro t-maximizing prophet.

(a) If h< p,what is the marginal product of heathens? 1,What

is the value of the marginal product of an additional heathen? s.

(b) If h> p,what is the marginal product of heathens? 0,What

is the value of the marginal product of an additional heathen in this case?

0.

(c) Sketch the shape of this production function in the graph below,Label

the axes,and indicate the amount of the input where h = p.

r

hp

p

_

(d) If w<s,how many heathens will be converted? p,If w>s,

how many heathens will be converted? 0.

19.4 (0) Allie’s Apples,Inc,purchases apples in bulk and sells two prod-

ucts,boxes of apples and jugs of cider,Allie’s has capacity limitations of

three kinds,warehouse space,crating facilities,and pressing facilities,A

box of apples requires 6 units of warehouse space,2 units of crating facili-

ties,and no pressing facilities,A jug of cider requires 3 units of warehouse

space,2 units of crating facilities,and 1 unit of pressing facilities,The

total amounts available each day are,1,200 units of warehouse space,600

units of crating facilities,and 250 units of pressing facilities.

(a) If the only capacity limitations were on warehouse facilities,and if all

warehouse space were used for the production of apples,how many boxes

of apples could be produced in one day? 200,How many jugs of cider

could be produced each day if,instead,all warehouse space were used in

NAME 245

the production of cider and there were no other capacity constraints?

400,Draw a blue line in the following graph to represent the warehouse

space constraint on production combinations.

(b) Following the same reasoning,draw a red line to represent the con-

straints on output to limitations on crating capacity,How many boxes of

apples could Allie produce if he only had to worry about crating capacity?

300,Howmanyjugsofcider? 300.

(c) Finally draw a black line to represent constraints on output combina-

tions due to limitations on pressing facilities,How many boxes of apples

could Allie produce if he only had to worry about the pressing capacity

and no other constraints? An infinite number,How many

jugs of cider? 250.

(d) Now shade the area that represents feasible combinations of daily

production of apples and cider for Allie’s Apples.

0 300 400 500

100

200

300

400

500

600

100

Cider

200

Apples

600

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Blue line

Black revenue line

Red line

Black line

(e) Allie’s can sell apples for $5 per box of apples and cider for $2 per

jug,Draw a black line to show the combinations of sales of apples and

cider that would generate a revenue of $1,000 per day,At the pro t-

maximizing production plan,Allie’s is producing 200 boxes of apples

and 0 jugs of cider,Total revenues are $1,000.

246 PROFIT MAXIMIZATION (Ch,19)

19.5 (0) A pro t-maximizing rm produces one output,y,and uses one

input,x,to produce it,The price per unit of the factor is denoted by

w and the price of the output is denoted by p,You observe the rm’s

behavior over three periods and nd the following:

Period y x w p

1 1 1 1 1

2 2.5 3,5 1

3 4 8,25 1

(a) Write an equation that gives the rm’s pro ts,,as a function of the

amount of inputxit uses,the amount of outputy it produces,the per-unit

cost of the input w,and the price of output p,= py?wx.

(b) In the diagram below,draw an isopro t line for each of the three

periods,showing combinations of input and output that would yield the

same pro ts that period as the combination actually chosen,What are

the equations for these three lines? y = x,y =1+:5x,

y =2+:25x,Using the theory of revealed pro tability,shade in

the region on the graph that represents input-output combinations that

could be feasible as far as one can tell from the evidence that is available.

How would you describe this region in words? The region that

is below all 3 isoprofit lines.

06810

2

4

6

8

10

12

2

Output

4

Input

12

Period 3

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

Period 2

Period 1

NAME 247

19.6 (0) T-bone Pickens is a corporate raider,This means that he looks

for companies that are not maximizing pro ts,buys them,and then tries

to operate them at higher pro ts,T-bone is examining the nancial

records of two re neries that he might buy,the Shill Oil Company and

the Golf Oil Company,Each of these companies buys oil and produces

gasoline,During the time period covered by these records,the price of

gasoline fluctuated signi cantly,while the cost of oil remained constant

at $10 a barrel,For simplicity,we assume that oil is the only input to

gasoline production.

Shill Oil produced 1 million barrels of gasoline using 1 million barrels

of oil when the price of gasoline was $10 a barrel,When the price of

gasoline was $20 a barrel,Shill produced 3 million barrels of gasoline

using 4 million barrels of oil,Finally,when the price of gasoline was $40

a barrel,Shill used 10 million barrels of oil to produce 5 million barrels

of gasoline.

Golf Oil (which is managed by Martin E,Lunch III) did exactly the

same when the price of gasoline was $10 and $20,but when the price

of gasoline hit $40,Golf produced 3.5 million barrels of gasoline using 8

million barrels of oil.

(a) Using black ink,plot Shill Oil’s isopro t lines and choices for the three

di erent periods,Label them 10,20,and 40,Using red ink draw Golf

Oil’s isopro t line and production choice,Label it with a 40 in red ink.

06810

2

4

6

8

10

12

2

Million barrels of gasoline

4

Million barrels of oil

12

10

20

40

Red 40

248 PROFIT MAXIMIZATION (Ch,19)

(b) How much pro ts could Golf Oil have made when the price of gasoline

was $40 a barrel if it had chosen to produce the same amount that it did

when the price was $20 a barrel? $80 million,What pro ts

did Golf actually make when the price of gasoline was $40? $60

million.

(c) Is there any evidence that Shill Oil is not maximizing pro ts? Explain.

No,The data satisfy WAPM.

(d) Is there any evidence that Golf Oil is not maximizing pro ts? Explain.

Yes,When price of gas was $40,Golf could

have made more money by acting as it did

when price of gas was $20.

19.7 (0) After carefully studying Shill Oil,T-bone Pickens decides that

it has probably been maximizing its pro ts,But he still is very interested

in buying Shill Oil,He wants to use the gasoline they produce to fuel his

delivery fleet for his chicken farms,Capon Truckin’,In order to do this

Shill Oil would have to be able to produce 5 million barrels of gasoline

from 8 million barrels of oil,Mark this point on your graph,Assuming

that Shill always maximizes pro ts,would it be technologically feasible

for it to produce this input-output combination? Why or why not?

No,If it could,then it would have made

more profits by choosing this combination

than what it chose when price of oil was

$40.

19.8 (0) Suppose that rms operate in a competitive market,attempt to

maximize pro ts,and only use one factor of production,Then we know

that for any changes in the input and output price,the input choice and

the output choice must obey the Weak Axiom of Pro t Maximization,

p y? w x 0.

Which of the following propositions can be proven by the Weak Ax-

iom of Pro t Maximizing Behavior (WAPM)? Respond yes or no,and

give a short argument.

NAME 249

(a) If the price of the input does not change,then a decrease in the price

of the output will imply that the rm will produce the same amount or

less output,Yes,If price of input doesn’t

change,w =0,so WAPM says p y 0.

(b) If the price of the output remains constant,then a decrease in the

input price will imply that the rm will use the same amount or more

of the input,Yes,If price of output doesn’t

change,p =0,so WAPM says? w x 0.

(c) If both the price of the output and the input increase and the rm

produces less output,then the rm will use more of the input,No.

Sign pattern is (+)(?)?(+)(+) 0,which

cannot happen.

19.9 (1) Farmer Hoglund has discovered that on his farm,he can get

30 bushels of corn per acre if he applies no fertilizer,When he applies N

pounds of fertilizer to an acre of land,the marginal product of fertilizer is

1?N=200 bushels of corn per pound of fertilizer.

(a) If the price of corn is $3 a bushel and the price of fertilizer is $p per

pound (where p<3),how many pounds of fertilizer should he use per

acre in order to maximize pro ts? 200?66:66p.

(b) (Only for those who remember a bit of easy integral calculus.) Write

down a function that states Farmer Hoglund’s yield per acre as a function

of the amount of fertilizer he uses,30 +N?N

2

=400.

(c) Hoglund’s neighbor,Skoglund,has better land than Hoglund,In fact,

for any amount of fertilizer that he applies,he gets exactly twice as much

corn per acre as Hoglund would get with the same amount of fertilizer.

How much fertilizer will Skoglund use per acre when the price of corn is

$3 a bushel and the price of fertilizer is $p a pound? 200?33:33p.

(Hint,Start by writing down Skoglund’s marginal product of fertilizer as

a function of N.)

250 PROFIT MAXIMIZATION (Ch,19)

(d) When Hoglund and Skoglund are both maximizing pro ts,will

Skoglund’s output be more than twice as much,less than twice as much

or exactly twice as much as Hoglund’s? Explain,More than

twice as much,S,would produce twice as

much as H,if they used equal amounts of

fertilizer,but S,uses more fertilizer

than H,does.

(e) Explain how someone who looked at Hoglund’s and Skoglund’s corn

yields and their fertilizer inputs but couldn’t observe the quality of

their land,would get a misleading idea of the productivity of fertil-

izer,Fertilizer did not cause the entire

difference in yield,The best land got the

most fertilizer.

19.10 (0) A rm has two variable factors and a production function,

f(x

1;x

2

)=x

1=2

1

x

1=4

2

,The price of its output is 4,Factor 1 receives a

wage of w

1

and factor 2 receives a wage of w

2

.

(a) Write an equation that says that the value of the marginal product

of factor 1 is equal to the wage of factor 1 2x

1=2

1

x

1=4

2

= w

1

and

an equation that says that the value of the marginal product of factor

2 is equal to the wage of factor 2,x

1=2

1

x

3=4

2

= w

2

,Solve two

equations in the two unknowns,x

1

and x

2

,to give the amounts of factors

1 and 2 that maximize the rm’s pro ts as a function of w

1

and w

2

.This

gives x

1

= 8=(w

3

1

w

2

) and x

2

= 4=(w

2

1

w

2

2

),(Hint,You could

use the rst equation to solve for x

1

as a function of x

2

and of the factor

wages,Then substitute the answer into the second equation and solve for

x

2

as a function of the two wage rates,Finally use your solution for x

2

to nd the solution for x

1

.)

(b) If the wage of factor 1 is 2,and the wage of factor 2 is 1,how many

units of factor 1 will the rm demand? 1,How many units of

factor 2 will it demand? 1,How much output will it produce?

1,How much pro t will it make? 1.

19.11 (0) A rm has two variable factors and a production function

f(x

1;x

2

)=x

1=2

1

x

1=2

2

,The price of its output is 4,the price of factor 1 is

w

1

,and the price of factor 2 is w

2

.

NAME 251

(a) Write the two equations that say that the value of the marginal prod-

uct of each factor is equal to its wage,2x

1=2

1

x

1=2

2

= w

1

and

2x

1=2

1

x

1=2

2

= w

2

,If w

1

=2w

2

,these two equations imply that

x

1

=x

2

= 1/2.

(b) For this production function,is it possible to solve the two marginal

productivity equations uniquely for x

1

and x

2

No.

19.12 (1) A rm has two variable factors and a production function

f(x

1;x

2

)=

p

2x

1

+4x

2

,On the graph below,draw production isoquants

corresponding to an ouput of 3 and to an output of 4.

(a) If the price of the output good is 4,the price of factor 1 is 2,and

the price of factor 2 is 3,nd the pro t-maximizing amount of factor 1

0,the pro t-maximizing amount of factor 2 16/9,andthe

pro t-maximizing output 8/3.

0481216

4

8

12

Factor 1

Factor 2

16

9

_

4

252 PROFIT MAXIMIZATION (Ch,19)

Chapter 20 NAME

Cost Minimization

Introduction,In the chapter on consumer choice,you studied a con-

sumer who tries to maximize his utility subject to the constraint that he

has a xed amount of money to spend,In this chapter you study the

behavior of a rm that is trying to produce a xed amount of output

in the cheapest possible way,In both theories,you look for a point of

tangency between a curved line and a straight line,In consumer theory,

there is an \indi erence curve" and a \budget line." In producer theory,

there is a \production isoquant" and an \isocost line." As you recall,

in consumer theory,nding a tangency gives you only one of the two

equations you need to locate the consumer’s chosen point,The second

equation you used was the budget equation,In cost-minimization theory,

again the tangency condition gives you one equation,This time you don’t

know in advance how much the producer is spending; instead you are told

how much output he wants to produce and must nd the cheapest way

to produce it,So your second equation is the equation that tells you that

the desired amount is being produced.

Example,A rm has the production function f(x

1;x

2

)=(

p

x

1

+

3

p

x

2

)

2

,The price of factor 1 is w

1

= 1 and the price of factor 2

is w

2

= 1,Let us nd the cheapest way to produce 16 units of out-

put,We will be looking for a point where the technical rate of sub-

stitution equals?w

1

=w

2

,If you calculate the technical rate of sub-

stitution (or look it up from the warm up exercise in Chapter 18),

you nd TRS(x

1;x

2

)=?(1=3)(x

2

=x

1

)

1=2

,Therefore we must have

(1=3)(x

2

=x

1

)

1=2

=?w

1

=w

2

=?1,This equation can be simpli ed

to x

2

=9x

1

,So we know that the combination of inputs chosen has to

lie somewhere on the line x

2

=9x

1

,We are looking for the cheapest way

to produce 16 units of output,So the point we are looking for must sat-

isfy the equation (

p

x

1

+3

p

x

2

)

2

= 16,or equivalently

p

x

1

+3

p

x

2

=4.

Since x

2

=9x

1

,we can substitute for x

2

in the previous equation to get

p

x

1

+3

p

9x

1

= 4,This equation simpli es further to 10

p

x

1

=4,Solving

this for x

1

,wehavex

1

=16=100,Then x

2

=9x

1

= 144=100.

The amounts x

1

and x

2

that we solved for in the previous para-

graph are known as the conditional factor demands for factors 1 and 2,

conditional on the wages w

1

=1,w

2

= 1,and output y = 16,We ex-

press this by saying x

1

(1;1;16) = 16=100 and x

2

(1;1;16) = 144=100.

Since we know the amount of each factor that will be used to pro-

duce 16 units of output and since we know the price of each factor,

we can now calculate the cost of producing 16 units,This cost is

c(w

1;w

2;16) = w

1

x

1

(w

1;w

2;16)+w

2

x

2

(w

1;w

2;16),In this instance since

w

1

= w

2

=1,wehavec(1;1;16) = x

1

(1;1;16) +x

2

(1;1;16) = 160=100.

In consumer theory,you also dealt with cases where the consumer’s

indi erence \curves" were straight lines and with cases where there were

254 COST MINIMIZATION (Ch,20)

kinks in the indi erence curves,Then you found that the consumer’s

choice might occur at a boundary or at a kink,Usually a careful look

at the diagram would tell you what is going on,The story with kinks

and boundary solutions is almost exactly the same in the case of cost-

minimizing rms,You will nd some exercises that show how this works.

20.1 (0) Nadine sells user-friendly software,Her rm’s production func-

tion is f(x

1;x

2

)=x

1

+2x

2

,wherex

1

is the amount of unskilled labor

and x

2

is the amount of skilled labor that she employs.

(a) In the graph below,draw a production isoquant representing input

combinations that will produce 20 units of output,Draw another isoquant

representing input combinations that will produce 40 units of output.

010203040

10

20

30

40

x2

x1

20 units

40 units

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Constant.

(c) If Nadine uses only unskilled labor,how much unskilled labor would

she need in order to produce y units of output? y.

(d) If Nadine uses only skilled labor to produce output,how much skilled

labor would she need in order to produce y units of output?

y

2

.

(e) If Nadine faces factor prices (1;1),what is the cheapest way for her

to produce 20 units of output? x

1

= 0,x

2

= 10.

NAME 255

(f) If Nadine faces factor prices (1;3),what is the cheapest way for her

to produce 20 units of output? x

1

= 20,x

2

= 0.

(g) If Nadine faces factor prices (w

1;w

2

),what will be the minimal cost

of producing 20 units of output? c =minf20w

1;10w

2

g =

10 minf2w

1;w

2

g.

(h) If Nadine faces factor prices (w

1;w

2

),what will be the mini-

mal cost of producing y units of output? c(w

1;w

2;y)=

minfw

1;w

2

=2gy.

20.2 (0) The Ontario Brassworks produces brazen e ronteries,As you

know brass is an alloy of copper and zinc,used in xed proportions,The

production function is given by,f(x

1;x

2

)=minfx

1;2x

2

g,wherex

1

is

theamountofcopperitusesandx

2

istheamountofzincthatitusesin

production.

(a) Illustrate a typical isoquant for this production function in the graph

below.

010203040

10

20

30

40

x2

x1

x

2

=

1_

2

x

1

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Constant.

(c) If the rm wanted to produce 10 e ronteries,how much copper would

it need? 10 units,How much zinc would it need? 5 units.

256 COST MINIMIZATION (Ch,20)

(d) If the rm faces factor prices (1;1),what is the cheapest way for it

to produce 10 e ronteries? How much will this cost? It can

only produce 10 units of output by using the

bundle (10;5),so this is the cheapest way.

It will cost $15.

(e) If the rm faces factor prices (w

1;w

2

),what is the cheapest cost to

produce 10 e ronteries? c(w

1;w

2;10) = 10w

1

+5w

2

.

(f) If the rm faces factor prices (w

1;w

2

),what will be the minimal cost

of producing y e ronteries? (w

1

+w

2

=2)y.

Calculus 20.3 (0) A rm uses labor and machines to produce output according to

the production function f(L;M)=4L

1=2

M

1=2

,whereL is the number of

units of labor used and M is the number of machines,The cost of labor

is $40 per unit and the cost of using a machine is $10.

(a) On the graph below,draw an isocost line for this rm,showing com-

binations of machines and labor that cost $400 and another isocost line

showing combinations that cost $200,What is the slope of these isocost

lines? -4.

(b) Suppose that the rm wants to produce its output in the cheapest

possible way,Find the number of machines it would use per worker.

(Hint,The rm will produce at a point where the slope of the production

isoquant equals the slope of the isocost line.) 4.

(c) On the graph,sketch the production isoquant corresponding to an

output of 40,Calculate the amount of labor 5 units and the

number of machines 20 that are used to produce 40 units of output

in the cheapest possible way,given the above factor prices,Calculate the

cost of producing 40 units at these factor prices,c(40;10;40) = 400.

(d) How many units of labor y/8 and how many machines y/2

would the rm use to produce y units in the cheapest possible way? How

much would this cost? 10y,(Hint,Notice that there are constant

returns to scale.)

NAME 257

010203040

10

20

30

40

Machines

Labour

$400 isocost line

$200 isocost line

20.4 (0) Earl sells lemonade in a competitive market on a busy street

corner in Philadelphia,His production function is f(x

1;x

2

)=x

1=3

1

x

1=3

2

,

where output is measured in gallons,x

1

is the number of pounds of lemons

he uses,and x

2

is the number of labor-hours spent squeezing them.

(a) Does Earl have constant returns to scale,decreasing returns to scale,

or increasing returns to scale? Decreasing.

(b) Where w

1

is the cost of a pound of lemons and w

2

is the wage rate

for lemon-squeezers,the cheapest way for Earl to produce lemonade is to

use w

1

=w

2

hours of labor per pound of lemons,(Hint,Set the slope

of his isoquant equal to the slope of his isocost line.)

(c) If he is going to produce y units in the cheapest way possible,

then the number of pounds of lemons he will use is x

1

(w

1;w

2;y)=

w

1=2

2

y

3=2

=w

1=2

1

and the number of hours of labor that he will use

is x

2

(w

1;w

2;y)= w

1=2

1

y

3=2

=w

1=2

2

,(Hint,Use the production func-

tion and the equation you found in the last part of the answer to solve

for the input quantities.)

(d) The cost to Earl of producing y units at factor prices w

1

and w

2

is

c(w

1;w

2;y)=w

1

x

1

(w

1;w

2;y)+w

2

x

2

(w

1;w

2;y)= 2w

1=2

1

w

1=2

2

y

3=2

.

20.5 (0) The prices of inputs (x

1;x

2;x

3;x

4

)are(4;1;3;2).

258 COST MINIMIZATION (Ch,20)

(a) If the production function is given by f(x

1;x

2

)=minfx

1;x

2

g,what

is the minimum cost of producing one unit of output? $5.

(b) If the production function is given by f(x

3;x

4

)=x

3

+x

4

,what is the

minimum cost of producing one unit of output? $2.

(c) If the production function is given by f(x

1;x

2;x

3;x

4

)=minfx

1

+

x

2;x

3

+x

4

g,what is the minimum cost of producing one unit of output?

$3.

(d) If the production function is given by f(x

1;x

2

)=minfx

1;x

2

g +

minfx

3;x

4

g,what is the minimum cost of producing one unit of output?

$5.

20.6 (0) Joe Grow,an avid indoor gardener,has found that the number

of happy plants,h,depends on the amount of light,l,and water,w.In

fact,Joe noticed that plants require twice as much light as water,and any

more or less is wasted,Thus,Joe’s production function is h =minfl;2wg.

(a) Suppose Joe is using 1 unit of light,what is the least amount of

water he can use and still produce a happy plant? 1=2 unit of

water.

(b) If Suppose Joe wants to produce 4 happy plants,what are the mini-

mum amounts of light and water required? (4;2).

(c) Joe’s conditional factor demand function for light is l(w

1;w

2;h)=

h and his conditional factor demand function for water is

w(w

1;w

2;h)= h=2.

(d) If each unit of light costs w

1

and each unit of water costs w

2

,Joe’s

cost function is c(w

1;w

2;h)= w

1

h+

w

2

2

h.

20.7 (1) Joe’s sister,Flo Grow,is a university administrator,She uses

an alternative method of gardening,Flo has found that happy plants

only need fertilizer and talk,(Warning,Frivolous observations about

university administrators’ talk being a perfect substitute for fertilizer is

in extremely poor taste.) Where f is the number of bags of fertilizer used

and t is the number of hours she talks to her plants,the number of happy

plants produced is exactly h = t+2f,Suppose fertilizer costs w

f

per bag

and talk costs w

t

per hour.

NAME 259

(a) If Flo uses no fertilizer,how many hours of talk must she devote if she

wants one happy plant? 1 hour,If she doesn’t talk to her plants

at all,how many bags of fertilizer will she need for one happy plant?

1=2 bag.

(b) If w

t

<w

f

=2,would it be cheaper for Flo to use fertilizer or talk to

raise one happy plant? It would be cheaper to talk.

(c) Flo’s cost function is c(w

f;w

t;h)= minf

w

f

2;w

t

gh.

(d) Her conditional factor demand for talk is t(w

f;w

t;h)= h if

w

t

<w

f

=2and 0 if w

t

>w

f

=2.

20.8 (0) Remember T-bone Pickens,the corporate raider? Now he’s con-

cerned about his chicken farms,Pickens’s Chickens,He feeds his chickens

on a mixture of soybeans and corn,depending on the prices of each,Ac-

cording to the data submitted by his managers,when the price of soybeans

was $10 a bushel and the price of corn was $10 a bushel,they used 50

bushels of corn and 150 bushels of soybeans for each coop of chickens.

When the price of soybeans was $20 a bushel and the price of corn was

$10 a bushel,they used 300 bushels of corn and no soybeans per coop

of chickens,When the price of corn was $20 a bushel and the price of

soybeans was $10 a bushel,they used 250 bushels of soybeans and no corn

for each coop of chickens.

(a) Graph these three input combinations and isocost lines in the following

diagram.

0 100 200 300 400

100

200

300

400

Corn

Soybeans

125

260 COST MINIMIZATION (Ch,20)

(b) How much money did Pickens’ managers spend per coop of chickens

when the prices were (10;10)? $2,000,When the prices were

(10;20)? $2,500,When the prices were (20;10)? $3,000.

(c) Is there any evidence that Pickens’s managers were not minimizing

costs? Why or why not?

There is no such evidence,since the data

satisfy WACM.

(d) Pickens wonders whether there are any prices of corn and soybeans at

which his managers will use 150 bushels of corn and 50 bushels of soybeans

to produce a coop of chickens,How much would this production method

cost per coop of chickens if the prices were p

s

=10andp

c

= 10?

$2,000,if the prices were p

s

= 10,p

c

= 20? $3,500,if the

prices were p

s

= 20,p

c

= 10? $2,500.

(e) If Pickens’s managers were always minimizing costs,can it be pos-

sible to produce a coop of chickens using 150 bushels and 50 bushels of

soybeans? No,At prices (20;10),this bundle

costs less than the bundle actually used

at prices (20;10),If it produced as much

as that bundle,the chosen bundle wouldn’t

have been chosen.

20.9 (0) A genealogical rm called Roots produces its output using only

one input,Its production function is f(x)=

p

x.

(a) Does the rm have increasing,constant,or decreasing returns to scale?

Decreasing.

(b) How many units of input does it take to produce 10 units of output?

100 units,If the input costs w per unit,what does it cost to

produce 10 units of output? 100w.

NAME 261

(c) How many units of input does it take to produce y units of output?

y

2

,If the input costs w per unit,what does it cost to produce y units

of output? y

2

w.

(d) If the input costs w per unit,what is the average cost of producing y

units? AC(w;y)= yw.

20.10 (0) A university cafeteria produces square meals,using only one

input and a rather remarkable production process,We are not allowed to

say what that ingredient is,but an authoritative kitchen source says that

\fungus is involved." The cafeteria’s production function is f(x)=x

2

,

where x is the amount of input and f(x) is the number of square meals

produced.

(a) Does the cafeteria have increasing,constant,or decreasing returns to

scale? Increasing.

(b) How many units of input does it take to produce 144 square meals?

12,If the input costs w per unit,what does it cost to produce 144

square meals? 12w.

(c) How many units of input does it take to produce y square meals?

p

y,If the input costs w per unit,what does it cost to produce y

square meals? w

p

y.

(d) If the input costs w per unit,what is the average cost of producing y

square meals? AC(w;y)= w=

p

y.

20.11 (0) Irma’s Handicrafts produces plastic deer for lawn ornaments.

\It’s hard work," says Irma,\but anything to make a buck." Her produc-

tion function is given by f(x

1;x

2

)=(minfx

1;2x

2

g)

1=2

,wherex

1

is the

amount of plastic used,x

2

is the amount of labor used,and f(x

1;x

2

)is

the number of deer produced.

(a) In the graph below,draw a production isoquant representing input

combinations that will produce 4 deer,Draw another production isoquant

representing input combinations that will produce 5 deer.

262 COST MINIMIZATION (Ch,20)

010203040

10

20

30

40

x2

x1

x

2

=

1_

2

x

1

Output

of 5

deer

Output

of 4

deer

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Decreasing returns to scale.

(c) If Irma faces factor prices (1;1),what is the cheapest way for her to

produce 4 deer? Use (16,8),How much does this cost? $24.

(d) At the factor prices (1;1),what is the cheapest way to produce 5 deer?

Use (25,12.5),How much does this cost? $37.50.

(e) At the factor prices (1;1),the cost of producing y deer with this

technology is c(1;1;y)= 3y

2

=2.

(f) At the factor prices (w

1;w

2

),the cost of producing y deer with this

technology is c(w

1;w

2;y)= (w

1

+w

2

=2)y

2

.

20.12 (0) Al Deardwarf also makes plastic deer for lawn ornaments.

Al has found a way to automate the production process completely,He

doesn’t use any labor{only wood and plastic,Al says he likes the business

\because I need the doe." Al’s production function is given by f(x

1;x

2

)=

(2x

1

+ x

2

)

1=2

,wherex

1

is the amount of plastic used,x

2

is the amount

of wood used,and f(x

1;x

2

) is the number of deer produced.

NAME 263

(a) In the graph below,draw a production isoquant representing input

combinations that will produce 4 deer,Draw another production isoquant

representing input combinations that will produce 6 deer.

010203040

10

20

30

40

x2

x1

Output

of 4

deer

Output

of 6

deer

36

16

8 18

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Decreasing returns to scale.

(c) If Al faces factor prices (1;1),what is the cheapest way for him to

produce 4 deer? (8;0),How much does this cost? $8.

(d) At the factor prices (1;1),what is the cheapest way to produce 6

deer? (18;0),How much does this cost? $18.

(e) At the factor prices (1;1),the cost of producing y deer with this

technology is c(1;1;y)= y

2

=2.

(f) At the factor prices (3;1),the cost of producing y deer with this

technology is c(3;1;y)= y

2

.

20.13 (0) Suppose that Al Deardwarf from the last problem cannot vary

the amount of wood that he uses in the short run and is stuck with using

20 units of wood,Suppose that he can change the amount of plastic that

he uses,even in the short run.

(a) How much plastic would Al need in order to make 100 deer? 4,990

units.

264 COST MINIMIZATION (Ch,20)

(b) If the cost of plastic is $1 per unit and the cost of wood is $1 per unit,

how much would it cost Al to make 100 deer? $5,010.

(c) Write down Al’s short-run cost function at these factor prices.

c(1;1;y)=20+(y

2

20)=2.

Chapter 21 NAME

Cost Curves

Introduction,Here you continue to work on cost functions,Total cost

can be divided into xed cost,the part that doesn’t change as output

changes,and variable cost,To get the average (total) cost,average xed

cost,and average variable cost,just divide the appropriate cost function

by y,the level of output,The marginal cost function is the derivative of

the total cost function with respect to output|or the rate of increase in

cost as output increases,if you don’t know calculus.

Remember that the marginal cost curve intersects both the average

cost curve and the average variable cost curve at their minimum points.

So to nd the minimum point on the average cost curve,you simply set

marginal cost equal to average cost and similarly for the minimum of

average variable cost.

Example,A rm has the total cost function C(y) = 100 + 10y.Letus

nd the equations for its various cost curves,Total xed costs are 100,so

the equation of the average xed cost curve is 100=y,Total variable costs

are 10y,so average variable costs are 10y=y = 10 for all y,Marginal cost

is 10 for all y,Average total costs are (100 + 10y)=y =10+10=y.Notice

that for this rm,average total cost decreases as y increases,Notice also

that marginal cost is less than average total cost for all y.

21.1 (0) Mr,Otto Carr,owner of Otto’s Autos,sells cars,Otto buys

autos for $c each and has no other costs.

(a) What is his total cost if he sells 10 cars? 10c,What if he sells 20

cars? 20c,Write down the equation for Otto’s total costs assuming

he sells y cars,TC(y)= cy.

(b) What is Otto’s average cost function? AC(y)= c,For every

additional auto Otto sells,by how much do his costs increase? c.

Write down Otto’s marginal cost function,MC(y)= c.

(c) In the graph below draw Otto’s average and marginal cost curves if

c = 20.

266 COST CURVES (Ch,21)

010203040

10

20

30

40

AC,MC

Red line

AC=MC=20

Output

(d) Suppose Otto has to pay $b a year to produce obnoxious television

commercials,Otto’s total cost curve is now TC(y)= cy + b,his

average cost curve is now AC(y)= c + b=y,and his marginal cost

curve is MC(y)= c.

(e) If b = $100,use red ink to draw Otto’s average cost curve on the

graph above.

21.2 (0) Otto’s brother,Dent Carr,is in the auto repair business,Dent

recently had little else to do and decided to calculate his cost conditions.

He found that the total cost of repairing s cars is TC(s)=2s

2

+ 10,But

Dent’s attention was diverted to other things,:,and that’s where you

come in,Please complete the following:

Dent’s Total Variable Costs,2s

2

.

Total Fixed Costs,10.

Average Variable Costs,2s.

Average Fixed Costs,10=s.

Average Total Costs,2s+10=s.

Marginal Costs,4s.

NAME 267

21.3 (0) A third brother,Rex Carr,owns a junk yard,Rex can use one

of two methods to destroy cars,The rst involves purchasing a hydraulic

car smasher that costs $200 a year to own and then spending $1 for every

car smashed into oblivion; the second method involves purchasing a shovel

that will last one year and costs $10 and paying the last Carr brother,

Scoop,to bury the cars at a cost of $5 each.

(a) Write down the total cost functions for the two methods,where y is

output per year,TC

1

(y)= y + 200,TC

2

(y)= 5y +10.

(b) The rst method has an average cost function 1 + 200=y and a

marginal cost function 1,For the second method these costs are

5+10=y and 5.

(c) If Rex wrecks 40 cars per year,which method should he use?

Method 2,If Rex wrecks 50 cars per year,which method should

he use? Method 1,What is the smallest number of cars per year

for which it would pay him to buy the hydraulic smasher? 48 cars

per year.

21.4 (0) Mary Magnolia wants to open a flower shop,the Petal Pusher,

in a new mall,She has her choice of three di erent floor sizes,200 square

feet,500 square feet,or 1,000 square feet,The monthly rent will be $1 a

square foot,Mary estimates that if she has F square feet of floor space

and sells y bouquets a month,her variable costs will be c

v

(y)=y

2

=F per

month.

(a) If she has 200 square feet of floor space,write down her marginal cost

function,MC =

y

100

and her average cost function,AC =

200

y

+

y

200

,At what amount of output is average cost minimized?

200,At this level of output,how much is average cost? $2.

(b) If she has 500 square feet,write down her marginal cost function:

MC = y=250 and her average cost function,AC =

(500=y)+y=500,At what amount of output is average cost min-

268 COST CURVES (Ch,21)

imized? 500,At this level of output,how much is average cost?

$2.

(c) If she has 1,000 square feet of floor space,write down her marginal

cost function,MC = y=500 and her average cost function:

AC =(1;000=y)+y=1;000,At what amount of output is

average cost minimized? 1,000,At this level of output,how much

is average cost? $2.

(d) Use red ink to show Mary’s average cost curve and her marginal cost

curves if she has 200 square feet,Use blue ink to show her average cost

curve and her marginal cost curve if she has 500 square feet,Use black

ink to show her average cost curve and her marginal cost curve if she has

1,000 square feet,Label the average cost curves AC and the marginal

cost curves MC.

0 400 600 800 1000

Bouquents

1

2

3

4

Dollars

200 1200

mc

ac

mc

ac

mc

ac

Red

lines

Blue

lines

Black

lines

LRMC=LRAC (yellow line)

(e) Use yellow marker to show Mary’s long-run average cost curve and

her long-run marginal cost curve in your graph,Label them LRAC and

LRMC.

21.5 (0) Touchie MacFeelie publishes comic books,The only inputs he

needs are old jokes and cartoonists,His production function is

Q =,1J

1

2

L

3=4;

NAME 269

whereJ is the number of old jokes used,L the number of hours of cartoon-

ists’ labor used as inputs,and Q is the number of comic books produced.

(a) Does this production process exhibit increasing,decreasing,or con-

stant returns to scale? Explain your answer,It exhibits

increasing returns to scale since f(tJ;tL)=

t

5=4

f(J;L) >tf(J;L).

(b) If the number of old jokes used is 100,write an expression for the

marginal product of cartoonists’ labor as a function of L,MP =

3

4L

1=4

Is the marginal product of labor decreasing or increasing as the

amount of labor increases? Decreasing.

21.6 (0) Touchie MacFeelie’s irascible business manager,Gander Mac-

Grope,announces that old jokes can be purchased for $1 each and that

the wage rate of cartoonists’ labor is $2.

(a) Suppose that in the short run,Touchie is stuck with exactly 100 old

jokes (for which he paid $1 each) but is able to hire as much labor as he

wishes,How much labor would he have to hire in order produce Q comic

books? Q

4=3

.

(b) Write down Touchie’s short-run total cost as a function of his output

2Q

4=3

+ 100.

(c) His short-run marginal cost function is 8Q

1=3

=3.

(d) His short-run average cost function is 2Q

1=3

+ 100=Q.

Calculus 21.7 (1) Touchie asks his brother,Sir Francis MacFeelie,to study the

long-run picture,Sir Francis,who has carefully studied the appendix to

Chapter 19 in your text,prepared the following report.

(a) If all inputs are variable,and if old jokes cost $1 each and car-

toonist labor costs $2 per hour,the cheapest way to produce exactly

one comic book is to use 10

4=5

(4=3)

3=5

7:4 jokes and

10

4=5

(3=4)

2=5

5:6 hours of labor,(Fractional jokes are cer-

tainly allowable.)

270 COST CURVES (Ch,21)

(b) This would cost 18.7 dollars.

(c) Given our production function,the cheapest proportions in which to

use jokes and labor are the same no matter how many comic books we

print,But when we double the amount of both inputs,the number of

comic books produced is multiplied by 2

5=4

.

21.8 (0) Consider the cost function c(y)=4y

2

+ 16.

(a) The average cost function is AC =4y +

16

y

.

(b) The marginal cost function is MC =8y.

(c) The level of output that yields the minimum average cost of production

is y =2.

(d) The average variable cost function is AVC =4y.

(e) At what level of output does average variable cost equal marginal

cost? At y =0.

21.9 (0) A competitive rm has a production function of the form

Y =2L +5K.Ifw =$2andr = $3,what will be the minimum cost of

producing 10 units of output? $6.

Chapter 22 NAME

Firm Supply

Introduction,The short-run supply curve of a competitive rm is the

portion of its short-run marginal cost curve that is upward sloping and

lies above its average variable cost curve,The long-run supply curve of a

competitive rm is the portion of its short-run marginal cost curve that

is upward-sloping and lies above its long-run average cost curve.

Example,A rm has the long-run cost function c(y)=2y

2

+ 200 for

y>0andc(0) = 0,Let us nd its long-run supply curve,The rm’s

marginal cost when its output is y is MC(y)=4y,If we graph output on

the horizontal axis and dollars on the vertical axis,then we nd that the

long-run marginal cost curve is an upward-sloping straight line through

the origin with slope 4,The long-run supply curve is the portion of this

curve that lies above the long-run average cost curve,When output is y,

long-run average costs of this rm are AC(y)=2y + 200=y.ThisisaU-

shaped curve,As y gets close to zero,AC(y) becomes very large because

200=y becomes very large,When y is very large,AC(y) becomes very

large because 2y is very large,When is it true that AC(y) <MC(y)?

This happens when 2y+ 200=y < 4y,Simplify this inequality to nd that

AC(y) <MC(y)wheny>10,Therefore the long-run supply curve is

the piece of the long-run marginal cost curve for which y>10,So the

long-run supply curve has the equation p =4y for y>10,If we want to

nd quantity supplied as a function of price,we just solve this expression

for y as a function of p.Thenwehavey = p=4 whenever p>40.

Suppose that p<40,For example,what if p = 20,how much will

the rm supply? At a price of 20,if the rm produces where price equals

long-run marginal cost,it will produce 5 = 20=4 units of output,When

the rm produces only 5 units,its average costs are 2 5 + 200=5 = 50.

Therefore when the price is 20,the best the rm can do if it produces a

positive amount is to produce 5 units,But then it will have total costs of

5 50 = 250 and total revenue of 5 20 = 100,It will be losing money,It

would be better o producing nothing at all,In fact,for any price p<40,

the rm will choose to produce zero output.

22.1 (0) Remember Otto’s brother Dent Carr,who is in the auto repair

business? Dent found that the total cost of repairing s cars is c(s)=

2s

2

+ 100.

(a) This implies that Dent’s average cost is equal to 2s + 100=s,

his average variable cost is equal to 2s,and his marginal cost is

equal to 4s,On the graph below,plot the above curves,and also plot

Dent’s supply curve.

272 FIRM SUPPLY (Ch,22)

0 5 10 15 20

20

40

60

Output

Dollars

80

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Supply

mc

ac

avc

Revenue

Costs

Profit

(b) If the market price is $20,how many cars will Dent be willing to

repair? 5,If the market price is $40,how many cars will Dent

repair? 10.

(c) Suppose the market price is $40 and Dent maximizes his pro ts,On

the above graph,shade in and label the following areas,total costs,total

revenue,and total pro ts.

Calculus 22.2 (0) A competitive rm has the following short-run cost function:

c(y)=y

3

8y

2

+30y +5.

(a) The rm’s marginal cost function is MC(y)= 3y

2

16y+30.

(b) The rm’s average variable cost function is AVC(y)= y

2

8y+

30,(Hint,Notice that total variable costs equal c(y)?c(0).)

(c) On the axes below,sketch and label a graph of the marginal cost

function and of the average variable cost function.

(d) Average variable cost is falling as output rises if output is less than

4 and rising as output rises if output is greater than 4.

(e) Marginal cost equals average variable cost when output is 4.

NAME 273

(f) The rm will supply zero output if the price is less than 14.

(g) The smallest positive amount that the rm will ever supply at any

price is 4,At what price would the rm supply exactly 6 units

of output? 42.

02468

10

20

30

y

Costs

40

mc

avc

Calculus 22.3 (0) Mr,McGregor owns a 5-acre cabbage patch,He forces his

wife,Flopsy,and his son,Peter,to work in the cabbage patch without

wages,Assume for the time being that the land can be used for nothing

other than cabbages and that Flopsy and Peter can nd no alternative

employment,The only input that Mr,McGregor pays for is fertilizer,If

he uses x sacks of fertilizer,the amount of cabbages that he gets is 10

p

x.

Fertilizer costs $1 per sack.

(a) What is the total cost of the fertilizer needed to produce 100 cabbages?

$100,What is the total cost of the amount of fertilizer needed to

produce y cabbages? y

2

=100.

(b) If the only way that Mr,McGregor can vary his output is by varying

the amount of fertilizer applied to his cabbage patch,write an expression

for his marginal cost,as a function of y,MC(y)= y=50.

(c) If the price of cabbages is $2 each,how many cabbages will Mr,Mc-

Gregor produce? 100,How many sacks of fertilizer will he buy?

100,How much pro t will he make? $100.

274 FIRM SUPPLY (Ch,22)

(d) The price of fertilizer and of cabbages remain as before,but Mr,Mc-

Gregor learns that he could nd summer jobs for Flopsy and Peter in

a local sweatshop,Flopsy and Peter would together earn $300 for the

summer,which Mr,McGregor could pocket,but they would have no time

to work in the cabbage patch,Without their labor,he would get no cab-

bages,Now what is Mr,McGregor’s total cost of producing y cabbages?

c(y) = 300 + (y=10)

2

.

(e) Should he continue to grow cabbages or should he put Flopsy and

Peter to work in the sweatshop? Sweatshop.

22.4 (0) Severin,the herbalist,is famous for his hepatica,His total cost

function is c(y)=y

2

+10 fory>0andc(0) = 0,(That is,his cost of

producing zero units of output is zero.)

(a) What is his marginal cost function? 2y,What is his average cost

function? y +10=y.

(b) At what quantity is his marginal cost equal to his average cost?

p

10,At what quantity is his average cost minimized?

p

10.

(c) In a competitive market,what is the lowest price at which he will

supply a positive quantity in long-run equilibrium? 2

p

10,How

much would he supply at that price?

p

10.

22.5 (1) Stanley Ford makes mountains out of molehills,He can do this

with almost no e ort,so for the purposes of this problem,let us assume

that molehills are the only input used in the production of mountains.

Suppose mountains are produced at constant returns to scale and that

it takes 100 molehills to make 1 mountain,The current market price of

molehills is $20 each,A few years ago,Stan bought an \option" that

permits him to buy up to 2,000 molehills at $10 each,His option contract

explicitly says that he can buy fewer than 2,000 molehills if he wishes,but

he can not resell the molehills that he buys under this contract,In or-

der to get governmental permission to produce mountains from molehills,

Stanley would have to pay $10,000 for a molehill-masher’s license.

(a) The marginal cost of producing a mountain for Stanley is $1,000

if he produces fewer than 20 mountains,The marginal cost of producing

a mountain is $2,000 if he produces more than 20 mountains.

NAME 275

(b) On the graph below,show Stanley Ford’s marginal cost curve (in blue

ink) and his average cost curve (in red ink).

010203040

1000

2000

3000

Output

Dollars

4000

Blue mc curve

Red ac

curve

Pencil

mc

curve

(c) If the price of mountains is $1,600,how many mountains will Stanley

produce? 20 mountains.

(d) The government is considering raising the price of a molehill-masher’s

license to $11,000,Stanley claims that if it does so he will have to go out

of business,Is Stanley telling the truth? No,What is the highest

fee for a license that the government could charge without driving him

out of business? The maximum they could charge

is the amount of his profits excluding the

license fee,$12,000.

(e) Stanley’s lawyer,Eliot Sleaze,has discovered a clause in Stanley’s

option contract that allows him to resell the molehills that he purchased

under the option contract at the market price,On the graph above,

use a pencil to draw Stanley’s new marginal cost curve,If the price of

mountains remains $1,600,how many mountains will Stanley produce

now? He will sell all of his molehills and

produce zero mountains.

22.6 (1) Lady Wellesleigh makes silk purses out of sows’ ears,She is

the only person in the world who knows how to do so,It takes one sow’s

ear and 1 hour of her labor to make a silk purse,She can buy as many

276 FIRM SUPPLY (Ch,22)

sows’ ears as she likes for $1 each,Lady Wellesleigh has no other source

of income than her labor,Her utility function is a Cobb-Douglas function

U(c;r)=c

1=3

r

2=3

,wherec is the amount of money per day that she has

to spend on consumption goods and r is the amount of leisure that she

has,Lady Wellesleigh has 24 hours a day that she can devote either to

leisure or to working.

(a) Lady Wellesleigh can either make silk purses or she can earn $5 an

hour as a seamstress in a sweatshop,If she worked in the sweat shop,how

many hours would she work? 8,(Hint,To solve for this amount,

write down Lady Wellesleigh’s budget constraint and recall how to nd

the demand function for someone with a Cobb-Douglas utility function.)

(b) If she could earn a wage of $w an hour as a seamstress,how much

would she work? 8 hours.

(c) If the price of silk purses is $p,how much money will Lady Wellesleigh

earn per purse after she pays for the sows’ ears that she uses? p?1.

(d) If she can earn $5 an hour as a seamstress,what is the lowest price

at which she will make any silk purses? $6.

(e) What is the supply function for silk purses? (Hint,The price of silk

purses determines the \wage rate" that Lady W,can earn by making silk

purses,This determines the number of hours she will choose to work and

hence the supply of silk purses.) S(p)=8for p>6,0

otherwise.

Calculus 22.7 (0) Remember Earl,who sells lemonade in Philadelphia? You

met him in the chapter on cost functions,Earl’s production function is

f(x

1;x

2

)=x

1=3

1

x

1=3

2

,wherex

1

is the number of pounds of lemons he

uses and x

2

is the number of hours he spends squeezing them,As you

found out,his cost function is c(w

1;w

2;y)=2w

1=2

1

w

1=2

2

y

3=2

,wherey is

the number of units of lemonade produced.

(a) If lemons cost $1 per pound,the wage rate is $1 per hour,and the

price of lemonade is p,Earl’s marginal cost function is MC(y)= 3y

1=2

and his supply function is S(p)= p

2

=9,If lemons cost $4 per pound

and the wage rate is $9 per hour,his supply function will be S(p)=

p

2

=324.

NAME 277

(b) In general,Earl’s marginal cost depends on the price of lemons and

the wage rate,At prices w

1

for lemons and w

2

for labor,his mar-

ginal cost when he is producing y units of lemonade is MC(w

1;w

2;y)=

3w

1=2

1

w

1=2

2

y

1=2

,The amount that Earl will supply depends on the

three variables,p,w

1

,w

2

,As a function of these three variables,Earl’s

supply is S(p;w

1;w

2

)= p

2

=9w

1

w

2

.

Calculus 22.8 (0) As you may recall from the chapter on cost functions,Irma’s

handicrafts has the production function f(x

1;x

2

)=(minfx

1;2x

2

g)

1=2

,

where x

1

is the amount of plastic used,x

2

is the amount of labor used,

and f(x

1;x

2

) is the number of lawn ornaments produced,Let w

1

be the

price per unit of plastic and w

2

be the wage per unit of labor.

(a) Irma’s cost function is c(w

1;w

2;y)= (w

1

+w

2

=2)y

2

.

(b) If w

1

= w

2

= 1,then Irma’s marginal cost of producing y units of

output is MC(y)= 3y,The number of units of output that she would

supply at price p is S(p)= p=3,At these factor prices,her average

cost per unit of output would be AC(y)= 3y=2.

(c) If the competitive price of the lawn ornaments she sells is p = 48,and

w

1

= w

2

= 1,how many will she produce? 16,How much pro t will

she make? 384.

(d) More generally,at factor prices w

1

and w

2

,her marginal cost is a

function MC(w

1;w

2;y)= (2w

1

+w

2

)y,At these factor prices and

an output price of p,the number of units she will choose to supply is

S(p;w

1;w

2

)= p=(2w

1

+w

2

).

22.9 (0) Jack Benny can get blood from a stone,If he has x stones,the

number of pints of blood he can extract from them is f(x)=2x

1

3

.Stones

cost Jack $w each,Jack can sell each pint of blood for $p.

(a) How many stones does Jack need to extract y pintsofblood?

y

3

=8.

(b) What is the cost of extracting y pints of blood? wy

3

=8.

278 FIRM SUPPLY (Ch,22)

(c) What is Jack’s supply function when stones cost $8 each? y =

(p=3)

1=2

,When stones cost $w each? y =(8p=3w)

1=2

.

(d) If Jack has 19 relatives who can also get blood from a stone in the

same way,what is the aggregate supply function for blood when stones

cost $w each? Y = 20(8p=3w)

1=2

.

22.10 (1) The Miss Manners Re nery in Dry Rock,Oklahoma,converts

crude oil into gasoline,It takes 1 barrel of crude oil to produce 1 barrel of

gasoline,In addition to the cost of oil there are some other costs involved

in re ning gasoline,Total costs of producing y barrels of gasoline are

described by the cost function c(y)=y

2

=2+p

o

y,wherep

o

is the price of

a barrel of crude oil.

(a) Express the marginal cost of producing gasoline as a function of p

o

and y,y +p

o

.

(b) Suppose that the re nery can buy 50 barrels of crude oil for $5 a

barrel but must pay $15 a barrel for any more that it buys beyond 50

barrels,The marginal cost curve for gasoline will be y +5 up to 50

barrels of gasoline and y +15 thereafter.

(c) Plot Miss Manners’ supply curve in the diagram below using blue ink.

0255075100

20

40

60

Barrels of gasoline

Price of gasoline

80

30

Red line

Black line Blue lines

NAME 279

(d) Suppose that Miss Manners faces a horizontal demand curve for gaso-

line at a price of $30 per barrel,Plot this demand curve on the graph

above using red ink,How much gasoline will she supply? 25

barrels.

(e) If Miss Manners could no longer get the rst 50 barrels of crude for

$5,but had to pay $15 a barrel for all crude oil,how would her output

change? It would decrease to 15 barrels.

(f) Now suppose that an entitlement program is introduced that permits

re neries to buy one barrel of oil at $5 for each barrel of oil that they

buy for $15,What will Miss Manners’ supply curve be now? S(p)=

p?10,Assume that it can buy fractions of a barrel in the same

manner,Plot this supply curve on the graph above using black ink,If

the demand curve is horizontal at $30 a barrel,how much gasoline will

Miss Manners supply now? 20 barrels.

22.11 (2) Suppose that a farmer’s cost of growing y bushels of corn is

given by the cost function c(y)=(y

2

=20) +y.

(a) If the price of corn is $5 a bushel,how much corn will this farmer

grow? 40 bushels.

(b) What is the farmer’s supply curve of corn as a function of the price

of corn? S(p)= 10p?10.

(c) The government now introduces a Payment in Kind (PIK) program,If

the farmer decides to grow y bushels of corn,he will get (40?y)=2 bushels

from the government stockpiles,Write an expression for the farmer’s

pro ts as a function of his output and the market price of corn,taking

into account the value of payments in kind received,py?c(y)+

p(40?y)=2=py?y

2

=20?y +p(40?y)=2.

(d) At the market price p,what will be the farmer’s pro t-maximizing

output of corn? S(p)=5p?10,Plot a supply curve for corn in

the graph below.

280 FIRM SUPPLY (Ch,22)

030450

1

2

3

4

5

6

10

Price

20

Bushels of corn

60

Red line

(e) If p = $2,how many bushels of corn will he produce? 0,How

many bushels will he get from the government stockpiles? 20.

(f) If p = $5,how much corn will he supply? 15 bushels,How

many bushels of corn will he get from the government stockpiles,assuming

he chooses to be in the PIK program? $12.50.

(g) At any price between p =$2andp = $5,write a formula for the

size of the PIK payment,His supply curve is S(p)=

5p?10,and his payment is (40?y)=2.So

he gets 25?2:5p.

(h) How much corn will he supply to the market,counting both pro-

duction and PIK payment,as a function of the market price p?

Sum supply curve and PIK payment to get

TS(p)=2:5p+15.

(i) Use red ink to illustrate the total supply curve of corn (including the

corn from the PIK payment) in your graph above.

Chapter 23 NAME

Industry Supply

Introduction,To nd the industry supply of output,just add up the

supply of output coming from each individual rm,Remember to add

quantities,not prices,The industry supply curve will have a kink in it

where the market price becomes low enough that some rm reduces its

quantity supplied to zero.

The last three questions of this chapter apply supply and demand

analysis to some problems in the economics of illegal activities,In these

examples,you will make use of your knowledge of where supply functions

come from.

23.0 Warm Up Exercise,Here are some drills for you on nding

market supply functions from linear rm supply functions,The trick here

is to remember that the market supply function may have kinks in it,For

example,if the rm supply functions are s

1

(p)=p and s

2

(p)=p?2,

then the market supply function is S(p)=p for p 2andS(p)=2p?2

for p>2; that is,only the rst rm supplies a positive output at prices

below $2,and both rms supply output at prices above $2,Now try to

construct the market supply function in each of the following cases.

(a) s

1

(p)=p;s

2

(p)=2p;s

3

(p)=3p,S(p)=6p.

(b) s

1

(p)=2p;s

2

(p)=p?1,S(p)=2p for p 1;S(p)=

3p?1 for p>1.

(c) 200 rms each have a supply function s

1

(p)=2p?8 and 100 rms

each have a supply function s

2

(p)=p?3,S(p)=0for

p<3,S(p) = 100p?300 for 3 p 4,S(p)=

500p?1;900 for p>4.

(d) s

1

(p)=3p?12;s

2

(p)=2p?8;s

3

(p)=p?4,S(p)=6p?24

for p>4.

23.1 (1) Al Deardwarf’s cousin,Zwerg,makes plaster garden gnomes.

The technology in the garden gnome business is as follows,You need a

gnome mold,plaster,and labor,A gnome mold is a piece of equipment

that costs $1,000 and will last exactly one year,After a year,a gnome

282 INDUSTRY SUPPLY (Ch,23)

mold is completely worn out and has no scrap value,With a gnome

mold,you can make 500 gnomes per year,For every gnome that you

make,you also have to use a total of $7 worth of plaster and labor,The

total amounts of plaster and labor used are variable in the short run,If

you want to produce only 100 gnomes a year with a gnome mold,you

spend only $700 a year on plaster and labor,and so on,The number

of gnome molds in the industry cannot be changed in the short run,To

get a newly built one,you have to special-order it from the gnome-mold

factory,The gnome-mold factory only takes orders on January 1 of any

given year,and it takes one whole year from the time a gnome mold is

ordered until it is delivered on the next January 1,When a gnome mold

is installed in your plant,it is stuck there,To move it would destroy it.

Gnome molds are useless for anything other than making garden gnomes.

For many years,the demand function facing the garden-gnome in-

dustry has been D(p)=60;000?5;000p,whereD(p) is the total number

of garden gnomes sold per year and p is the price,Prices of inputs have

been constant for many years and the technology has not changed,No-

body expects any changes in the future,and the industry is in long-run

equilibrium,The interest rate is 10%,When you buy a new gnome mold,

you have to pay for it when it is delivered,For simplicity of calculations,

we will assume that all of the gnomes that you build during the one-year

life of the gnome mold are sold at Christmas and that the employees and

plaster suppliers are paid only at Christmas for the work they have done

during the past year,Also for simplicity of calculations,let us approxi-

mate the date of Christmas by December 31.

(a) If you invested $1,000 in the bank on January 1,how much money

could you expect to get out of the bank one year later? $1,100,If

you received delivery of a gnome mold on January 1 and paid for it at that

time,by how much would your revenue have to exceed the costs of plaster

and labor if it is to be worthwhile to buy the machine? (Remember that

the machine will be worn out and worthless at the end of the year.)

$1,100.

(b) Suppose that you have exactly one newly installed gnome mold in

your plant; what is your short-run marginal cost of production if you

produce up to 500 gnomes? $7,What is your average variable cost

for producing up to 500 gnomes? $7,With this equipment,is it

possible in the short run to produce more than 500 gnomes? No.

(c) If you have exactly one newly installed gnome mold,you would pro-

duce 500 gnomes if the price of gnomes is above 7 dollars,You

would produce no gnomes if the price of gnomes is below 7 dol-

NAME 283

lars,You would be indi erent between producing any number of gnomes

between 0 and 500 if the price of gnomes is 7 dollars.

(d) If you could sell as many gnomes as you liked for $10 each and none

at a higher price,what rate of return would you make on your $1,000 by

investing in a gnome mold? 50%,Is this higher than the return from

putting your money in the bank? Yes,What is the lowest price for

gnomes at which investing in a gnome mold gives the same rate of return

as you get from the bank? $9.20,Could the long-run equilibrium

price be lower than this? No.

(e) At the price you found in the last section,how many gnomes would

be demanded each year? 14,000,How many molds would be

purchased each year? 28,Is this a long-run equilibrium price?

Yes.

23.2 (1) We continue our study of the garden-gnome industry,Suppose

that initially everything was as described in the previous problem,To

the complete surprise of everyone in the industry,on January 1,1993,the

invention of a new kind of plaster was announced,This new plaster made

it possible to produce garden gnomes using the same molds,but it reduced

the cost of the plaster and labor needed to produce a gnome from $7 to $5

per gnome,Assume that consumers’ demand function for gnomes in 1993

was not changed by this news,The announcement came early enough in

the day for everybody to change his order for gnome molds to be delivered

on January 1,1994,but of course,the number of molds available to be

used in 1993 is already determined from orders made one year ago,The

manufacturer of garden gnome molds contracted to sell them for $1,000

a year ago,so it can’t change the price it charges on delivery.

(a) In 1993,what will be the equilibrium total output of garden gnomes?

14,000,What will be the equilibrium price of garden gnomes?

$9.20,Cousin Zwerg bought a gnome mold that was delivered on

January 1,1993,and,as had been agreed,he paid $1,000 for it on that

day,On January 1,1994,when he sold the gnomes he had made during

the year and when he paid the workers and the suppliers of plaster,he

received a net cash flow of $ 2,100,Did he make more than a 10%

rate of return on his investment in the gnome mold? Yes,What rate

of return did he make? 110%.

284 INDUSTRY SUPPLY (Ch,23)

(b) Zwerg’s neighbor,Munchkin,also makes garden gnomes,and he has

a gnome mold that is to be delivered on January 1,1993,On this day,

Zwerg,who is looking for a way to invest some more money,is considering

buying Munchkin’s new mold from Munchkin and installing it in his own

plant,If Munchkin keeps his mold,he will get a net cash flow of $

2,100 in one year,If the interest rate that Munchkin faces,both

for borrowing and lending is 10%,then should he be willing to sell his

mold for $1,000? No,What is the lowest price that he would be

willing to sell it for? $1,909,If the best rate of return that Zwerg

can make on alternative investments of additional funds is 10%,what is

the most that Zwerg would be willing to pay for Munchkin’s new mold?

$1,909.

(c) What do you think will happen to the number of garden gnomes or-

dered for delivery on January 1,1994? Will it be larger,smaller,or the

same as the number ordered the previous year? Larger,After the

passage of su cient time,the industry will reach a new long-run equilib-

rium,What will be the new equilibrium price of gnomes? $7.20.

23.3 (1) On January 1,1993,there were no changes in technology or

demand functions from that in our original description of the industry,

but the government astonished the garden gnome industry by introducing

a tax on the production of garden gnomes,For every garden gnome

produced,the manufacturer must pay a $1 tax,The announcement came

early enough in the day so that there was time for gnome producers to

change their orders of gnome molds for 1994,Of course the gnome molds

to be used in 1993 had been already ordered a year ago,Gnome makers

had signed contracts promising to pay $1,000 for each gnome mold that

they ordered,and they couldn’t back out of these promises.

(a) Recalling from previous problems the number of gnome molds ordered

for delivery on January 1,1993,we see that if gnome makers produce up

to capacity in 1993,they will produce 14,000 gnomes,Given the

demand function,we see that the market price would then have to be

$9.20.

(b) If you have a garden gnome mold,the marginal cost of producing a

garden gnome,including the tax,is $8,Therefore all gnome molds

(will,will not) will be used up to capacity in 1993.

NAME 285

(c) In 1993,what will be the total output of garden gnomes?

14,000,What will be the price of garden gnomes? $9.20.

What rate of return will Deardwarf’s cousin Zwerg make on his invest-

ment in a garden gnome mold that he ordered a year ago and paid $1,000

foratthattime40%.

(d) Remember that Zwerg’s neighbor,Munchkin,also has a gnome mold

that is to be delivered on January 1,1993,Knowing about the tax makes

Munchkin’s mold a less attractive investment than it was without the

tax,but still Zwerg would buy it if he can get it cheap enough so that he

makes a 10% rate of return on his investment,How much should he be

willing to pay for Munchkin’s new mold? $545.45.

(e) What do you think will happen to the number of gnome molds ordered

for delivery on January 1,1994? Will it be larger,smaller,or the same

as the number ordered the previous year? Smaller.

(f) The tax on garden gnomes was left in place for many years,and no-

body expected any further changes in the tax or in demand or supply con-

ditions,After the passage of su cient time,the industry reached a new

long-run equilibrium,What was the new equilibrium price of gnomes?

$10.20.

(g) In the short run,who would end up paying the tax on garden gnomes,

the producers or the consumers? Producers,In the long run,did

the price of gnomes go up by more,less,or the same amount as the tax

per gnome? Same amount.

(h) Suppose that early in the morning of January 1,1993,the government

had announced that there would be a $1 tax on garden gnomes,but

that the tax would not go into e ect until January 1,1994,Would the

producers of garden gnomes necessarily be worse o than if there were

no tax? Why or why not? No,The producers would

anticipate the tax increase and restrict

supply,thereby raising prices.

286 INDUSTRY SUPPLY (Ch,23)

(i) Is it reasonable to suppose that the government could introduce \sur-

prise" taxes without making rms suspicious that there would be similar

\surprises" in the future? Suppose that the introduction of the tax in Jan-

uary 1993 makes gnome makers suspicious that there will be more taxes

introduced in later years,Will this a ect equilibrium prices and supplies?

How? If a surprise tax makes gnome makers

expect similar ‘‘surprises’’ in future,it

will take a higher current price to get

them to enter the industry,This will raise

the price paid by consumers.

23.4 (0) Consider a competitive industry with a large number of rms,

all of which have identical cost functions c(y)=y

2

+1 fory>0and

c(0) = 0,Suppose that initially the demand curve for this industry is

given by D(p)=52?p,(The output of a rm does not have to be an

integer number,but the number of rms does have to be an integer.)

(a) What is the supply curve of an individual rm? S(p)= p=2,If

there are n rms in the industry,what will be the industry supply curve?

Y = np=2.

(b) What is the smallest price at which the product can be sold? p

=

2.

(c) What will be the equilibrium number of rms in the industry? (Hint:

Take a guess at what the industry price will be and see if it works.)

Guess at p

=2,This gives D(p)=52?2=

n2=2,which says n

=50.

(d) What will be the equilibrium price? p

=2,What will be the

equilibrium output of each rm? y

=1.

(e) What will be the equilibrium output of the industry? Y

=50.

NAME 287

(f) Now suppose that the demand curve shifts to D(p)=52:5?p.

What will be the equilibrium number of rms? (Hint,Can a new rm

enter the market and make nonnegative pro ts?) If a new

firm entered,there would be 51 firms,The

supply-demand equation would be 52:5?p =

51p=2,Solve for p

= 105=53 < 2,A new firm

would lose money,Therefore in equilibrium

there would be 50 firms.

(g) What will be the equilibrium price? Solve 52:5?p =

50p=2 to get p

=2:02,What will be the equilibrium

output of each rm? y

=1:01,What will be the equilibrium

pro ts of each rm? Around,02.

(h) Now suppose that the demand curve shifts to D(p)=53?p,What will

be the equilibrium number of rms? 51,What will be the equilibrium

price? 2.

(i) What will be the equilibrium output of each rm? y =1,What

will be the equilibrium pro ts of each rm? Zero.

23.5 (3) In 1990,the town of Ham Harbor had a more-or-less free market

in taxi services,Any respectable rm could provide taxi service as long

as the drivers and cabs satis ed certain safety standards.

Let us suppose that the constant marginal cost per trip of a taxi ride

is $5,and that the average taxi has a capacity of 20 trips per day,Let

the demand function for taxi rides be given by D(p)=1;200?20p,where

demand is measured in rides per day,and price is measured in dollars.

Assume that the industry is perfectly competitive.

(a) What is the competitive equilibrium price per ride? (Hint,In com-

petitive equilibrium,price must equal marginal cost.) 5,What

is the equilibrium number of rides per day? 1,100,How many

taxicabs will there be in equilibrium? 55.

288 INDUSTRY SUPPLY (Ch,23)

(b) In 1990 the city council of Ham Harbor created a taxicab licensing

board and issued a license to each of the existing cabs,The board stated

that it would continue to adjust the taxicab fares so that the demand for

rides equals the supply of rides,but no new licenses will be issued in the

future,In 1995 costs had not changed,but the demand curve for taxicab

rides had become D(p)=1;220?20p,What was the equilibrium price

of a ride in 1995? $6.

(c) What was the pro t per ride in 1995,neglecting any costs associated

with acquiring a taxicab license? $1,What was the pro t per taxicab

license per day? 20,If the taxi operated every day,what was the

pro t per taxicab license per year? $7,300.

(d) If the interest rate was 10% and costs,demand,and the number of

licenses were expected to remain constant forever,what would be the

market price of a taxicab license? $73,000.

(e) Suppose that the commission decided in 1995 to issue enough new

licenses to reduce the taxicab price per ride to $5,How many more

licenses would this take? 1.

(f) Assuming that demand in Ham Harbor is not going to grow any

more,how much would a taxicab license be worth at this new fare?

Nothing.

(g) How much money would each current taxicab owner be willing to

pay to prevent any new licenses from being issued? $73,000

each,What is the total amount that all taxicab owners together would

be willing to pay to prevent any new licences from ever being issued?

$4,015,000,The total amount that consumers would be willing

to pay to have another taxicab license issued would be (more than,less

than,the same as) more than this amount.

23.6 (2) In this problem,we will determine the equilibrium pattern

of agricultural land use surrounding a city,Think of the city as being

located in the middle of a large featureless plain,The price of wheat at

the market at the center of town is $10 a bushel,and it only costs $5 a

bushel to grow wheat,However,it costs 10 cents a mile to transport a

bushel of wheat to the center of town.

NAME 289

(a) If a farm is located t miles from the center of town,write down

a formula for its pro t per bushel of wheat transported to market.

Profit per bushel =5?:10t.

(b) Suppose you can grow 1,000 bushels on an acre of land,How much

will an acre of land located t miles from the market rent for? Rent =

5;000?100t.

(c) How far away from the market do you have to be for land to be worth

zero? 50 miles.

23.7 (1) Consider an industry with three rms,Suppose the rms have

the following supply functions,S

1

(p)=p,S

2

(p)=p?5,and S

3

(p)=2p

respectively,On the graph below plot each of the three supply curves and

the resulting industry supply curve.

010203040

5

10

15

Quantity

Price

20

S

2

S

1

S

3

Industry

supply

(a) If the market demand curve has the form D(p) = 15,what is the

resulting market price? 5,Output? 15,What is the output

level for rm 1 at this price? 5,Firm 2? 0,Firm 3?

10.

23.8 (0) Suppose all rms in a given industry have the same supply

curve given by S

i

(p)=p=2,Plot and label the four industry supply

curves generated by these rms if there are 1,2,3,or 4 rms operating

in the industry.

290 INDUSTRY SUPPLY (Ch,23)

010203040

5

10

15

Quantity

Price

20

S

2

S

1

S

3

S

4

(a) If all of the rms had a cost structure such that if the price was below

$3,they would be losing money,what would be the equilibrium price and

output in the industry if the market demand was equal to D(p)=3:5?

Answer,price = $3.50,quantity= 3.5,How many rms would

exist in such a market? 2.

(b) What if the identical conditions as above held except that the market

demand was equal to D(p)=8?p? Now,what would be the equilibrium

price and output? $3.20 and 4.8,How many rms would

operate in such a market? 3.

23.9 (0) There is free entry into the pollicle industry,Anybody can

enter this industry and have the same U-shaped average cost curve as all

of the other rms in the industry.

(a) On the diagram below,draw a representative rm’s average and mar-

ginal cost curves using blue ink,Also,indicate the long-run equilibrium

level of the market price.

NAME 291

010203040

5

10

15

Quantity

Price

20

P

P+t

P+l

Blue

mc

Blue ac

Red ac

Red mc

Black ac

(b) Suppose the government imposes a tax,t,on every unit of output sold

by the industry,Use red ink to draw the new conditions on the above

graph,After the industry has adjusted to the imposition of the tax,the

competitive model would predict the following,the market price would

(increase,decrease) increase by amount t,there would

be (more,the same,fewer) fewer rms operating in the industry,and

the output level for each rm operating in the industry would Stay

the same,(increase,stay the same,decrease).

(c) What if the government imposes a tax,l,onevery rm in the in-

dustry,Draw the new cost conditions on the above graph using black

ink,After the industry has adjusted to the imposition of the tax the

competitive model would predict the following,the market price would

(increase,decrease) increase,there would be (more,the same,

fewer) fewer rms operating in the industry,and the output level

for each rm operating in the industry would increase (increase,

stay the same,decrease).

23.10 (0) In many communities,a restaurant that sells alcoholic bev-

erages is required to have a license,Suppose that the number of licenses

is limited and that they may be easily transferred to other restaurant

owners,Suppose that the conditions of this industry closely approximate

perfect competition,If the average restaurant’s revenue is $100,000 a

year,and if a liquor license can be leased for a year for $85,000 from an

existing restaurant,what is the average variable cost in the industry?

$15,000.

292 INDUSTRY SUPPLY (Ch,23)

23.11 (2) In order to protect the wild populations of cockatoos,the

Australian authorities have outlawed the export of these large parrots.

An illegal market in cockatoos has developed,The cost of capturing an

Australian cockatoo and shipping him to the United States is about $40

per bird,Smuggled parrots are drugged and shipped in suitcases,This is

extremely traumatic for the birds and about 50% of the cockatoos shipped

die in transit,Each smuggled cockatoo has a 10% chance of being discov-

ered,in which case the bird is con scated and a ne of $500 is charged.

Con scated cockatoos that are alive are returned to the wild,Con scated

cockatoos that are found dead are donated to university cafeterias.

(a) The probability that a smuggled parrot will reach the buyer alive and

uncon scated is,45,Therefore when the price of smuggled parrots is

p,what is the expected gross revenue to a parrot-smuggler from shipping

a parrot?,45p.

(b) What is the expected cost,including expected nes and the cost of

capturing and shipping,per parrot? $:10 500 + 40 = $90.

(c) The supply schedule for smuggled parrots will be a horizontal line at

the market price $200,(Hint,At what price does a parrot-smuggler

just break even?)

(d) The demand function for smuggled cockatoos in the United States is

D(p)=7;200?20p per year,How many smuggled cockatoos will be sold

in the United States per year at the equilibrium price? 3,200,How

many cockatoos must be caught in Australia in order that this number of

live birds reaches U.S,buyers? 3;200=:45 = 7;111.

(e) Suppose that instead of returning live con scated cockatoos to the

wild,the customs authorities sold them in the American market,The

pro ts from smuggling a cockatoo do not change from this policy change.

Since the supply curve is horizontal,it must be that the equilibrium price

of smuggled cockatoos will have to be the same as the equilibrium price

when the con scated cockatoos were returned to nature,How many live

cockatoos will be sold in the United States in equilibrium? 3,200.

How many cockatoos will be permanently removed from the Australian

wild? 6,400.

The story behind this problem is based on actual fact,but the num-

bers we use are just made up for illustration,It would be very interesting

to have some good estimates of the actual demand functions and cost

functions.

NAME 293

(f) Suppose that the trade in cockatoos is legalized,Suppose that it

costs about $40 to capture and ship a cockatoo to the United States

in a comfortable cage and that the number of deaths in transit by this

method is negligible,What would be the equilibrium price of cockatoos

in the United States? $40,How many cockatoos would be sold in

the United States? 6,400,How many cockatoos would have to be

caught in Australia for the U.S,market? 6,400.

23.12 (0) The horn of the rhinoceros is prized in Japan and China for its

alleged aphrodisiac properties,This has proved to be most unfortunate for

the rhinoceroses of East Africa,Although it is illegal to kill rhinoceroses

in the game parks of Kenya,the rhinoceros population of these parks has

been almost totally depleted by poachers,The price of rhinoceros horns

in recent years has risen so high that a poacher can earn half a year’s

wages by simply killing one rhinoceros,Such high rewards for poaching

have made laws against poaching almost impossible to enforce in East

Africa,There are also large game parks with rhinoceros populations in

South Africa,Game wardens there were able to prevent poaching almost

completely and the rhinoceros population of South Africa has prospered.

In a recent program from the television series Nova,a South African game

warden explained that some rhinoceroses even have to be \harvested" in

order to prevent overpopulation of rhinoceroses,\What then," asked the

interviewer,\do you do with the horns from the animals that are harvested

or that die of natural causes?" The South African game warden proudly

explained that since international trade in rhinoceros horns was illegal,

South Africa did not contribute to international crime by selling these

horns,Instead the horns were either destroyed or stored in a warehouse.

(a) Suppose that all of the rhinoceros horns produced in South Africa

are destroyed,Label the axes below and draw world supply and demand

curves for rhinoceros horns with blue ink,Label the equilibrium price

and quantity.

294 INDUSTRY SUPPLY (Ch,23)

Price

Quantity

P

P

a

b

Q

a

Q

b

D (Blue)

S (Blue) S (Red)

(b) If South Africa were to sell its rhinoceros horns on the world mar-

ket,which of the curves in your diagram would shift and in what di-

rection? Supply curve to the right,Use red ink to

illustrate the shifted curve or curves,If South Africa were to do this,

would world consumption of rhinoceros horns be increased or decreased?

Increased,Would the world price of rhinoceros horns be increased

or decreased? Decreased,Would the amount of rhinoceros poach-

ing be increased or decreased? Decreased.

23.13 (1) The sale of rhinoceros horns is not prohibited because of con-

cern about the wicked pleasures of aphrodisiac imbibers,but because the

supply activity is bad for rhinoceroses,Similarly,the Australian reason

for restricting the exportation of cockatoos to the United States is not be-

cause having a cockatoo is bad for you,Indeed it is legal for Australians

to have cockatoos as pets,The motive for the restriction is simply to

protect the wild populations from being overexploited,In the case of

other commodities,it appears that society has no particular interest in

restricting the supply activities but wishes to restrict consumption,A

good example is illicit drugs,The growing of marijuana,for example,is a

simple pastoral activity,which in itself is no more harmful than growing

sweet corn or brussels sprouts,It is the consumption of marijuana to

which society objects.

Suppose that there is a constant marginal cost of $5 per ounce for

growing marijuana and delivering it to buyers,But whenever the mari-

juana authorities nd marijuana growing or in the hands of dealers,they

seize the marijuana and ne the supplier,Suppose that the probability

NAME 295

that marijuana is seized is,3 and that the ne if you are caught is $10

per ounce.

(a) If the \street price" is $p per ounce,what is the expected revenue net

of nes to a dealer from selling an ounce of marijuana?,7p?3.

What then would be the equilibrium price of marijuana? $11.4.

(b) Suppose that the demand function for marijuana has the equation

Q = A?Bp,If all con scated marijuana is destroyed,what will be the

equilibrium consumption of marijuana? A?11:4B,Suppose that

con scated marijuana is not destroyed but sold on the open market,What

will be the equilibrium consumption of marijuana? A?11:4B.

(c) The price of marijuana will (increase,decrease,stay the same)

Stay the same.

(d) If there were increasing rather than constant marginal cost in mar-

ijuana production,do you think that consumption would be greater

if con scated marijuana were sold than if it were destroyed? Ex-

plain,Consumption will increase because

the supply curve will shift to the right,

lowering the price.

296 INDUSTRY SUPPLY (Ch,23)

Chapter 24 NAME

Monopoly

Introduction,The pro t-maximizing output of a monopolist is found by

solving for the output at which marginal revenue is equal to marginal cost.

Having solved for this output,you nd the monopolist’s price by plugging

the pro t-maximizing output into the demand function,In general,the

marginal revenue function can be found by taking the derivative of the

total revenue function with respect to the quantity,But in the special case

of linear demand,it is easy to nd the marginal revenue curve graphically.

With a linear inverse demand curve,p(y)=a?by,the marginal revenue

curve always takes the form MR(y)=a?2by.

24.1 (0) Professor Bong has just written the rst textbook in Punk

Economics,It is called Up Your Isoquant,Market research suggests that

the demand curve for this book will be Q =2;000?100P,whereP is

its price,It will cost $1,000 to set the book in type,This setup cost is

necessary before any copies can be printed,In addition to the setup cost,

there is a marginal cost of $4 per book for every book printed.

(a) The total revenue function for Professor Bong’s book is R(Q)=

20Q?Q

2

=100.

(b) The total cost function for producing Professor Bong’s book is C(Q)=

1;000 + 4Q.

(c) The marginal revenue function is MR(Q)= 20?Q=50 and

the marginal cost function is MC(Q)= 4,The pro t-maximizing

quantity of books for professor Bong to sell is Q

= 800.

24.2 (0) Peter Morgan sells pigeon pies from a pushcart in Central Park.

Morgan is the only supplier of this delicacy in Central Park,His costs are

zero due to the abundant supplies of raw materials available in the park.

(a) When he rst started his business,the inverse demand curve for pigeon

pies was p(y) = 100?y,where the price is measured in cents and y

measures the number of pies sold,Use black ink to plot this curve in

the graph below,On the same graph,use red ink to plot the marginal

revenue curve.

298 MONOPOLY (Ch,24)

0 50 75 100 125

Pigeon pies

25

50

75

100

Cents

25 150

Black

lines

Blue line

Red line

(b) What level of output will maximize Peter’s pro ts? 50,What

price will Peter charge per pie? 50 cents.

(c) After Peter had been in business for several months,he noticed that

the demand curve had shifted to p(y)=75?y=2,Useblueinktoplot

this curve in the graph above,Plot the new marginal revenue curve on

the same graph with black ink.

(d) What is his pro t-maximizing output at this new price? 75,What

is the new pro t-maximizing price? 37.5 cents per pie.

24.3 (0) Suppose that the demand function for Japanese cars in the

United States is such that annual sales of cars (in thousands of cars) will

be 250?2P,whereP is the price of Japanese cars in thousands of dollars.

(a) If the supply schedule is horizontal at a price of $5,000 what will

be the equilibrium number of Japanese cars sold in the United States?

240 thousand,How much money will Americans spend in total on

Japanese cars? 1.2 billion dollars.

(b) Suppose that in response to pressure from American car manufactur-

ers,the United States imposes an import duty on Japanese cars in such a

way that for every car exported to the United States the Japanese man-

ufacturers must pay a tax to the U.S,government of $2,000,How many

Japanese automobiles will now be sold in the United States? 236

thousand,At what price will they be sold? 7 thousand dollars.

NAME 299

(c) How much revenue will the U.S,government collect with this tari?

472 million dollars.

(d) On the graph below,the price paid by American consumers is mea-

sured on the vertical axis,Use blue ink to show the demand and supply

schedules before the import duty is imposed,After the import duty is

imposed,the supply schedule shifts and the demand schedule stays as

before,Use red ink to draw the new supply schedule.

0 100 150 200 250

Japanese autos (thousands)

2

4

6

8

Price (thousands)

50 300

7

5

Blue

lines

Red line

Demand

Supply

Supply with duty

(e) Suppose that instead of imposing an import duty,the U.S,government

persuades the Japanese government to impose \voluntary export restric-

tions" on their exports of cars to the United States,Suppose that the

Japanese agree to restrain their exports by requiring that every car ex-

ported to the United States must have an export license,Suppose further

that the Japanese government agrees to issue only 236,000 export licenses

and sells these licenses to the Japanese rms,If the Japanese rms know

the American demand curve and if they know that only 236,000 Japanese

cars will be sold in America,what price will they be able to charge in

America for their cars? 7 thousand dollars.

(f) How much will a Japanese rm be willing to pay the Japanese govern-

ment for an export license? 2 thousand dollars,(Hint,Think

about what it costs to produce a car and how much it can be sold for if

youhaveanexportlicense.)

(g) How much will be the Japanese government’s total revenue from the

sale of export licenses? 472 million dollars.

300 MONOPOLY (Ch,24)

(h) How much money will Americans spend on Japanese cars? 1.652

billion dollars.

(i) Why might the Japanese \voluntarily" submit to export controls?

Total revenue of Japanese companies and

government is greater with export controls

than without them,Since there is less

output,costs are lower,Higher revenue,

lower costs imply more profit.

24.4 (0) A monopolist has an inverse demand curve given by p(y)=

12?y and a cost curve given by c(y)=y

2

.

(a) What will be its pro t-maximizing level of output? 3.

(b) Suppose the government decides to put a tax on this monopolist so

that for each unit it sells it has to pay the government $2,What will be

its output under this form of taxation? 2.5.

(c) Suppose now that the government puts a lump sum tax of $10 on the

pro ts of the monopolist,What will be its output? 3.

24.5 (1) In Gomorrah,New Jersey,there is only one newspaper,the

Daily Calumny,The demand for the paper depends on the price and the

amount of scandal reported,The demand function is Q =15S

1=2

P

3

,

where Q is the number of issues sold per day,S is the number of column

inches of scandal reported in the paper,and P is the price,Scandals

are not a scarce commodity in Gomorrah,However,it takes resources to

write,edit,and print stories of scandal,The cost of reporting S units

of scandal is $10S,These costs are independent of the number of papers

sold,In addition it costs money to print and deliver the paper,These

cost $:10 per copy and the cost per unit is independent of the amount

of scandal reported in the paper,Therefore the total cost of printing Q

copies of the paper with S column inches of scandal is $10S +:10Q.

(a) Calculate the price elasticity of demand for the Daily Calumny.

3,Does the price elasticity depend on the amount of scandal re-

ported? No,Is the price elasticity constant over all prices? Yes.

NAME 301

(b) Remember that MR = P(1 +

1

),To maximize pro ts,the Daily

Calumny will set marginal revenue equal to marginal cost,Solve for

the pro t-maximizing price for the Calumny to charge per newspaper.

$.15,When the newspaper charges this price,the di erence between

the price and the marginal cost of printing and delivering each newspaper

is $.05.

(c) If the Daily Calumny charges the pro t-maximizing price and prints

100 column inches of scandal,how many copies would it sell? (Round

to the nearest integer.) 44,444,Write a general expression

for the number of copies sold as a function of S,Q(S)= Q =

15S

1=2

(:15)

3

=4;444:44S

1=2

.

(d) Assuming that the paper charges the pro t-maximizing price,write

an expression for pro ts as a function of Q and S,Profits=

:15Q?:10Q?10S,Using the solution for Q(S) that you found

in the last section,substitute Q(S)forQ to write an expression for pro ts

as a function of S alone,Profits =:05(4;444:44S

1=2

)?

10S = 222:22S

1=2

10S.

(e) If the Daily Calumny charges its pro t-maximizing price,and prints

the pro t-maximizing amount of scandal,how many column inches of

scandal should it print? 123.456 inches,How many copies

are sold 49,383 and what is the amount of pro t for the Daily

Calumny if it maximizes its pro ts? 1,234.5.

24.6 (0) In the graph below,use black ink to draw the inverse demand

curve,p

1

(y) = 200?y.

(a) If the monopolist has zero costs,where on this curve will it choose to

operate? At y = 100,p = 100.

(b) Now draw another demand curve that passes through the pro t-

maximizing point and is flatter than the original demand curve,Use

a red pen to mark the part of this new demand curve on which the mo-

nopolist would choose to operate,(Hint,Remember the idea of revealed

preference?)

302 MONOPOLY (Ch,24)

(c) The monopolist would have (larger,smaller) pro ts at the new demand

curve than it had at the original demand curve,Larger.

0 50 100 150 200

50

100

150

Quantity

Price

200

Red

Line

Black Line

Chapter 25 NAME

Monopoly Behavior

Introduction,Problems in this chapter explore the possibilities of price

discrimination by monopolists,There are also problems related to spatial

markets,where transportation costs are accounted for and we show that

lessons learned about spatial models give us a useful way of thinking about

competition under product di erentiation in economics and in politics.

Remember that a price discriminator wants the marginal revenue in

each market to be equal to the marginal cost of production,Since he

produces all of his output in one place,his marginal cost of production

is the same for both markets and depends on his total output,The trick

for solving these problems is to write marginal revenue in each market as

a function of quantity sold in that market and to write marginal cost as

a function of the sum of quantities sold in the two markets,The pro t-

maximizing conditions then become two equations that you can solve

for the two unknown quantities sold in the two markets,Of course,if

marginal cost is constant,your job is even easier,since all you have to do

is nd the quantities in each market for which marginal revenue equals

the constant marginal cost.

Example,A monopolist sells in two markets,The inverse demand curve

in market 1 is p

1

= 200?q

1

,The inverse demand curve in market 2 is

p

2

= 300?q

2

,The rm’s total cost function is C(q

1

+q

2

)=(q

1

+q

2

)

2

.The

rm is able to price discriminate between the two markets,Let us nd the

prices that it will charge in each market,In market 1,the rm’s marginal

revenue is 200?2q

1

,In market 2,marginal revenue is 300?2q

2

.The

rm’s marginal costs are 2(q

1

+q

2

),To maximize its pro ts,the rm sets

marginal revenue in each market equal to marginal cost,This gives us the

two equations 200?2q

1

=2(q

1

+q

2

) and 300?2q

2

=2(q

1

+q

2

),Solving

these two equations in two unknowns for q

1

and q

2

,we nd q

1

=16:67

and q

2

=66:67,We can nd the price charged in each market by plugging

these quantities into the demand functions,The price charged in market

1 will be 183.33,The price charged in market 2 will be 233.33.

25.1 (0) Ferdinand Sludge has just written a disgusting new book,Orgy

in the Piggery,His publisher,Graw McSwill,estimates that the demand

for this book in the United States is Q

1

=50;000? 2;000P

1

,where

P

1

is the price in America measured in U.S,dollars,The demand for

Sludge’s opus in England is Q

2

=10;000?500P

2

,whereP

2

is its price

in England measured in U,S,dollars,His publisher has a cost function

C(Q) = $50;000 + $2Q,whereQ is the total number of copies of Orgy

that it produces.

(a) If McSwill must charge the same price in both countries,how many

copies should it sell 27,500,and what price should it charge

304 MONOPOLY BEHAVIOR (Ch,25)

$13 to maximize its pro ts,and how much will those pro ts be?

$252,500.

(b) If McSwill can charge a di erent price in each country,and wants to

maximize pro ts,how many copies should it sell in the United States?

23,000,What price should it charge in the United States?

$13.50,How many copies should it sell in England? 4,500.

What price should it charge in England? $11,How much will its

total pro ts be? $255,000.

25.2 (0) A monopoly faces an inverse demand curve,p(y) = 100?2y,

and has constant marginal costs of 20.

(a) What is its pro t-maximizing level of output? 20.

(b) What is its pro t-maximizing price? $60.

(c) What is the socially optimal price for this rm? $20.

(d) What is the socially optimal level of output for this rm? 40.

(e) What is the deadweight loss due to the monopolistic behavior of this

rm? 400.

(f) Suppose this monopolist could operate as a perfectly discriminating

monopolist and sell each unit of output at the highest price it would fetch.

The deadweight loss in this case would be 0.

Calculus 25.3 (1) Banana Computer Company sells Banana computers both in

the domestic and foreign markets,Because of di erences in the power

supplies,a Banana purchased in one market cannot be used in the other

market,The demand and marginal revenue curves associated with the

two markets are as follows:

P

d

=20;000?20QP

f

=25;000?50Q

MR

d

=20;000?40QMR

f

=25;000?100Q:

Banana’s production process exhibits constant returns to scale and it

takes $1,000,000 to produce 100 computers.

NAME 305

(a) Banana’s long-run average cost function is AC(Q)= $10,000

and its long-run marginal cost function is MC(Q)= $10,000.

(Hint,If there are constant returns to scale,does long-run average cost

change as output changes?) Draw the average and marginal cost curves

on the graph.

(b) Draw the demand curve for the domestic market in black ink and

the marginal revenue curve for the domestic market in pencil,Draw the

demand curve for the foreign market in red ink and the marginal revenue

curve for the foreign market in blue ink.

0 100 200 300 400 500 600 700 800

10

20

30

40

50

60

Dollars (1,000s)

Red line

Blue line

Black line

Pencil line

LRAC

LRMC

Banana Computers

(c) If Banana is maximizing its pro ts,it will sell 250 computers in

the domestic market at 15,000 dollars each and 150 computers

in the foreign market at 17,500 dollars each,What are Banana’s

total pro ts? $2,375,000.

306 MONOPOLY BEHAVIOR (Ch,25)

(d) At the pro t-maximizing price and quantity,what is the price elas-

ticity of demand in the domestic market3,What is the price

elasticity of demand in the foreign market2:33,Is demand more

or less elastic in the market where the higher price is charged? Less

elastic.

(e) Suppose that somebody gures out a wiring trick that allows a Banana

computer built for either market to be costlessly converted to work in the

other,(Ignore transportation costs.) On the graph below,draw the new

inverse demand curve (with blue ink) and marginal revenue curve (with

black ink) facing Banana.

0 100 200 300 400 500 600 700 800

10

20

30

40

Dollars (1,000s)

Blue line

LRAC

LRMC

Banana Computers

Black line

(f) Given that costs haven’t changed,how many Banana computers

should Banana sell? 400,What price will it charge? $15,714.

How will Banana’s pro ts change now that it can no longer practice price

discrimination? Decrease by $89,284.

25.4 (0) A monopolist has a cost function given by c(y)=y

2

and faces

a demand curve given by P(y) = 120?y.

(a) What is his pro t-maximizing level of output? 30,What price

will the monopolist charge? $90.

NAME 307

(b) If you put a lump sum tax of $100 on this monopolist,what would its

output be? 30.

(c) If you wanted to choose a price ceiling for this monopolist so as to

maximize consumer plus producer surplus,what price ceiling should you

choose? $80.

(d) How much output will the monopolist produce at this price ceiling?

40.

(e) Suppose that you put a speci c tax on the monopolist of $20 per unit

output,What would its pro t-maximizing level of output be? 25.

25.5 (1) The Grand Theater is a movie house in a medium-sized college

town,This theater shows unusual lms and treats early-arriving movie

goers to live organ music and Bugs Bunny cartoons,If the theater is

open,the owners have to pay a xed nightly amount of $500 for lms,

ushers,and so on,regardless of how many people come to the movie.

For simplicity,assume that if the theater is closed,its costs are zero,The

nightly demand for Grand Theater movies by students is Q

S

= 220?40P

S

,

where Q

S

is the number of movie tickets demanded by students at price

P

S

,The nightly demand for nonstudent moviegoers is Q

N

= 140?20P

N

.

(a) If the Grand Theater charges a single price,P

T

,toeverybody,then

at prices between 0 and $5.50,the aggregate demand function for movie

tickets is Q

T

(P

T

)= 360?60P

T

,Over this range of prices,the

inverse demand function is then P

T

(Q

T

)= 6?Q

T

=60.

(b) What is the pro t-maximizing number of tickets for the Grand The-

ater to sell if it charges one price to everybody? 180,At what price

would this number of tickets be sold? $3,How much pro ts would

the Grand make? $40,How many tickets would be sold to students?

100,To nonstudents? 80.

(c) Suppose that the cashier can accurately separate the students from

the nonstudents at the door by making students show their school ID

cards,Students cannot resell their tickets and nonstudents do not have

access to student ID cards,Then the Grand can increase its pro ts by

charging students and nonstudents di erent prices,What price will be

charged to students? $2.75,How many student tickets will be sold?

308 MONOPOLY BEHAVIOR (Ch,25)

110,What price will be charged to nonstudents? $3.50,How

many nonstudent tickets will be sold? 70,How much pro t will the

Grand Theater make? $47.50.

(d) If you know calculus,see if you can do this part,Suppose that

the Grand Theater can hold only 150 people and that the manager

wants to maximize pro ts by charging separate prices to students and

to nonstudents,If the capacity of the theater is 150 seats and Q

S

tickets are sold to students,what is the maximum number of tickets

that can be sold to nonstudents? Q

N

= 150?Q

S

,Write

an expression for the price of nonstudent tickets as a function of the

number of student tickets sold,(Hint,First nd the inverse nonstu-

dent demand function.) P

N

=?1=2+Q

S

=20,Write

an expression for Grand Theater pro ts as a function of the number

Q

S

only,(Hint,Make substitutions using your previous answers.)

Q

S

(11=2?Q

S

=40) + (?1=2+20=Q

S

)(150?Q

S

)?

500 =?3Q

2

S

=40 + 27Q

S

=2?575,How many student

tickets should the Grand sell to maximize pro ts? 90,What price

is charged to students? $3.25,How many nonstudent tickets are

sold? 60,What price is charged to nonstudents? $4,How much

pro t does the Grand make under this arrangement? $32.50.

25.6 (2) The Mall Street Journal is considering o ering a new service

which will send news articles to readers by email,Their market research

indicates that there are two types of potential users,impecunious under-

graduates studying microeconomics and high-level executives,Let x be

the number of articles that a user requests per year,The executives have

an inverse demand function P

E

(x) = 100?x and the undergraduates

have an inverse demand function P

U

(x)=80?x,(Prices are measured

in cents.) The Journal has a zero marginal cost of sending articles via

email,Please draw these demand functions in the graph below and label

them.

NAME 309

20 40 60 80 100 120

20

40

60

80

100

120

Quantity

Price

0

P (X) = 100 - X

E

P (X) = 80 - X

U

(a) Suppose that the Journal can identify which of the users are under-

graduates and which are executives,It decides to o er a plan where users

can buy a xed number of articles per year for a xed price per year.

If it wants to maximize total pro ts it will o er 100 articles to the

executives and 80 articles per year to the students.

(b) It will charge $50 per year to the executives and $32 per year

to the students.

(c) Suppose that the Journal cannot identify which users are executives

and which are undergraduates,In this case it simply o ers two packages,

and lets the users self-select the one that is optimal for them,Suppose

that it o ers two packages,one that allows up to 80 articles per year the

other that allows up to 100 articles per year,What’s the highest price

that the undergraduates will pay for the 80-article subscription? $32.

(d) What (gross) consumer surplus would the executives get if they con-

sumed 80 articles per year? $48.

(e) What is the the maximum price that the Journal can charge for 100

articles per year if it o ers 80 a year at the highest price the undergradu-

ates are willing to pay? Solve 50?p =48?32 to find

p = $34.

310 MONOPOLY BEHAVIOR (Ch,25)

(f) Suppose that the Mall Street Journal decides to include only 60 articles

in the student package,What is the most it could charge and still get

student to buy this package? $30.

(g) If the Mall Street Journal o ers a \student package" of 60 articles

at this price,how much net consumer surplus would executives get from

buying the student package? $12.

(h) What is the most that the Mall Street Journal could charge for 100

article package and expect executives to buy this package rather than the

student package? $38.

(i) If the number of executives in the population equals the number of

students,would the Mall Street Journal make higher pro ts by o ering a

student package of 80 articles or a student package of 60 articles? 60.

25.7 (2) Bill Barriers,CEO of MightySoft software,is contemplating

a new marketing strategy,bundling their best-selling wordprocessor and

their spreadsheet together and selling the pair of software products for

one price.

From the viewpoint of the company,bundling software and selling it

at a discounted price has two e ects on sales,1) revenues go up due to

to additional sales of the bundle; and 2) revenues go down since there is

less of a demand for the individual components of the bundle.

The pro tability of bundling depends on which of these two e ects

dominates,Suppose that MightySoft sells the wordprocessor for $200 and

the spreadsheet for $250,A marketing survey of 100 people who purchased

either of these packages in the last year turned up the following facts:

1) 20 people bought both.

2) 40 people bought only the wordprocessor,They would be willing to

spend up to $120 more for the spreadsheet.

3) 40 people bought only the spreadsheet,They would be willing to

spend up to $100 more for the wordprocessor.

In answering the following questions you may assume the following:

1) New purchasers of MightySoft products will have the same charac-

teristics as this group.

2) There is a zero marginal cost to producing extra copies of either

software package.

3) There is a zero marginal cost to creating a bundle.

(a) Let us assume that MightySoft also o ers the products separately

as well as bundled,In order to determine how to price the bundle,Bill

Barriers asks himself the following questions,In order to sell the bundle

to the wordprocessor purchasers,the price would have to be less than

200 + 120 = 320.

NAME 311

(b) In order to sell to the spreadsheet users,the price would have to be

less than 250 + 100 = 350.

(c) What would MightySoft’s pro ts be on a group of 100 users if it priced

the bundle at $320? Everyone buys the bundle so

profits are 100 320 = $32;000.

(d) What would MightySoft’s pro ts be on a group of 100 users if it

priced the bundle at $350? 20 people would buy both

anyway,40 people bought spreadsheet only

and would be willing to buy the bundle,

40 people buy the wordprocessor,but not

the spreadsheet,Total profits are 20

350 + 40 350 + 40 200 = 29;000.

(e) If MightySoft o ers the bundle,what price should it set? $320

is the more profitable price.

(f) What would pro ts be without o ering the bundle? Without

the bundle,profits would be 20 (200+250)+

40 200 + 40 250 = 27;000.

(g) What would be the pro ts with the bundle? 100 320 =

32;000

(h) Is it more pro table to bundle or not bundle? bundle.

(i) Suppose that MightySoft worries about the reliability of their market

survey and decides that they believe that without bundling t of the 100

people will buy both products,and (100?t)=2 will buy the wordprocessor

only and (100?t)=2 will buy the spreadsheet only,Calculate pro ts as a

function of t if there is no bundling,225 (100?t)+450 t.

312 MONOPOLY BEHAVIOR (Ch,25)

(j) What are pro ts with the bundle? $32000.

(k) At what values of t would it be unpro table to o er the bundle?

Solve for the t that equates the two

profits to find t =42:22,So if more than

42 of the 100 new purchasers would buy both

products anyway,it is not profitable to

bundle them.

(l) This analysis so far has been concerned only with customers who

would purchase at least one of the programs at the original set of prices.

Is there any additional source of demand for the bundle? What does

this say about the calculations we have made about the pro tability of

bundling? Yes,it may be that there are

some consumers who were not willing to pay

$200 for the wordprocessor or $250 for

the spreadsheet,but would be willing to

pay $320 for the bundle,This means that

bundling would be even more profitable

than the calculations above indicate.

25.8 (0) Col,Tom Barker is about to open his newest amusement park,

Elvis World,Elvis World features a number of exciting attractions,you

can ride the rapids in the Blue Suede Chutes,climb the Jailhouse Rock

and eat dinner in the Heartburn Hotel,Col,Tom gures that Elvis World

will attract 1,000 people per day,and each person will take x =50?50p

rides,where p is the price of a ride,Everyone who visits Elvis World is

pretty much the same and negative rides are not allowed,The marginal

cost of a ride is essentially zero.

(a) What is each person’s inverse demand function for rides? p(x)=

1?x=50.

(b) If Col,Tom sets the price to maximize pro t,how many rides will be

taken per day by a typical visitor? 25.

NAME 313

(c) What will the price of a ride be? 50 cents.

(d) What will Col,Tom’s pro ts be per person? $12.50

(e) What is the Pareto e cient price of a ride? Zero.

(f) If Col,Tom charged the Pareto e cient price for a ride,how many

rides would be purchased? 50.

(g) How much consumers’ surplus would be generated at this price and

quantity? 25.

(h) If Col,Tom decided to use a two-part tari,he would set an admission

fee of $25 and charge a price per ride of 0.

25.9 (1) The city of String Valley is squeezed between two mountains

and is 36 miles long,running from north to south,and only about 1

block wide,Within the town,the population has a uniform density of

100 people per mile,Because of the rocky terrain,nobody lives outside

the city limits on either the north or the south edge of town,Because of

strict zoning regulations,the city has only three bowling alleys,One of

these is located at the city limits on the north edge of town,one of them is

located at the city limits on the south edge of town,and one is located at

the exact center of town,Travel costs including time and gasoline are $1

per mile,All of the citizens of the town have the same preferences,They

are willing to bowl once a week if the cost of bowling including travel

costs and the price charged by the bowling alley does not exceed $15.

(a) Consider one of the bowling alleys at either edge of town,If it charges

$10 for a night of bowling,how far will a citizen of String Valley be willing

to travel to bowl there? Up to 5 miles,How many customers

would this bowling alley have per week if it charged $10 per night of

bowling? 500.

(b) Write a formula for the number of customers that a bowling alley

at the edge of town will have if it charges $p per night of bowling.

100 (15?p).

(c) Write a formula for this bowling alley’s inverse demand function.

p =15?q=100.

314 MONOPOLY BEHAVIOR (Ch,25)

(d) Suppose that the bowling alleys at the end of town have a marginal

cost of $3 per customer and set their prices to maximize pro ts,(For

the time being assume that these bowling alleys face no competition from

the other bowling alleys in town.) How many customers will they have?

600,What price will they charge? $9,How far away from the edge

of town does their most distant customer live? 6 miles.

(e) Now consider the bowling alley in the center of town,If it charges a

price of $p,how many customers will it have per week? 2*(15-p).

(f) If the bowling alley in the center of town also has marginal costs

of $3 per customer and maximizes its pro ts,what price will it charge?

$9,How many customers will it have per week? 1,200,How far

away from the center of town will its most distant customers live? 6

miles.

(g) Suppose that the city relaxes its zoning restrictions on where the

bowling alleys can locate,but continues to issue operating licenses to

only 3 bowling alleys,Both of the bowling alleys at the end of town

are about to lose their leases and can locate anywhere in town that they

like at about the same cost,The bowling alley in the center of town is

committed to stay where it is,Would either of the alleys at the edge of

town improve its pro ts by locating next to the existing bowling alley in

the center of town? No,What would be a pro t-maximizing location

for each of these two bowling alleys? One would be located

12 miles north of the town center and one

12 miles south.

25.10 (1) In a congressional district somewhere in the U.S,West a

new representative is being elected,The voters all have one-dimensional

political views that can be neatly arrayed on a left-right spectrum,We

can de ne the \location" of a citizen’s political views in the following way.

The citizen with the most extreme left-wing views is said to be at point

0 and the citizen with the most estreme right-wing views is said to be at

point 1,If a citizen has views that are to the right of the views of the

fraction x of the state’s population,that citizen’s views are said to be

located at the point x,Candidates for o ce are forced to publically state

their own political position on the zero-one left-right scale,Voters always

vote for the candidate whose stated position is nearest to their own views.

NAME 315

(If there is a tie for nearest candidate,voters flip a coin to decide which

to vote for.)

(a) There are two candidates for the congressional seat,Suppose that

each candidate cares only about getting as many votes as possible,Is

there an equilibrium in which each candidate chooses the best position

given the position of the other candidate? If so,describe this equilibrium.

The only equilibrium is one in which both

candidates choose the same position,and

that position is at the point 1/2.

25.11 (2) In the congressional district described by the previous problem,

let us investigate what will happen if the two candidates do not care

about the number of votes that they get but only about the amount

of campaign contributions that they receive,Therefore each candidate

chooses his ideological location in such a way as to maximize the amount

of campaign contributions he receives,given the position of the other.*

Let us de ne a left-wing extremist as a voter whose political views

lie to the left of the leftmost candidate,a right-wing extremist as a voter

whose political views lie to the right of the rightmost candidate,and a

moderate voter as one whose political views lie between the positions

of the two candidates,Assume that each extremist voter contributes to

the candidate whose position is closest to his or her own views and that

moderate voters make no campaign contributions,The number of dol-

lars that an extremist voter contributes to his or her favorite candidate

is proportional to the distance between the two candidates,Speci cally,

we assume that there is some constant C such that if the left-wing can-

didate is located at x and the right-wing candidate is located at y,then

total campaign contributions received by the left-wing candidate will be

$Cx(y?x) and total campaign contributions received by the right-wing

candidate will be $C(1?y)(y?x).

(a) If the right-wing candidate is located at y,the contribution-

maximizing position for the left-wing candidate is x = y=2 If the

left-wing candidate is located at x,the contribution-maximizing position

for the right-wing candidate is y = ((1 + x)=2 (Hint,Take a

derivative and set it equal to zero.)

(b) Solve for the unique pair of ideological positions for the two can-

didates such that each takes the position that maximizes his campaign

contributions given the position of the other,x =1=3,y =2=3

* This assumption is a bit extreme,Candidates typically spend at least

some of their campaign contributions on advertising for votes,and this

advertising a ects the voting outcomes.

316 MONOPOLY BEHAVIOR (Ch,25)

(c) Suppose that in addition to collecting contributions from extremists

on his side,candidates can also collect campaign contributions from mod-

erates whose views are closer to their position than to that of their rival’s

position,Suppose that moderates,like extremists,contribute to their

preferred candidate and that they contribute in proportion to the dif-

ference between their own ideological distance from their less-preferred

candidate and their ideological distance from their more-preferred can-

didate,Show that in this case the unique positions in which the left-

and right-wing candidates are each maximizing their campaign contribu-

tions,given the position of the other candidate,occurs where x =1=4

and y =3=4,Total contributions received

by the left wing candidate will be

C

parenleftbig

x(y?x)+(y?x)

2

=4

Total contributions

received by the right-wing candidate

will be C

parenleftbig

(1?y)(y?x)+(y?x)

2

=4

.

Differentiating the former expression with

respect to x and the latter with respect

to y and solving the resulting simultaneous

equations yields x =1=4 and y =3=4.

Chapter 26 NAME

Factor Markets

Introduction,In this chapter you will examine the factor demand de-

cision of a monopolist,If a rm is a monopolist in some industry,it

will produce less output than if the industry were competitively orga-

nized,Therefore it will in general want to use less inputs than does a

competitive rm,The value marginal product is just the value of the ex-

tra output produced by hiring an extra unit of the factor,The ordinary

logic of competitive pro t maximization implies that a competitive rm

will hire a factor up until the point where the value marginal product

equals the price of the factor.

The marginal revenue product is the extra revenue produced by

hiring an extra unit of a factor,For a competitive rm,the marginal

revenue product is the same as the value of the marginal product,but

they di er for monopolist,A monopolist has to take account of the fact

that increasing its production will force the price down,so the marginal

revenue product of an extra unit of a factor will be less than the value

marginal product.

Another thing we study in this chapters is monopsony,whichisthe

case of a market dominated by a single buyer of some good,The case of

monopsony is very similar to the case of a monopoly,The monopsonist

hires less of a factor than a similar competitive rm because the monop-

sony recognizes that the price it has to pay for the factor depends on how

much it buys.

Finally,we consider an interesting example of factor supply,in which

a monopolist produces a good that is used by another monopolist.

Example,Suppose a monopolist faces a demand curve for output of the

form p(y) = 100?2y,The production function takes the simple form

y =2x,and the factor costs $4 per unit,How much of the factor of

production will the monopolist want to employ? How much of the factor

would a competitive industry employ if all the rms in the industry had

the same production function?

Answer,The monopolist will employ the factor up to the point where

the marginal revenue product equals the price of the factor,Revenue as

a function of output is R(y)=p(y)y = (100?2y)y,To nd revenue as a

function of the input,we substitute y =2x:

R(x) = (100?4x)2x = (200?8x)x:

The marginal revenue product function will have the form MRP

x

= 200?

16x,Setting marginal revenue product equal to factor price gives us the

equation

200?16x =4:

Solving this equation gives us x

=12:25:

318 FACTOR MARKETS (Ch,26)

If the industry were competitive,then the industry would employ the

factor up to the point where the value of the marginal product was equal

to 4,This gives us the equation

p2=4;

so p

= 2,How much output would be demanded at this price? We plug

this into the demand function to get the equation 2 = 100?2y,which

implies y

= 49,Since the production function is y =2x,wecansolve

for x

= y

=2=24:5:

26.1 (0) Gargantuan Enterprises has a monopoly in the production of

antimacassars,Its factory is located in the town of Pantagruel,There is

no other industry in Pantagruel,and the labor supply equation there is

W =10+:1L,whereW is the daily wage and L is the number of person-

days of work performed,Antimacassars are produced with a production

function,Q =10L,whereL is daily labor supply and Q is daily output.

The demand curve for antimacassars is P =41?

Q

1;000

,whereP is the

price and Q is the number of sales per day.

(a) Find the pro t-maximizing output for Gargantuan,(Hint,Use the

production function to nd the labor input requirements for any level of

output,Make substitutions so you can write the rm’s total costs as a

function of its output and then its pro t as a function of output,Solve

for the pro t{maximizing output.) 10,000.

(b) How much labor does it use? 1,000,What is the wage rate that

it pays? $110.

(c) What is the price of antimacassars? $31,How much pro t is

made? $200,000.

26.2 (0) The residents of Seltzer Springs,Michigan,consume bottles of

mineral water according to the demand function D(p)=1;000?p.Here

D(p) is the demand per year for bottles of mineral water if the price per

bottle is p.

The sole distributor of mineral water in Seltzer Springs,Bubble Up,

purchases mineral water at c per bottle from their supplier Perry Air.

Perry Air is the only supplier of mineral water in the area and behaves

as a pro t-maximizing monopolist,For simplicity we suppose that it has

zero costs of production.

(a) What is the equilibrium price charged by the distributor Bubble Up?

p

=

1;000+c

2

.

NAME 319

(b) What is the equilibrium quantity sold by Bubble Up? D(p

)=

1;000?c

2

.

(c) What is the equilibrium price charged by the producer Perry Air?

c

= 500.

(d) What is the equilibrium quantity sold by Perry Air? D(c

)=

250.

(e) What are the pro ts of Bubble Up?

b

= (500?250)(750?

500) = 250

2

.

(f) What are the pro ts of Perry Air?

p

= 500 250.

(g) How much consumer’s surplus is generated in this market? CS

e

=

250

2

=2.

(h) Suppose that this situation is expected to persist forever and that

the interest rate is expected to be constant at 10% per year,What is the

minimum lump sum payment that Perry Air would need to pay to Bubble

Up to buy it out? 10 250

2

.

(i) Suppose that Perry Air does this,What will be the new price and

quantity for mineral water? p

= 500 and D(p

) = 500.

(j) What are the pro ts of the new merged rm?

p

= 500

2

.

(k) What is the total amount of consumers’ surplus generated? How does

this compare with the previous level of consumers’ surplus? CS

i

=

500

2

=2 >CS

e

.

Calculus 26.3 (0) Upper Peninsula Underground Recordings (UPUR) has a mon-

opoly on the recordings of the famous rock group Moosecake,Moosecake’s

music is only provided on digital tape,and blank digital tapes cost them

c per tape,There are no other manufacturing or distribution costs,Let

p(x) be the inverse demand function for Moosecake’s music as a function

of x,the number of tapes sold.

320 FACTOR MARKETS (Ch,26)

(a) What is the rst-order condition for pro t maximization? For future

reference,let x

be the pro t-maximizing amount produced and p

be the

price at which it sells,(In this part,assume that tapes cannot be copied.)

p(x

)+p

0

(x

)x

= c.

Now a new kind of consumer digital tape recorder becomes widely

available that allows the user to make 1 and only 1 copy of a prerecorded

digital tape,The copies are a perfect substitute in consumption value for

the original prerecorded tape,and there are no barriers to their use or

sale,However,everyone can see the di erence between the copies and the

orginals and recognizes that the copies cannot be used to make further

copies,Blank tapes cost the consumers c per tape,the same price the

monopolist pays.

(b) All Moosecake fans take advantage of the opportunity to make a single

copy of the tape and sell it on the secondary market,How is the price of an

original tape related to the price of a copy? Derive the inverse demand

curve for original tapes facing UPUR,(Hint,There are two sources of

demand for a new tape,the pleasure of listening to it,and the pro ts

from selling a copy.) If UPUR produces x tapes,2x

tapes reach the market,so UPUR can sell

a single tape for p(2x)+[p(2x)?c],The

first term is the willingness-to-pay for

listening; the second term is profit from

selling a copy.

(c) Write an expression for UPUR’s pro ts if it produces x tapes.

[p(2x)+p(2x)?c]x?cx =2p(2x)x?2cx.

(d) Let x

be the pro t-maximizing level of production by UPUR,How

does it compare to the former pro t-maximizing level of production?

From the two profit functions,one sees

that 2x

= x

,so x

= x

=2.

(e) How does the price of a copy of a Moosecake tape compare to the

price determined in Part (a)? The prices are the same.

(f) If p

is the price of a copy of a Moosecake tape,how much will a new

Moosecake tape sell for? 2p

c.

Chapter 27 NAME

Oligopoly

Introduction,In this chapter you will solve problems for rm and indus-

try outcomes when the rms engage in Cournot competition,Stackelberg

competition,and other sorts of oligopoly behavior,In Cournot competi-

tion,each rm chooses its own output to maximize its pro ts given the

output that it expects the other rm to produce,The industry price de-

pends on the industry output,say,q

A

+q

B

,where A and B are the rms.

To maximize pro ts,rm A sets its marginal revenue (which depends on

the output of rm A and the expected output of rm B since the expected

industry price depends on the sum of these outputs) equal to its marginal

cost,Solving this equation for rm A’s output as a function of rm B’s

expected output gives you one reaction function; analogous steps give you

rm B’s reaction function,Solve these two equations simultaneously to

get the Cournot equilibrium outputs of the two rms.

Example,In Heifer’s Breath,Wisconsin,there are two bakers,Anderson

and Carlson,Anderson’s bread tastes just like Carlson’s|nobody can

tell the di erence,Anderson has constant marginal costs of $1 per loaf of

bread,Carlson has constant marginal costs of $2 per loaf,Fixed costs are

zero for both of them,The inverse demand function for bread in Heifer’s

Breath is p(q)=6?:01q,whereq is the total number of loaves sold per

day.

Let us nd Anderson’s Cournot reaction function,If Carlson bakes

q

C

loaves,then if Anderson bakes q

A

loaves,total output will be q

A

+

q

C

and price will be 6?:01(q

A

+ q

C

),For Anderson,the total cost of

producing q

A

units of bread is just q

A

,so his pro ts are

pq

A

q

A

=(6?:01q

A

:01q

C

)q

A

q

A

=6q

A

:01q

2

A

:01q

C

q

A

q

A

:

Therefore if Carlson is going to bake q

C

units,then Anderson will choose

q

A

to maximize 6q

A

:01q

2

A

:01q

C

q

A

q

A

,This expression is maximized

when 6?:02q

A

:01q

C

= 1,(You can nd this out either by setting

A’s marginal revenue equal to his marginal cost or directly by setting

the derivative of pro ts with respect to q

A

equal to zero.) Anderson’s

reaction function,R

A

(q

C

) tells us Anderson’s best output if he knows

that Carlson is going to bake q

C

,We solve from the previous equation to

nd R

A

(q

C

)=(5?:01q

C

)=:02 = 250?:5q

C

.

We can nd Carlson’s reaction function in the same way,If Carlson

knows that Anderson is going to produce q

A

units,then Carlson’s pro ts

will be p(q

A

+q

C

)?2q

C

=(6?:01q

A

:01q

C

)q

C

2q

C

=6q

C

:01q

A

q

C

:01q

2

C

2q

C

,Carlson’s pro ts will be maximized if he chooses q

C

to satisfy

the equation 6?:01q

A

:02q

C

= 2,Therefore Carlson’s reaction function

is R

C

(q

A

)=(4?:01q

A

)=:02 = 200?:5q

A

.

322 OLIGOPOLY (Ch,27)

Let us denote the Cournot equilibrium quantities by q

A

and q

C

.The

Cournot equilibrium conditions are that q

A

= R

A

( q

C

)and q

C

= R

C

( q

A

).

Solving these two equations in two unknowns we nd that q

A

= 200 and

q

C

= 100,Now we can also solve for the Cournot equilibrium price and for

the pro ts of each baker,The Cournot equilibrium price is 6?:01(200 +

100) = $3,Then in Cournot equilibrium,Anderson makes a pro t of $2

on each of 200 loaves and Carlson makes $1 on each of 100 loaves.

In Stackelberg competition,the follower’s pro t-maximizing output

choice depends on the amount of output that he expects the leader to

produce,His reaction function,R

F

(q

L

),is constructed in the same way

as for a Cournot competitor,The leader knows the reaction function of

the follower and gets to choose her own output,q

L

,rst,So the leader

knows that the industry price depends on the sum of her own output and

the follower’s output,that is,on q

L

+ R

F

(q

L

),Since the industry price

can be expressed as a function of q

L

only,so can the leader’s marginal

revenue,So once you get the follower’s reaction function and substitute it

into the inverse demand function,you can write down an expression that

depends on just q

L

and that says marginal revenue equals marginal cost

for the leader,You can solve this expression for the leader’s Stackelberg

output and plug in to the follower’s reaction function to get the follower’s

Stackelberg output.

Example,Suppose that one of the bakers of Heifer’s Breath plays the role

of Stackelberg leader,Perhaps this is because Carlson always gets up an

hour earlier than Anderson and has his bread in the oven before Anderson

gets started,If Anderson always nds out how much bread Carlson has

in his oven and if Carlson knows that Anderson knows this,then Carlson

can act like a Stackelberg leader,Carlson knows that Anderson’s reaction

function is R

A

(q

C

) = 250?:5q

c

,Therefore Carlson knows that if he bakes

q

C

loaves of bread,then the total amount of bread that will be baked in

Heifer’s Breath will be q

C

+R

A

(q

C

)=q

C

+250?:5q

C

= 250+:5q

C

.Since

Carlson’s production decision determines total production and hence the

price of bread,we can write Carlson’s pro t simply as a function of his

own output,Carlson will choose the quantity that maximizes this pro t.

If Carlson bakes q

C

loaves,the price will be p =6?:01(250 +,5q

C

)=

3:5?:005q

C

,Then Carlson’s pro ts will be pq

C

2q

C

=(3:5?:005q

C

)q

C

2q

C

=1:5q

C

:005q

2

C

,His pro ts are maximized when q

C

= 150,(Find

this either by setting marginal revenue equal to marginal cost or directly

by setting the derivative of pro ts to zero and solving for q

C

.) If Carlson

produces 150 loaves,then Anderson will produce 250?:5 150 = 175

loaves,The price of bread will be 6?:01(175 + 150) = 2:75,Carlson will

now make $.75 per loaf on each of 150 loaves and Anderson will make

$1.75 on each of 175 loaves.

27.1 (0) Carl and Simon are two rival pumpkin growers who sell their

pumpkins at the Farmers’ Market in Lake Witchisit,Minnesota,They are

the only sellers of pumpkins at the market,where the demand function

for pumpkins is q =3;200?1;600p,The total number of pumpkins sold

at the market is q = q

C

+ q

S

,whereq

C

is the number that Carl sells

NAME 323

and q

S

is the number that Simon sells,The cost of producing pumpkins

for either farmer is $.50 per pumpkin no matter how many pumpkins he

produces.

(a) The inverse demand function for pumpkins at the Farmers’ Market is

p = a?b(q

C

+ q

S

),where a = 2 and b = 1=1;600,The

marginal cost of producing a pumpkin for either farmer is $.50.

(b) Every spring,each of the farmers decides how many pumpkins to

grow,They both know the local demand function and they each know

how many pumpkins were sold by the other farmer last year,In fact,

each farmer assumes that the other farmer will sell the same number this

year as he sold last year,So,for example,if Simon sold 400 pumpkins

last year,Carl believes that Simon will sell 400 pumpkins again this year.

If Simon sold 400 pumpkins last year,what does Carl think the price of

pumpkins will be if Carl sells 1,200 pumpkins this year? 1,If

Simon sold q

t?1

S

pumpkins in year t?1,then in the spring of year t,Carl

thinks that if he,Carl,sells q

t

C

pumpkins this year,the price of pumpkins

this year will be 2?(q

t?1

S

+q

t

C

)=1;600.

(c) If Simon sold 400 pumpkins last year,Carl believes that if he sells

q

t

C

pumpkins this year then the inverse demand function that he faces is

p =2?400=1;600?q

t

C

=1;600 = 1:75?q

t

C

=1;600,Therefore if Simon

sold 400 pumpkins last year,Carl’s marginal revenue this year will be

1:75?q

t

C

=800,More generally,if Simon sold q

t?1

S

pumpkins last year,

then Carl believes that if he,himself,sells q

t

C

pumpkins this year,his

marginal revenue this year will be 2?q

t?1

S

=1;600?q

t

C

=800.

(d) Carl believes that Simon will never change the amount of pumpkins

that he produces from the amount q

t?1

S

that he sold last year,Therefore

Carl plants enough pumpkins this year so that he can sell the amount

that maximizes his pro ts this year,To maximize this pro t,he chooses

the output this year that sets his marginal revenue this year equal to

his marginal cost,This means that to nd Carl’s output this year when

Simon’s output last year was q

t?1

S

,Carl solves the following equation.

2?q

t?1

S

=1;600?q

t

C

=800 =,5.

(e) Carl’s Cournot reaction function,R

t

C

(q

t?1

S

),is a function that tells us

what Carl’s pro t-maximizing output this year would be as a function of

Simon’s output last year,Use the equation you wrote in the last answer to

nd Carl’s reaction function,R

t

C

(q

t?1

S

)= 1;200?q

t?1

S

=2,(Hint:

This is a linear expression of the form a?bq

t?1

S

,You have to nd the

constants a and b.)

324 OLIGOPOLY (Ch,27)

(f) Suppose that Simon makes his decisions in the same way that Carl

does,Notice that the problem is completely symmetric in the roles played

by Carl and Simon,Therefore without even calculating it,we can guess

that Simon’s reaction function is R

t

S

(q

t?1

C

)= 1;200?q

t?1

C

=2,(Of

course,if you don’t like to guess,you could work this out by following

similar steps to the ones you used to nd Carl’s reaction function.)

(g) Suppose that in year 1,Carl produced 200 pumpkins and Simon pro-

duced 1,000 pumpkins,In year 2,how many would Carl produce?

700,How many would Simon produce? 1,100,In year 3,how

many would Carl produce? 650,How many would Simon produce?

850,Use a calculator or pen and paper to work out several more

terms in this series,To what level of output does Carl’s output appear

to be converging? 800 How about Simon’s? 800.

(h) Write down two simultaneous equations that could be solved to nd

outputs q

S

and q

C

such that,if Carl is producing q

C

and Simon is produc-

ing q

S

,then they will both want to produce the same amount in the next

period,(Hint,Use the reaction functions.) q

s

=1;200?q

C

=2

and q

C

=1;200?q

S

=2.

(i) Solve the two equations you wrote down in the last part for an equi-

librium output for each farmer,Each farmer,in Cournot equilibrium,

produces 800 units of output,The total amount of pumpkins brought

to the Farmers’ Market in Lake Witchisit is 1,600,The price of

pumpkins in that market is $1,How much pro t does each farmer

make? $400.

27.2 (0) Suppose that the pumpkin market in Lake Witchisit is as

we described it in the last problem except for one detail,Every spring,

the snow thaws o of Carl’s pumpkin eld a week before it thaws o of

Simon’s,Therefore Carl can plant his pumpkins one week earlier than

Simon can,Now Simon lives just down the road from Carl,and he can

tell by looking at Carl’s elds how many pumpkins Carl planted and how

many Carl will harvest in the fall,(Suppose also that Carl will sell every

pumpkin that he produces.) Therefore instead of assuming that Carl will

sell the same amount of pumpkins that he did last year,Simon sees how

many Carl is actually going to sell this year,Simon has this information

before he makes his own decision about how many to plant.

NAME 325

(a) If Carl plants enough pumpkins to yield q

t

C

this year,then Simon

knows that the pro t-maximizing amount to produce this year is q

t

S

=

Hint,Remember the reaction functions you found in the last problem.

1;200?q

t

C

=2.

(b) When Carl plants his pumpkins,he understands how Simon will make

his decision,Therefore Carl knows that the amount that Simon will

produce this year will be determined by the amount that Carl produces.

In particular,if Carl’s output is q

t

C

,then Simon will produce and sell

1;200?q

t

C

=2 and the total output of the two producers will be

1;200 +q

t

C

=2,Therefore Carl knows that if his own output is q

C

,

the price of pumpkins in the market will be 1:25?q

t

C

=3;200.

(c) In the last part of the problem,you found how the price of pumpkins

this year in the Farmers’ Market is related to the number of pumpkins

that Carl produces this year,Now write an expression for Carl’s total

revenue in year t as a function of his own output,q

t

C

,1:25q

t

C

(q

t

C

)

2

=3;200,Write an expression for Carl’s marginal revenue in

year t as a function of q

t

C

,1:25?q

t

C

=1;600.

(d) Find the pro t-maximizing output for Carl,1,200,Find the

pro t-maximizing output for Simon,600,Find the equilibrium price

of pumpkins in the Lake Witchisit Farmers’ Market,$7/8,How

much pro t does Carl make? $450,How much pro t does Simon

make? $225,An equilibrium of the type we discuss here is known

as a Stackleberg equilibrium.

(e) If he wanted to,it would be possible for Carl to delay his plant-

ing until the same time that Simon planted so that neither of them

would know the other’s plans for this year when he planted,Would

it be in Carl’s interest to do this? Explain,(Hint,What are Carl’s

pro ts in the equilibrium above? How do they compare with his prof-

its in Cournot equilibrium?) No,Carl’s profits in

Stackleberg equilibrium are larger than

in Cournot equilibrium,So if the output

326 OLIGOPOLY (Ch,27)

when neither knows the other’s output this

year until after planting time is a Cournot

equilibrium,Carl will want Simon to know

his output.

27.3 (0) Suppose that Carl and Simon sign a marketing agreement.

They decide to determine their total output jointly and to each produce

the same number of pumpkins,To maximize their joint pro ts,how many

pumpkins should they produce in toto? 1,200,How much does each

one of them produce? 600,How much pro t does each one of them

make? 450.

27.4 (0) The inverse market demand curve for bean sprouts is given by

P(Y) = 100?2Y,and the total cost function for any rm in the industry

is given by TC(y)=4y.

(a) The marginal cost for any rm in the industry is equal to $4,The

change in price for a one-unit increase in output is equal to $?2.

(b) If the bean-sprout industry were perfectly competitive,the industry

output would be 48,and the industry price would be $4.

(c) Suppose that two Cournot rms operated in the market,The reaction

function for Firm 1 would be y

1

=24?y

2

=2,(Reminder,Unlike

the example in your textbook,the marginal cost is not zero here.) The

reaction function of Firm 2 would be y

2

=24?y

1

=2,If the rms

were operating at the Cournot equilibrium point,industry output would

be 32,each rm would produce 16,and the market price

would be $36.

(d) For the Cournot case,draw the two reaction curves and indicate the

equilibrium point on the graph below.

NAME 327

0 6 12 18 24

6

12

18

y1

y2

24

e

Firm 1's reaction

function

Firm 2's

reaction

function

(e) If the two rms decided to collude,industry output would be 24

and the market price would equal $52.

(f) Suppose both of the colluding rms are producing equal amounts of

output,If one of the colluding rms assumes that the other rm would

not react to a change in industry output,what would happen to a rm’s

own pro ts if it increased its output by one unit? Profits would

increase by $22.

(g) Suppose one rm acts as a Stackleberg leader and the other rm

behaves as a follower,The maximization problem for the leader can be

written as max

y

1

[100?2(y

1

+24?y

1

=2)]y

1

4y

1

.

Solving this problem results in the leader producing an output of

24 and the follower producing 12,This implies an industry

output of 36 and price of $28.

27.5 (0) Grinch is the sole owner of a mineral water spring that costlessly

burbles forth as much mineral water as Grinch cares to bottle,It costs

Grinch $2 per gallon to bottle this water,The inverse demand curve for

Grinch’s mineral water is p = $20?:20q,wherep is the price per gallon

and q is the number of gallons sold.

328 OLIGOPOLY (Ch,27)

(a) Write down an expression for pro ts as a function of q,(q)=

(20?:20q)q?2q,Find the pro t-maximizing choice of q for

Grinch,45.

(b) What price does Grinch get per gallon of mineral water if he produces

the pro t-maximizing quantity? $11,How much pro t does he make?

$405.

(c) Suppose,now,that Grinch’s neighbor,Grubb nds a mineral spring

that produces mineral water that is just as good as Grinch’s water,but

that it costs Grubb $6 a bottle to get his water out of the ground and

bottle it,Total market demand for mineral water remains as before.

Suppose that Grinch and Grubb each believe that the other’s quantity

decision is independent of his own,What is the Cournot equilibrium out-

put for Grubb? 50=3,What is the price in the Cournot equilibrium?

$9.33.

27.6 (1) Albatross Airlines has a monopoly on air travel between Peoria

and Dubuque,If Albatross makes one trip in each direction per day,the

demand schedule for round trips is q = 160?2p,whereq is the number of

passengers per day,(Assume that nobody makes one-way trips.) There

is an \overhead" xed cost of $2,000 per day that is necessary to fly the

airplane regardless of the number of passengers,In addition,there is a

marginal cost of $10 per passenger,Thus,total daily costs are $2;000+10q

if the plane flies at all.

(a) On the graph below,sketch and label the marginal revenue curve,and

the average and marginal cost curves.

020406080

20

40

60

Q

MR,MC

80

mc

mr

ac

NAME 329

(b) Calculate the pro t-maximizing price and quantity and total daily

pro ts for Albatross Airlines,p = 45,q = 70,=

$450 per day.

(c) If the interest rate is 10% per year,how much would someone be will-

ing to pay to own Albatross Airlines’s monopoly on the Dubuque-Peoria

route,(Assuming that demand and cost conditions remain unchanged

forever.) About $1.6 million.

(d) If another rm with the same costs as Albatross Airlines were to enter

the Dubuque-Peoria market and if the industry then became a Cournot

duopoly,would the new entrant make a pro t? No; losses

would be about $900 per day.

(e) Suppose that the throbbing night life in Peoria and Dubuque becomes

widely known and in consequence the population of both places doubles.

As a result,the demand for airplane trips between the two places dou-

bles to become q = 320?4p,Suppose that the original airplane had a

capacity of 80 passengers,If AA must stick with this single plane and if

no other airline enters the market,what price should it charge to maxi-

mize its output and how much pro t would it make? p = $60,=

$2,000.

(f) Let us assume that the overhead costs per plane are constant regardless

of the number of planes,If AA added a second plane with the same costs

and capacity as the rst plane,what price would it charge? $45.

How many tickets would it sell? 140,How much would its pro ts

be? $900,If AA could prevent entry by another competitor,would

it choose to add a second plane? No.

(g) Suppose that AA stuck with one plane and another rm entered the

market with a plane of its own,If the second rm has the same cost

function as the rst and if the two rms act as Cournot oligopolists,what

will be the price,$40,quantities,80,and pro ts? $400.

27.7 (0) Alex and Anna are the only sellers of kangaroos in Sydney,

Australia,Anna chooses her pro t-maximizing number of kangaroos to

sell,q

1

,based on the number of kangaroos that she expects Alex to sell.

Alex knows how Anna will react and chooses the number of kangaroos that

330 OLIGOPOLY (Ch,27)

she herself will sell,q

2

,after taking this information into account,The

inverse demand function for kangaroos is P(q

1

+q

2

)=2;000?2(q

1

+q

2

).

It costs $400 to raise a kangaroo to sell.

(a) Alex and Anna are Stackelberg competitors,Alex is the leader

and Anna is the follower.

(b) If Anna expects Alex to sell q

2

kangaroos,what will her own marginal

revenue be if she herself sells q

1

kangaroos? MR(q

1

+ q

2

)=

2;000?4q

1

2q

2

.

(c) What is Anna’s reaction function,R(q

2

)? R(q

2

) = 400?

1=2q

2

.

(d) Now if Alex sells q

2

kangaroos,what is the total number of kangaroos

that will be sold? 400 + 1=2q

2

,What will be the market price as

a function of q

2

only? P(q

2

)=1;200?q

2

.

(e) What is Alex’s marginal revenue as a function of q

2

only?

MR(q

2

)=1;200? 2q

2

,How many kangaroos will Alex

sell? 400,How many kangaroos will Anna sell? 200,What will

the industry price be? $800.

27.8 (0) Consider an industry with the following structure,There are

50 rms that behave in a competitive manner and have identical cost

functions given by c(y)=y

2

=2,There is one monopolist that has 0

marginal costs,The demand curve for the product is given by

D(p)=1;000?50p:

(a) What is the supply curve of one of the competitive rms? y = p.

The total supply from the competitive sector at price p is S(p)= 50p.

(b) If the monopolist sets a price p,the amount that it can sell is D

m

(p)=

1;000?100p.

NAME 331

(c) The monopolist’s pro t-maximizing output is y

m

= 500,What

is the monopolist’s pro t-maximizing price? p =5.

(d) How much output will the competitive sector provide at this price?

50 5 = 250,What will be the total amount of output sold in

this industry? y

m

+y

c

= 750.

27.9 (0) Consider a market with one large rm and many small rms.

The supply curve of the small rms taken together is

S(p) = 100 +p:

The demand curve for the product is

D(p) = 200?p:

The cost function for the one large rm is

c(y)=25y:

(a) Suppose that the large rm is forced to operate at a zero level of

output,What will be the equilibrium price? 50,What will be the

equilibrium quantity? 150.

(b) Suppose now that the large rm attempts to exploit its market power

and set a pro t-maximizing price,In order to model this we assume that

customers always go rst to the competitive rms and buy as much as

they are able to and then go to the large rm,In this situation,the

equilibrium price will be $37.50,The quantity supplied by the

large rm will be 25,and the equilibrium quantity supplied by the

competitive rms will be 137.5.

(c) What will be the large rm’s pro ts? $312.50.

(d) Finally suppose that the large rm could force the competitive rms

out of the business and behave as a real monopolist,What will be the

equilibrium price? 225=2,What will be the equilibrium quantity?

175=2,What will be the large rm’s pro ts? (175=2)

2

.

332 OLIGOPOLY (Ch,27)

Calculus 27.10 (2) In a remote area of the American Midwest before the railroads

arrived,cast iron cookstoves were much desired,but people lived far apart,

roads were poor,and heavy stoves were expensive to transport,Stoves

could be shipped by river boat to the town of Bouncing Springs,Missouri.

Ben Kinmore was the only stove dealer in Bouncing Springs,He could

buy as many stoves as he wished for $20 each,delivered to his store.

The only farmers who traded in Bouncing Springs lived along a road that

ran east and west through town,Along that road,there was one farm

every mile and the cost of hauling a stove was $1 per mile,There were

no other stove dealers on the road in either direction,The owners of

every farm along the road had a reservation price of $120 for a cast iron

cookstove,That is,any of them would be willing to pay up to $120 to

have a stove rather than to not have one,Nobody had use for more than

one stove,Ben Kinmore charged a base price of $p for stoves and added

to the price the cost of delivery,For example,if the base price of stoves

was $40 and you lived 45 miles west of Bouncing Springs,you would have

to pay $85 to get a stove,$40 base price plus a hauling charge of $45.

Since the reservation price of every farmer was $120,it follows that if the

base price were $40,any farmer who lived within 80 miles of Bouncing

Springs would be willing to pay $40 plus the price of delivery to have a

cookstove,Therefore at a base price of $40,Ben could sell 80 cookstoves

to the farmers living west of him,Similarly,if his base price is $40,he

could sell 80 cookstoves to the farmers living within 80 miles to his east,

for a total of 160 cookstoves.

(a) If Ben set a base price of $p for cookstoves where p<120,and if he

charged $1 a mile for delivering them,what would be the total number of

cookstoves he could sell? 2(120?p),(Remember to count the ones

he could sell to his east as well as to his west.) Assume that Ben has no

other costs than buying the stoves and delivering them,Then Ben would

make a pro t of p?20 per stove,Write Ben’s total pro t as a function

of the base price,$p,that he charges,2(120?p)(p?20) =

2(140p?p

2

2;400).

(b) Ben’s pro t-maximizing base price is $70,(Hint,You just wrote

pro ts as a function of prices,Now di erentiate this expression for pro ts

with respect to p.) Ben’s most distant customer would be located at a

distance of 50 miles from him,Ben would sell 100 cookstoves

and make a total pro t of $5,000.

(c) Suppose that instead of setting a single base price and making all

buyers pay for the cost of transportation,Ben o ers free delivery of cook-

stoves,He sets a price $p and promises to deliver for free to any farmer

who lives within p?20 miles of him,(He won’t deliver to anyone who lives

NAME 333

further than that,because it then costs him more than $p to buy a stove

and deliver it.) If he is going to price in this way,how high should he set

p? $120,How many cookstoves would Ben deliver? 200,How

much would his total revenue be? $24,000 How much would his

total costs be,including the cost of deliveries and the cost of buying the

stoves? $14,000,(Hint,What is the average distance that he has

to haul a cookstove?) How much pro t would he make? $10,000.

Can you explain why it is more pro table for Ben to use this pricing

scheme where he pays the cost of delivery himself rather than the scheme

where the farmers pay for their own deliveries? If Ben pays

for delivery,he can price-discriminate

between nearby farmers and faraway ones.

He charges a higher price,net of transport

cost,to nearby farmers and a lower net

price to faraway farmers,who are willing

to pay less net of transport cost.

Calculus 27.11 (2) Perhaps you wondered what Ben Kinmore,who lives o in

the woods quietly collecting his monopoly pro ts,is doing in this chapter

on oligopoly,Well,unfortunately for Ben,before he got around to selling

any stoves,the railroad built a track to the town of Deep Furrow,just 40

miles down the road,west of Bouncing Springs,The storekeeper in Deep

Furrow,Huey Sunshine,was also able to get cookstoves delivered by train

to his store for $20 each,Huey and Ben were the only stove dealers on

the road,Let us concentrate our attention on how they would compete

for the customers who lived between them,We can do this,because Ben

can charge di erent base prices for the cookstoves he ships east and the

cookstoves he ships west,So can Huey.

Suppose that Ben sets a base price,p

B

,for stoves he sends west

and adds a charge of $1 per mile for delivery,Suppose that Huey sets

a base price,p

H

,for stoves he sends east and adds a charge of $1 per

mile for delivery,Farmers who live between Ben and Huey would buy

from the seller who is willing to deliver most cheaply to them (so long as

the delivered price does not exceed $120),If Ben’s base price is p

B

and

Huey’s base price is p

H

,somebody who lives x miles west of Ben would

have to pay a total of p

B

+ x to have a stove delivered from Ben and

p

H

+(40?x) to have a stove delivered by Huey.

(a) If Ben’s base price is p

B

and Huey’s is p

H

,write down an equation that

could be solved for the distance x

to the west of Bouncing Springs that

334 OLIGOPOLY (Ch,27)

Ben’s market extends,p

B

+x

= p

H

+(40?x

),If Ben’s base

price is p

B

and Huey’s is p

H

,then Ben will sell 20 + (p

H

p

B

)=2

cookstoves and Huey will sell 20 + (p

B

p

H

)=2 cookstoves.

(b) Recalling that Ben makes a pro t of p

B

20 on every cookstove that

he sells,Ben’s pro ts can be expressed as the following function of p

B

and p

H

,(20 + (p

H

p

B

)=2)(p

B

20).

(c) If Ben thinks that Huey’s price will stay at p

H

,no matter what price

Ben chooses,what choice of p

B

will maximize Ben’s pro ts? p

B

=

30 + p

H

=2,(Hint,Set the derivative of Ben’s pro ts with respect

to his price equal to zero.) Suppose that Huey thinks that Ben’s price

will stay at p

B

,no matter what price Huey chooses,what choice of p

H

will maximize Huey’s pro ts? p

H

=30+p

B

=2,(Hint,Use the

symmetry of the problem and the answer to the last question.)

(d) Can you nd a base price for Ben and a base price for Huey such that

each is a pro t-maximizing choice given what the other guy is doing?

(Hint,Find prices p

B

and p

H

that simultaneously solve the last two

equations.) p

B

= p

H

=60,How many cookstoves does Ben sell

to farmers living west of him? 20,How much pro t does he make on

these sales? $800.

(e) Suppose that Ben and Huey decided to compete for the customers

who live between them by price discriminating,Suppose that Ben o ers

to deliver a stove to a farmer who lives x miles west of him for a price

equal to the maximum of Ben’s total cost of delivering a stove to that

farmer and Huey’s total cost of delivering to the same farmer less 1 penny.

Suppose that Huey o ers to deliver a stove to a farmer who lives x miles

west of Ben for a price equal to the maximum of Huey’s own total cost of

delivering to this farmer and Ben’s total cost of delivering to him less a

penny,For example,if a farmer lives 10 miles west of Ben,Ben’s total cost

of delivering to him is $30,$20 to get the stove and $10 for hauling it 10

miles west,Huey’s total cost of delivering it to him is $50,$20 to get the

stove and $30 to haul it 30 miles east,Ben will charge the maximum of

his own cost,which is $30,and Huey’s cost less a penny,which is $49.99.

The maximum of these two numbers is $49.99,Huey will charge the

maximum of his own total cost of delivering to this farmer,which is $50,

and Ben’s cost less a penny,which is $29.99,Therefore Huey will charge

$50.00 to deliver to this farmer,This farmer will buy from Ben

NAME 335

whose price to him is cheaper by one penny,When the two merchants

have this pricing policy,all farmers who live within 20 miles of

Ben will buy from Ben and all farmers who live within 20 miles

of Huey will buy from Huey,A farmer who lives x miles west of Ben

and buys from Ben must pay 59:99?x dollars to have a cookstove

delivered to him,A farmer who lives x miles east of Huey and buys from

Huey must pay 59:99?x for delivery of a stove,On the graph

below,use blue ink to graph the cost to Ben of delivering to a farmer who

lives x miles west of him,Use red ink to graph the total cost to Huey

of delivering a cookstove to a farmer who lives x miles west of Ben,Use

pencil to mark the lowest price available to a farmer as a function of how

far west he lives from Ben.

010203040

20

40

60

Miles west of Ben

Dollars

80

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Blue line

Red line

Pencil line

Ben's profit

Huey's profit

(f) With the pricing policies you just graphed,which farmers get stoves

delivered most cheaply,those who live closest to the merchants or those

who live midway between them? Those who live midway

between them,On the graph you made,shade in the area rep-

resenting each merchant’s pro ts,How much pro ts does each merchant

make? $400,If Ben and Huey are pricing in this way,is there any

way for either of them to increase his pro ts by changing the price he

charges to some farmers? No.

336 OLIGOPOLY (Ch,27)

Chapter 28 NAME

Game Theory

Introduction,In this introduction we o er three examples of two-person

games,The rst game has a dominant strategy equilibrium,The second

has a Nash equilibrium in pure strategies that is not a dominant strategy

equilibrium,The third has no pure strategy Nash equilibrium,but it does

have a mixed strategy equilibrium.

Example,Albert and Victoria are roommates,Each of them prefers a

clean room to a dirty room,but neither likes to clean the room,If both

clean the room,they each get a payo of 5,If one cleans and the other

doesn’t clean the room,the person who does the cleaning has a utility of

0,and the person who doesn’t clean the room has a utility of 8,If neither

cleans the room,the room stays a mess and each has a utility of 1,The

payo s from the strategies \Clean" and \Don’t Clean" are shown in the

box below.

Clean Room{Dirty Room

Albert

Victoria

Clean Don’t Clean

Clean 5;5 0;8

Don’t Clean 8;0 1;1

In this game,notice that if Victoria chooses to clean,then Albert

will be better o not cleaning than he would be if he chose to clean.

Likewise if Victoria chooses not to clean,Albert is better o not clean-

ing than cleaning,Therefore \Don’t Clean" is a dominant strategy for

Albert,Similar reasoning shows that no matter what Albert chooses to

do,Victoria is better o if she chooses \Don’t Clean." Therefore the out-

come where both roommates choose \Don’t Clean" is a dominant strategy

equilibrium,It is interesting to notice that this is true,even though both

persons would be better o if they both chose the strategy \Clean."

Example,This game is set in the South Paci c in 1943,Admiral Imamura

must transport Japanese troops from the port of Rabaul in New Britain,

across the Bismarck Sea to New Guinea,The Japanese fleet could either

travel north of New Britain,where it is likely to be foggy,or south of

New Britain,where the weather is likely to be clear,U.S,Admiral Ken-

ney hopes to bomb the troop ships,Kenney has to choose whether to

338 GAME THEORY (Ch,28)

concentrate his reconnaissance aircraft on the Northern or the Southern

route,Once he nds the convoy,he can bomb it until its arrival in New

Guinea,Kenney’s sta has estimated the number of days of bombing

time for each of the outcomes,The payo s to Kenney and Imamura from

each outcome are shown in the box below,The game is modeled as a

\zero-sum game." For each outcome,Imamura’s payo is the negative of

Kenney’s payo,

TheBattleoftheBismarckSea

Kenney

Imamura

North South

North 2;?2 2;?2

South 1;?1 3;?3

This game does not have a dominant strategy equilibrium,since there

is no dominant strategy for Kenney,His best choice depends on what Ima-

mura does,The only Nash equilibrium for this game is where Imamura

chooses the northern route and Kenney concentrates his search on the

northern route,To check this,notice that if Imamura goes North,then

Kenney gets an expected two days of bombing if he (Kenney) chooses

North and only one day if he (Kenney) chooses South,Furthermore,if

Kenney concentrates on the north,Imamura is indi erent between go-

ing north or south,since he can be expected to be bombed for two days

either way,Therefore if both choose \North," then neither has an incen-

tive to act di erently,You can verify that for any other combination of

choices,one admiral or the other would want to change,As things actually

worked out,Imamura chose the Northern route and Kenney concentrated

his search on the North,After about a day’s search the Americans found

the Japanese fleet and inflicted heavy damage on it.

Some two-player games do not have a \Nash equilibrium in pure

strategies." But every two-player game of the kind we look at has a

Nash equilibrium in mixed strategies,If a player is indi erent between

two strategies,then he is also willing to choose randomly between them.

Sometimes this is just what is needed to give an equilibrium.

Example,A soccer player has been awarded a free kick,The only player

allowed to defend against his kick is the opposing team’s goalie,The

kicker has two possible strategies,He can try to kick the ball into the

right side of the goal or he can try to kick the ball into the left side of the

This example is discussed in Luce and Rai a’s Games and Decisions,

John Wiley,1957 or Dover,1989,We recommend this book to anyone

interested in reading more about game theory.

NAME 339

goal,There is not time for the goalie to determine where the ball is going

before he must commit himself by jumping either to the left or to the

right side of the net,Let us suppose that if the goalie guesses correctly

where the kicker is going to kick,then the goalie always stops the ball.

The kicker has a very accurate shot to the right side of the net,but is not

so good at shooting left,If he kicks to the right side of the net and the

goalie jumps left,the kicker will always score,But the kicker kicks to the

left side of the net and the goalie jumps to the right,then the kicker will

score only half of the time,This story leads us to the following payo

matrix,where if the kicker makes the goal,the kicker gets a payo of 1

and the goalie a payo of 0 and if the kicker does not make the goal,the

goalie gets a payo of 1 and the kicker a payo of 0.

The Free Kick

Goalie

Kicker

Kick Left Kick Right

Jump Left 1;0 0;1

Jump Right,5;:5 1;0

This game has no Nash equilibrium in pure strategies,There is no

combination of actions taken with certainty such that each is making the

best response to the other’s action,The goalie always wants to be where

the kicker is kicking and the kicker always wants to kick where the goalie

isn’t,What we can nd is a pair of equilibrium mixed strategies.

In this mixed strategy equilibrium each player’s strategy is chosen at

random,The kicker will be willing to choose a random strategy only if

the expected payo is the same from kicking to either side,The payo s

from kicking to the right and to the left depend on what the goalie is

doing,Let

G

be the probability that the goalie will jump left and 1?

G

be the probability that he will jump right,The kicker realizes that if he

kicks to the right,he will score when the goalie goes left and he will not

score when the goalie goes right,The expected payo to the kicker if he

kicks to the right is therefore just

G

,If the kicker kicks to the left,then

the only way that he can score is if the goalie jumps right,This happens

with probability 1?

G

,Even then he will only score half the time,So

the expected payo to the kicker from kicking left is,5(1?

G

),These

two expected payo s are equalized when

G

=,5(1?

G

),If we solve

this equation,we nd

G

=1=3,This has to be the probability that the

goalie goes left in a mixed strategy equilibrium.

Now let us nd the probability that the kicker kicks left in a mixed

strategy equilibrium,In equilibrium,the kicker’s probability

K

of kick-

ing left must be such that the goalie gets the same expected payo from

jumping left as from jumping right,The expected payo to the goalie is

340 GAME THEORY (Ch,28)

the probability that the kicker does not score,If the goalie jumps left,

then the kicker will not score if he kicks left and will score if he kicks

right,so the expected payo to the goalie from going left is

K

.Ifthe

goalie jumps right,then with probability (1?

K

),the kicker will kick

right and the goalie will stop the ball,When the kicker is kicking to the

undefended left side of the net,he only makes it half the time,so if the

goalie jumps right,the probability that the kicker kicks left and makes

the kick is only,5

K

,Therefore the expected payo to the goalie from

jumping right is (1?

K

)+:5

K

=1?:5

K

,Equalizing the payo to the

goalie from jumping left or jumping right requires

K

=1?:5

K

.Solving

this equation we nd that in the equilibrium mixed strategy,

K

=2=3.

28.1 (0) Perhaps you have wondered what it could mean that \the meek

shall inherit the earth." While we don’t claim this is always the case,here

is an example where it is true,In a famous experiment,two psychologists

put two pigs|a little one and a big one|into a pen that had a lever at

one end and a trough at the other end,When the lever was pressed,a

serving of pigfeed would appear in a trough at the other end of the pen.

If the little pig would press the lever,then the big pig would eat all of

the pigfeed and keep the little pig from getting any,If the big pig pressed

the lever,there would be time for the little pig to get some of the pigfeed

before the big pig was able to run to the trough and push him away.

Let us represent this situation by a game,in which each pig has two

possible strategies,One strategy is Press the Lever,The other strategy

is Wait at the Trough,If both pigs wait at the trough,neither gets any

feed,If both pigs press the lever,the big pig gets all of the feed and the

little pig gets a poke in the ribs,If the little pig presses the lever and

the big pig waits at the trough,the big pig gets all of the feed and the

little pig has to watch in frustration,If the big pig presses the lever and

the little pig waits at the trough,then the little pig is able to eat 2=3

of the feed before the big pig is able to push him away,The payo s are

as follows,(These numbers are just made up,but their relative sizes are

consistent with the payo s in the Baldwin-Meese experiment.)

Big Pig{Little Pig

Little Pig

Big Pig

Press Wait

Press?1;9?1;10

Wait 6;4 0;0

Baldwin and Meese (1979),\Social Behavior in Pigs Studied by

Means of Operant Conditioning," Animal Behavior

NAME 341

(a) Is there a dominant strategy for the little pig? Yes,Wait,Is

there a dominant strategy for the big pig? No.

(b) Find a Nash equilibrium for this game,Does the game have more than

one Nash equilibrium? The only Nash equilibrium is

where little pig waits and big pig presses.

(Incidentally,while Baldwin and Meese did not interpret this experiment

as a game,the result they observed was the result that would be predicted

by Nash equilibrium.)

(c) Which pig gets more feed in Nash equilibrium? Little pig.

28.2 (0) Consider the following game matrix.

A Game Matrix

Player A

Player B

Left Right

Top a;b c;d

Bottom e;f g;h

(a) If (top,left) is a dominant strategy equilibrium,then we know that

a> e,b> d,c >g,and f >h.

(b) If (top,left) is a Nash equilibrium,then which of the above inequalities

must be satis ed? a>e; b>d.

(c) If (top,left) is a dominant strategy equilibrium must it be a Nash

equilibrium? Why? Yes,A dominant strategy

equilibrium is always a Nash equilibrium.

28.3 (1) This problem is based on an example developed by the biologist

John Maynard Smith to illustrate the uses of game theory in the theory

of evolution,Males of a certain species frequently come into conflict with

other males over the opportunity to mate with females,If a male runs

into a situation of conflict,he has two alternative \strategies." A male

342 GAME THEORY (Ch,28)

can play \Hawk" in which case he will ght the other male until he either

wins or is badly hurt,Or he can play \Dove," in which case he makes

a display of bravery but retreats if his opponent starts to ght,If an

animal plays Hawk and meets another male who is playing Hawk,they

both are seriously injured in battle,If he is playing Hawk and meets an

animal who is playing Dove,the Hawk gets to mate with the female and

the Dove slinks o to celibate contemplation,If an animal is playing Dove

and meets another Dove,they both strut around for a while,Eventually

the female either chooses one of them or gets bored and wanders o,The

expected payo s to each of two males in a single encounter depend on

which strategy each adopts,These payo s are depicted in the box below.

The Hawk-Dove Game

Animal A

Animal B

Hawk Dove

Hawk?5;?5 10;0

Dove 0;10 4;4

(a) Now while wandering through the forest,a male will encounter many

conflict situations of this type,Suppose that he cannot tell in advance

whether another animal that he meets will behave like a Hawk or like

a Dove,The payo to adopting either strategy oneself depends on the

proportion of the other guys that is Hawks and the proportion that is

Doves,For example,suppose all of the other males in the forest act

like Doves,Any male that acted like a Hawk would nd that his rival

always retreated and would therefore enjoy a payo of 10 on every

encounter,If a male acted like a Dove when all other males acted like

Doves,he would receive an average payo of 4.

(b) If strategies that are more pro table tend to be chosen over strategies

that are less pro table,explain why there cannot be an equilibrium in

which all males act like Doves,If you know that you

are meeting a Dove,it pays to be a Hawk.

(c) If all the other males acted like Hawks,then a male who adopted the

Hawk strategy would be sure to encounter another Hawk and would get

a payo of?5,If instead,this male adopted the Dove strategy,he

would again be sure to encounter a Hawk,but his payo would be 0.

NAME 343

(d) Explain why there could not be an equilibrium where all of the an-

imals acted like Hawks,If everyone plays Hawk,it

would be profitable to play Dove.

(e) Since there is not an equilibrium in which everybody chooses the same

strategy,we might ask whether there might be an equilibrium in which

some fraction of the males chose the Hawk strategy and the rest chose

the Dove strategy,Suppose that the fraction of a large male population

that chooses the Hawk strategy is p,Then if one acts like a Hawk,the

fraction of one’s encounters in which he meets another Hawk is about p

and the fraction of one’s encounters in which he meets a Dove is about

1?p,Therefore the average payo to being a Hawk when the fraction of

Hawks in the population is p,mustbep (?5) + (1?p) 10 = 10?15p.

Similarly,if one acts like a Dove,the probability of meeting a Hawk is

about p and the probability of meeting another Dove is about (1?p).

Therefore the average payo to being a Dove when the proportion of

Hawks in the population is p will be p 0+(1?p) 4.

(f) Write an equation that states that when the proportion of the popu-

lation that acts like Hawks is p,the payo to Hawks is the same as the

payo s to Doves,4?4p =10?15p.

(g) Solve this equation for the value of p such that at this value Hawks

do exactly as well as Doves,This requires that p = 6=11.

(h) On the axes below,use blue ink to graph the average payo to the

strategy Dove when the proportion of the male population who are Hawks

is p,Use red ink to graph the average payo to the strategy,Hawk,when

the proportion of the male population who are Hawks is p,Label the

equilibrium proportion in your diagram by E.

344 GAME THEORY (Ch,28)

0255075100

2

4

6

Percentage of hawks

Payoff

8

Blue

Line

Red Line

e

(i) If the proportion of Hawks is slightly greater than E,whichstrat-

egy does better? Dove,If the proportion of Hawks is slightly less

than E,which strategy does better? Hawk,If the more pro table

strategy tends to be adopted more frequently in future plays,then if the

strategy proportions are out of equilibrium,will changes tend to move the

proportions back toward equilibrium or further away from equilibrium?

Closer.

28.4 (1) Evangeline and Gabriel met at a freshman mixer,They want

desperately to meet each other again,but they forgot to exchange names

or phone numbers when they met the rst time,There are two possible

strategies available for each of them,These are Go to the Big Party or

Stay Home and Study,They will surely meet if they both go to the party,

and they will surely not otherwise,The payo to meeting is 1,000 for

each of them,The payo to not meeting is zero for both of them,The

payo s are described by the matrix below.

Close Encounters of the Second Kind

Evangeline

Gabriel

Go to Party Stay Home

Go to Party 1000;1000 0;0

Stay Home 0;0 0;0

NAME 345

(a) A strategy is said to be a weakly dominant strategy for a player if

the payo from using this strategy is at least as high as the payo from

using any other strategy,Is there any outcome in this game where both

players are using weakly dominant strategies? The only one is

(top,left).

(b) Find all of the pure-strategy Nash equilibria for this game,There

are two,(top,left) and (bottom,right).

(c) Do any of the pure Nash equilibria that you found seem more rea-

sonable than others? Why or why not? Although (bottom,

right) is a Nash equilibrium,it seems a

silly one,If either player believes that

there is any chance that the other will go

to the party,he or she will also go.

(d) Let us change the game a little bit,Evangeline and Gabriel are still

desperate to nd each other,But now there are two parties that they

might go to,There is a little party at which they would be sure to meet

if they both went there and a huge party at which they might never see

each other,The expected payo to each of them is 1,000 if they both go

to the little party,Since there is only a 50-50 chance that they would nd

each other at the huge party,the expected payo to each of them is only

500,If they go to di erent parties,the payo to both of them is zero.

The payo matrix for this game is:

More Close Encounters

Evangeline

Gabriel

Little Party Big Party

Little Party 1000;1000 0;0

Big Party 0;0 500;500

346 GAME THEORY (Ch,28)

(e) Does this game have a dominant strategy equilibrium? No,What

are the two Nash equilibria in pure strategies? (1) Both go

to the little party,(2) Both go to the big

party.

(f) One of the Nash equilibria is Pareto superior to the other,Suppose

that each person thought that there was some slight chance that the

other would go to the little party,Would that be enough to convince

them both to attend the little party? No,Can you think of any rea-

son why the Pareto superior equilibrium might emerge if both players

understand the game matrix,if both know that the other understands

it,and each knows that the other knows that he or she understands the

game matrix? If both know the game matrix and

each knows that the other knows it,then

each may predict the other will choose the

little party.

28.5 (1) This is a famous game,known to game theorists as \The Battle

of the Sexes." The story goes like this,Two people,let us call them

Michelle and Roger,although they greatly enjoy each other’s company,

have very di erent tastes in entertainment,Roger’s tastes run to ladies’

mud wrestling,while Michelle prefers Italian opera,They are planning

their entertainment activities for next Saturday night,For each of them,

there are two possible actions,go to the wrestling match or go to the

opera,Roger would be happiest if both of them went to see mud wrestling.

His second choice would be for both of them to go to the opera,Michelle

would prefer if both went to the opera,Her second choice would be that

they both went to see the mud wrestling,They both think that the worst

outcome would be that they didn’t agree on where to go,If this happened,

they would both stay home and sulk.

BattleoftheSexes

Roger

Michelle

Wrestling Opera

Wrestling 2;1 0;0

Opera 0;0 1;2

NAME 347

(a) Is the sum of the payo s to Michelle and Roger constant over all

outcomes? No,(If so,this is called a \zero-sum game." Otherwise it is

called a \nonzero sum game.") Does this game have a dominant strategy

equilibrium? No.

(b) Find two Nash equilibria in pure strategies for this game,Both

go to opera,Both go to mud wrestling.

(c) Find a Nash equilibrium in mixed strategies,Michele

chooses opera with probability 2=3 and

wrestling with probability 1=3,Roger chooses

opera with probability 1=3 and mud wrestling

with probability 2=3.

28.6 (1) This is another famous two-person game,known to game the-

orists as \Chicken." Two teenagers in souped-up cars drive toward each

other at great speed,The rst one to swerve out of the road is \chicken."

The best thing that can happen to you is that the other guy swerves and

you don’t,Then you are the hero and the other guy is the chicken,If you

both swerve,you are both chickens,If neither swerves,you both end up

in the hospital,A payo matrix for a chicken-type game is the following.

Chicken

Joe Bob

Leroy

Swerve Don’t Swerve

Swerve 1;1 1;2

Don’t Swerve 2;1 0;0

(a) Does this game have a dominant strategy? No,What are the two

Nash equilibria in pure strategies? The two outcomes where

one teenager swervesand the does not.

348 GAME THEORY (Ch,28)

(b) Find a Nash equilibrium in mixed strategies for this game,Play

each strategy with probability 1=2.

28.7 (0) I propose the following game,I flip a coin,and while it is in the

air,you call either heads or tails,If you call the coin correctly,you get

to keep the coin,Suppose that you know that the coin always comes up

heads,What is the best strategy for you to pursue? Always call

heads.

(a) Suppose that the coin is unbalanced and comes up heads 80% of

the time and tails 20% of the time,Now what is your best strategy?

Always call heads.

(b) What if the coin comes up heads 50% of the time and tails 50% of the

time? What is your best strategy? It doesn’t matter.

You can call heads always,tails always,or

randomize your calls.

(c) Now,suppose that I am able to choose the type of coin that I will toss

(where a coin’s type is the probability that it comes up heads),and that

you will know my choice,What type of coin should I choose to minimize

my losses? A fair coin.

(d) What is the Nash mixed strategy equilibrium for this game? (It may

help to recognize that a lot of symmetry exists in the game.) I

choose a fair coin,and you randomize with

50% heads and 50% tails.

28.8 (0) Ned and Ruth love to play \Hide and Seek." It is a simple

game,but it continues to amuse,It goes like this,Ruth hides upstairs or

downstairs,Ned can look upstairs or downstairs but not in both places.

If he nds Ruth,Ned gets one scoop of ice cream and Ruth gets none,If

he does not nd Ruth,Ruth gets one scoop of ice cream and Ned gets

none,Fill in the payo s in the matrix below.

NAME 349

Hide and Seek

Ned

Ruth

Upstairs Downstairs

Upstairs 1;0 0;1

Downstairs 0;1 1;0

(a) Is this a zero-sum game? Yes,What are the Nash equilibria in

pure strategies? There are none.

(b) Find a Nash equilibrium in mixed strategies for this game.

Ruth hides upstairs and Ned searches

upstairs with probability 1/2; Ruth hides

downstairs and Ned searches downstairs with

probability 1/2.

(c) After years of playing this game,Ned and Ruth think of a way to

liven it up a little,Now if Ned nds Ruth upstairs,he gets two scoops of

ice cream,but if he nds her downstairs,he gets one scoop,If Ned nds

Ruth,she gets no ice cream,but if he doesn’t nd her she gets one scoop.

Fill in the payo s in the graph below.

Advanced Hide and Seek

Ned

Ruth

Upstairs Downstairs

Upstairs 2;0 0;1

Downstairs 0;1 1;0

350 GAME THEORY (Ch,28)

(d) Are there any Nash equilibria in pure strategies? No,What mixed

strategy equilibrium can you nd? Ruth hides downstairs

2/3 of the time,Ned looks downstairs 1/2

the time,If both use equilibrium strategies,what fraction of the

time will Ned nd Ruth? 1/2.

28.9 (1) Let’s have another look at the soccer example that was discussed

in the introduction to this section,But this time,we will generalize the

payo matrix just a little bit,Suppose the payo matrix is as follows.

The Free Kick

Goalie

Kicker

Kick Left Kick Right

Jump Left 1;0 0;1

Jump Right 1-p,p 1;0

Now the probability that the kicker will score if he kicks to the left

and the goalie jumps to the right is p,Wewillwanttoseehowthe

equilibrium probabilities change as p changes.

(a) If the goalie jumps left with probability

G

,then if the kicker kicks

right,his probability of scoring is

G

.

(b) If the goalie jumps left with probability

G

,then if the kicker kicks

left,his probability of scoring is p(1?

G

).

(c) Find the probability

G

that makes kicking left and kicking right lead

to the same probability of scoring for the kicker,(Your answer will be a

function of p.)

G

=

p

1+p

.

(d) If the kicker kicks left with probability

K

,then if the goalie jumps

left,the probability that the kicker will not score is

K

.

NAME 351

(e) If the kicker kicks left with probability

K

,then if the goalie jumps

right,the probability that the kicker will not score is (1?p)

K

+

(1?

K

).

(f) Find the probability

K

that makes the payo to the goalie equal from

jumping left or jumping right.

1

1+p

.

(g) The variable p tells us how good the kicker is at kicking the ball

into the left side of the goal when it is undefended,As p increases,does

the equilibrium probability that the kicker kicks to the left increase or

decrease? Decreases,Explain why this happens in a way that

even a TV sports announcer might understand,The better

the kicker’s weak side gets,the less often

the goalie defends the kicker’s good side.

So kicker can kick to good side more often.

28.10 (0) Maynard’s Cross is a trendy bistro that specializes in carpac-

cio and other uncooked substances,Most people who come to Maynard’s

come to see and be seen by other people of the kind who come to May-

nard’s,There is,however,a hard core of 10 customers per evening who

come for the carpaccio and don’t care how many other people come,The

number of additional customers who appear at Maynard’s depends on

how many people they expect to see,In particular,if people expect that

the number of customers at Maynard’s in an evening will be X,then

the number of people who actually come to Maynard’s is Y =10+:8X:

In equilibrium,it must be true that the number of people who actually

attend the restaurant is equal to the number who are expected to attend.

(a) What two simultaneous equations must you solve to nd the equilib-

rium attendance at Maynard’s? y =10+:8x and x = y.

(b) What is the equilibrium nightly attendance? 50.

(c) On the following axes,draw the lines that represent each of the

two equations you mentioned in Part (a),Label the equilibrium atten-

352 GAME THEORY (Ch,28)

dance level.

020406080

20

40

60

x

y

80

X=Y

Y=10+.8X

e

Y=11+.8X

(d) Suppose that one additional carpaccio enthusiast moves to the area.

Like the other 10,he eats at Maynard’s every night no matter how many

others eat there,Write down the new equations determining attendance

at Maynard’s and solve for the new equilibrium number of customers.

y =11+:8x and y = x,sox = y =55.

(e) Use a di erent color ink to draw a new line representing the equa-

tion that changed,How many additional customers did the new steady

customer attract (besides himself)? 4.

(f) Suppose that everyone bases expectations about tonight’s attendance

on last night’s attendance and that last night’s attendance is public knowl-

edge,Then X

t

= Y

t?1

,whereX

t

is expected attendance on day t and

Y

t?1

is actual attendance on day t?1,At any time t,Y

t

=10+:8X

t

.

Suppose that on the rst night that Maynard’s is open,attendance is 20.

What will be attendance on the second night? 26.

(g) What will be the attendance on the third night? 30.8.

(h) Attendance will tend toward some limiting value,What is it? 50.

28.11 (0) Yogi’s Bar and Grill is frequented by unsociable types who

hate crowds,If Yogi’s regular customers expect that the crowd at Yogi’s

will beX,then the number of people who show up at Yogi’s,Y,will be the

larger of the two numbers,120?2X and 0,Thus Y =maxf120?2X;0g:

NAME 353

(a) Solve for the equilibrium attendance at Yogi’s,Draw a diagram de-

picting this equilibrium on the axes below.

020406080

20

40

60

x

y

80

e

X=Y

Y=120-2X

(b) Suppose that people expect the number of customers on any given

night to be the same as the number on the previous night,Suppose that

50 customers show up at Yogi’s on the rst day of business,How many

will show up on the second day? 20,The third day? 80,The

fourth day? 0,The fth day? 120,The ninety-ninth day?

120,The hundredth day? 0.

(c) What would you say is wrong with this model if at least some of Yogi’s

customers have memory spans of more than a day or two?

They’d notice that last night’s attendance

is not a good predictor of tonight’s,If

attendance is low on odd-numbered days and

high on even-numbered days,it would be

smart to adjust by coming on odd-numbered

days.

28.12 (2) Economic ideas and equilibrium analysis have many fascinat-

ing applications in biology,Popular discussions of natural selection and

biological tness often take it for granted that animal traits are selected

for the bene t of the species,Modern thinking in biology emphasizes that

individuals (or strictly speaking,genes) are the unit of selection,A mu-

tant gene that induces an animal to behave in such a way as to help the

354 GAME THEORY (Ch,28)

species at the expense of the individuals that carry that gene will soon

be eliminated,no matter how bene cial that behavior is to the species.

A good illustration is a paper in the Journal of Theoretical Biology,

1979,by H,J,Brockmann,A,Grafen,and R,Dawkins,called \Evo-

lutionarily Stable Nesting Strategy in a Digger Wasp." They maintain

that natural selection results in behavioral strategies that maximize an

individual animal’s expected rate of reproduction over the course of its

lifetime,According to the authors,\Time is the currency which an animal

spends."

Females of the digger wasp Sphex ichneumoneus nest in underground

burrows,Some of these wasps dig their own burrows,After she has dug

her burrow,a wasp goes out to the elds and hunts katydids,These

she stores in her burrow to be used as food for her o spring when they

hatch,When she has accumulated several katydids,she lays a single egg

in the burrow,closes o the food chamber,and starts the process over

again,But digging burrows and catching katydids is time-consuming,An

alternative strategy for a female wasp is to sneak into somebody else’s

burrow while she is out hunting katydids,This happens frequently in

digger wasp colonies,A wasp will enter a burrow that has been dug by

another wasp and partially stocked with katydids,The invader will start

catching katydids,herself,to add to the stock,When the founder and

the invader nally meet,they ght,The loser of the ght goes away and

never comes back,The winner gets to lay her egg in the nest.

Since some wasps dig their own burrows and some invade burrows

begun by others,it is likely that we are observing a biological equilibrium

in which each strategy is as e ective a way for a wasp to use its time for

producing o spring as the other,If one strategy were more e ective than

the other,then we would expect that a gene that led wasps to behave

in the more e ective way would prosper at the expense of genes that led

them to behave in a less e ective way.

Suppose the average nesting episode takes 5 days for a wasp that

digs its own burrow and tries to stock it with katydids,Suppose that the

average nesting episode takes only 4 days for invaders,Suppose that when

they meet,half the time the founder of the nest wins the ght and half

the time the invader wins,Let D be the number of wasps that dig their

own burrows and let I be the number of wasps that invade the burrows

of others,The fraction of the digging wasps that are invaded will be

about

5

4

I

D

,(Assume for the time being that

5

4

I

D

< 1.) Half of the diggers

who are invaded will win their ght and get to keep their burrows,The

fraction of digging wasps who lose their burrows to other wasps is then

1

2

5

4

I

D

=

5

8

I

D

,Assume also that all the wasps who are not invaded by other

wasps will successfully stock their burrows and lay their eggs.

(a) Then the fraction of the digging wasps who do not lose their burrows

is just 1?

5

8

I

D

.

Therefore over a period of 40 days,a wasp who dug her own bur-

row every time would have 8 nesting episodes,Her expected number of

successes would be 8?5

I

D

.

NAME 355

(b) In 40 days,a wasp who chose to invade every time she had a chance

would have time for 10 invasions,Assuming that she is successful half the

time on average,her expected number of successes would be 5.

Write an equation that expresses the condition that wasps who always dig

their own burrows do exactly as well as wasps who always invade burrows

dug by others,8?5

I

D

=5.

(c) The equation you have just written should contain the expression

I

D

.

Solve for the numerical value of

I

D

that just equates the expected number

of successes for diggers and invaders,The answer is

3

5

.

(d) But there is a problem here,the equilibrium we found doesn’t appear

to be stable,On the axes below,use blue ink to graph the expected num-

ber of successes in a 40-day period for wasps that dig their own burrows

every time where the number of successes is a function of

I

D

.Useblack

ink to graph the expected number of successes in a 40-day period for in-

vaders,Notice that this number is the same for all values of

I

D

,Label the

point where these two lines cross and notice that this is equilibrium,Just

to the right of the crossing,where

I

D

is just a little bit bigger than the

equilibrium value,which line is higher,the blue or the black? Black.

At this level of

I

D

,which is the more e ective strategy for any individ-

ual wasp? Invade,Suppose that if one strategy is more e ective

than the other,the proportion of wasps adopting the more e ective one

increases,If,after being in equilibrium,the population got joggled just

a little to the right of equilibrium,would the proportions of diggers and

invaders return toward equilibrium or move further away? Further

away.

Success

e

Blue line

8-5(I/D)

5

Black line

I_

D

356 GAME THEORY (Ch,28)

(e) The authors noticed this likely instability and cast around for possible

changes in the model that would lead to stability,They observed that

an invading wasp does help to stock the burrow with katydids,This may

save the founder some time,If founders win their battles often enough

and get enough help with katydids from invaders,it might be that the

expected number of eggs that a founder gets to lay is an increasing rather

than a decreasing function of the number of invaders,On the axes below,

show an equilibrium in which digging one’s own burrow is an increasingly

e ective strategy as

I

D

increases and in which the payo to invading is

constant over all ratios of

I

D

,Is this equilibrium stable? Yes.

Success

I_

D

e

Chapter 29 NAME

Exchange

Introduction,The Edgeworth box is a thing of beauty,An amazing

amount of information is displayed with a few lines,points and curves,In

fact one can use an Edgeworth box to tell just about everything there is to

say about the case of two traders dealing in two commodities,Economists

know that the real world has more than two people and more than two

commodities,But it turns out that the insights gained from this model

extend nicely to the case of many traders and many commodities,So

for the purpose of introducing the subject of exchange equilibrium,the

Edgeworth box is exactly the right tool,We will start you out with an

example of two gardeners engaged in trade.

Example,Alice and Byron consume two goods,camelias and dahlias.

Alice has 16 camelias and 4 dahlias,Byron has 8 camelias and 8 dahlias.

They consume no other goods,and they trade only with each other,To

describe the possible allocations of flowers,we rst draw a box whose

width is the total number of camelias and whose height is the total number

of dahlias that Alice and Byron have between them,The width of the

box is therefore 16 + 8 = 24 and the height of the box is 4 + 8 = 12.

Dahlias Byron

12

6

0 6 121824

Alice Camelias

Any feasible allocation of flowers between Alice and Byron is fully

described by a single point in the box,Consider,for example,the alloca-

tion where Alice gets the bundle (15;9) and Byron gets the bundle (9;3).

This allocation is represented by the point A =(15;9) in the Edgeworth

box,The distance 15 from A to the left side of the box is the number of

camelias for Alice and the distance 9 from A to the bottom of the box is

the number of dahlias for Alice,This point also determines Byron’s con-

sumption of camelias and dahlias,The distance 9 from A to the right side

of the box is the total number of camelias consumed by Byron,and the

distance from A to the top of the box is the number of dahlias consumed

by Byron,Since the width of the box is the total supply of camelias and

the height of the box is the total supply of dahlias,these conventions en-

sure that any point in the box represents a feasible allocation of the total

358 EXCHANGE (Ch,29)

supply of camelias and dahlias.

It is useful to mark the initial allocation on the Edgeworth box,In

this case,the initial allocation is represented by the point E =(16;4).

Now suppose that Alice’s utility function is U(c;d)=c+2d and Byron’s

utility funtion is U(c;d)=cd,Alice’s indi erence curves will be straight

lines with slope?1=2,The indi erence curve that passes through her

initial endowment,for example,will be a line that runs from the point

(24;0) to the point (0;12),Since Byron has Cobb-Douglas utility,his

indi erence curves will be rectangular hyperbolas,but since quantities

for Byron are measured from the upper right corner of the box,these

indi erence curves will be flipped over as in the diagram.

The Pareto set or contract curve is the set of points where Alice’s

indi erence curves are tangent to Byron’s,There will be tangency if the

slopes are the same,The slope of Alice’s indi erence curve at any point is

1=2,The slope of Byron’s indi erence curve depends on his consumption

of the two goods,When Byron is consuming the bundle (c

B;d

B

),the slope

of his indi erence curve is equal to his marginal rate of substitution,which

is?d

B

=c

B

,Therefore Alice’s and Byron’s indi erence curves will nuzzle

up in a nice tangency whenever?d

B

=c

B

=?1=2,So the Pareto set in

this example is just the diagonal of the Edgeworth box.

Some problems ask you to nd a competitive equilibrium,For an

economy with two goods,the following procedure is often a good way to

calculate equilibrium prices and quantities.

Since demand for either good depends only on the ratio of prices of

good 1 to good 2,it is convenient to set the price of good 1 equal to

1andletp

2

be the price of good 2.

With the price of good 1 held at 1,calculate each consumer’s demand

for good 2 as a function of p

2

.

Write an equation that sets the total amount of good 2 demanded by

all consumers equal to the total of all participants’ initial endowments

of good 2.

Solve this equation for the value of p

2

that makes the demand for

good 2 equal to the supply of good 2,(When the supply of good 2

equals the demand of good 2,it must also be true that the supply of

good 1 equals the demand for good 1.)

Plug this price into the demand functions to determine quantities.

Example,Frank’s utility function is U(x

1;x

2

)=x

1

x

2

and Maggie’s is

U(x

1;x

2

)=minfx

1;x

2

g,Frank’s initial endowment is 0 units of good 1

and 10 units of good 2,Maggie’s initial endowment is 20 units of good 1

and 5 units of good 2,Let us nd a competitive equilibrium for Maggie

and Frank.

Set p

1

= 1 and nd Frank’s and Maggie’s demand functions for good

2 as a function of p

2

,Using the techniques learned in Chapter 6,we

nd that Frank’s demand function for good 2 is m=2p

2

,wherem is his

income,Since Frank’s initial endowment is 0 units of good 1 and 10 units

of good 2,his income is 10p

2

,Therefore Frank’s demand for good 2 is

10p

2

=2p

2

= 5,Since goods 1 and 2 are perfect complements for Maggie,

she will choose to consume where x

1

= x

2

,This fact,together with her

budget constraint implies that Maggie’s demand function for good 2 is

NAME 359

m=(1 + p

2

),Since her endowment is 20 units of good 1 and 5 units of

good 2,her income is 20 + 5p

2

,Therefore at price p

2

,Maggie’s demand

is (20 + 5p

2

)=(1 +p

2

),Frank’s demand plus Maggie’s demand for good 2

adds up to 5 + (20 + 5p

2

)=(1 +p

2

),The total supply of good 2 is Frank’s

10 unit endowment plus Maggie’s 5 unit endowment,which adds to 15

units,Therefore demand equals supply when

5+

(20 + 5p

2

)

(1 +p

2

)

=15:

Solving this equation,one nds that the equilibrium price is p

2

=2,At

the equilibrium price,Frank will demand 5 units of good 2 and Maggie

will demand 10 units of good 2.

29.1 (0) Morris Zapp and Philip Swallow consume wine and books.

Morris has an initial endowment of 60 books and 10 bottles of wine,Philip

has an initial endowment of 20 books and 30 bottles of wine,They have

no other assets and make no trades with anyone other than each other.

For Morris,a book and a bottle of wine are perfect substitutes,His utility

function is U(b;w)=b+w,whereb is the number of books he consumes

and w is the number of bottles of wine he consumes,Philip’s preferences

are more subtle and convex,He has a Cobb-Douglas utility function,

U(b;w)=bw,In the Edgeworth box below,Morris’s consumption is

measured from the lower left,and Philip’s is measured from the upper

right corner of the box.

020406080

20

40

Books

PhilipWine

Morris

e

Blue curve

Red curve

Black

line

(a) On this diagram,mark the initial endowment and label it E.Usered

ink to draw Morris Zapp’s indi erence curve that passes through his initial

endowment,Use blue ink to draw in Philip Swallow’s indi erence curve

that passes through his initial endowment,(Remember that quantities

for Philip are measured from the upper right corner,so his indi erence

curves are \Phlipped over.")

360 EXCHANGE (Ch,29)

(b) At any Pareto optimum,where both people consume some of each

good,it must be that their marginal rates of substitution are equal,No

matter what he consumes,Morris’s marginal rate of substitution is equal

to -1,When Philip consumes the bundle,(b

P;w

P

),his MRS is

w

P

=b

P

,Therefore every Pareto optimal allocation where both

consume positive amounts of both goods satis es the equation w

P

=

b

P

,Use black ink on the diagram above to draw the locus of Pareto

optimal allocations.

(c) At a competitive equilibrium,it will have to be that Morris consumes

some books and some wine,But in order for him to do so,it must be that

the ratio of the price of wine to the price of books is 1,Therefore

we know that if we make books the numeraire,then the price of wine in

competitive equilibrium must be 1.

(d) At the equilibrium prices you found in the last part of the question,

what is the value of Philip Swallow’s initial endowment? 50,At these

prices,Philip will choose to consume 25 books and 25

bottles of wine,If Morris Zapp consumes all of the books and all of the

wine that Philip doesn’t consume,he will consume 55 books and

15 bottles of wine.

(e) At the competitive equilibrium prices that you found above,Morris’s

income is 70,Therefore at these prices,the cost to Morris of con-

suming all of the books and all of the wine that Philip doesn’t consume

is (the same as,more than,less than) the same as his income.

At these prices,can Morris a ord a bundle that he likes better than the

bundle (55;15)? No.

(f) Suppose that an economy consisted of 1,000 people just like Morris

and 1,000 people just like Philip,Each of the Morris types had the same

endowment and the same tastes as Morris,Each of the Philip types had

the same endowment and tastes as Philip,Would the prices that you

found to be equilibrium prices for Morris and Philip still be competitive

equilibrium prices? Yes,If each of the Morris types and each of the

Philip types behaved in the same way as Morris and Philip did above,

would supply equal demand for both wine and books? Yes.

NAME 361

29.2 (0) Consider a small exchange economy with two consumers,Astrid

and Birger,and two commodities,herring and cheese,Astrid’s initial

endowment is 4 units of herring and 1 unit of cheese,Birger’s initial en-

dowment has no herring and 7 units of cheese,Astrid’s utility function is

U(H

A;C

A

)=H

A

C

A

,Birger is a more inflexible person,His utility func-

tion is U(H

B;C

B

)=minfH

B;C

B

g.(HereH

A

and C

A

are the amounts

of herring and cheese for Astrid,and H

B

and C

B

are amounts of herring

and cheese for Birger.)

(a) Draw an Edgeworth box,showing the initial allocation and sketching

in a few indi erence curves,Measure Astrid’s consumption from the lower

left and Birger’s from the upper right,In your Edgeworth box,draw two

di erent indi erence curves for each person,using blue ink for Astrid’s

and red ink for Birger’s.

02468

2

4

Cheese

BirgerHerring

Astrid

e

Blue curves

Red curves

Black

line

(b) Use black ink to show the locus of Pareto optimal allocations,(Hint:

Since Birger is kinky,calculus won’t help much here,But notice that

because of the rigidity of the proportions in which he demands the two

goods,it would be ine cient to give Birger a positive amount of either

good if he had less than that amount of the other good,What does that

tell you about where the Pareto e cient locus has to be?) Pareto

efficient allocations lie on the line with

slope 1 extending from Birger’s corner of

the box.

29.3 (0) Dean Foster Z,Interface and Professor J,Fetid Nightsoil ex-

change bromides and platitudes,Dean Interface’s utility function is

U

I

(B

I;P

I

)=B

I

+2

p

P

I

:

Professor Nightsoil’s utility function is

U

N

(B

N;P

N

)=B

N

+4

p

P

N

:

362 EXCHANGE (Ch,29)

Dean Interface’s initial endowment is 8 bromides and 12 platitudes,Pro-

fessor Nightsoil’s initial endowment is 8 bromides and 4 platitudes.

0481216

4

8

12

Bromides

Platitudes

16

Nightsoil

Interface

e

Red curve

Pencil curve

Blue line

3.2

(a) If Dean Interface consumes P

I

platitudes and B

I

bromides,his mar-

ginal rate of substitution will be?P

1=2

I

,If Professor Nightsoil

consumes P

N

platitudes and B

N

bromides,his marginal rate of substitu-

tion will be?2P

1=2

N

.

(b) On the contract curve,Dean Interface’s marginal rate of substitution

equals Professor Nightsoil’s,Write an equation that states this condition.

p

P

I

=

p

P

N

=2,This equation is especially simple because each

person’s marginal rate of substitution depends only on his consumption

of platitudes and not on his consumption of bromides.

(c) From this equation we see that P

I

=P

N

= 1=4 at all points on the

contract curve,This gives us one equation in the two unknowns P

I

and

P

N

.

(d) But we also know that along the contract curve it must be that P

I

+

P

N

= 16,since the total consumption of platitudes must equal

the total endowment of platitudes.

(e) Solving these two equations in two unknowns,we nd that everywhere

on the contract curve,P

I

and P

N

are constant and equal to P

I

=3:2

and P

N

=12:8.

NAME 363

(f) In the Edgeworth box,label the initial endowment with the letter

E,Dean Interface has thick gray penciled indi erence curves,Profes-

sor Nightsoil has red indi erence curves,Draw a few of these in the

Edgeworth box you made,Use blue ink to show the locus of Pareto op-

timal points,The contract curve is a (vertical,horizontal,diagonal)

horizontal line in the Edgeworth box.

(g) Find the competitive equilibrium prices and quantities,You know

what the prices have to be at competitive equilibrium because you know

what the marginal rates of substitution have to be at every Pareto

optimum,P

I

=3:2,P

N

=12:8,platitude

price/bromide price =

1

p

3:2

.

29.4 (0) A little exchange economy contains just two consumers,named

Ken and Barbie,and two commodities,quiche and wine,Ken’s initial

endowment is 3 units of quiche and 2 units of wine,Barbie’s initial en-

dowment is 1 unit of quiche and 6 units of wine,Ken and Barbie have iden-

tical utility functions,We write Ken’s utility function as,U(Q

K;W

K

)=

Q

K

W

K

and Barbie’s utility function as U(Q

B;W

B

)=Q

B

W

B

,whereQ

K

and W

K

are the amounts of quiche and wine for Ken and Q

B

and W

B

are amounts of quiche and wine for Barbie.

(a) Draw an Edgeworth box below,to illustrate this situation,Put quiche

on the horizontal axis and wine on the vertical axis,Measure goods for

Ken from the lower left corner of the box and goods for Barbie from the

upper right corner of the box,(Be sure that you make the length of the

box equal to the total supply of quiche and the height equal to the total

supply of wine.) Locate the initial allocation in your box,and label it W.

On the sides of the box,label the quantities of quiche and wine for each

of the two consumers in the initial endowment.

364 EXCHANGE (Ch,29)

24

2

4

6

8

0

Ken Quiche

Wine Barbie

w

ce

Black line

Red

curve

Blue

curve

Pareto

efficient

points

(b) Use blue ink to draw an indi erence curve for Ken that shows alloca-

tions in which his utility is 6,Use red ink to draw an indi erence curve

for Barbie that shows allocations in which her utility is 6.

(c) At any Pareto optimal allocation where both consume some of each

good,Ken’s marginal rate of substitution between quiche and wine must

equal Barbie’s,Write an equation that states this condition in terms

of the consumptions of each good by each person,W

B

=Q

B

=

W

K

=Q

K

.

(d) On your graph,show the locus of points that are Pareto e cient.

(Hint,If two people must each consume two goods in the same proportions

as each other,and if together they must consume twice as much wine as

quiche,what must those proportions be?)

(e) In this example,at any Pareto e cient allocation,where both persons

consume both goods,the slope of Ken’s indi erence curve will be?2.

Therefore,since we know that competitive equilibrium must be Pareto

e cient,we know that at a competitive equilibrium,p

Q

=p

W

= 2.

(f) What must be Ken’s consumption bundle in competitive equilibrium?

2 quiche,4 wine,How about Barbie’s consumption bundle?

2 quiche,4 wine,(Hint,You found competitive equilib-

rium prices above,You know Ken’s initial endowment and you know the

NAME 365

equilibrium prices,In equilibrium Ken’s income will be the value of his

endowment at competitive prices,Knowing his income and the prices,

you can compute his demand in competitive equilibrium,Having solved

for Ken’s consumption and knowing that total consumption by Ken and

Barbie equals the sum of their endowments,it should be easy to nd

Barbie’s consumption.)

(g) On the Edgeworth box for Ken and Barbie,draw in the competitive

equilibrium allocation and draw Ken’s competitive budget line (with black

ink).

29.5 (0) Linus Straight’s utility function is U(a;b)=a +2b,wherea

is his consumption of apples and b is his consumption of bananas,Lucy

Kink’s utility function isU(a;b)=minfa;2bg,Lucy initially has 12 apples

and no bananas,Linus initially has 12 bananas and no apples,In the

Edgeworth box below,goods for Lucy are measured from the upper right

corner of the box and goods for Linus are measured from the lower left

corner,Label the initial endowment point on the graph with the letter

E,Draw two of Lucy’s indi erence curves in red ink and two of Linus’s

indi erence curves in blue ink,Use black ink to draw a line through all

of the Pareto optimal allocations.

612

6

12

0

Linus Apples

Bananas Lucy

e

Red curves

Blue

curves

Black line

(a) In this economy,in competitive equilibrium,the ratio of the price of

apples to the price of bananas must be 1/2.

(b) Let a

S

be Linus’s consumption of apples and let b

S

be his consumption

of bananas,At competititive equilibrium,Linus’s consumption will have

to satisfy the budget constraint,a

s

+ 2 b

S

= 24,This gives us

one equation in two unknowns,To nd a second equation,consider Lucy’s

366 EXCHANGE (Ch,29)

consumption,In competitive equilibrium,total consumption of apples

equals the total supply of apples and total consumption of bananas equals

the total supply of bananas,Therefore Lucy will consume 12?a

s

apples

and 12?b

s

bananas,At a competitive equilibrium,Lucy will be

consuming at one of her kink points,The kinks occur at bundles where

Lucy consumes 2 apples for every banana that she consumes.

Therefore we know that

12?a

s

12?b

s

= 2.

(c) You can solve the two equations that you found above to nd the

quantities of apples and bananas consumed in competitive equilibrium

by Linus and Lucy,Linus will consume 6 units of apples and

9 units of bananas,Lucy will consume 6 units of apples

and 3 units of bananas.

29.6 (0) Consider a pure exchange economy with two consumers and

two goods,At some given Pareto e cient allocation it is known that both

consumers are consuming both goods and that consumer A has a marginal

rate of substitution between the two goods of 2,What is consumer B’s

marginal rate of substitution between these two goods? 2.

29.7 (0) Charlotte loves apples and hates bananas,Her utility function

is U(a;b)=a?

1

4

b

2

,wherea is the number of apples she consumes and

b is the number of bananas she consumes,Wilbur likes both apples and

bananas,His utility function is U(a;b)=a+2

p

b,Charlotte has an initial

endowment of no apples and 8 bananas,Wilbur has an initial endowment

of 16 apples and 8 bananas.

(a) On the graph below,mark the initial endowment and label it E.Use

red ink to draw the indi erence curve for Charlotte that passes through

this point,Use blue ink to draw the indi erence curve for Wilbur that

passes through this point.

NAME 367

0481216

4

8

12

Bananas

Apples

16

Wilbur

Charlotte

e

Red

line

Blue line

Black line

(b) If Charlotte hates bananas and Wilbur likes them,how many bananas

can Charlotte be consuming at a Pareto optimal allocation? 0.

On the graph above,use black ink to mark the locus of Pareto optimal

allocations of apples and bananas between Charlotte and Wilbur.

(c) We know that a competitive equilibrium allocation must be Pareto

optimal and the total consumption of each good must equal the total

supply,so we know that at a competitive equilibrium,Wilbur must be

consuming 16 bananas,If Wilbur is consuming this number of

bananas,his marginal utility for bananas will be 1/4 and his marginal

utility of apples will be 1,If apples are the numeraire,then

the only price of bananas at which he will want to consume exactly 16

bananas is 1/4,In competitive equilibrium,for the Charlotte-Wilbur

economy,Wilbur will consume 16 bananas and 14 apples

and Charlotte will consume 0 bananas and 2 apples.

29.8 (0) Mutt and Je have 8 cups of milk and 8 cups of juice to

divide between themselves,Each has the same utility function given by

u(m;j)=maxfm;jg,wherem is the amount of milk and j is the amount

of juice that each has,That is,each of them cares only about the larger

of the two amounts of liquid that he has and is indi erent to the liquid

of which he has the smaller amount.

368 EXCHANGE (Ch,29)

(a) Sketch an Edgeworth box for Mutt and Je,Use blue ink to show a

couple of indi erence curves for each,Use red ink to show the locus of

Pareto optimal allocations,(Hint,Look for boundary solutions.)

02468

2

4

6

Milk

Juice

8

Jeff

Mutt

Red

point

Red

point

Blue

curves

(Jeff)

Blue

curves

(Mutt)

29.9 (1) Remember Tommy Twit from Chapter 3,Tommy is happiest

when he has 8 cookies and 4 glasses of milk per day and his indi erence

curves are concentric circles centered around (8,4),Tommy’s mother,

Mrs,Twit,has strong views on nutrition,She believes that too much

of anything is as bad as too little,She believes that the perfect diet for

Tommy would be 7 glasses of milk and 2 cookies per day,In her view,

a diet is healthier the smaller is the sum of the absolute values of the

di erences between the amounts of each food consumed and the ideal

amounts,For example,if Tommy eats 6 cookies and drinks 6 glasses of

milk,Mrs,Twit believes that he has 4 too many cookies and 1 too few

glasses of milk,so the sum of the absolute values of the di erences from

her ideal amounts is 5,On the axes below,use blue ink to draw the locus

of combinations that Mrs,Twit thinks are exactly as good for Tommy

as (6;6),Also,use red ink to draw the locus of combinations that she

thinks is just as good as (8;4),On the same graph,use red ink to draw an

indi erence \curve" representing the locus of combinations that Tommy

likes just as well as 7 cookies and 8 glasses of milk.

NAME 369

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Cookies

Milk

0

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Black

line

Red curve

Blue curve

Tommy's red

curve

(a) On the graph,shade in the area consisting of combinations of cookies

and milk that both Tommy and his mother agree are better than 7 cookies

and 8 glasses of milk,where \better" for Mrs,Twit means she thinks it

is healthier,and where \better" for Tommy means he likes it better.

(b) Use black ink to sketch the locus of \Pareto optimal" bundles of

cookies and milk for Tommy,In this situation,a bundle is Pareto optimal

if any bundle that Tommy prefers to this bundle is a bundle that Mrs.

Twit thinks is worse for him,The locus of Pareto optimal points that you

just drew should consist of two line segments,These run from the point

(8,4) to the point 5,7 and from that point to the point 2,7.

29.10 (2) This problem combines equilibrium analysis with some of the

things you learned in the chapter on intertemporal choice,It concerns the

economics of saving and the life cycle on an imaginary planet where life

is short and simple,In advanced courses in macroeconomics,you would

study more-complicated versions of this model that build in more earthly

realism,For the present,this simple model gives you a good idea of how

the analysis must go.

370 EXCHANGE (Ch,29)

On the planet Drongo there is just one commodity,cake,and two

time periods,There are two kinds of creatures,\old" and \young." Old

creatures have an income of I units of cake in period 1 and no income in

period 2,Young creatures have no income in period 1 and an income of I

units of cake in period 2,There are N

1

old creatures and N

2

young crea-

tures,The consumption bundles of interest to creatures are pairs (c

1;c

2

),

where c

1

is cake in period 1 and c

2

is cake in period 2,All creatures,old

and young,have identical utility functions,representing preferences over

cake in the two periods,This utility function is U(c

1;c

2

)=c

a

1

c

1?a

2

,where

a is a number such that 0 a 1.

(a) If current cake is taken to be the numeraire,(that is,its price is

set at 1),write an expression for the present value of a consumption

bundle (c

1;c

2

),c

1

+ c

2

=(1 + r),Write down the present value

of income for old creatures I and for young creatures

I

=(1 + r),The budget line for any creature is determined by the

condition that the present value of its consumption bundle equals the

present value of its income,Write down this budget equation for old

creatures,c

1

+ c

2

=(1 + r)=I and for young creatures:

c

1

+c

2

=(1 +r)=I

=(1 +r).

(b) If the interest rate is r,write down an expression for an old creature’s

demand for cake in period 1 c

1

= aI andinperiod2 c

2

=

(1?a)I(1+r),Write an expression for a young creature’s demand

for cake in period 1 c

1

= aI

=(1 + r) andinperiod2 c

2

=

(1?a)I

,(Hint,If its budget line is p

1

c

1

+p

2

c

2

= W and its utility

function is of the form proposed above,then a creature’s demand function

for good 1 is c

1

= aW=p and demand for good 2 is c

2

=(1?a)W=p.) If

the interest rate is zero,how much cake would a young creature choose in

period 1? aI

,For what value of a would it choose the same amount

in each period if the interest rate is zero? a =1=2,If a =,55,

what would r have to be in order that young creatures would want to

consume the same amount in each period?,22.

(c) The total supply of cake in period 1 equals the total cake earnings of

all old creatures,since young creatures earn no cake in this period,There

are N

1

old creatures and each earns I units of cake,so this total is N

1

I.

Similarly,the total supply of cake in period 2 equals the total amount

earned by young creatures,This amount is N

2

I

.

NAME 371

(d) At the equilibrium interest rate,the total demand of creatures for

period-1 cake must equal total supply of period-1 cake,and similarly the

demand for period-2 cake must equal supply,If the interest rate is r,then

the demand for period-1 cake by each old creature is aI and the

demand for period-1 cake by each young creature is aI

=(1 + r).

Since there areN

1

old creatures and N

2

young creatures,the total demand

for period-1 cake at interest rate r is N

1

aI +N

2

aI

=(1 +r).

(e) Using the results of the last section,write an equation that sets the

demand for period-1 cake equal to the supply,N

1

aI+N

2

aI

=(1+

r)=N

1

I,Write a general expression for the equilibrium value of r,

given N

1

,N

2

,I,andI

,r =

N

2

I

a

N

1

I(1?a)

1,Solve this equation

for the special case when N

1

= N

2

and I = I

and a =11=21.

r = 10%.

(f) In the special case at the end of the last section,show that the interest

rate that equalizes supply and demand for period-1 cake will also equalize

supply and demand for period-2 cake,(This illustrates Walras’s law.)

Supply = demand for period 2 if N

1

(1?

a)I(1 + r)+N

2

(1?a)I

= N

2

I

.IfN

1

= N

2

and I = I

,then (1?a)(1 +r)+(1?a)=1.

If a =11=21,then r = 10%.

372 EXCHANGE (Ch,29)

Chapter 30 NAME

Production

Introduction,In this section we explore economywide production pos-

sibility sets,We pay special attention to the principle of comparative

advantage,The principle is simply that e ciency suggests that people

should specialize according to their relative abilities in di erent activities

rather than absolute abilities.

Example,For simplicity,let us imagine an island with only two people

on it,both of them farmers,They do not trade with the outside world.

Farmer A has 100 acres and is able to grow two crops,wheat and hay.

Each acre of his land that he plants to wheat will give him 50 bushels

of wheat,Each acre of his land that he plants to hay will give him 2

tons of hay,Farmer B also has 100 acres,but his land is not so good.

Each acre of his land yields only 20 bushels of wheat and only 1 ton of

hay,Notice that,although Farmer A’s land is better for both wheat and

hay,Farmer B’s land has comparative advantage in the production of hay.

This is true because the ratio of tons of hay to bushels of wheat per acre

2=50 =,04 for Farmer A and 1=20 =,05 for Farmer B,Farmer A,on the

other hand,has comparative advantage in the production of wheat,since

the ratio of bushels of wheat to tons of hay is 50=2 = 25 for Farmer A

and 20=1 = 20 for Farmer B,The e cient way to arrange production is to

have Farmer A \specialize" in wheat and farmer B \specialize" in hay,If

Farmer A devotes all of his land to wheat and Farmer B devotes all of his

land to hay,then total wheat production will be 5,000 bushels and total

hay production will be 100 tons,Suppose that they decide to produce

only 4,000 bushels of wheat,Given that they are going to produce 4,000

bushels of wheat,the most hay they can possibly produce together will

be obtained if Farmer A devotes 80 acres to wheat and 20 acres to hay

while Farmer B devotes all of his land to hay,Suppose that they decide to

produce 6,000 bushels of wheat,Then they will get the most hay possible

given that they are producing 6,000 bushels of wheat if Farmer A puts

all of his land into wheat and Farmer B puts 50 acres into wheat and the

remaining 50 acres into hay.

30.1 (0) Tip and Spot nally got into college,Tip can write term papers

at the rate of 10 pages per hour and solve workbook problems at the rate

of 3 per hour,Spot can write term papers at the rate of 6 pages per

hour and solve workbook problems at the rate of 2 per hour,Which of

these two has comparative advantage in solving workbook problems?

Spot.

374 PRODUCTION (Ch,30)

0 2040608010

20

40

60

80

Problems

Pages

120

Spot

Tip

Joint

30

18

12

36 96

(a) Tip and Spot each work 6 hours a day,They decide to work together

and to produce a combination of term papers and workbook problems

that lies on their joint production possibility frontier,On the above graph

plot their joint production possibility frontier,If they produce less than

60 pages of term papers,then Tip will write all of the term papers.

If they produce more than 60 pages of term papers,then Tip

will continue to specialize in writing term papers and Spot will also

write some term papers.

30.2 (0) Robinson Crusoe has decided that he will spend exactly 8

hours a day gathering food,He can either spend this time gathering

coconuts or catching sh,He can catch 1 sh per hour and he can gather

2 coconuts per hour,On the graph below,show Robinson’s production

possibility frontier between sh and coconuts per day,Write an equation

for the line segment that is Robinson’s production possibility frontier.

F +C=2=8.

NAME 375

0481216

4

8

12

Fish

Coconuts

16

Utility of 4

Utility of 8

Production

possibility frontier

(a) Robinson’s utility function is U(F;C)=FC,whereF is his daily

sh consumption and C is his daily coconut consumption,On the graph

above,sketch the indi erence curve that gives Robinson a utility of 4,

and also sketch the indi erence curve that gives him a utility of 8,How

many sh will Robinson choose to catch per day? 4,How many

coconuts will he collect? 8,(Hint,Robinson will choose a bundle

that maximizes his utility subject to the constraint that the bundle lies

in his production possibility set,But for this technology,his production

possibility set looks just like a budget set.)

(b) Suppose Robinson is not isolated on an island in the Paci c,but is

retired and lives next to a grocery store where he can buy either sh or

coconuts,If sh cost $1 per sh,how much would coconuts have to cost in

order that he would choose to consume twice as many coconuts as sh?

$.50,Suppose that a social planner decided that he wanted Robinson

to consume 4 sh and 8 coconuts per day,He could do this by setting

the price of sh equal to $1,the price of coconuts equal to $.50 and

giving Robinson a daily income of $ 8,

(c) Back on his island,Robinson has little else to do,so he pretends that

he is running a competitive rm that produces sh and coconuts,He

wonders,\What would the price have to be to make me do just what I

am actually doing? Let’s assume that sh are the numeraire and have a

price of $1,And let’s pretend that I have access to a competitive labor

market where I can hire as much labor as I want at some given wage.

There is a constant returns to scale technology,An hour’s labor produces

376 PRODUCTION (Ch,30)

one sh or 2 coconuts,At wages above $ 1 per hour,I wouldn’t

produce any sh at all,because it would cost me more than $1 to produce

a sh,At wages below $ 1 per hour,I would want to produce

in nitely many sh since I would make a pro t on every one,So the

only possible wage rate that would make me choose to produce a positive

nite amount of sh is $ 1 per hour,Now what would the price

of coconuts have to be to induce me to produce a positive number of

coconuts,At the wage rate I just found,the cost of producing a coconut

is $.50,At this price and only at this price,would I be willing to

produce a nite positive number of coconuts."

30.3 (0) We continue the story of Robinson Crusoe from the previous

problem,One day,while walking along the beach,Robinson Crusoe saw

a canoe in the water,In the canoe was a native of a nearby island,The

native told Robinson that on his island there were 100 people and that

they all lived on sh and coconuts,The native said that on his island,it

takes 2 hours to catch a sh and 1 hour to nd a coconut,The native said

that there was a competitive economy on his island and that sh were

the numeraire,The price of coconuts on the neighboring island must

have been $.50,The native o ered to trade with Crusoe at these

prices,\I will trade you either sh for coconuts or coconuts for sh at

the exchange rate of 2 coconuts for a sh," said he,\But you

will have to give me 1 sh as payment for rowing over to your island."

Would Robinson gain by trading with him? No,If so,would he buy

sh and sell coconuts or vice versa? Neither,Since their

prices are the same as the rate at which

he can transform the two goods,he can gain

nothing by trading.

(a) Several days later,Robinson saw another canoe in the water on the

other side of his island,In this canoe was a native who came from a

di erent island,The native reported that on his island,one could catch

only 1 sh for every 4 hours of shing and that it takes 1 hour to nd a

coconut,This island also had a competitive economy,The native o ered

to trade with Robinson at the same exchange rate that prevailed on his

own island,but said that he would have to have 2 sh in return for rowing

between the islands,If Robinson decides to trade with this island,he

chooses to produce only fish and will get his coconuts from

the other island,On the graph above,use black ink to draw Robinson’s

production possibility frontier if he doesn’t trade and use blue ink to

NAME 377

show the bundles he can a ord if he chooses to trade and specializes

appropriately,Remember to take away 2 sh to pay the trader.

0481216

4

8

12

Fish

Coconuts

16

Utility of 4

Utility of 8

Production

possibility frontier

(b) Write an equation for Crusoe’s \budget line" if he specializes appro-

priately and trades with the second trader,If he does this,what bundle

will he choose to consume? 3 fish,12 coconuts,Does he

like this bundle better than the bundle he would have if he didn’t trade?

Yes.

30.4 (0) The Isle of Veritas has made it illegal to trade with the outside

world,Only two commodities are consumed on this island,milk and

wheat,On the north side of the island are 40 farms,Each of these

farms can produce any combination of non-negative amounts of milk and

wheat that satis es the equation m =60?6w,On the south side of the

island are 60 farms,Each of these farms can produce any combination

of non-negative amounts of milk and wheat that satis es the equation

m =40?2w,The economy is in competitive equilibrium and 1 unit of

wheat exchanges for 4 units of milk.

(a) On the diagram below,use black ink to draw the production possibility

set for a typical farmer from the north side of the island,Given the

equilibrium prices,will this farmer specialize in milk,specialize in wheat,

or produce both goods? Specialize in milk,Use blue ink

to draw the budget that he faces in his role as a consumer if he makes

the optimal choice of what to produce.

378 PRODUCTION (Ch,30)

020406080

20

40

60

Wheat

Milk

80

Black line

Blue line

Red line

Pencil line

15

10

(b) On the diagram below,use black ink to draw the production possibility

set for a typical farmer from the south side of the island,Given the

equilibrium prices,will this farmer specialize in milk,specialize in wheat,

or produce both goods? Specialize in wheat,Use blue

ink to draw the budget that he faces in his role as a consumer if he makes

the optimal choice of what to produce.

020406080

20

40

60

Wheat

Milk

80

Black line

Blue line

Red line

Pencil line

(c) Suppose that peaceful Viking traders discover Veritas and o er to

exchange either wheat for milk or milk for wheat at an exchange rate of

NAME 379

1 unit of wheat for 3 units of milk,If the Isle of Veritas allows free trade

with the Vikings,then this will be the new price ratio on the island,At

this price ratio,would either type of farmer change his output? No.

(d) On the rst of the two graphs above,use red ink to draw the budget

for northern farmers if free trade is allowed and the farmers make the

right choice of what to produce,On the second of the two graphs,use

red ink to draw the budget for southern farmers if free trade is allowed

and the farmers make the right choice of what to produce.

(e) The council of elders of Veritas will meet to vote on whether to accept

the Viking o er,The elders from the north end of the island get 40

votes and the elders from the south end get 60 votes,Assuming that

everyone votes in the sel sh interest of his end of the island,how will

the northerners vote? In favor,How will the southerners vote?

Against,How is it that you can make a de nite answer to the last

two questions without knowing anything about the farmers’s consumption

preferences? The change strictly enlarges the

budget set for northerners and strictly

shrinks it for southerners.

(f) Suppose that instead of o ering to make exchanges at the rate of 1 unit

of wheat for 3 units of milk,the Vikings had o ered to trade at the price

of 1 unit of wheat for 1 unit of milk and vice versa,Would either type

of farmer change his output? Yes,Southerners would

now switch to specializing in milk,Use pencil

to sketch the budget line for each kind of farmer at these prices if he

makes the right production decision,How will the northerners vote now?

In favor,How will the southerners vote now? Depends

on their preferences about consumption.

Explain why it is that your answer to one of the last two questions has

to be \it depends." The two alternative budget

lines for southerners are not nested.

30.5 (0) Recall our friends the Mungoans of Chapter 2,They have a

strange two-currency system consisting of Blue Money and Red Money.

Originally,there were two prices for everything,a blue-money price and

a red-money price,The blue-money prices are 1 bcu per unit of ambrosia

and 1 bcu per unit of bubble gum,The red-money prices are 2 rcu’s per

unit of ambrosia and 4 rcu’s per unit of bubble gum.

380 PRODUCTION (Ch,30)

(a) Harold has a blue income of 9 and a red income of 24,If it has to

pay in both currencies for any purchase,draw its budget set in the graph

below,(Hint,You answered this question a few months ago.)

0 5 10 15 20

5

10

15

Ambrosia

Bubble gum

20

Part j budget set

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

(12,9)

9

12

6

9

Part a budget set

(b) The Free Choice party campaigns on a platform that Mungoans should

be allowed to purchase goods at either the blue-money price or the red-

money price,whichever they prefer,We want to construct Harold’s bud-

get set if this reform is instituted,To begin with,how much bubble gum

could Harold consume if it spent all of its blue money and its red money

on bubble gum? 15 units of bubble gum.

(c) How much ambrosia could it consume if it spent all of its blue

money and all of its red money on ambrosia? 21 units of

ambrosia.

(d) If Harold were spending all of its money of both colors on bubble gum

and it decided to purchase a little bit of ambrosia,which currency would

it use? The red currency.

(e) How much ambrosia could it buy before it ran out of that color money?

12 units of ambrosia.

(f) What would be the slope of this budget line before it ran out of that

kind of money? The slope would be?

1

2

.

NAME 381

(g) If Harold were spending all of its money of both colors on ambrosia

and it decided to purchase a little bit of bubble gum,which currency

would it use? The blue currency.

(h) How much bubble gum could it buy before it ran out of that color

money? He could buy 9 units of bubble gum.

(i) What would be the slope of this budget line before it ran out of that

kind of money? The slope would be?1.

(j) Use your answers to the above questions to draw Harold’s budget set

in the above graph if it could purchase bubble gum and ambrosia using

either currency.

382 PRODUCTION (Ch,30)

Chapter 31 NAME

Welfare

Introduction,Here you will look at various ways of determining social

preferences,You will check to see which of the Arrow axioms for ag-

gregating individual preferences are satis ed by these welfare relations.

You will also try to nd optimal allocations for some given social welfare

functions,The method for solving these last problems is analogous to

solving for a consumer’s optimal bundle given preferences and a budget

constraint,Two hints,Remember that for a Pareto optimal allocation

inside the Edgeworth box,the consumers’ marginal rates of substitution

will be equal,Also,in a \fair allocation," neither consumer prefers the

other consumer’s bundle to his own.

Example,A social planner has decided that she wants to allocate income

between 2 people so as to maximize

p

Y

1

+

p

Y

2

where Y

i

is the amount of

income that person i gets,Suppose that the planner has a xed amount

of money to allocate and that she can enforce any income distribution

such that Y

1

+ Y

2

= W,whereW is some xed amount,This planner

would have ordinary convex indi erence curves between Y

1

and Y

2

and

a \budget constraint" where the \price" of income for each person is 1.

Therefore the planner would set her marginal rate of substitution between

income for the two people equal to the relative price which is 1,When you

solve this,you will nd that she sets Y

1

= Y

2

= W=2,Suppose instead

that it is \more expensive" for the planner to give money to person 1 than

to person 2,(Perhaps person 1 is forgetful and loses money,or perhaps

person 1 is frequently robbed.) For example,suppose that the planner’s

budget is 2Y

1

+Y

2

= W,Then the planner maximizes

p

Y

1

+

p

Y

2

subject

to 2Y

1

+Y

2

= W,Setting her MRS equal to the price ratio,we nd that

p

Y

2

p

Y

1

=2,SoY

2

=4Y

1

,Therefore the planner makes Y

1

= W=5and

Y

2

=4W=5.

31.1 (2) One possible method of determining a social preference relation

is the Borda count,also known as rank-order voting,Each voter is asked

to rank all of the alternatives,If there are 10 alternatives,you give your

rst choice a 1,your second choice a 2,and so on,The voters’ scores for

each alternative are then added over all voters,The total score for an

alternative is called its Borda count,For any two alternatives,x and y,

if the Borda count of x is smaller than or the same as the Borda count

for y,thenx is \socially at least as good as" y,Suppose that there are

a nite number of alternatives to choose from and that every individual

has complete,reflexive,and transitive preferences,For the time being,

let us also suppose that individuals are never indi erent between any two

di erent alternatives but always prefer one to the other.

384 WELFARE (Ch,31)

(a) Is the social preference ordering de ned in this way complete? Yes.

Reflexive? Yes,Transitive? Yes.

(b) If everyone prefers x to y,will the Borda count rank x as socially

preferred to y? Explain your answer,Yes,If everybody

ranks x ahead of y,then everyone must give

x a higher rank than y,Then the sum of

the ranks of x must be larger than the sum

of the ranks of y.

(c) Suppose that there are two voters and three candidates,x,y,and

z,Suppose that Voter 1 ranks the candidates,x rst,z second,and y

third,Suppose that Voter 2 ranks the candidates,y rst,x second,and z

third,What is the Borda count for x? 3,For y? 4,For

z? 5,Now suppose that it is discovered that candidate z once

lifted a beagle by the ears,Voter 1,who has rather large ears himself,

is appalled and changes his ranking to x rst,y second,z third,Voter

2,who picks up his own children by the ears,is favorably impressed and

changes his ranking to y rst,z second,x third,Now what is the Borda

count for x? 4,For y? 3,For z? 5.

(d) Does the social preference relation de ned by the Borda count have

the property that social preferences between x and y depend only on how

people rank x versus y and not on how they rank other alternatives? Ex-

plain,No,In the above example,the ranking

of z changed,but nobody changed his mind

about whether x was better than y or vice

versa,Before the change x beat y,and after

the change y beat x.

31.2 (2) Suppose the utility possibility frontier for two individuals is

given by U

A

+2U

B

= 200,On the graph below,plot the utility frontier.

NAME 385

0 50 100 150 200

50

100

150

UA

UB

200

Blue line

Black line

Red line

Utility frontier

(a) In order to maximize a \Nietzschean social welfare function,"

W(U

A;U

B

)=maxfU

A;U

B

g,on the utility possibility frontier shown

above,one would set U

A

equal to 200 and U

B

equal to 0.

(b) If instead we use a Rawlsian criterion,W(U

A;U

B

)=minfU

A;U

B

g,

then the social welfare function is maximized on the above utility possi-

bility frontier where U

A

equals 66.66 and U

B

equals 66.66.

(c) Suppose that social welfare is given by W(U

A;U

B

)=U

1=2

A

U

1=2

B

.In

this case,with the above utility possibility frontier,social welfare is max-

imized where U

A

equals 100 and U

B

is 50,(Hint,You might

want to think about the similarities between this maximization problem

and the consumer’s maximization problem with a Cobb-Douglas utility

function.)

(d) Show the three social maxima on the above graph,Use black ink

to draw a Nietzschean isowelfare line through the Nietzschean maximum.

Use red ink to draw a Rawlsian isowelfare line through the Rawlsian

maximum,Use blue ink to draw a Cobb-Douglas isowelfare line through

the Cobb-Douglas maximum.

31.3 (2) A parent has two children named A and B and she loves both

of them equally,She has a total of $1,000 to give to them.

386 WELFARE (Ch,31)

(a) The parent’s utility function is U(a;b)=

p

a +

p

b,wherea is the

amount of money she gives to A and b istheamountofmoneyshegives

to B,How will she choose to divide the money? a = b = $500.

(b) Suppose that her utility function is U(a;b)=?

1

a

1

b

,How will she

choose to divide the money? a = b = $500.

(c) Suppose that her utility function is U(a;b)=loga +logb,How will

she choose to divide the money? a = b = $500.

(d) Suppose that her utility function is U(a;b)=minfa;bg,How will she

choose to divide the money? a = b = $500.

(e) Suppose that her utility function is U(a;b)=maxfa;bg,How will she

choose to divide the money? a =$1;000,b =0,or vice

versa.

(Hint,In each of the above cases,we notice that the parent’s problem is

to maximize U(a;b) subject to the constraint that a+b =1;000,This is

just like the consumer problems we studied earlier,It must be that the

parent sets her marginal rate of substitution between a and b equal to 1

since it costs the same to give money to each child.)

(f) Suppose that her utility function is U(a;b)=a

2

+ b

2

,How will

she choose to divide the money between her children? Explain why she

doesn’t set her marginal rate of substitution equal to 1 in this case.

She gives everything to one child,Her

preferences are not convex,indifference

curves are quarter circles.

31.4 (2) In the previous problem,suppose that A is a much more e cient

shopper than B so that A is able to get twice as much consumption

goods as B can for every dollar that he spends,Let a be the amount of

consumption goods that A gets and b the amount that B gets,We will

measure consumption goods so that one unit of consumption goods costs

$1 for A and $2 for B,Thus the parent’s budget constraint is a +2b =

1;000.

(a) If the mother’s utility function is U(a;b)=a+b,which child will get

more money? A,Which child will consume more goods? A.

NAME 387

(b) If the mother’s utility function is U(a;b)=a b,which child will get

more money? They get the same amount of money.

Which child will get to consume more? A consumes more.

(c) If the mother’s utility function is U(a;b)=?

1

a

1

b

,which child will

get more money? B gets more money,Which child will get

to consume more? They consume the same amount.

(d) If the mother’s utility function is U(a;b)=maxfa;bg,which child

will get more money? A,Which child will get to consume more?

A.

(e) If the mother’s utility function is U(a;b)=minfa;bg,which child will

get more money? B,Which child will get to consume more?

They consume the same amount.

Calculus 31.5 (1) Norton and Ralph have a utility possibility frontier that is given

by the following equation,U

R

+U

2

N

= 100 (where R and N signify Ralph

and Norton respectively).

(a) If we set Norton’s utility to zero,what is the highest possible utility

Ralph can achieve? 100,If we set Ralph’s utility to zero,what is

the best Norton can do? 10.

(b) Plot the utility possibility frontier on the graph below.

0 5 10 15 20

25

50

75

Norton's utility

Ralph's utility

100

388 WELFARE (Ch,31)

(c) Derive an equation for the slope of the above utility possibility curve.

dU

R

dU

N

=?2U

N

.

(d) Both Ralph and Norton believe that the ideal allocation is given by

maximizing an appropriate social welfare function,Ralph thinks that

U

R

= 75,U

N

= 5 is the best distribution of welfare,and presents the

maximization solution to a weighted-sum-of-the-utilities social welfare

function that con rms this observation,What was Ralph’s social welfare

function? (Hint,What is the slope of Ralph’s social welfare function?)

W = U

R

+10U

N

.

(e) Norton,on the other hand,believes that U

R

= 19,U

N

=9isthe

best distribution,What is the social welfare function Norton presents?

W = U

R

+18U

N

.

31.6 (2) Roger and Gordon have identical utility functions,U(x;y)=

x

2

+y

2

,There are 10 units of x and 10 units of y to be divided between

them,Roger has blue indi erence curves,Gordon has red ones.

(a) Draw an Edgeworth box showing some of their indi erence curves and

mark the Pareto optimal allocations with black ink,(Hint,Notice that

the indi erence curves are nonconvex.)

010

10

Roger

Gordon

Black lines

Black lines

Red curves

Blue

curves

Fair

Fair

y

x

(b) What are the fair allocations in this case? See diagram.

31.7 (2) Paul and David consume apples and oranges,Paul’s util-

ity function is U

P

(A

P;O

P

)=2A

P

+ O

P

and David’s utility function is

NAME 389

U

D

(A

D;O

D

)=A

D

+2O

D

,whereA

P

and A

D

are apple consumptions for

Paul and David,and O

P

and O

D

are orange consumptions for Paul and

David,There are a total of 12 apples and 12 oranges to divide between

Paul and David,Paul has blue indi erence curves,David has red ones.

Draw an Edgeworth box showing some of their indi erence curves,Mark

the Pareto optimal allocations on your graph.

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

12

Apples

Oranges

0

Red curves

Blue

curves

Blue shading

Red shading

Pareto

optimal

Pareto optimal

Fair

Paul

David

12

(a) Write one inequality that says that Paul likes his own bundle as well

as he likes David’s and write another inequality that says that David likes

his own bundle as well as he likes Paul’s,2A

P

+O

P

2A

D

+O

D

and A

D

+2O

D

A

P

+2O

P

.

(b) Use the fact that at feasible allocations,A

P

+A

D

=12andO

P

+O

D

=

12 to eliminate A

D

and O

D

from the rst of these equations,Write the

resulting inequality involving only the variables A

P

and O

P

.Nowinyour

Edgeworth box,use blue ink to shade in all of the allocations such that

Paul prefers his own allocation to David’s,2A

P

+O

P

18.

(c) Use a procedure similar to that you used above to nd the allocations

where David prefers his own bundle to Paul’s,Describe these points

with an inequality and shade them in on your diagram with red ink.

A

D

+2O

D

18.

(d) On your Edgeworth box,mark the fair allocations.

31.8 (3) Romeo loves Juliet and Juliet loves Romeo,Besides love,

they consume only one good,spaghetti,Romeo likes spaghetti,but he

390 WELFARE (Ch,31)

also likes Juliet to be happy and he knows that spaghetti makes her

happy,Juliet likes spaghetti,but she also likes Romeo to be happy and

she knows that spaghetti makes Romeo happy,Romeo’s utility function

is U

R

(S

R;S

J

)=S

a

R

S

1?a

J

and Juliet’s utility function is U

J

(S

J;S

R

)=

S

a

J

S

1?a

R

,whereS

J

and S

R

are the amount of spaghetti for Romeo and

the amount of spaghetti for Juliet respectively,There is a total of 24 units

of spaghetti to be divided between Romeo and Juliet.

(a) Suppose that a =2=3,If Romeo got to allocate the 24 units of

spaghetti exactly as he wanted to,how much would he give himself?

16,How much would he give Juliet? 8,(Hint,Notice that this

problem is formally just like the choice problem for a consumer with a

Cobb-Douglas utility function choosing between two goods with a budget

constraint,What is the budget constraint?)

(b) If Juliet got to allocate the spaghetti exactly as she wanted to,how

much would she take for herself? 16,How much would she give

Romeo? 8.

(c) What are the Pareto optimal allocations? (Hint,An allocation

will not be Pareto optimal if both persons’ utility will be increased by

a gift from one to the other.) The Pareto optimal

allocations are all of the allocations in

which each person gets at least 8 units of

spaghetti.

(d) When we had to allocate two goods between two people,we drew an

Edgeworth box with indi erence curves in it,When we have just one

good to allocate between two people,all we need is an \Edgeworth line"

and instead of indi erence curves,we will just have indi erence dots.

Consider the Edgeworth line below,Let the distance from left to right

denote spaghetti for Romeo and the distance from right to left denote

spaghetti for Juliet.

(e) On the Edgeworth line you drew above,show Romeo’s favorite point

and Juliet’s favorite point.

NAME 391

(f) Suppose that a =1=3,If Romeo got to allocate the spaghetti,how

much would he choose for himself? 8,If Juliet got to allocate

the spaghetti,how much would she choose for herself? 8,Label

the Edgeworth line below,showing the two people’s favorite points and

the locus of Pareto optimal points.

(g) When a =1=3,at the Pareto optimal allocations what do Romeo and

Juliet disagree about? Romeo wants to give spaghetti

to Juliet,but she doesn’t want to take it.

Juliet wants to give spaghetti to Romeo,

but he doesn’t want to take it,Both like

spaghetti for themselves,but would rather

the other had it.

31.9 (2) Hat eld and McCoy hate each other but love corn whiskey.

Because they hate for each other to be happy,each wants the other to

have less whiskey,Hat eld’s utility function isU

H

(W

H;W

M

)=W

H

W

2

M

and McCoy’s utility function is U

M

(W

M;W

H

)=W

M

W

2

H

,whereW

M

is McCoy’s daily whiskey consumption and W

H

is Hat eld’s daily whiskey

consumption (both measured in quarts),There are 4 quarts of whiskey

to be allocated.

(a) If McCoy got to allocate all of the whiskey,how would he allocate it?

All for himself,If Hat eld got to allocate all of the whiskey,

how would he allocate it? All for himself.

(b) If each of them gets 2 quarts of whiskey,what will the utility of each

of them be2,If a bear spilled 2 quarts of their whiskey and they

divided the remaining 2 quarts equally between them,what would the

utility of each of them be? 0,If it is possible to throw away some

of the whiskey,is it Pareto optimal for them each to consume 2 quarts of

whiskey? No.

392 WELFARE (Ch,31)

(c) If it is possible to throw away some whiskey and they must consume

equal amounts of whiskey,how much should they throw away? 3

quarts.

Chapter 32 NAME

Externalities

Introduction,When there are externalities,the outcome from indepen-

dently chosen actions is typically not Pareto e cient,In these exercises,

you explore the consequences of alternative mechanisms and institutional

arrangements for dealing with externalities.

Example,A large factory pumps its waste into a nearby lake,The lake

is also used for recreation by 1,000 people,Let X betheamountofwaste

that the rm pumps into the lake,Let Y

i

be the number of hours per day

that person i spends swimming and boating in the lake,and let C

i

be the

number of dollars that person i spends on consumption goods,If the rm

pumps X units of waste into the lake,its pro ts will be 1;200X?100X

2

.

Consumers have identical utility functions,U(Y

i;C

i;X)=C

i

+9Y

i

Y

2

i

XY

i

,and identical incomes,Suppose that there are no restrictions

on pumping waste into the lake and there is no charge to consumers for

using the lake,Also,suppose that the factory and the consumers make

their decisions independently,The factory will maximize its pro ts by

choosing X = 6,(Set the derivative of pro ts with respect to X equal

to zero.) When X = 6,each consumer maximizes utility by choosing

Y

i

=1:5,(Set the derivative of utility with respect to Y

i

equal to zero.)

Notice from the utility functions that when each person is spending 1.5

hours a day in the lake,she will be willing to pay 1.5 dollars to reduce

X by 1 unit,Since there are 1,000 people,the total amount that people

will be willing to pay to reduce the amount of waste by 1 unit is $1,500.

If the amount of waste is reduced from 6 to 5 units,the factory’s pro ts

will fall from $3,600 to $3,500,Evidently the consumers could a ord to

bribe the factory to reduce its waste production by 1 unit.

32.1 (2) The picturesque village of Horsehead,Massachusetts,lies on a

bay that is inhabited by the delectable crustacean,homarus americanus,

also known as the lobster,The town council of Horsehead issues permits

to trap lobsters and is trying to determine how many permits to issue.

The economics of the situation is this:

1,It costs $2,000 dollars a month to operate a lobster boat.

2,If there are x boats operating in Horsehead Bay,the total revenue

from the lobster catch per month will be f(x)=$1;000(10x?x

2

).

(a) In the graph below,plot the curves for the average product,AP(x)=

f(x)=x,and the marginal product,MP(x)=10;000?2;000x.Inthe

same graph,plot the line indicating the cost of operating a boat.

394 EXTERNALITIES (Ch,32)

2 4 6 8 10 12

2

4

6

8

10

12

x

AP,MP

0

AP

Cost

MP

(b) If the permits are free of charge,how many boats will trap lobsters

in Horsehead,Massachusetts? (Hint,How many boats must enter before

there are zero pro ts?) 8 boats.

(c) What number of boats maximizes total pro ts? Set MP

equal to cost to give 10?2x =2,orx =4

boats.

(d) If Horsehead,Massachusetts,wants to restrict the number of boats to

the number that maximizes total pro ts,how much should it charge per

month for a lobstering permit? (Hint,With a license fee of F thousand

dollars per month,the marginal cost of operating a boat for a month

would be (2 + F) thousand dollars per month.) $4,000 per

month.

32.2 (2) Suppose that a honey farm is located next to an apple orchard

and each acts as a competitive rm,Let the amount of apples produced

be measured by A and the amount of honey produced be measured by H.

The cost functions of the two rms are c

H

(H)=H

2

=100 and c

A

(A)=

A

2

=100?H,The price of honey is $2 and the price of apples is $3.

(a) If the rms each operate independently,the equilibrium amount of

honey produced will be 100 and the equilibrium amount of apples

produced will be 150.

NAME 395

(b) Suppose that the honey and apple rms merged,What would be

the pro t-maximizing output of honey for the combined rm? 150.

What would be the pro t-maximizing amount of apples? 150.

(c) What is the socially e cient output of honey? 150,If the rms

stayed separate,how much would honey production have to be subsidized

to induce an e cient supply? $1 per unit.

32.3 (2) In El Carburetor,California,population 1,001,there is not

much to do except to drive your car around town,Everybody in town

is just like everybody else,While everybody likes to drive,everybody

complains about the congestion,noise,and pollution caused by tra c,A

typical resident’s utility function is U(m;d;h)=m+16d?d

2

6h=1;000,

where m is the resident’s daily consumption of Big Macs,d is the number

of hours per day that he,himself,drives,and h is the total amount of

driving (measured in person-hours per day) done by all other residents

of El Carburetor,The price of Big Macs is $1 each,Every person in El

Carburetor has an income of $40 per day,To keep calculations simple,

suppose it costs nothing to drive a car.

(a) If an individual believes that the amount of driving he does won’t af-

fect the amount that others drive,how many hours per day will he choose

to drive? 8,(Hint,What value of d maximizes U(m;d;h)?)

(b) If everybody chooses his best d,then what is the total amount h of

driving by other persons? 8,000.

(c) What will be the utility of each resident? 56.

(d) If everybody drives 6 hours a day,what will be the utility level of a

typical resident of El Carburetor? 64.

(e) Suppose that the residents decided to pass a law restricting the total

number of hours that anyone is allowed to drive,How much driving

should everyone be allowed if the objective is to maximize the utility of

the typical resident? (Hint,Rewrite the utility function,substituting

1;000d for h,and maximize with respect to d.) 5 hours per

day.

396 EXTERNALITIES (Ch,32)

(f) The same objective could be achieved with a tax on driving,How

much would the tax have to be per hour of driving? (Hint,This price

would have to equal an individual’s marginal rate of substitution between

driving and Big Macs when he is driving the \right" amount.) $6.

32.4 (3) Tom and Jerry are roommates,They spend a total of 80 hours

a week together in their room,Tom likes loud music,even when he sleeps.

His utility function is U

T

(C

T;M)=C

T

+ M,whereC

T

is the number

of cookies he eats per week and M is the number of hours of loud music

per week that is played while he is in their room,Jerry hates all kinds

of music,His utility function is U(C

J;M)=C

J

M

2

=12,Every week,

Tom and Jerry each get two dozen chocolate chip cookies sent from home.

They have no other source of cookies,We can describe this situation with

a box that looks like an Edgeworth box,The box has cookies on the

horizontal axis and hours of music on the vertical axis,Since cookies are

private goods,the number of cookies that Tom consumes per week plus

the number that Jerry consumes per week must equal 48,But music in

their room is a public good,Each must consume the same number of

hours of music,whether he likes it or not,In the box,let the height of a

point represent the total number of hours of music played in their room

per week,Let the distance of the point from the left side of the box be

\cookies for Tom" and the distance of the point from the right side of the

box be \cookies for Jerry."

012243648

20

40

60

Cookies

Music

80

Blue Line

Red Line

Blue Shading

a

b

Blue Line

Red Line

Tom

Jerry

(a) Suppose the dorm’s policy is that you must have your roommate’s

permission to play music,The initial endowment in this case denotes the

situation if Tom and Jerry make no deals,There would be no music,and

each person would consume 2 dozen cookies a week,Mark this initial

endowment on the box above with the label A,Use red ink to sketch

the indi erence curve for Tom that passes through this point,and use

NAME 397

blue ink to sketch the indi erence curve for Jerry that passes through

this point,[Hint,When you draw Jerry’s indi erence curve,remember

two things,(1) He hates music,so he prefers lower points on the graph

to higher ones,(2) Cookies for Jerry are measured from the right side

of the box,so he prefers points that are toward the left side of the box

to points that are toward the right.] Use blue ink to shade in the points

representing situations that would make both roommates better o than

they are at point A.

(b) Suppose,alternatively,that the dorm’s policy is \rock-n-roll is good

for the soul." You don’t need your roommate’s permission to play music.

Then the initial endowment is one in which Tom plays music for all of

the 80 hours per week that they are in the room together and where each

consumes 2 dozen cookies per week,Mark this endowment point in the

box above and label it B,Use red ink to sketch the indi erence curve

for Tom that passes through this point,and use blue ink to sketch the

indi erence curve for Jerry that passes through this point,Given the

available resources,can both Tom and Jerry be made better o than they

areatpointB? Yes.

Calculus 32.5 (0) A clothing store and a jewelry store are located side by side

in a small shopping mall,The number of customers who come to the

shopping mall intending to shop at either store depends on the amount

of money that the store spends on advertising per day,Each store also

attracts some customers who came to shop at the neighboring store,If

the clothing store spends $x

C

per day on advertising,and the jeweler

spends $x

J

on advertising per day,then the total pro ts per day of the

clothing store are

C

(x

C;x

J

)=(60+x

J

)x

C

2x

2

C

,and the total pro ts

per day of the jewelry store are

J

(x

C;x

J

) = (105 + x

C

)x

J

2x

2

J

.(In

each case,these are pro ts net of all costs,including advertising.)

(a) If each store believes that the other store’s amount of advertising

is independent of its own advertising expenditure,then we can nd the

equilibrium amount of advertising for each store by solving two equations

in two unknowns,One of these equations says that the derivative of the

clothing store’s pro ts with respect to its own advertising is zero,The

other equation requires that the derivative of the jeweler’s pro ts with

respect to its own advertising is zero,These two equations are written as

60 +x

J

4x

C

=0 and 105 +x

C

4x

J

=0,The

equilibrium amounts of advertising are x

C

=23 and x

J

=32.

Pro ts of the clothing store are $1,058 and pro ts of the jeweler

are 2,048.

398 EXTERNALITIES (Ch,32)

(b) The extra pro t that the jeweler would get from an extra dollar’s

worth of advertising by the clothing store is approximately equal to the

derivative of the jeweler’s pro ts with respect to the clothing store’s ad-

vertising expenditure,When the two stores are doing the equilibrium

amount of advertising that you calculated above,a dollar’s worth of ad-

vertising by the clothing store would give the jeweler an extra pro t of

about $32 and an extra dollar’s worth of advertising by the jeweler

would give the clothing store an extra pro t of about $23.

(c) Suppose that the owner of the clothing store knows the pro t functions

of both stores,She reasons to herself as follows,Suppose that I can decide

how much advertising I will do before the jeweler decides what he is going

to do,When I tell him what I am doing,he will have to adjust his behavior

accordingly,I can calculate his reaction function to my choice of x

C

,by

setting the derivative of his pro ts with respect to his own advertising

equal to zero and solving for his amount of advertising as a function of

my own advertising,When I do this,I nd thatx

J

= 105=4+x

C

=4.

If I substitute this value of x

J

into my pro t function and then choose x

C

to maximize my own pro ts,I will choose x

C

= 24.64 and he will

choose x

J

= 32.41,In this case my pro ts will be $1,062.72

and his pro ts will be $2,100.82.

(d) Suppose that the clothing store and the jewelry store have the same

pro t functions as before but are owned by a single rm that chooses

the amounts of advertising so as to maximize the sum of the two stores’

pro ts,The single rm would choose x

C

= $37.50 and x

J

=

$45,Without calculating actual pro ts,can you determine whether

total pro ts will be higher,lower,or the same as total pro ts would be

when they made their decisions independently? Yes,they

would be higher,How much would the total pro ts be?

$3,487.50.

32.6 (2) The cottagers on the shores of Lake Invidious are an unsavory

bunch,There are 100 of them,and they live in a circle around the lake.

Each cottager has two neighbors,one on his right and one on his left.

There is only one commodity,and they all consume it on their front

lawns in full view of their two neighbors,Each cottager likes to consume

the commodity but is very envious of consumption by the neighbor on

his left,Curiously,nobody cares what the neighbor on his right is doing.

In fact every consumer has a utility function U(c;l)=c?l

2

,wherec is

NAME 399

his own consumption and l is consumption by his neighbor on the left.

Suppose that each consumer owns 1 unit of the consumption good and

consumes it.

(a) Calculate his utility level,0.

(b) Suppose that each consumer consumes only 3=4 of a unit,Will all

individuals be better o or worse o? Better off.

(c) What is the best possible consumption if all are to consume the same

amount? 1=2.

(d) Suppose that everybody around the lake is consuming 1 unit,Can

any two people make themselves both better o either by redistributing

consumption between them or by throwing something away? No.

(e) How about a group of three people? No.

(f) How large is the smallest group that could cooperate to bene t all its

members? 100.

32.7 (0) Jim and Tammy are partners in Business and in Life,As

is all too common in this imperfect world,each has a little habit that

annoys the other,Jim’s habit,we will call activity X,and Tammy’s

habit,activity Y.Letx be the amount of activity X that Jim pursues

and y be the amount of activity Y that Tammy pursues,Due to a series

of unfortunate reverses,Jim and Tammy have a total of only $1,000,000

a year to spend,Jim’s utility function is U

J

= c

J

+ 500 lnx?10y,where

c

J

is the money he spends per year on goods other than his habit,x is

the number of units of activity X that he consumes per year,and y is the

number of units of activity Y that Tammy consumes per year,Tammy’s

utility function is U

T

= c

T

+ 500 lny?10x,wherec

T

istheamountof

money she spends on goods other than activity Y,y is the number of

units of activity Y that she consumes,and x is the number of units of

activity X that Jim consumes,Activity X costs $20 per unit,Activity

Y costs $100 per unit.

(a) Suppose that Jim has a right to half their joint income and Tammy

has a right to the other half,Suppose further that they make no bargains

with each other about how much activity X and Y they will consume.

How much of activity X will Jim choose to consume? 25 units.

How much of activity Y will Tammy consume? 5 units.

400 EXTERNALITIES (Ch,32)

(b) Because Jim and Tammy have quasilinear utility functions,their util-

ity possibility frontier includes a straight line segment,Furthermore,this

segment can be found by maximizing the sum of their utilities,Notice

that

U

J

(c

J;x;y)+U

T

(c

T;x;y)

= c

J

+ 500 lnx?20y +c

T

+ 500 lny?10x

= c

J

+c

T

+ 500 lnx?10x+ 500 lny?10y:

But we know from the family budget constraint that c

J

+c

T

=1;000;000?

20x?100y,Therefore we can write

U

J

(c

J;x;y)+U

T

(c

T;x;y)=1;000;000?20x?100y + 500 lnx?10x

+ 500 lny?10y

=1;000;000 + 500 lnx+ 500 lny?30x?110y:

Let us now choose x and y so as to maximize U

J

(c

J;x;y)+U

T

(c

T;x;y).

Setting the partial derivatives with respect to x and y equal to zero,we

nd the maximum where x = 16.67 and y = 4.54,Ifweplug

these numbers into the equation U

J

(c

J;x;y)+U

T

(c

T;x;y)=1;000;000+

500 lnx+500 lny?30x?110y,we nd that the utility possibility frontier is

described by the equation U

J

+U

C

= 1,001,163.86,(You need

a calculator or a log table to nd this answer.) Along this frontier,the

total expenditure on the annoying habits X and Y by Jim and Tammy is

787:34,The rest of the $1,000,000 is spent on c

J

and c

T

,Each possible

way of dividing this expenditure corresponds to a di erent point on the

utility possibility frontier,The slope of the utility possibility frontier

constructed in this way is -1.

32.8 (0) An airport is located next to a large tract of land owned by a

housing developer,The developer would like to build houses on this land,

but noise from the airport reduces the value of the land,The more planes

that fly,the lower is the amount of pro ts that the developer makes,Let

X be the number of planes that fly per day and let Y be the number of

houses that the developer builds,The airport’s total pro ts are 48X?X

2

,

and the developer’s total pro ts are 60Y?Y

2

XY,Let us consider the

outcome under various assumptions about institutional rules and about

bargaining between the airport and the developer.

(a) \Free to Choose with No Bargaining",Suppose that no bargains can

be struck between the airport and the developer and that each can decide

on its own level of activity,No matter how many houses the developer

builds,the number of planes per day that maximizes pro ts for the airport

is 24,Given that the airport is landing this number of planes,the

NAME 401

number of houses that maximizes the developer’s pro ts is 18,Total

pro ts of the airport will be 576 and total pro ts of the developer

will be 324,The sum of their pro ts will be 900.

(b) \Strict Prohibition",Suppose that a local ordinance makes it illegal

to land planes at the airport because they impose an externality on the

developer,Then no planes will fly,The developer will build 30

houses and will have total pro ts of 900.

(c) \Lawyer’s Paradise",Suppose that a law is passed that makes the

airport liable for all damages to the developer’s property values,Since the

developer’s pro ts are 60Y?Y

2

XY and his pro ts would be 60Y?Y

2

if no planes were flown,the total amount of damages awarded to the

developer will be XY,Therefore if the airport flies X planes and the

developer builds Y houses,then the airport’s pro ts after it has paid

damages will be 48X?X

2

XY,The developer’s pro ts including the

amount he receives in payment of damages will be 60Y?Y

2

XY+XY =

60Y?Y

2

,To maximize his net pro ts,the developer will choose to build

30 houses no matter how many planes are flown,To maximize its

pro ts,net of damages,the airport will choose to land 9 planes.

Total pro ts of the developer will be 900 and total pro ts of the

airport will be 81,The sum of their pro ts will be 981.

Calculus 32.9 (1) This problem concerns the airport and the developer from the

previous problem.

(a) \The Conglomerate",Suppose that a single rm bought the de-

veloper’s land and the airport and managed both to maximize joint

pro ts,Total pro ts,expressed as a function of X and Y would be

48X?X

2

+60Y?Y

2

XY Total pro ts are maximized

when X = 12 and Y = 24,Total pro ts are then equal to

1,008.

(b) \Dealing",Suppose that the airport and the developer remain in-

dependent,If the original situation was one of \free to choose," could

the developer increase his net pro ts by bribing the airport to cut back

one flight per day if the developer has to pay for all of the airport’s lost

pro ts? Yes,The developer decides to get the airport to reduce its

402 EXTERNALITIES (Ch,32)

flights by paying for all lost pro ts coming from the reduction of flights.

To maximize his own net pro ts,how many flights per day should he pay

the airport to eliminate? 12.

32.10 (1) Every morning,6,000 commuters must travel from East Potato

to West Potato,Commuters all try to minimize the time it takes to get to

work,There are two ways to make the trip,One way is to drive straight

across town,throught the heart of Middle Potato,The other way is to take

the Beltline Freeway that circles the Potatoes,The Beltline Freeway is

entirely uncongested,but the drive is roundabout and it takes 45 minutes

to get from East Potato to West Potato by this means,The road through

Middle Potato is much shorter,and if it were uncongested,it would take

only 20 minutes to travel from East Potato to West Potato by this means.

But this road can get congested,In fact,if the number of commuters who

use this road is N,then the number of minutes that it takes to drive from

East Potato to West Potato through Middle Potato is 20 +N=100.

(a) Assuming that no tolls are charged for using either road,in equilib-

rium how many commuters will use the road through Middle Potato?

2,500,What will be the total number of person-minutes per

day spent by commuters traveling from East Potato to West Potato?

45 6;000 = 270;000.

(b) Suppose that a social planner controlled access to the road through

Middle Potato and set the number of persons permitted to travel this

way so as to minimize the total number of person-minutes per day spent

by commuters traveling from East Potato to West Potato,Write an

expression for the total number of person-minutes per day spent by

commuters traveling from East Potato to West Potato as a function of

the number N of commuters permitted to travel on the Middle Potato

road,N(20 +

N

100

)+(6;000?N)45,How many com-

muters per day would the social planner allow to use the road through

Middle Potato? 1,250,In this case,how long would it take com-

muters who drove through Middle Potato to get to work? 32.5

minutes,What would be the total number of person-minutes per

day spent by commuters traveling from East Potato to West Potato?

1;250 32:5+4;750 45 = 254;375

NAME 403

(c) Suppose that commuters value time saved from commuting at $w per

minute and that the Greater Potato metropolitan government charges a

toll for using the Middle Potato road and divides the revenue from this

toll equally among all 6,000 commuters,If the government chooses the

toll in such a way as to minimize the total amount of time that people

spend commuting from East Potato to West Potato,how high should it

set the toll? $12:5w,How much revenue will it collect per day from

this toll? $15;625w,Show that with this policy every commuter is

better o than he or she was without the tolls and evaluate the gain per

consumer in dollars,Before the toll was in place,

all commuters spent 45 minutes traveling

to work,With the toll in place,commuters

who travel on the Beltline still spend

45 minutes traveling to work and commuters

who travel through Middle Potato are

indifferent between spending 45 minutes

traveling on the Beltline and paying the

toll to go through Middle Potato,Thus

nobody would be worse off even if toll

revenue were wasted,But everybody gets

back about $2:6w per day from the toll

revenue,so all are better off.

32.11 (2) Suppose that the Greater Potato metropolitan government

rejects the idea of imposing tra c tolls and decides instead to rebuild the

Middle Potato highway so as to double its capacity,With the doubled

capacity,the amount of time it takes to travel from East Potato to West

Potato on the Middle Potato highway is given by 20 + N=200,where

N is the number of commuters who use the Middle Potato highway,In

the new equilibrium,with expanded capacity and no tolls,how many

commuters will use the Middle Potato highway? 5,000 How long

will it take users of the Middle Potato highway to get to work? 45

minutes How many person-minutes of commuting time will be saved

404 EXTERNALITIES (Ch,32)

by expanding the capacity of the Middle Potato highway? 0 Do

you think people will think that this capacity expansion will be a good

use of their tax dollars? No.

Chapter 33 NAME

Law

Introduction,These problems are based on the survey of law and eco-

nomics found in your text,We hope that you will be pleased to see that

the techniques you learned in earlier chapters can provide useful insights

into issues that arise in law.

33.1 (2) Madame Norrell makes her living in Florida by stealing gold

buttons from designer jackets in expensive boutiques,She can sell each

button to a fence for $10,The maximum number of buttons she can steal

in a day is 50,Florida has a law against button theft,There is a ne of F

dollars if someone is caught stealing any number of buttons,The police

catch about 10 percent of all button thieves,and these must pay the ne

and forfeit any buttons they have stolen.

(a) Suppose that the only thing that Madame Norrell cares about is her

expected pro ts,What is the smallest ne that will discourage Madame

Norrell from stealing buttons? 4,500.

(b) Due to an oversupply of buttons,Madame Norrell’s fence announces

that he will no longer pay her a flat price for buttons,If Madame Norrell

delivers x buttons,she will be paid 5 lnx,(Assume that Madame Norrell

will take at least 1 button if she takes any at all.) Initially Madame

Norrell has $100,and the ne if she is caught stealing x buttons is $3

per button,However,she only has to pay the ne if she is caught,in

which case all her buttons are con scated and she collects zero from the

fence,How many buttons will Madame Norrell try to take,assuming she

maximizes her expected pro t? 15.

(c) What does the ne per button have to be to induce Madame Norrell

to limit herself to taking 10 buttons? 4.50.

(d) Now assume that Madame Norrell is an expected utility maximizer.

With probability,10,she is caught with x buttons and pays a ne of 3x.

With probability,90,she gets away with x buttons,which she can sell for

$10 each,She cares about the expected utility of her wealth,with von

Neumann-Morgenstern utility function lnx,Initially her wealth is $100.

How many buttons will she take? 29.

33.2 (2) Jim Levson rides his bike through the forest with reckless

abandon,while Dick Stout likes to hike in the woods,Let s be the speed

in miles per hour that James rides and w the speed with which Dick walks.

406 LAW (Ch,33)

Jim’s utility depends on how fast he rides and how many dollars he has,

while Dick’s utility depends on how fast he walks and how much money

he has.

U

Jim

=6

p

s?s+m

U

Dick

=4

p

w?w +m:

(a) How fast will Dick walk? 4 miles per hour,How fast will

Jim ride? 9 miles per hour.

(b) Alas,since Jim and Dick are both moving in the same forest,there

is some chance that Jim will run into Dick,Suppose that the expected

cost to Dick of such an accident depends on the speed that each moves:

c(s;w)=

s

2

16

+

w

2

2

,(Assume that Jim is tter than Dick and will incur

negligable costs in an accident.) If Dick has to pay the entire cost of an

accident,how fast will he walk? 1 mile per hour,How fast

will Jim ride? 9 miles per hour.

(c) Suppose that Jim now has full liability and must pay any costs that he

imposes on Dick,How fast will Dick walk? 4 miles per hour.

How fast will Jim ride? 4 miles per hour.

(d) What are the socially optimal speeds for Jim and Dick to move? Dick

should walk 1 mile per hour and Jim should ride 4 miles

per hour.

33.3 (2) Derri Bottled Water of Christchurch,New Zealand,sells bottled

water from \the bottom of the world." Due to a number of fortuitous

circumstances,Derri has a monopoly on bottled water in the South Island.

The demand for bottled water in the South Island is p(x)=10?x=200,

and the cost of producing x bottles of water is c(x)=x

2

=200,Here the

price is measured in New Zealand dollars and the quantity is measured

in 1;000 cases per month.

(a) Draw the demand curve,the marginal revenue curve,and the marginal

cost curve in the graph below,The pro t-maximizing quantity is 500

cases of water,and the pro t maximizing price is 7.5 dollars per case.

NAME 407

200 400 600 800 1000 1200

2

4

6

8

10

12

Quantity

Price

0

5

500

mc

Demand

mr

7.5

(b) The New Zealand antitrust authorities now bring action against Derri

waters for monopolizing the bottled water industry,They announce that

during the coming year they will con scate 50 percent of Derri’s prof-

its,Part of these con scated pro ts will be used to distribute rebates

to the consumers of bottled water,In particular,each purchaser of bot-

tled water will receive $2 per case from the government,How does this

rebate influence the demand for bottled water? Shifts it up

by $2 What is the equation for the new inverse demand curve?

p(x)=12?x=200.

(c) Solve for the new levels of output and price,Draw the marginal

revenue curve,marginal cost curve,and inverse demand curve in the

following graph.

408 LAW (Ch,33)

200 400 600 800 1000 1200

2

4

6

8

10

12

Quantity

Price

0

mc

Demand

mr

625

8

1_

8

Chapter 34 NAME

Information Technology

Introduction,We all recognize that information technology has revolu-

tionized the way we produce and consume,Some think that it is necessary

to have a \new economics" to understand this New Economy,We think

not,The economic tools that you have learned in this course can o er

very powerful insights into the economics of information technology,as

we illustrate in this set of problems.

34.1 (2) Bill Barriers,the president of MightySoft software company is

about to introduce a new computer operating system called DoorKnobs.

Because it is easier to swap les with people who have the same operating

system,the amount people are willing to pay to have DoorKnobs on their

computers is greater the larger they believe DoorKnobs’s market share to

be.

The perceived market share for DoorKnobs is the fraction of all com-

puters that the public believes is using DoorKnobs,When the price of

DoorKnobs is p,thenitsactual market share is the fraction of all com-

puter owners that would be willing to pay at least $p to have DoorKnobs

installed on their computers,Market researchers have discovered that if

DoorKnobs’s perceived market share is s and the price of DoorKnobs is

$p,then its actual market share will be x,wherex is related to the price

p and perceived market share s by the formula

p = 256s(1?x),(1)

In the short run,MightySoft can influence the perceived market share

of DoorKnobs by publicity,advertising,giving liquor and gifts to friendly

journalists,and giving away copies in conspicuous ways,In the long run,

the truth will emerge,and DoorKnobs’s perceived market share s must

equal its actual market share x.

(a) If the perceived market share is s,then the demand curve for Door-

Knobs is given by Equation 1,On the graph below,draw the demand

curve relating price to actual market share in the case in which Door-

Knobs’s perceived market share is s =1=2,Label this curve s =1=2.

(b) On the demand curve that you just drew with s =1=2,mark a

red dot on the point at which the actual market share of DoorKnobs is

1/2,(This is the point on the demand curve directly above x =1=2.)

What is the price at which half of the computer owners actually want to

buy DoorKnobs,given that everybody believes that half of all computer

owners want to buy DoorKnobs? $64

410 INFORMATION TECHNOLOGY (Ch,34)

(c) On the same graph,draw and label a separate demand curve for the

case where DoorKnobs’s perceived market share s takes on each of the

following values,s =1/8,1/4,3/4,7/8,1.

2 4 6 8 10 12 14 16

32

64

96

128

160

192

224

256

Actual Market Share (in sixteenths)

Willingness to Pay

0

S=1/8

S=1/4

S=1/2

S=3/4

S=7/8

S=1

(d) On the demand curve for a perceived market share of s =1=4,put

a red dot on the point at which the actual market share of DoorKnobs

is 1/4,(This is the point on this demand curve directly above x =1=4.)

If the perceived market share of DoorKnobs is 1/4,at what price is the

actual market share of DoorKnobs also 1/4? $48

(e) Just as you did for s =1=2ands =1=4,make red marks on the

demand curves corresponding to s = 1/8,3/4,7/8,and 1,showing the

price at which the actual market share is s,given that the perceived

market share is s.

(f) Let us now draw the long-run demand curve for DoorKnobs,where we

assume that computer owners’ perceived market shares s are the same as

the actual market shares x,If this is the case,it must be that s = x,so

the demand curve is given by p = 256x(1?x),On the graph above,plot

a few points on this curve and sketch in an approximation of the curve.

(Hint,Note that the curve you draw must go through all the red points

that you have already plotted.)

(g) Suppose that MightySoft sets a price of $48 for DoorKnobs and sticks

with that price,There are three di erent perceived market shares such

that the fraction of consumers who would actually want to buy Door-

Knobs for $48 is equal to the perceived market share,One such perceived

NAME 411

market share is 0,What are the other two possibilities? s =1=4

and s =3=4

(h) Suppose that by using its advertising and media influence,MightySoft

can temporarily set its perceived market share at any number between

0 and 1,If DoorKnobs’s perceived market share is x and if MightySoft

charges a price p = 256x(1?x),the actual market fraction will also be x

and the earlier perceptions will be reinforced and maintained,Assuming

that MightySoft chooses a perceived market share x and a price that

makes the actual market share equal to the perceived market share,what

market share x should MightySoft choose in order to maximize its revenue

and what price should it charge in order to maintain this market share?

(Hint,Revenue is px = 256x

2

(1?x).) Use calculus and show your

work,x =2=3,The first-order condition is

d

dx

256(x

2

x

3

)=0,This implies 2x =3x

2

,

which implies that x =2=3 or x =0,The

second order condition is satisfied only

when x =2=3,Price should be $256 1=3

2=3 = $56:89.

34.2 (1) Suppose that demand for DoorKnobs is as given in the previous

problem,and assume that the perceived market share in any period is

equal to the actual market share in the previous period,Then where x

t

is the actual market share in period t,the equation p = 256x

t?1

(1?x

t

)

is satis ed,Rearranging this equation,we nd that x

t

=1?(p=256x

t?1

)

whenever p=256x

t?1

1,If p=256x

t?1

0,then x

t

=0,Withthis

formula,if we know actual market share for any time period,we can

calculate market share for the next period.

Let us assume that DoorKnobs sets the price at p = $32 and never

changes this price,(To answer the following questions,you will nd a

calculator useful.)

(a) If the actual market share in the rst period was 1/2,nd the actual

market share in the second period,75,the third period,833.

Write down the actual market shares for the next few periods,8529,

8534,Do they seem to be approaching a limit? If so,what?

.853553.

412 INFORMATION TECHNOLOGY (Ch,34)

(b) Notice that when price is held constant at p,if DoorKnobs’s mar-

ket share converges to a constant x,itmustbethat x =1?(p=256 x).

Solve this equation for x in the case where p = $32,What do you make

of the fact that there are two solutions? This equation

implies x

2

x +1=8=0,Solutions are

x =0:85355 and x =0:14645,Both are

equilibrium market shares with a price of

$32.

34.3 (1) A group of 13 consumers are considering whether to connect to a

new computer network,Consumer 1 has an initial value of $1 for hooking

up to the network,consumer 2 has an initial value of $2,consumer 3 has

an iinitial value of $3,and so on up to consumer 13,Each consumer’s

willingness to pay to connect to the network depends on the total number

of persons who are connected to it,In fact,for each i,consumeri’s

willingness to pay to connect to the network is i times the total number

of persons connected,Thus if 5 people are connected to the network,

consumer 1’s willingness to pay is $5,consumer 2’s willingness to pay is

$10 and so on.

(a) What is the highest price at which 9 customers could hook up to the

market and all of them either make a pro t or break even? $45

(b) Suppose that the industry that supplies the computer network is com-

petitive and that the cost of hooking up each consumer to the network is

$45,Suppose that consumers are very conservative and nobody will sign

up for the network unless her buyer value will be at least as high as the

price she paid as soon as she signs up,How many people will sign up if

the price is $45? 0

(c) Suppose that the government o ers to subsidize \pioneer users" of the

system,The rst two users are allowed to connect for $10 each,After

the rst two users are hooked up,the government allows the next two

to connect for $25,After that,everyone who signs up will have to pay

the full cost of $45,Assume that users remain so conservative that will

sign up only if their buyer values will be at least equal to the price they

are charged when they connect,With the subsidy in place,how many

consumers in toto will sign up for the network? 9

34.4 (2) Professor Kremepu has written a new,highly simpli ed eco-

nomics text,Microeconomics for the Muddleheaded,which will be pub-

lished by East Frisian Press,The rst edition of this book will be in print

for two years,at which time it will be replaced by a new edition,East

NAME 413

Frisian Press has already made all its xed cost investments in the book

and must pay a constant marginal cost of $c for each copy that it sells.

Let p

1

be the price charged for new copies sold in the rst year of

publication and let p

2

be the price charged for new copies sold in the

second year of publication,The publisher and the students who buy

the book are aware that there will be an active market for used copies of

Microeconomics for the Muddleheaded one year after publication and that

used copies of the rst edition will have zero resale value two years after

publication,At the end of the rst year of publication,students can resell

their used textbooks to bookstores for 40% of the second-year price,p

2

.

The net cost to a student of buying the book in the rst year,using it

for class,and reselling it at the end of the year is p

1

0:4p

2

.Thenumber

of copies demanded in the rst year of publication is given by a demand

function,q

1

= D

1

(p

1

0:4p

2

).

Some of the students who use the book in the rst year of publication

will want to keep their copies for future reference,and some will damage

their books so that they cannot be resold,The cost of keeping one’s old

copy or of damaging it is the resale price 0:4p

2

,The number of books that

are either damaged or kept for reference is given by a \keepers" demand

function,D

k

(0:4p

2

),It follows that the number of used copies available

at the end of the rst year will be D

1

(p

1

0:4p

2

)?D

k

(0:4p

2

).

Students who buy Microeconomics for the Muddleheaded in the sec-

ond year of publication will not be able to resell their used copies,since

a new edition will then be available,These students can,however,buy

either a new copy or a used copy of the book,For simplicity of calcula-

tions,let us assume that students are indi erent between buying a new

copy or a used copy and that used copies cost the same as new copies in

the book store,(The results would be the same if students preferred new

to used copies,but bookstores priced used copies so that students were

indi erent between buying new and used copies.) The total number of

copies,new and used,that are purchased in the second year of publication

is q

2

= D

2

(p

2

).

(a) Write an expression for the number of new copies that East Frisian

Press can sell in the second year after publication if it sets prices p

1

in

year 1 and p

2

in year 2,D

2

(p

2

)?D

1

(p

1

:4p

2

)+D

k

(:4p

2

).

(b) Write an expression for the total number of new copies of Microeco-

nomics for the Muddleheaded that East Frisian can sell over two years at

prices p

1

and p

2

in years 1 and 2,D

1

(p

1

:4p

2

)+D

2

(p

2

)?

D

1

(p

1

:4p

2

)+D

k

(:4p

2

)=D

2

(p

2

)+D

k

(:4p

2

).

(c) Would the total number of copies sold over two years increase,de-

crease,or remain constant if p

1

were increased and p

2

remained constant?

It would remain constant.

414 INFORMATION TECHNOLOGY (Ch,34)

(d) Write an expression for the total revenue that East Frisian Press will

receive over the next two years if it sets prices p

1

and p

2

,p

1

D

1

(p

1

:4p

2

)+p

2

(D

2

(p

2

)?D

1

(p

1

:4p

2

)+D

k

(:4p

2

)) = (p

1

p

2

)D

1

(p

1

:4p

2

)+p

2

(D

2

(p

2

)+D

k

(:4p

2

)):

(e) To maximize its total pro ts over the next two years,East Frisian

must maximize the di erence between its total revenue and its variable

costs,Show that this di erence can be written as

(p

1

p

2

)D

1

(p

1

:4p

2

)+(p

2

c)

D

2

(p

2

)+D

k

(:4p

2

)

:

Variable cost is c(D

2

(p

2

)+D

k

(:4p

2

)).

Subtract this from previous answer.

(f) Suppose that East Frisian has decided that it must charge the same

price for the rst edition in both years that it is sold,Thus it must

set p = p

1

= p

2

,Write an expression for East Frisian’s revenue net

of variable costs over the next two years as a function of p.

(p?c)(D

2

(p)+D

k

(:4p))

34.5 (2) Suppose that East Frisian Press,discussed in the previous

problem,has a constant marginal cost of c = $10 for each copy of Micro-

economics for the Muddleheaded that it sells and let the demand functions

be

D

1

(p

1

0:4p

2

) = 100 (90?p

1

+0:4p

2

)

D

2

(p

2

) = 100(90?p

2

):

The number of books that people either damage or keep for reference

after the rst year is

D

k

(0:4p

2

) = 100(90?0:8p

2

):

(This assumption is consistent with the assumption that everyone’s will-

ingness to pay for keeping the book is half as great as her willingness to

pay to have the book while she is taking the course.) Assume that East

Frisian Press is determined to charge the same price in both years,so that

p

1

= p

2

= p.

NAME 415

(a) If East Frisian Press charges the same price p for Microeconomics for

the Muddleheaded in the rst and second years,show that the total sales

of new copies over the two years are equal to

18;000?180p:

Total sales are D

2

(p)+D

k

(:4p

2

) = 100(90?

p) + 100(90?:8p)) = 18;000?180p

(b) Write an expression for East Frisian’s total revenue,net of variable

costs,over the rst two years as a function of the price p,(p?

10)(18;000?180p)=19;800p?180p

2

180;000

(c) Solve for the price p that maximizes its total revenue net of variable

costs over the rst two years,p = $55,At this price,the net cost

to students in the rst year of buying the text and reselling it is $33.

The total number of copies sold in the rst year will be 5,700,The

total number of copies that are resold as used books is 1,100,The

total number of copies purchased by students in the second year will be

3,500,(Remember students in the second year know that they

cannot resell the book,so they have to pay the full price p for using it.)

The total number of new copies purchased by students in the second year

will be 2,400,Total revenue net of variable costs over the two years

will be $364,500.

34.6 (2) East Frisian Press is trying to decide whether it would be prof-

itable to produce a new edition of Microeconomics for the Muddleheaded

after one year rather than after two years,If it produces a new edition

after one year,it will destroy the used book market and all copies that

are purchased will be new copies,In this case,the number of new copies

that will be demanded in each of the two years will be 100(90?p),where

p is the price charged,The variable cost of each copy sold remains $10.

(a) Write an expression for the total number of copies sold over the course

of two years if the price is p in each year 200(90-p),Also,write

an expression for total revenue net of variable costs as a function of p.

200(p?10)(90?p).

416 INFORMATION TECHNOLOGY (Ch,34)

(b) Find the price that maximizes total revenue net of variable costs.

$50.

(c) The total number of new books sold in the rst year would be

4,000,and the total number of books sold in the second year would

be 4,000.

(d) East Frisian’s total revenue net of variable costs,if it markets a new

edition after one year,will be $320,000.

(e) Would it be more pro table for East Frisian Press to produce a new

edition after one year or after two years? After two years.

Which would be better for students? (Hint,The answer is not the same

for all students.) After two years is better for

students who take the course in the first

year of publication and plan to sell.

After one year is better for the other

students.

34.7 (3) Suppose that East Frisian Press publishes a new edition only af-

ter two years and that demands and costs are as in the previous problems.

Suppose that it sets two di erent prices p

1

and p

2

in the two periods.

(a) Write an expression for the total number of new copies sold at prices

p

1

and p

2

and show that this number depends on p

2

but not on p

1

.

100 ((70?p

1

+:4p

2

) + (140?1:8p

2

)?(70?p

1

+:4p

2

)) =

100(140?1:8p

2

)

(b) Show that at prices p

1

and p

2

,the di erence between revenues and

variablecostsisequalto

100

parenleftbig

90p

1

+ 108p

2

+1:4p

1

p

2

p

2

1

2:2p

2

2

1;800

:

This difference is 100(p

1

p

2

)(90?p

1

+

:4p

2

)+(p

2

10)(180?1:8p

2

),Expand this

expression.

NAME 417

(c) Calculate the prices p

1

and p

2

that maximize the di erence between

total revenue and variable costs and hence maximize pro ts,p

1

=

$80,p

2

= $50

(d) If East Frisian Press chooses its pro t-maximizing p

1

and p

2

,compare

the cost of using Microeconomics for the Muddleheaded for a student who

buys the book when it is rst published and resells it at the end of the

rst year with the cost for a student who buys the book at the beginning

of the second year and then discards it,The former has a

net cost of $80?:4 50 = $60 and the latter

has a cost of $50.

34.8 (2) The Silicon Valley company Intoot produces checkwriting soft-

ware,The program itself,Fasten,sells for $50 and includes a package of

checks,Check re ll packets for Fasten cost $20 to produce and Intoot sells

the checks at cost,Suppose that a consumer purchases Fasten for $50 in

period 1 and spends $20 on checks in each subsequent period,Assume

for simplicity that the consumer uses the program for an in nite number

of periods.

(a) If the interest rate is r =,10 per period,what is the present value

of the stream of payments made by the consumer? (Hint,a stream of

payments of x starting next period has a present value of x=r.) The total

cost of ownership of Fasten is 50+20/.10 = $250.

(b) Fasten’s competitor produces an equally e ective product called

Czechwriter,Czechwriter can do everything Fasten can do and vice versa

except that Fasten cannot use check re ll packets that are sold by anyone

other than Fasten,Czechwriter also sells for $50 and sells its checks for

$20 per period,A Fasten customer can switch to Czechwriter simply by

purchasing the program,This means his switching costs are $50

(c) Fasten is contemplating raising the price of checks to $30 per period.

If so,will its customers switch to Czechwriter? Explain,Yes,the

present value of continuing to use Fasten

are $300 while the costs of switching to

Czechwriter are $250.

418 INFORMATION TECHNOLOGY (Ch,34)

(d) Fasten contemplates raising the price of checks to $22 per period,Will

its customers switch? No,The present value of

continuing to use Fasten are $220 while the

present value of switching to Czechwriter

is $250.

(e) At what price for checks will Fasten’s customers just be indi erent to

switching? (Hint,Let x be this amount,Compare the present value of

staying with Fasten with the present value of switching to Czechwriter.)

Solve the equation x=:10 = 250 to find

x =25.

(f) If it charges the highest price that it can without making its customers

switch,what pro t does Fasten make on checks from each of its customers

per period? $5,What is the present value of the pro t per customer

that Fasten gets if it sets the price of checks equal to the number deter-

minedinthelastquestion? PV =5=:10 = 50,How does this

compare to the customer switching cost? It is the same.

(g) Suppose now that the cost of switching also involves several hours

of data conversion that the consumer values at $100,The total cost of

switching is the cost of the new program plus the data conversion cost

which is $150.

(h) Making allowances for the cost of data conversion,what is the highest

pricethatIntootcanchargeforitschecks? Solve x=:10 =

250 + 100 for x =35,What is the present value of pro t

from this price? $150,How does this compare to total switching

costs? It is the same.

(i) Suppose that someone writes a computer program that eliminates

the cost of converting data and makes this program available for free.

Suppose that Intoot continues to price its check re ll packages at $25,A

new customer is contemplating buying Fasten at a price of $50 and paying

$25 per period for checks,versus paying $50 for Czechwriter and paying

$20 for checks,If the functionality of the software is identical,which will

the consumer buy? Czechwriter.

NAME 419

(j) Intoot decides to distribute a coupon that o ers a discount of $50

o of the regular purchase price,What price would it have to set to

make consumers indi erent between purchasing Fasten and Czechwriter?

Solve 50 + 25=:10?d =50+20=:10 to find

that the discount should be $50.

(k) Suppose that consumers are shortsighted and only look at the cost

of the software itself,neglecting the cost of the checks,Which program

would they buy if Intoot o ered this coupon? Fasten,How might

Czechwriter respond to the Fasten o er? Issue its own

coupon for $50 and raise the price of its

checks to $35.

34.9 (2) Sol Microsystems has recently invented a new language,Guava,

which runs on a proprietary chip,the Guavachip,The chip can only be

used to run Guava,and Guava can only run on the Guavachip,Sol

estimates that if it sells the chip for a price p

c

and the language for a

price p

g

,the demand for the chip-language system will be

x = 120?(p

c

+p

g

):

(a) Sol initially sets up two independent subsidiaries,one to produce the

chip and one to produce the language,Each of the subsidiaries will price

its product so as to maximize its pro ts,while assuming that a change in

its own price will not a ect the pricing decision of the other subsidiary.

Assume that marginal costs are negligible for each company,If the price

of the language is set at p

g

,the chip company’s pro t function (neglecting

xed costs) is [120?p

c

p

g

]p

c

.

(b) Di erentiate this pro t function with respect to p

c

and set the result

equal to zero to calculate the optimal choice of p

c

as a function of p

g

.

p

c

= 120?2p

g

:

(c) Now consider the language subsidiary’s pricing decision,The optimal

choice of p

g

as a function of p

c

is p

g

= 120?2p

c

:

(d) Solving these two equations in two unknowns,we nd that p

c

=

40 and p

g

= 40,sothatp

c

+p

g

= 80

420 INFORMATION TECHNOLOGY (Ch,34)

(e) Sol Microsystems decides that the independent subsidiary system is

cumbersome,so it sets up Guava Computing which sells a bundled system

consisting of the chip and the language,Let p be the price of the bundle.

Guava Computing’s pro t function is [120?p]p.

(f) Di erentiate this pro t with respect to p and set the resulting expres-

sion to zero to determine p = 60.

(g) Compare the prices charged by the integrated system and the separate

subsidiaries,Which is lower? Integrated system,Which

is better for consumers? Integrated system,Which makes

more pro t? Integrated system.

34.10 (2) South Belgium Press produces the academic journal Nano-

economics,which has a loyal following among short microeconomists,and

Gigaeconomics,a journal for tall macroeconomists,It o ers a license for

the electronic version of each journal to university libraries at a subscrip-

tion cost per journal of $1,000 per year,The 200 top universities all

subscribe to both journals,each paying $2,000 per year to South Bel-

gium,By revealed preference,their willingness to pay for each journal is

at least $1,000.

(a) In an attempt to lower costs,universities decide to form pairs,with

one member of each pair subscribing to Nanoeconomics and one member

of each pair subscribing to Gigaeconomics,They agree to use interlibrary

loan to share the other journal,Since the copies are electronic,there is

no incremental cost to doing this,Under this pairing scheme,how many

subscriptions of each journal will South Belgium sell? 100

.

(b) In order to stem the revenue hemorrhage,South Belgium raises the

price of each journal,Assuming library preferences and budgets haven’t

changed,how high can they set this price? They can raise

the price to $2,000,since libraries have

already indicated that they are willing

to pay this much for the pair of journals.

(c) How does library expenditure and South Belgium’s revenue compare

to those of the previous regime? They remain the same.

NAME 421

(d) If there were a cost of interlibrary loan,how would your an-

swer change? Assuming they still bought

both journals,libraries would be worse

off since they would have to pay the

transactions cost for interlibrary loan.

422 INFORMATION TECHNOLOGY (Ch,34)

Chapter 35 NAME

Public Goods

Introduction,In previous chapters we studied sel sh consumers con-

suming private goods,A unit of private goods consumed by one person

cannot be simultaneously consumed by another,If you eat a ham sand-

wich,Icannoteatthesamehamsandwich,(Ofcoursewecanbotheat

ham sandwiches,but we must eat di erent ones.) Public goods are a dif-

ferent matter,They can be jointly consumed,You and I can both enjoy

looking at a beautiful garden or watching reworks at the same time,The

conditions for e cient allocation of public goods are di erent from those

for private goods,With private goods,e ciency demands that if you and

I both consume ham sandwiches and bananas,then our marginal rates of

substitution must be equal,If our tastes di er,however,we may consume

di erent amounts of the two private goods.

If you and I live in the same town,then when the local reworks

show is held,there will be the same amount of reworks for each of us.

E ciency does not require that my marginal rate of substitution between

reworks and ham sandwiches equal yours,Instead,e ciency requires

that the sum of the amount that viewers are willing to pay for a marginal

increase in the amount of reworks equal the marginal cost of reworks.

This means that the sum of the absolute values of viewers’ marginal rates

of substitution between reworks and private goods must equal the mar-

ginal cost of public goods in terms of private goods.

Example,A quiet midwestern town has 5,000 people,all of whom are in-

terested only in private consumption and in the quality of the city streets.

The utility function of person i is U(X

i;G)=X

i

+A

i

G?B

i

G

2

,whereX

i

is the amount of money that person i has to spend on private goods and

G is the amount of money that the town spends on xing its streets,To

nd the Pareto optimal amount of money for this town to spend on xing

its streets,we must set the sum of the absolute values of marginal rates of

substitution between public and private goods equal to the relative prices

of public and private goods,In this example we measure both goods in

dollar values,so the price ratio is 1,The absolute value of person i’s

marginal rate of substitution between public goods and private goods is

the ratio of the marginal utility of public goods to the marginal utility of

private goods,The marginal utility of private goods is 1 and the marginal

utility of public goods for person i is A

i

B

i

G,Therefore the absolute

value of person i’s MRS is A

i

B

i

G and the sum of absolute values

of marginal rates of substitution is

P

i

(A

i

B

i

G)=

P

i

A

i

(

P

B

i

)G.

Therefore Pareto e ciency requires that

P

i

A

i

(

P

i

B

i

)G =1,Solving

this for G,wehaveG =(

P

i

A

i

1)=

P

i

B

i

.

35.1 (0) Muskrat,Ontario,has 1,000 people,Citizens of Muskrat con-

sume only one private good,Labatt’s ale,There is one public good,the

town skating rink,Although they may di er in other respects,inhabitants

424 PUBLIC GOODS (Ch,35)

have the same utility function,This function is U(X

i;G)=X

i

100=G,

where X

i

is the number of bottles of Labatt’s consumed by citizen i and

G is the size of the town skating rink,measured in square meters,The

price of Labatt’s ale is $1 per bottle and the price of the skating rink is

$10 per square meter,Everyone who lives in Muskrat has an income of

$1,000 per year.

(a) Write down an expression for the absolute value of the marginal rate

of substitution between skating rink and Labatt’s ale for a typical citizen.

100=G

2

What is the marginal cost of an extra square meter of skating

rink (measured in terms of Labatt’s ale)? 10.

(b) Since there are 1,000 people in town,all with the same marginal

rate of substitution,you should now be able to write an equation that

states the condition that the sum of absolute values of marginal rates of

substitution equals marginal cost,Write this equation and solve it for the

Pareto e cient amount of G,1;000

100

G

2

=10.SoG = 100.

(c) Suppose that everyone in town pays an equal share of the cost of

the skating rink,Total expenditure by the town on its skating rink will

be $10G,Then the tax bill paid by an individual citizen to pay for the

skating rink is $10G=1;000 = $G=100,Every year the citizens of Muskrat

vote on how big the skating rink should be,Citizens realize that they will

have to pay their share of the cost of the skating rink,Knowing this,a

citizen realizes that if the size of the skating rink is G,then the amount

of Labatt’s ale that he will be able to a ord is 1;000?G=100.

(d) Therefore we can write a voter’s budget constraint as X

i

+G=100 =

1;000,In order to decide how big a skating rink to vote for,a voter simply

solves for the combination of X

i

and G that maximizes his utility subject

to his budget constraint and votes for that amount of G.HowmuchG is

that in our example? G = 100.

(e) If the town supplies a skating rink that is the size demanded by the

voters will it be larger than,smaller than,or the same size as the Pareto

optimal rink? The same.

(f) Suppose that the Ontario cultural commission decides to promote

Canadian culture by subsidizing local skating rinks,The provincial gov-

ernment will pay 50% of the cost of skating rinks in all towns,The costs

of this subsidy will be shared by all citizens of the province of Ontario.

There are hundreds of towns like Muskrat in Ontario,It is true that to

pay for this subsidy,taxes paid to the provincial government will have

to be increased,But there are hundreds of towns from which this tax

NAME 425

is collected,so that the e ect of an increase in expenditures in Muskrat

on the taxes its citizens have to pay to the state can be safely neglected.

Now,approximately how large a skating rink would citizens of Muskrat

vote for? G = 100

p

2,(Hint,Rewrite the budget constraint for

individuals observing that local taxes will be only half as large as before

and the cost of increasing the size of the rink only half as much as before.

Then solve for the utility-maximizing combination.)

(g) Does this subsidy promote economic e ciency? No.

35.2 (0) Ten people have dinner together at an expensive restaurant

and agree that the total bill will be divided equally among them.

(a) What is the additional cost to any one of them of ordering an appetizer

that costs $20? $2.

(b) Explain why this may be an ine cient system,Each pays

less than full cost of own meal,so all

overindulge.

35.3 (0) Cowflop,Wisconsin,has 1,000 people,Every year they have

a reworks show on the Fourth of July,The citizens are interested in

only two things|drinking milk and watching reworks,Fireworks cost 1

gallon of milk per unit,People in Cowflop are all pretty much the same.

In fact,they have identical utility functions,The utility function of each

citizen i is U

i

(x

i;g)=x

i

+

p

g=20,where x

i

is the number of gallons

of milk per year consumed by citizen i and g is the number of units of

reworks exploded in the town’s Fourth of July extravaganza,(Private

use of reworks is outlawed.)

(a) Solve for the absolute value of each citizen’s marginal rate of substi-

tution between reworks and milk,1=(40

p

g).

(b) Find the Pareto optimal amount of reworks for Cowflop,625.

35.4 (0) Bob and Ray are two hungry economics majors who are sharing

an apartment for the year,In a flea market they spot a 25-year-old sofa

that would look great in their living room.

Bob’s utility function is u

B

(S;M

B

)=(1+S)M

B

,and Ray’s utility

function is u

R

(S;M

R

)=(2+S)M

R

,In these expressions M

B

and M

R

are

the amounts of money that Bob and Ray have to spend on other goods,

S = 1 if they get the sofa,and S = 0 if they don’t get the sofa,Bob has

W

B

dollars to spend,and Ray has W

R

dollars.

426 PUBLIC GOODS (Ch,35)

(a) What is Bob’s reservation price for the sofa? Solve W

B

=

2(W

B

p

B

) to get p

B

= W

B

=2.

(b) What is Ray’s reservation price for the sofa? Solve 2W

R

=

3(W

R

p

R

),which gives p

R

= W

R

=3.

(c) If Bob has a total of W

B

= $100 and Ray has a total of W

R

= $75

to spend on sofas and other stu,they could buy the sofa and have a

Pareto improvement over not buying it so long as the cost of the sofa is

no greater than $75.

35.5 (0) Bonnie and Clyde are business partners,Whenever they work,

they have to work together,Their only source of income is pro t from

their partnership,Their total pro t per year is 50H,whereH is the

number of hours that they work per year,Since they must work together,

they both must work the same number of hours,so the variable \hours of

labor" is like a public \bad" for the two person community consisting of

Bonnie and Clyde,Bonnie’s utility function is U

B

(C

B;H)=C

B

:02H

2

and Clyde’s utility function is U

C

(C

C;H)=C

C

:005H

2

,whereC

B

and

C

C

are the annual amounts of money spent on consumption for Bonnie

and for Clyde.

(a) If the number of hours that they both work is H,what is the ratio

of Bonnie’s marginal utility of hours of work to her marginal utility of

private goods:04H,What is the ratio of Clyde’s marginal utility

of hours of work to his marginal utility of private goods:01H.

(b) If Bonnie and Clyde are both working H hours,then the total amount

of money that would be needed to compensate them both for having to

work an extra hour is the sum of what is needed to compensate Bonnie

and the amount that is needed to compensate Clyde,This amount is

approximately equal to the sum of the absolute values of their marginal

rates of substitution between work and money,Write an expression for

this amount as a function of H.,05H,How much extra money will

they make if they work an extra hour? $50.

(c) Write an equation that can be solved for the Pareto optimal number

of hours for Bonnie and Clyde to work.,05H =50.

Find the Pareto optimal H,H =1;000,(Hint,Notice that

this model is formally the same as a model with one public good H and

one private good,income.)

NAME 427

35.6 (0) Lucy and Melvin share an apartment,They spend some of

their income on private goods like food and clothing that they consume

separately and some of their income on public goods like the refrigerator,

the household heating,and the rent,which they share,Lucy’s utility

function is 2X

L

+G and Melvin’s utility function is X

M

G,whereX

L

and

X

M

are the amounts of money spent on private goods for Lucy and for

Melvin and where G is the amount of money that they spend on public

goods,Lucy and Melvin have a total of $8,000 per year between them to

spend on private goods for each of them and on public goods.

(a) What is the absolute value of Lucy’s marginal rate of substitution

between public and private goods? 1=2,What is the absolute value

of Melvin’s? X

M

=G.

(b) Write an equation that expresses the condition for provision of the

Pareto e cient quantity of the public good,1=2+X

M

=G =1.

(c) Suppose that Melvin and Lucy each spend $2,000 on private goods

for themselves and they spend the remaining $4,000 on public goods,Is

this a Pareto e cient outcome? Yes.

(d) Give an example of another Pareto optimal outcome in which Melvin

gets more than $2,000 and Lucy gets less than $2,000 worth of private

goods,One example,Melvin gets $2,500; Lucy

gets $500 and G =$5;000.

(e) Give an example of another Pareto optimum in which Lucy gets

more than $2,000,Lucy gets $5,000; Melvin gets

$1;000 and G =$2;000.

(f) Describe the set of Pareto optimal allocations,The allocations

that satisfy the equations X

M

=G =1=2 and

X

L

+X

M

+G =$8;000.

(g) The Pareto optima that treat Lucy better and Melvin worse will have

(more of,less of,the same amount of) public good as the Pareto optimum

that treats them equally,Less of.

428 PUBLIC GOODS (Ch,35)

35.7 (0) This problem is set in a fanciful location,but it deals with a

very practical issue that concerns residents of this earth,The question

is,\In a Democracy,when can we expect that a majority of citizens will

favor having the government supply pure private goods publicly?" This

problem also deals with the e ciency issues raised by public provision

of private goods,We leave it to you to see whether you can think of

important examples of publicly supplied private goods in modern Western

economies.

On the planet Jumpo there are two goods,aerobics lessons and

bread,The citizens all have Cobb-Douglas utility functions of the form

U

i

(A

i;B

i

)=A

1=2

i

B

1=2

i

,whereA

i

and B

i

are i’s consumptions of aerobics

lessons and bread,Although tastes are all the same,there are two di er-

ent income groups,the rich and the poor,Each rich creature on Jumpo

has an income of 100 fondas and every poor creature has an income of

50 fondas (the currency unit on Jumpo),There are two million poor

creatures and one million rich creatures on Jumpo,Bread is sold in the

usual way,but aerobics lessons are provided by the state despite the fact

that they are private goods,The state gives the same amount of aerobics

lessons to every creature on Jumpo,The price of bread is 1 fonda per

loaf,The cost to the state of aerobics lessons is 2 fondas per lesson,This

cost of the state-provided lessons is paid for by taxes collected from the

citizens of Jumpo,The government has no other expenses than providing

aerobics lessons and collects no more or less taxes than the amount needed

to pay for them,Jumpo is a democracy,and the amount of aerobics to

be supplied will be determined by majority vote.

(a) Suppose that the cost of the aerobics lessons provided by the state

is paid for by making every creature on Jumpo pay an equal amount of

taxes,On planets,such as Jumpo,where every creature has exactly one

head,such a tax is known as a \head tax." If every citizen of Jumpo gets

20 lessons,how much will be total government expenditures on lessons?

120 million fondas,How much taxes will every citizen

have to pay? 40 fondas,If 20 lessons are given,how much will a

rich creature have left to spend on bread after it has paid its taxes? 60

fondas,How much will a poor creature have left to spend on bread

after it has paid its taxes? 10 fondas.

(b) More generally,when everybody pays the same amount of taxes,if x

lessons are provided by the government to each creature,the total cost

to the government is 6 million times x and the taxes that one

creature has to pay is 2 times x.

NAME 429

(c) Since aerobics lessons are going to be publicly provided with every-

body getting the same amount and nobody able to get more lessons from

another source,each creature faces a choice problem that is formally the

same as that faced by a consumer,i,who is trying to maximize a Cobb-

Douglas utility function subject to the budget constraint 2A + B = I,

whereI is its income,Explain why this is the case,If A lessons

are provided,your taxes are 2A fondas.

After taxes,you have I?2A fondas to

spend on B.

(d) Suppose that the aerobics lessons are paid for by a head tax and all

lessons are provided by the government in equal amounts to everyone.

How many lessons would the rich people prefer to have supplied? 25.

How many would the poor people prefer to have supplied? 12.5.

(Hint,In each case you just have to solve for the Cobb-Douglas demand

with an appropriate budget.)

(e) If the outcome is determined by majority rule,how many aerobics

lessons will be provided? 12.5,How much bread will the rich get?

75,How much bread will the poor get? 25.

(f) Suppose that aerobics lessons are \privatized," so that no lessons are

supplied publicly and no taxes are collected,Every creature is allowed to

buy as many lessons as it likes and as much bread as it likes,Suppose

that the price of bread stays at 1 fonda per unit and the price of lessons

stays at 2 fondas per unit,How many aerobics lessons will the rich get?

25,How many will the poor get? 12.5,How much bread will the

rich get? 50,How much bread will the poor get? 25.

(g) Suppose that aerobics lessons remain publicly supplied but are paid

for by a proportional income tax,The tax rate is set so that tax rev-

enue pays for the lessons,If A aerobics lessons are o ered to each

creature on Jumpo,the tax bill for a rich person will be 3A fondas

and the tax bill for a poor person will be 1:5A fondas,If A lessons

are given to each creature,show that total tax revenue collected will

be the total cost of A lessons,There are 2,000,000

poor and 1,000,000 rich,total revenue is

2;000;000 1:5A+1;000;000 3A =6;000;000A.

430 PUBLIC GOODS (Ch,35)

There are 3,000,000 people in all,If each

gets A lessons and lessons cost 2 fondas,

total cost is 6;000;000A.

(h) With the proportional income tax scheme discussed above,what bud-

get constraint would a rich person consider in deciding how many aerobics

lessons to vote for? 3A + B = 100,What is the relevant bud-

get constraint for a poor creature? 1:5A + B =50,With these

tax rates,how many aerobics lessons per creature would the rich favor?

50=3,How many would the poor favor? 50=3,What quantity of

aerobics lessons per capita would be chosen under majority rule? 50=3.

How much bread would the rich get? 50,How much bread would the

poor get? 25.

(i) Calculate the utility of a rich creature under a head tax.

p

937:5

Under privatization.

p

1;250,Under a proportional income tax.

p

833:3,(Hint,In each case,solve for the consumption of bread and

the consumption of aerobics lessons that a rich person gets,and plug these

into the utility function.) Now calculate the utility of each poor creature

under the head tax.

p

312:5,Under privatization.

p

312:5,Un-

der the proportional income tax.

p

416:67,(Express these utilities

as square roots rather than calculating out the roots.)

(j) Is privatization Pareto superior to the head tax? Yes,Is a propor-

tional income tax Pareto superior to the head tax? No,Is privatization

Pareto superior to the proportional income tax? No,Explain the last

two answers,Rich prefer privatization,poor

prefer proportional income tax.

Chapter 36 NAME

Information

Introduction,The economics of information and incentives is a rela-

tively new branch of microeconomics,in which much intriguing work is

going on,This chapter shows you a sample of these problems and the

way that economists think about them.

36.1 (0) There are two types of electric pencil-sharpener producers.

\High-quality" manufacturers produce very good sharpeners that con-

sumers value at $14,\Low-quality" manufacturers produce less good ones

that are valued at $8,At the time of purchase,customers cannot distin-

guish between a high-quality product and a low-quality product; nor can

they identify the manufacturer,However,they can determine the quality

of the product after purchase,The consumers are risk neutral; if they

have probability q of getting a high-quality product and 1?q of getting

a low-quality product,then they value this prospect at 14q +8(1?q).

Each type of manufacturer can manufacture the product at a constant

unit cost of $11.50,All manufacturers behave competitively.

(a) Suppose that the sale of low-quality electric pencil-sharpeners is ille-

gal,so that the only items allowed to appear on the market are of high

quality,What will be the equilibrium price? $11.50.

(b) Suppose that there were no high-quality sellers,How many low-quality

sharpeners would you expect to be sold in equilibrium? Sellers

won’t sell for less than $11.50,consumers

won’t pay that much for low-quality product.

So in equilibrium there would be no sales.

(c) Could there be an equilibrium in which equal (positive) quantities

of the two types of pencil sharpeners appear in the market? No.

Average willingness to pay would be $11,

which is less than the cost of production.

So there would be zero trade.

432 INFORMATION (Ch,36)

(d) Now we change our assumptions about the technology,Suppose

that each producer can choose to manufacture either a high-quality or

a low-quality pencil-sharpener,with a unit cost of $11.50 for the for-

mer and $11 for the latter,what would we expect to happen in equilib-

rium? No trade,Producers would produce the

low-quality product since it has a lower

production cost,If all producers produce

low-quality output,costs will be $11 and

the willingness-to-pay for low quality is

$8.

(e) Assuming that each producer is able to make the production choice

described in the last question,what good would it do if the government

banned production of low-quality electric pencil-sharpeners? If

there is no ban,there will be no output

and no consumers’ surplus,If low-quality

products are banned,then in equilibrium

there is output and positive consumers’

surplus.

36.2 (0) In West Bend,Indiana,there are exactly two kinds of workers.

One kind has a (constant) marginal product worth $10 and the other kind

has a (constant) marginal product worth $15,There are equal numbers

of workers of each kind,A rm cannot directly tell the di erence between

the two kinds of workers,Even after it has hired them,it won’t be able

to monitor their work closely enough to determine which workers are of

which type.

(a) If the labor market is competitive,workers will be paid the average

value of their marginal product,This amount is $12.50.

(b) Suppose that the local community college o ers a microeconomics

course in night school,taught by Professor M,De Sade,The high-

productivity workers think that taking this course is just as bad as a

$3 wage cut,and the low-productivity workers think it is just as bad as

a $6 wage cut,The rm can observe whether or not an individual takes

the microeconomics course,Suppose that the high-productivity workers

all choose to take the microeconomics course and the low-productivity

NAME 433

workers all choose not to,The competitive wage for people who take the

microeconomics course will be $15 and the wage for people who don’t

take the microeconomics course will be $10.

(c) If there is a separating equilibrium,with high-productivity workers

taking the course and low-productivity workers not taking it,then the

net bene ts from taking the microeconomics course will be $2

for the high-productivity workers and $?1 for the low-productivity

workers,Therefore there (will be,won’t be) will be a separating

equilibrium of this type.

(d) Suppose that Professor De Sade is called o to Washington,to lec-

ture wayward representaatives on the economics of family values,His

replacement is Professor Morton Kremepu,Kremepu prides himself on

his ability to make economics \as easy as political science and as fun as

the soaps on TV." Professor Kremepu ’s claims are exaggerated,but at

least students like him better than De Sade,High-productivity workers

think that taking Kremepu ’s course is as bad as a $1 wage cut,and

low-productivity workers think that taking Kremepu ’s course is as bad

as a $4 wage cut,If the high-productivity workers all choose to take the

microeconomics course and the low-productivity workers all choose not to,

the competitive wage for people who take the microeconomics course will

be $15 and the wage for people who don’t take the microeconomics

course will be $10.

(e) If there is a separating equilibrium with high-productivity workers

taking the course and low-productivity workers not taking it,then the net

bene ts from taking Kremepu ’s microeconomics course will be $4

for the high-productivity workers and $1 for the low-productivity

workers,Therefore there (will be,won’t be) won’t be a separating

equilibrium of this type.

36.3 (1) In Enigma,Ohio,there are two kinds of workers,Klutzes

whose labor is worth $1,000 per month and Kandos,whose labor is worth

$2,500 per month,Enigma has exactly twice as many Klutzes as Kandos.

Klutzes look just like Kandos and are accomplished liars,If you ask,

they will claim to be Kandos,Kandos always tell the truth,Monitoring

individual work accomplishments is too expensive to be worthwhile,In

the old days,there was no way to distinguish the two types of labor,so

everyone was paid the same wage,If labor markets were competitive,

what was this wage? $1,500

434 INFORMATION (Ch,36)

(a) A professor who loves to talk o ered to give a free monthly lecture

on macroeconomics and personal hygiene to the employees of one small

rm,These lectures had no e ect on productivity,but both Klutzes and

Kandos found them to be excruciatingly dull,To a Klutz,each hour’s

lecture was as bad as losing $100,To a Kando,each hour’s lecture was as

bad as losing $50,Suppose that the rm gave each of its employees a pay

raise of $55 a month but insisted that he attend the professor’s lectures.

What would happen to the rm’s labor force? All Klutzes

would leave,Kandos would stay on,More

Kandos could be hired at these terms.

Klutzes would not accept job,What would happen

to the average productivity of the rm’s employees? Rise by

$1,000--from $1,500 to $2,500.

(b) Other rms noticed that those who had listened to the professor’s

lectures were more productive than those who had not,So they tried to

bid them away from their original employer,Since all those who agreed

to listen to the original lecture series were Kandos,their wage was bid up

to $2,500.

(c) After observing the \e ect of his lectures on labor productivity," the

professor decided to expand his e orts,He found a huge auditorium where

he could lecture to all the laborers in Enigma who would listen to him.

If employers believed that listening to the professor’s lectures improved

productivity by the improvement in productivity in the rst small rm

and o ered bonuses for attending the lectures accordingly,who would

attend the lectures? Everybody,Having observed this outcome,

how much of a wage premium would rms pay for those who had attended

the professor’s lectures? 0.

(d) The professor was disappointed by the results of his big lecture and

decided that if he gave more lectures per month,his pupils might \learn

more." So he decided to give a course of lectures for 20 hours a month.

Would there now be an equilibrium in which the Kandos all took his

course and none of the Klutzes took it and where those who took the

course were paid according to their true productivity? Yes,If

those who take the course get $2,500 and

people who do not get $1,000 a month,then

Kandos would take the course,since the

NAME 435

pain of 20 hours of lecture costs $1,000,

but the wage premium is $1,500,Klutzes

would not take the course,since the pain

of lectures costs $2,000 a month and the

wage premium is $1,500.

(e) What is the smallest number of hours the professor could lecture and

still maintain a separating equilibrium? 15 hours

36.4 (1) Old MacDonald produces hay,He has a single employee,Jack.

If Jack works for x hours he can produce x bales of hay,Each bale of hay

sells for $1,The cost to Jack of working x hours is c(x)=x

2

=10.

(a) What is the e cient number of bales of hay for Jack to cut? 5.

(b) If the most that Jack could earn elsewhere is zero,how much would

MacDonald have to pay him to get him to work the e cient amount?

5

2

=10 = $2:50.

(c) What is MacDonald’s net pro t? 5?2:50 = $2:50.

(d) Suppose that Jack would receive $1 for passing out leaflets,an activity

that involves no e ort whatsoever,How much would he have to receive

from MacDonald for producing the e cient number of bales of hay?

$3.50.

(e) Suppose now that the opportunity for passing out leaflets is no longer

available,but that MacDonald decides to rent his hay eld out to Jack for

a flat fee,How much would he rent it for? $2.50.

36.5 (0) In Rustbucket,Michigan,there are 200 people who want to sell

their used cars,Everybody knows that 100 of these cars are \lemons"

and 100 of these cars are \good." The problem is that nobody except the

original owners know which are which,Owners of lemons will be happy

to get rid of their cars for any price greater than $200,Owners of good

used cars will be willing to sell them for any price greater than $1,500,

but will keep them if they can’t get $1,500,There are a large number of

buyers who would be willing to pay $2,500 for a good used car,but would

pay only $300 for a lemon,When these buyers are not sure of the quality

of the car they buy,they are willing to pay the expected value of the car,

given the knowledge they have.

436 INFORMATION (Ch,36)

(a) If all 200 used cars in Rustbucket were for sale,how much would

buyers be willing to pay for a used car? $1,400,Would owners

of good used cars be willing to sell their used cars at this price? No.

Would there be an equilibrium in which all used cars are sold? No.

Describe the equilibrium that would take place in Rustbucket,Good

car owners won’t sell,Lemon owners will

sell,Price of a used car will be $300.

(b) Suppose that instead of there being 100 cars of each kind,everyone

in town is aware that there are 120 good cars and 80 lemons,How much

would buyers be willing to pay for a used car? $1,620,Would

owners of good used cars be willing to sell their used cars at this price?

Yes,Would there be an equilibrium in which all used cars are sold?

Yes,Would there be an equilibrium in which only the lemons were

sold? Yes,Describe the possible equilibrium or equilibria that would

take place in Rustbucket,One equilibrium has all

cars sold at a price of $1,620,There is

also an equilibrium where only the lemons

are sold.

36.6 (1) Each year,1,000 citizens of New Crankshaft,Pennsylvania,sell

their used cars and buy new cars,The original owners of the old cars

have no place to keep second cars and must sell them,These used cars

vary a great deal in quality,Their original owners know exactly what is

good and what is bad about their cars,but potential buyers can’t tell

them apart by looking at them,Lamentably,though they are in other

respects model citizens,the used-car owners in New Crankshaft have no

scruples about lying about their old jalopies,Each car has a value,V,

which a buyer who knew all about its qualities would be willing to pay.

There is a very large number of potential buyers,any one of which would

be willing to pay $V foracarofvalue$V:

The distribution of values of used cars on the market is quite simply

described,In any year,for any V between 0 and $2,000,the number of

used cars available for sale that are worth less than $V is V=2,Potential

used-car buyers are all risk-neutral,That is if they don’t know the value of

a car for certain,they value it at its expected value,given the information

they have.

NAME 437

Rod’s Garage in New Crankshaft will test out any used car and nd

its true value V,Rod’s Garage is known to be perfectly accurate and

perfectly honest in its appraisals,The only problem is that getting an

accurate appraisal costs $200,People with terrible cars are not going to

want to pay $200 to have Rod tell the world how bad their cars are,But

people with very good cars will be willing to pay Rod the $200 to get

their cars appraised,so they can sell them for their true values.

Let’s try to gure our exactly how the equilibrium works,which cars

get appraised,and what the unappraised cars sell for.

(a) If nobody had their car appraised,what would the market price

for used cars in North Crankshaft be and what would be the total

revenue received by used-car owners for their cars? They’d

all sell for $1,000 for total revenue of

$1,000,000.

(b) If all the cars that are worth more than $X are appraised and all

the cars that are worth less than $X are sold without appraisal,what

will the market price of unappraised used cars be? (Hint,What is the

expected value of a random draw from the set of cars worth less than

$X?) $X=2.

(c) If all the cars that are worth more than $X are appraised and all

thecarsthatareworthlessthan$X are sold without appraisal,then if

your car is worth $X,how much money would you have left if you had

it appraised and then sold it for its true value? $X?200,How

much money would you get if you sold it without having it appraised?

$X=2.

(d) In equilibrium,there will be a car of marginal quality such that all

cars better than this car will be appraised and all cars worse than this car

will be sold without being appraised,The owner of this car will be just

indi erent between selling his car unappraised and having it appraised.

What will be the value of this marginal car? Solve X=2=

X?200 to get X = $400.

(e) In equilibrium,how many cars will be sold unappraised and what

will they sell for? The worst 200 cars will be

unappraised and will sell for $200.

438 INFORMATION (Ch,36)

(f) In equilibrium,what will be the total net revenue of all owners

of used cars,after Rod’s Garage has been paid for its appraisals?

$1;000;000?800 200 = 840;000.

36.7 (2) In Pot Hole,Georgia,1,000 people want to sell their used cars.

These cars vary in quality,Original owners know exactly what their cars

are worth,All used cars look the same to potential buyers until they have

bought them; then they nd out the truth,For any number X between

0 and 2,000,the number of cars of quality lower than X is X=2,If a car

is of quality X,its original owner will be willing to sell it for any price

greater than X,If a buyer knew that a car was of quality X,she would

be willing to pay X + 500 for it,When buyers are not sure of the quality

of a car,they are willing to pay its expected value,given their knowledge

of the distribution of qualities on the market.

(a) Suppose that everybody knows that all the used cars in Pot Hole are

for sale,What would used cars sell for? $1,500,Would every

used car owner be willing to sell at this price? No,Which used

cars would appear on the market? Those worth less than

$1,500.

(b) Let X

be some number between 0 and 2,000 and suppose that all

cars of quality lower than X

are sold,but original owners keep all cars

of quality higher than X

,What would buyers be willing to pay for a

used car? X

=2 + 500,At this price,which used cars would be

for sale? Cars worth less than X

=2 + 500.

(c) Write an equation for the equilibrium value of X

,atwhichtheprice

that buyers are willing to pay is exactly enough to induce all cars of

quality less than X

into the market,X

=2 + 500 = X

,Solve

this equation for the equilibrium value of X

,X

=$1;000.

QUIZZES

This section contains short multiple-choice quizzes based on the workbook

problems in each chapter,Typically the questions are slight variations on

the workbook problems,so that if you have worked and understood the

corresponding workbook problem,the quiz question will be pretty easy.

Instructors who have adopted Workouts for their course can make use

of the test-item le o ered with the textbook,The test-item le contains

alternative versions of each quiz question in the back of Workouts,The

questions in these quizzes use di erent numerical values but the same in-

ternal logic,They can be used to provide additional problems for student

practice or for in-class quizzes.

When we teach this course we tell the students to work through all

the quiz questions in Workouts for each chapter,either by themselves

or with a study group,During the term we have a short in-class quiz

every other week or so,using the alternative versions from the test-item

le,These are essentially the Workouts quizzes with di erent numbers.

Hence,students who have done their homework nd it easy to do well on

the quizzes.

440 QUIZZES (Ch,36)

Quiz 2 NAME

The Budget Set

2.1 In Problem 2.1,if you have an income of $12 to spend,if commodity 1

costs $2 per unit,and if commodity 2 costs $6 per unit,then the equation

for your budget line can be written as

(a) x

1

=2+x

2

=6 = 12.

(b) (x

1

+x

2

)=(8) = 12.

(c) x

1

+3x

2

=6.

(d) 3x

1

+7x

2

= 13.

(e) 8(x

1

+x

2

) = 12.

2.2 In Problem 2.3,if you could exactly a ord either 6 units of x and 14

units of y,or 10 units of x and 6 units of y,then if you spent all of your

income on y,how many units of y could you buy?

(a) 26.

(b) 18.

(c) 34.

(d) 16.

(e) None of the other options are correct.

2.3 In Problem 2.4,Murphy used to consume 100 units of x and 50 units

of y when the price of x was 2 and the price of y was 4,If the price of x

rose to 5 and the price of y rose to 8,how much would Murphy’s income

have to rise so that he could still a ord his original bundle?

(a) 700.

(b) 500.

(c) 350.

(d) 1,050.

442 THE BUDGET SET (Ch,2)

(e) None of the other options are correct.

2.4 In Problem 2.7,Edmund must pay $6 each for punk rock video

casettes,If Edmund is paid $48 per sack for accepting garbage and if

his relatives send him an allowance of $384,then his budget line is de-

scribed by the equation:

(a) 6V =48G.

(b) 6V +48G = 384.

(c) 6V?48G = 384.

(d) 6V = 384?G.

(e) None of the other options are correct.

2.5 InProblem2.10,ifinthesameamountoftimethatittakesher

to read 40 pages of economics and 30 pages of sociology,Martha could

read 30 pages of economics and 50 pages of sociology,then which of these

equations describes combinations of pages of economics,E,and sociology,

S,that she could read in the time it takes to read 40 pages of economics

and 30 pages of sociology?

(a) E +S = 70.

(b) E=2+S = 50.

(c) 2E +S = 110.

(d) E +S = 80.

(e) All of the above.

2.6 In Problem 2.11,ads in the boring business magazine are read by

300 lawyers and 1,000 M.B.As,Ads in the consumer publication are

read by 250 lawyers and 300 M.B.A.’s,If Harry had $3,000 to spend

on advertising,if the price of ads in the boring business magazine were

$600 and the price of ads in the consumer magazine were $300,then the

combinations of recent M.B.A’s and lawyers with hot tubs whom he could

reach with his advertising budget would be represented by the integer

values along a line segment that runs between the two points

(a) (2,500,3,000) and (1,500,5,000).

(b) (3,000,3,500) and (1,500,6,000).

(c) (0,3,000) and (1,500,0).

NAME 443

(d) (3,000,0) and (0,6,000).

(e) (2,000,0) and (0,5,000).

2.7 In the economy of Mungo,discussed in Problem 2.12,there is a third

creature called Ike,Ike has a red income of 40 and a blue income of

10,(Recall that blue prices are 1 bcu (blue currency unit) per unit of

ambrosia and 1 bcu per unit of bubble gum,Red prices are 2 rcus (red

currency units) per unit of ambrosia and 6 rcus per unit of bubble gum.

You have to pay twice for what you buy,once in red currency,once in

blue currency.) If Ike spends all of its blue income,but not all of its red

income,then it must be that

(a) it consumes at least 5 units of bubble gum.

(b) it consumes at least 5 units of ambrosia.

(c) it consumes exactly twice as much bubblegum as ambrosia.

(d) it consumes at least 15 units of bubble gum.

(e) it consumes equal amounts of ambrosia and bubble gum.

444 THE BUDGET SET (Ch,2)

Quiz 3 NAME

Preferences

3.1 In Problem 3.1,Charlie’s indi erence curves have the equation

x

B

= constant=x

A

,where larger constants correspond to better indif-

ference curves,Charlie strictly prefers the bundle (7,15) to the following

bundle:

(a) (15,7).

(b) (8,14).

(c) (11,11).

(d) all three of these bundles.

(e) none of these bundles.

3.2 In Problem 3.2,Ambrose has indi erence curves with the equation

x

2

= constant?4x

1=2

1

,where larger constants correspond to higher indif-

ference curves,If good 1 is drawn on the horizontal axis and good 2 on

the vertical axis,what is the slope of Ambrose’s indi erence curve when

his consumption bundle is (1,6)?

(a)?1=6

(b)?6=1

(c)?2

(d)?7

(e)?1

3.3 In Problem 3.8,Nancy Lerner is taking a course from Professor Good-

heart who will count only her best midterm grade and from Professor

Stern who will count only her worst midterm grade,In one of her classes,

Nancy has scores of 50 on her rst midterm and 30 on her second midterm.

When the rst midterm score is measured on the horizontal axis and her

second midterm score on the vertical,her indi erence curve has a slope

of zero at the point (50,30),Therefore it must be that

(a) this class could be Professor Goodheart’s but couldn’t be Professor

Stern’s.

446 PREFERENCES (Ch,3)

(b) this class could be Professor Stern’s but couldn’t be Professor Good-

heart’s.

(c) this class couldn’t be either Goodheart’s or Stern’s.

(d) this class could be either Goodheart’s or Stern’s.

3.4 In Problem 3.9,if we graph Mary Granola’s indi erence curves with

avocados on the horizontal axis and grapefruits on the vertical axis,then

whenever she has more grapefruits than avocados,the slope of her indif-

ference curve is?2,Whenever she has more avocados than grapefruits,

the slope is?1=2,Mary would be indi erent between a bundle with 24

avocados and 36 grapefruits and another bundle that has 34 avocados and

(a) 28 grapefruits.

(b) 32 grapefruits.

(c) 22 grapefruits.

(d) 25 grapefruits.

(e) 26.50 grapefruits.

3.5 In Problem 3.12,recall that Tommy Twit’s mother measures the de-

parture of any bundle from her favorite bundle for Tommy by the sum

of the absolute values of the di erences,Her favorite bundle for Tommy

is (2,7){that is,2 cookies and 7 glasses of milk,Tommy’s mother’s in-

di erence curve that passes through the point (c;m)=(3;6) also passes

through

(a) (4,5).

(b) the points (2,5),(4,7),and (3,8).

(c) (2,7).

(d) the points (3,7),(2,6),and (2,8).

(e) None of the other options are correct.

3.6 In Problem 3.1,Charlie’s indi erence curves have the equation

x

B

= constant=x

A

,where larger constants correspond to better indif-

ference curves,Charlie strictly prefers the bundle (9,19) to the following

bundle:

(a) (19,9).

(b) (10,18).

(c) (15,17).

(d) More than one of these options are correct.

(e) None of the above are correct.

Quiz 4 NAME

Utility

4.1 In Problem 4.1,Charlie has the utility function U(x

A;x

B

)=x

A

x

B

.

His indi erence curve passing through 10 apples and 30 bananas will also

pass through the point where he consumes 2 apples and

(a) 25 bananas.

(b) 50 bananas.

(c) 152 bananas.

(d) 158 bananas.

(e) 150 bananas.

4.2 In Problem 4.1,Charlie’s utility function is U(A;B)=AB where

A and B are the numbers of apples and bananas,respectively,that he

consumes,When Charlie is consuming 20 apples and 100 bananas,then

if we put apples on the horizontal axis and bananas on the vertical axis,

the slope of his indi erence curve at his current consumption is

(a)?20.

(b)?5.

(c)?10.

(d)?1=5.

(e)?1=10.

4.3 In Problem 4.2,Ambrose has the utility function U(x

1;x

2

)=4x

1=2

1

+

x

2

,If Ambrose is initially consuming 81 units of nuts and 14 units of

berries,then what is the largest number of berries that he would be

willing to give up in return for an additional 40 units of nuts?

(a) 11

(b) 25

(c) 8

448 UTILITY (Ch,4)

(d) 4

(e) 2

4.4 Joe Bob,from Problem 4.12 has a cousin Jonas who consume goods

1 and 2,Jonas thinks that 2 units of good 1 is always a perfect substitute

for 3 units of good 2,Which of the following utility functions is the only

one that would NOT represent Jonas’s preferences?

(a) U(x

1;x

2

)=3x

1

+2x

2

+1;000.

(b) U(x

1;x

2

)=9x

2

1

+12x

1

x

2

+4x

2

2

.

(c) U(x

1;x

2

)=minf3x

1;2x

2

g.

(d) U(x

1;x

2

)=30x

1

+20x

2

10;000.

(e) More than one of the above does NOT represent Jonas’s preferences.

4.5 In Problem 4.7,Harry Mazzola has the utility function U(x

1;x

2

)=

minfx

1

+2x

2;2x

1

+ x

2

g,He has $40 to spend on corn chips and french

fries,If the price of corn chips is 5 dollar(s) per unit and the price of

french fries is 5 dollars per unit,then Harry will

(a) de nitely spend all of his income on corn chips.

(b) de nitely spend all of his income on french fries.

(c) consume at least as much corn chips as french fries,but might consume

both.

(d) consume at least as much french fries as corn chips,but might consume

both.

(e) consume equal amounts of french fries and corn chips.

4.6 Phil Rupp’s sister Ethel has the utility function U(x;y)=minf2x+

y;3yg.Wherex is measured on the horizontal axis and y on the vertical

axis,her indi erence curves

(a) consist of a vertical line segment and a horizontal line segment which

meet in a kink along the line y =2x.

(b) consist of a vertical line segment and a horizontal line segment which

meet in a kink along the line x =2y.

(c) consist of a horizontal line segment and a negatively sloped line seg-

ment which meet in a kink along the line x = y.

(d) consist of a positively sloped line segment and a negatively sloped line

segment which meet along the line x = y.

(e) consist of a horizontal line segment and a positively sloped line seg-

ment which meet in a kink along the line x =2y.

Quiz 5 NAME

Choice

5.1 In Problem 5.1,Charlie has a utility function U(x

A;x

B

)=x

A

x

B

,

the price of apples is 1 and the price of bananas is 2,If Charlie’s income

were 240,how many units of bananas would he consume if he chooses the

bundle that maximizes his utility subject to his budget constraint?

(a) 60

(b) 30

(c) 120

(d) 12

(e) 180

5.2 In Problem 5.1,if Charlie’s income is 40,the price of apples is 5

and the price of bananas is 6,how many apples are contained in the best

bundle that Charlie can a ord?

(a) 8

(b) 15

(c) 10

(d) 11

(e) 4

5.3 In Problem 5.2,Clara’s utility function is U(X;Y)=(X +2)(Y +1).

If Clara’s marginal rate of substitution is?2 and she is consuming 10

units of good X,how many units of good Y is she consuming?

(a) 2

(b) 24

(c) 12

(d) 23

450 CHOICE (Ch,5)

(e) 5

5.4 In Problem 5.3,Ambrose’s utility function is U(x

1;x

2

)=4x

1=2

1

+x

2

.

If the price of nuts is 1,the price of berries is 4,and his income is 72,how

many units of nuts will Ambrose choose?

(a) 2

(b) 64

(c) 128

(d) 67

(e) 32

5.5 Ambrose’s utility function is 4x

1=2

1

+x

2

,If the price of nuts is 1,the

price of berries is 4,and his income is 100,how many units of berries will

Ambrose choose?

(a) 65

(b) 9

(c) 18

(d) 8

(e) 12

5.6 In Problem 5.6,Elmer’s utility function is U(x;y)=minfx;y

2

g.If

the price of x is 15,the price of y is 10,and Elmer chooses to consume 7

units of y,what must Elmer’s income be?

(a) 1,610

(b) 175

(c) 905

(d) 805

(e) There is not enough information to tell.

Quiz 6 NAME

Demand

6.1 (See Problem 6.1,) If Charlie’s utility function is X

4

A

X

B

,apples

cost 90 cents each,and bananas cost 10 cents each,then Charlie’s budget

line is tangent to one of his indi erence curves whenever the following

equation is satis ed:

(a) 4X

B

=9X

A

.

(b) X

B

= X

A

.

(c) X

A

=4X

B

.

(d) X

B

=4X

A

.

(e) 90X

A

+10X

B

= M.

6.2 (See Problem 6.1.) If Charlie’s utility function is X

4

A

X

B

,the price

of apples is p

A

,the price of bananas is p

B

,and his income is m,then

Charlie’s demand for apples is

(a) m=(2p

A

).

(b) 0:25p

A

m.

(c) m=(p

A

+p

B

).

(d) 0:80m=p

A

.

(e) 1:25p

B

m=p

A

.

6.3 Ambrose’s brother Bartholomew has a utility function U(x

1;x

2

)=

24x

1=2

1

+ x

2

,His income is 51,the price of good 1 (nuts) is 4,and the

price of good 2 (berries) is 1,How many units of nuts will Bartholomew

demand?

(a) 19

(b) 5

(c) 7

(d) 9

452 DEMAND (Ch,6)

(e) 16

6.4 Ambrose’s brother Bartholomew has a utility function U(x

1;x

2

)=

8x

1=2

1

+x

2

,His income is 23,the price of nuts is 2,and the price of berries

is 1,How many units of berries will Bartholomew demand?

(a) 15

(b) 4

(c) 30

(d) 10

(e) There is not enough information to determine the answer.

6.5 In Problem 6.6,recall that Miss Mu et insists on consuming 2 units

of whey per unit of curds,If the price of curds is 3 and the price of whey

is 6,then if Miss Mu ett’s income is m,her demand for curds will be

(a) m=3.

(b) 6m=3.

(c) 3C +6W = m.

(d) 3m.

(e) m=15.

6.6 In Problem 6.8,recall that Casper’s utility function is 3x+y,where

x is his consumption of cocoa and y is his consumption of cheese,If the

total cost of x units of cocoa is x

2

,the price of cheese is 8,and Casper’s

income is $174,how many units of cocoa will he consume?

(a) 9

(b) 12

(c) 23

(d) 11

(e) 24

6.7 (See Problem 6.13.) Kinko’s utility function is U(w;j)=

minf7w;3w +12jg,wherew is the number of whips that he owns and j

is the number of leather jackets,If the price of whips is $20 and the price

of leather jackets is $60,Kinko will demand:

NAME 453

(a) 6 times as many whips as leather jackets.

(b) 5 times as many leather jackets as whips.

(c) 3 times as many whips as leather jackets.

(d) 4 times as many leather jackets as whips.

(e) only leather jackets.

454 DEMAND (Ch,6)

Quiz 7 NAME

Revealed Preference

7.1 In Problem 7.1,if the only information we had about Goldie were

that she chooses the bundle (6,6) when prices are (6,3) and she chooses

the bundle (10,0) when prices are (5,5),then we could conclude that

(a) the bundle (6,6) is revealed preferred to (10,0) but there is no evidence

that she violates WARP.

(b) neither bundle is revealed preferred to the other.

(c) Goldie violates WARP.

(d) the bundle (10,0) is revealed preferred to (6,6) and she violates WARP.

(e) the bundle (10,0) is revealed preferred to (6,6) and there is no evidence

that she violates WARP.

7.2 In Problem 7.3,Pierre’s friend Henri lives in a town where he has

to pay 3 francs per glass of wine and 6 francs per loaf of bread,Henri

consumes 6 glasses of wine and 4 loaves of bread per day,Recall that Bob

has an income of $15 per day and pays $.50 per loaf of bread and $2 per

glass of wine,If Bob has the same tastes as Henri,and if the only thing

that either of them cares about is consumption of bread and wine,we can

deduce

(a) nothing about whether one is better than the other.

(b) that Henri is better o than Bob.

(c) that Bob is better o than Henri.

(d) that both of them violate the weak axiom of revealed preferences.

(e) that Bob and Henri are equally well o,

7.3 Let us reconsider the case of Ronald in Problem 7.4,Let the prices

and consumptions in the base year be as in Situation D,where p

1

=3,

p

2

=1,x

1

=5,andx

2

= 15,If in the current year,the price of good 1 is

1 and the price of good 2 is 3,and his current consumptions of good 1 and

good 2 are 25 and 10 respectively,what is the Laspeyres price index of

current prices relative to base-year prices? (Pick the most nearly correct

answer.)

456 REVEALED PREFERENCE (Ch,7)

(a) 1.67

(b) 1.83

(c) 1

(d) 0.75

(e) 2.50

7.4 On the planet Homogenia,every consumer who has ever lived con-

sumes only two goods x and y and has the utility function U(x;y)=xy.

The currency in Homogenia is the fragel,On this planet in 1900,the

price of good 1 was 1 fragel and the price of good 2 was 2 fragels,Per

capita income was 120 fragels,In 1990,the price of good 1 was 5 fragels

and the price of good 2 was 5 fragels,The Laspeyres price index for the

price level in 1990 relative to the price level in 1900 is

(a) 3.75.

(b) 5.

(c) 3.33.

(d) 6.25.

(e) not possible to determine from this information.

7.5 On the planet Hyperion,every consumer who has ever lived has a

utility function U(x;y)=minfx;2yg,The currency of Hyperion is the

doggerel,In 1850 the price of x was 1 doggerel per unit,and the price of

y was 2 doggerels per unit,In 1990,the price of x was 10 doggerels per

unit and the price of y was 4 doggerels per unit,Paasche price index of

prices in 1990 relative to prices in 1850 is

(a) 6.

(b) 4.67.

(c) 2.50.

(d) 3.50.

(e) not possible to determine without further information.

Quiz 8 NAME

Slutsky Equation

8.1 In Problem 8.1,Charlie’s utility function is x

A

x

B

,The price of

apples used to be $1 per unit and the price of bananas was $2 per unit.

His income was $40 per day,If the price of apples increased to $1.25 and

the price of bananas fell to $1.25,then in order to be able to just a ord

his old bundle,Charlie would have to have a daily income of

(a) $37.50.

(b) $76.

(c) $18.75.

(d) $56.25.

(e) $150.

8.2 In Problem 8.1,Charlie’s utility function isx

A

x

B

,The price of apples

used to be $1 and the price of bananas used to be $2,and his income used

to be $40,If the price of apples increased to 8 and the price of bananas

stayed constant,the substitution e ect on Charlie’s apple consumption

reduces his consumption by

(a) 17.50 apples.

(b) 7 apples.

(c) 8.75 apples.

(d) 13.75 apples.

(e) None of the other options are correct.

8.3 Neville,in Problem 8.2,has a friend named Colin,Colin has the same

demand function for claret as Neville,namely q =,02m?2p,wherem

is income and p is price,Colin’s income is 6,000 and he initially had to

pay a price of 30 per bottle of claret,The price of claret rose to 40,The

substitution e ect of the price change

(a) reduced his demand by 20.

(b) increased his demand by 20.

458 SLUTSKY EQUATION (Ch,8)

(c) reduced his demand by 8.

(d) reduced his demand by 32.

(e) reduced his demand by 18.

8.4 Goods 1 and 2 are perfect complements and a consumer always con-

sumes them in the ratio of 2 units of Good 2 per unit of Good 1,If a

consumer has income 120 and if the price of good 2 changes from 3 to 4,

while the price of good 1 stays at 1,then the income e ect of the price

change

(a) is 4 times as strong as the substitution e ect.

(b) does not change demand for good 1.

(c) accounts for the entire change in demand.

(d) is exactly twice as strong as the substitution e ect.

(e) is 3 times as strong as the substitution e ect.

8.5 Suppose that Agatha in Problem 8.10 had $570 to spend on tickets

for her trip,She needs to travel a total of 1,500 miles,Suppose that the

price of rst-class tickets is $0.50 per mile and the price of second-class

tickets is $0.30 per mile,How many miles will she travel by second class?

(a) 900

(b) 1,050

(c) 450

(d) 1,000

(e) 300

8.6 In Problem 8.4,Maude thinks delphiniums and hollyhocks are perfect

substitutes,one for one,If delphiniums currently cost $5 per unit and

hollyhocks cost $6 per unit,and if the price of delphiniums rises to $9 per

unit,

(a) the income e ect of the change in demand for delphiniums will be

bigger than the substitution e ect.

(b) there will be no change in the demand for hollyhocks.

(c) the entire change in demand for delphiniums will be due to the sub-

stitution e ect.

(d) the fraction 1=4 of the change will be due to the income e ect.

(e) the fraction 3=4 of the change will be due to the income e ect.

Quiz 9 NAME

Buying and Selling

9.1 In Problem 9.1,if Abishag owned 9 quinces and 10 kumquats,and if

the price of kumquats is 3 times the price of quinces,how many kumquats

could she a ord if she spent all of her money on kumquats?

(a) 26

(b) 19

(c) 10

(d) 13

(e) 10

9.2 Suppose that Mario in Problem 9.2 consumes eggplant and tomatoes

in the ratio of one bushel of eggplant per bushel of tomatoes,His garden

yields 30 bushels of eggplant and 10 bushels of tomatoes,He initially faced

prices of $10 per bushel for each vegetable,but the price of eggplant rose

to $30 per bushel,while the price of tomatoes stayed unchanged,After

the price change,he would

(a) increase his eggplant consumption by 5 bushels.

(b) decrease his eggplant consumption by at least 5 bushels.

(c) increase his consumption of eggplant by 7 bushels.

(d) decrease his consumption of eggplant by 7 bushels.

(e) decrease his tomato consumption by at least 1 bushel.

9.3 (See Problem 9.9(b).) Dr,Johnson earns $5 per hour for his labor

and has 80 hours to allocate between labor and leisure,His only other

income besides his earnings from labor is a lump sum payment of $50 per

week,Suppose that the rst $200 per week of his labor income is untaxed,

but all labor income above $200 is taxed at a rate of 40 percent.

(a) Dr,J.’s budget line has a kink in it at the point where he takes 50

units of leisure.

(b) Dr,J.’s budget line has a kink where his income is 250 and his leisure

is 40.

460 BUYING AND SELLING (Ch,9)

(c) The slope of Dr,J.’s budget line is everywhere?3.

(d) Dr,J.’s budget line has no kinks in the part of it that corresponds to

a positive labor supply.

(e) Dr,J.’s budget line has a piece that is a horizontal straight line.

9.4 Dudley,in Problem 9.15,has a utility function U(C;R)=C?(12?

R)

2

,whereR is leisure and C is consumption per day,He has 16 hours

per day to divide between work and leisure,If Dudley has a nonlabor

income of $40 per day and is paid a wage of $6 per hour,how many hours

of leisure will he choose per day?

(a) 6

(b) 7

(c) 8

(d) 10

(e) 9

9.5 Mr,Cog in Problem 9.7 has 18 hours a day to divide between labor

and leisure,His utility function is U(C;R)=CR where C is the number

of dollars per day that he spends on consumption and R is the number of

hours per day that he spends at leisure,If he has 16 dollars of nonlabor

income per day and gets a wage rate of 13 dollars per hour when he works,

his budget equation,expressing combinations of consumption and leisure

that he can a ord to have,can be written as:

(a) 13R +C = 16.

(b) 13R +C = 250.

(c) R+C=13 = 328.

(d) C = 250 + 13R.

(e) C = 298 + 13R.

9.6 Mr,Cog in Problem 9.7 has 18 hours per day to divide between labor

and leisure,His utility function is U(C;R)=CR where C is the number

of dollars per day that he spends on consumption and R is the number of

hours per day that he spends at leisure,If he has a nonlabor income of

42 dollars per day and a wage rate of 13 dollars per hour,he will choose

a combination of labor and leisure that allows him to spend

(a) 276 dollars per day on consumption.

NAME 461

(b) 128 dollars per day on consumption.

(c) 159 dollars per day on consumption.

(d) 138 dollars per day on consumption.

(e) 207 dollars per day on consumption.

462 BUYING AND SELLING (Ch,9)

Quiz 10 NAME

Intertemporal Choice

10.1 If Peregrine in Problem 10.1 consumes (1,000,1,155) and earns

(800,1365) and if the interest rate is 0.05,the present value of his endow-

ment is

(a) 2,165.

(b) 2,100.

(c) 2,155.

(d) 4,305.

(e) 5,105.

10.2 Suppose that Molly from Problem 10.2 had income $400 in period 1

and income 550 in period 2,Suppose that her utility function werec

a

1

c

1?a

2

,

where a =0:40 and the interest rate were 0:10,If her income in period 1

doubled and her income in period 2 stayed the same,her consumption in

period 1 would

(a) double.

(b) increase by 160.

(c) increase by 80

(d) stay constant.

(e) increase by 400.

10.3 Mr,O,B,Kandle,of Problem 10.8 has a utility function c

1

c

2

where

c

1

is his consumption in period 1 and c

2

is his consumption in period 2.

He will have no income in period 2,If he had an income of 30,000 in

period 1 and the interest rate increased from 10% to 12%,

(a) his savings would increase by 2% and his consumption in period 2

would also increase.

(b) his savings would not change,but his consumption in period 2 would

increase by 300.

(c) his consumption in both periods would increase.

464 INTERTEMPORAL CHOICE (Ch,10)

(d) his consumption in both periods would decrease.

(e) his consumption in period 1 would decrease by 12% and his consump-

tion in period 2 would also decrease.

10.4 Harvey Habit in Problem 10.9 has a utility function U(c

1;c

2

)=

minfc

1;c

2

g,If he had an income of 1,025 in period 1,and 410 in period

2,and if the interest rate were 0.05,how much would Harvey choose to

spend on bread in period 1?

(a) 1,087.50

(b) 241.67

(c) 362.50

(d) 1,450

(e) 725

10.5 In the village in Problem 10.10,if the harvest this year is 3,000 and

the harvest next year will be 1,100,and if rats eat 50% of any grain that

is stored for a year,how much grain could the villagers consume next year

if they consume 1,000 bushels of grain this year?

(a) 2,100.

(b) 1,000.

(c) 4,100.

(d) 3,150.

(e) 1,200.

10.6 Patience has a utility function U(c

1;c

2

)=c

1=2

1

+0:83c

1=2

2

,c

1

is her

consumption in period 1 and c

2

is her consumption in period 2,Her

income in period 1 is 2 times as large as her income in period 2,At what

interestratewillshechoosetoconsumethesameamountinperiod1as

in period 2?

(a) 0.40

(b) 0.10

(c) 0.20

(d) 0

(e) 0.30

Quiz 11 NAME

Asset Markets

11.1 Ashley,in Problem 11.6,has discovered another wine,Wine D,Wine

drinkers are willing to pay 40 dollars to drink it right now,The amount

that wine drinkers are willing to pay will rise by 10 dollars each year

that the wine ages,The interest rate is 10%,How much would Ashley

be willing to pay for the wine if he buys it as an investment? (Pick the

closest answer.)

(a) $56

(b) $40

(c) $100

(d) $440

(e) $61

11.2 Chillingsworth,from Problem 11.10 has a neighbor,Shivers,who

faces the same options for insulating his house as Chillingsworth,But

Shivers has a larger house,Shivers’s annual fuel bill for home heating is

1,000 dollars per year,Plan A will reduce his annual fuel bill by 15%,plan

B will reduce it by 20%,and plan C will eliminate his need for heating fuel

altogether,The Plan A insulation job would cost Shivers 1,000 dollars,

Plan B would cost him 1,900 dollars,and Plan C would cost him 11,000

dollars,If the interest rate is 10% and his house and the insulation job

last forever,which plan is the best for Shivers?

(a) Plan A.

(b) Plan B.

(c) Plan C.

(d) Plans A and B are equally good.

(e) He is best o using none of the plans.

11.3 The price of an antique is expected to rise by 2% during the next

year,The interest rate is 6%,You are thinking of buying an antique

and selling it a year from now,You would be willing to pay a total of

200 dollars for the pleasure of owning the antique for a year,How much

would you be willing to pay to buy this antique? (See Problem 11.5.)

466 ASSET MARKETS (Ch,11)

(a) $3,333.33

(b) $4,200

(c) $200

(d) $5,000

(e) $2,000

11.4 A bond has a face value of 9,000 dollars,It will pay 900 dollars in

interest at the end of every year for the next 46 years,At the time of the

nal interest payment,46 years from now,the company that issued the

bond will \redeem the bond at face value." That is,the company buys

back the bond from its owner at a price equal to the face value of the

bond,If the interest rate is 10% and is expected to remain at 10%,how

much would a rational investor pay for this bond right now?

(a) $9,000

(b) $50,400

(c) $41,400

(d) More than any of the above numbers.

(e) Less than any of the above numbers.

11.5 The sum of the in nite geometric series 1;0:86;0:86

2;0:86

3;::,is

closest to which of the following numbers?

(a) in nity.

(b) 1.86.

(c) 7.14.

(d) 0.54.

(e) 116.28.

11.6 If the interest rate is 11%,and will remain 11% forever,how much

would a rational investor be willing to pay for an asset that will pay him

5,550 dollars one year from now,1,232 dollars two years from now,and

nothing at any other time?

(a) $6,000

(b) $5,000

(c) $54,545.45

(d) $72,000

(e) $7,000

Quiz 12 NAME

Uncertainty

12.1 In Problem 12.9,Billy has a von Neumann-Morgenstern utility func-

tion U(c)=c

1=2

,If Billy is not injured this season,he will receive an

income of 25 million dollars,If he is injured,his income will be only

$10,000,The probability that he will be injured is,1 and the probability

that he will not be injured is,9,His expected utility is

(a) 4,510.

(b) between 24 million and 25 million dollars.

(c) 100,000.

(d) 9,020.

(e) 18,040.

12.2 (See Problem 12.2.) Willy’s only source of wealth is his chocolate

factory,He has the utility function pc

1=2

f

+(1?p)c

1=2

nf

where p is the

probability of a flood and 1?p is the probability of no flood,Let c

f

and

c

n

f be his wealth contingent on a flood and on no flood,respectively,The

probability of a flood is p =1=15,The value of Willy’s factory is $600,000

if there is no flood and 0 if there is a flood,Willy can buy insurance where

if he buys $x worth of insurance,he must pay the insurance company

$3x=17 whether there is a flood or not,but he gets back $x from the

company if there is a flood,Willy should buy

(a) no insurance since the cost per dollar of insurance exceeds the prob-

ability of a flood.

(b) enough insurance so that if there is a flood,after he collects his insur-

ance his wealth will be 1/9 of what it would be if there is no flood.

(c) enough insurance so that if there is a flood,after he collects his insur-

ance,his wealth will be the same whether there is a flood or not.

(d) enough insurance so that if there is a flood,after he collects his in-

surance,his wealth will be 1/4 of what it would be if there is no flood.

(e) enough insurance so that if there is a flood,after he collects his insur-

ance his wealth will be 1/7 of what it would be if there is no flood.

12.3 Sally Kink is an expected utility maximizer with utility function

pu(c

1

)+(1?p)u(c

2

)whereforanyx<4;000,u(x)=2x and where

u(x)=8;000 +x for x greater than or equal to 4,000.

468 UNCERTAINTY (Ch,12)

(a) Sally will be risk averse if her income is less than 4,000 but risk loving

if her income is more than 4,000.

(b) Sally will be risk neutral if her income is less than 4,000 and risk

averse if her income is more than 4,000.

(c) For bets that involve no chance of her wealth exceeding 4,000,Sally

will take any bet that has a positive expected net payo,

(d) Sally will never take a bet if there is a chance that it leaves her with

wealth less than 8,000.

(e) None of the above are true.

12.4 (See Problem 12.11.) Martin’s expected utility function is pc

1=2

1

+

(1?p)c

1=2

2

where p is the probability that he consumes c

1

and 1?p is

the probability that he consumes c

2

,Wilbur is o ered a choice between

getting a sure payment of $Z or a lottery in which he receives $2,500 with

probability 0.40 and he receives $900 with probability 0.60,Wilbur will

choose the sure payment if

(a) Z>1;444 and the lottery if Z<1;444.

(b) Z>1;972 and the lottery if Z<1;972.

(c) Z>900 and the lottery if Z<900.

(d) Z>1;172 and the lottery if Z<1;172.

(e) Z>1;540 and the lottery if Z<1;540.

12.5 Clancy has $4,800,He plans to bet on a boxing match between

Sullivan and Flanagan,He nds that he can buy coupons for $6 that

will pay o $10 each if Sullivan wins,He also nds in another store some

coupons that will pay o $10 if Flanagan wins,The Flanagan tickets cost

$4 each,Clancy believes that the two ghters each have a probability

of 1/2 of winning,Clancy is a risk averter who tries to maximize the

expected value of the natural log of his wealth,Which of the following

strategies would maximize his expected utility?

(a) Don’t gamble at all.

(b) Buy 400 Sullivan tickets and 600 Flanagan tickets.

(c) Buy exactly as many Flanagan tickets as Sullivan tickets.

(d) Buy 200 Sullivan tickets and 300 Flanagan tickets.

(e) Buy 200 Sullivan tickets and 600 Flanagan tickets.

Quiz 13 NAME

Risky Assets

13.1 Suppose that Ms,Lynch in Problem 13.1 can make up her portfolio

using a risk-free asset that o ers a sure- re rate of return of 15% and a

risky asset with expected rate of return 30%,with standard deviation 5.

If she chooses a portfolio with expected rate of return 18.75%,then the

standard deviation of her return on this portfolio will be:

(a) 0.63%.

(b) 4.25%.

(c) 1.25%.

(d) 2.50%,

(e) None of the other options are correct.

13.2 Suppose that Fenner Smith of Problem 13.2 must divide his portfolio

between two assets,one of which gives him an expected rate of return of

15 with zero standard deviation and one of which gives him an expected

rate of return of 30 and has a standard deviation of 5,He can alter the

expected rate of return and the variance of his portfolio by changing the

proportions in which he holds the two assets,If we draw a \budget line"

with expected return on the vertical axis and standard deviation on the

horizontal axis,depicting the combinations that Smith can obtain,the

slope of this budget line is

(a) 3.

(b)?3.

(c) 1.50.

(d)?1:50.

(e) 4.50.

470 RISKY ASSETS (Ch,13)

Quiz 14 NAME

Consumer’s Surplus

14.1 In Problem 14.1,Sir Plus has a demand function for mead that is

given by the equation D(p) = 100?p,If the price of mead is 75,how

much is Sir Plus’s net consumer surplus?

(a) 312.50

(b) 25

(c) 625

(d) 156.25

(e) 6,000

14.2 Ms,Quasimodo in Problem 14.3 has the utility function U(x;m)=

100x?x

2

=2+m where x is his consumption of earplugs and m is money

left over to spend on other stu,If she has $10,000 to spend on earplugs

and other stu,and if the price of earplugs rises from $50 to $95,then

her net consumer’s surplus

(a) falls by 1,237.50.

(b) falls by 3237.50.

(c) falls by 225.

(d) increases by 618.75.

(e) increases by 2,475.

14.3 Bernice in Problem 14.5 has the utility function u(x;y)=minfx;yg

where x is the number of pairs of earrings she buys per week and y is the

number of dollars per week she has left to spend on other things,(We

allow the possibility that she buys fractional numbers of pairs of earrings

per week.) If she originally had an income of $13 per week and was paying

a price of $2 per pair of earrings,then if the price of earrings rose to $4,

the compensating variation of that price change (measured in dollars per

week) would be closest to

(a) $5.20.

(b) $8.67.

472 CONSUMER’S SURPLUS (Ch,14)

(c) $18.33.

(d) $17.33.

(e) $16.33.

14.4 If Bernice (whose utility function is minfx;yg where x is her con-

sumption of earrings and y is money left for other stu ) had an income

of $16 and was paying a price of $1 for earrings when the price of earrings

went up to $8,then the equivalent variation of the price change was

(a) $12.44.

(b) $56.

(c) $112.

(d) $6.22.

(e) $34.22.

14.5 In Problem 14.7,Lolita’s utility function is U(x;y)=x?x

2

=2+y

where x is her consumption of cow feed and y is her consumption of hay.

If the price of cow feed is 0.40,the price of hay is 1,and her income is 4,

and if Lolita chooses the combination of hay and cow feed that she likes

best from among those combinations she can a ord,her utility will be

(a) 4.18.

(b) 3.60.

(c) 0.18.

(d) 6.18.

(e) 2.18.

Quiz 15 NAME

Market Demand

15.1 In Gas Pump,South Dakota,every Buick owner’s demand for gaso-

line is 20?5p for p less than or equal to 4 and 0 for p>4,Every Dodge

owner’sdemandis15?3p for p less than or equal to 5 and 0 for p>5.

Suppose that Gas Pump,S.D.,has 100 Buick owners and 50 Dodge own-

ers,If the price of gasoline is 4,what is the total amount of gasoline

demanded in Gas Pump?

(a) 300

(b) 75

(c) 225

(d) 150

(e) None of the other options are correct.

15.2 In Problem 15.5,the demand function for drangles is given by

D(p)=(p +1)

2

,If the price of drangles is 10,then the price elasticity

of demand is

(a)?7:27.

(b)?3:64.

(c)?5:45.

(d)?0:91.

(e)?1:82.

15.3 In Problem 15.6,the only quantities of good 1 that Barbie can buy

are 1 unit or zero units,For x

1

equal to zero or 1 and for all positive values

of x

2

,suppose that Barbie’s preferences were represented by the utility

function (x

1

+4)(x

2

+ 2),Then if her income were 28,her reservation

price for good 1 would be

(a) 12.

(b) 1.50.

(c) 6.

474 MARKET DEMAND (Ch,15)

(d) 2.

(e) 0.40.

15.4 In the same football conference as the university in Problem 15.9

is another university where the demand for football tickets at each game

is 80;000?12;000p,If the capacity of the stadium at that university is

50,000 seats,what is the revenue-maximizing price for this university to

charge per ticket?

(a) 3.33

(b) 2.50

(c) 6.67

(d) 1.67

(e) 10

15.5 In Problem 15.9,the demand for tickets is given byD(p) = 200;000?

10;000p,wherep is the price of tickets,If the price of tickets is 4,then

the price elasticity of demand for tickets is

(a)?0:50.

(b)?0:38.

(c)?0:75.

(d)?0:13.

(e)?0:25.

Quiz 16 NAME

Equilibrium

16.1 This problem will be easier if you have done Problem 16.3.The

inverse demand function for grapefruit is de ned by the equation p =

296?7q,whereq is the number of units sold,The inverse supply function

is de ned by p =17+2q,A tax of 27 is imposed on suppliers for each

unit of grapefruit that they sell,When the tax is imposed,the quantity

of grapefruit sold falls to

(a) 31.

(b) 17.50.

(c) 26.

(d) 28.

(e) 29.50.

16.2 In a crowded city far away,the civic authorities decided that rents

were too high,The long-run supply function of two-room rental apart-

ments was given by q =18+2p and the long run demand function was

given by q = 114?4p where p is the rental rate in crowns per week.

The authorities made it illegal to rent an apartment for more than 10

crowns per week,To avoid a housing shortage,the authorities agreed to

pay landlords enough of a subsidy to make supply equal to demand,How

much would the weekly subsidy per apartment have to be to eliminate

excess demand at the ceiling price?

(a) 9

(b) 15

(c) 18

(d) 36

(e) 27

16.3 Suppose that King Kanuta from Problem 16.11 demands that each

of his subjects gives him 4 coconuts for every coconut that the subject

consumes,The king puts all of the coconuts that he collects in a large

pile and burns them,The supply of coconuts is given by S(p

s

) = 100p

s

,

where p

s

is the price received by suppliers,The demand for coconuts by

the king’s subjects is given by D(p

d

)=8;320?100p

d

,wherep

d

is the

price paid by consumers,In equilibrium,the price received by suppliers

will be

476 EQUILIBRIUM (Ch,16)

(a) 16.

(b) 24.

(c) 41.60.

(d) 208.

(e) None of the other options are correct.

16.4 In Problem 16.6,the demand function for Schrecklichs is 200?4P

S

2P

L

and the demand function for LaMerdes is 200?3P

L

P

S

,whereP

S

and P

L

are respectively the price of Schrecklichs and LaMerdes,If the

world supply of Schrecklichs is 100 and the world supply of Lamerdes is

90,then the equilibrium price of Schrecklichs is

(a) 8.

(b) 25.

(c) 42.

(d) 34.

(e) 16.

Quiz 17 NAME

Auctions

17.1 First Fiddler’s Bank has foreclosed on a home mortgage and is selling

the house at auction,There are three bidders for the house,Jesse,Sheila,

and Elsie,First Fiddler’s does not know the willingness to pay of any

of these bidders but on the basis of its previous experience believes that

each of them has a probability of 1/3 of valuing the house at $700,000,

a probability of 1/3 of valuing it at $500,000,and a probability of 1/3 of

valuing it at $200,000,First Fiddlers believes that these probabilities are

independent between buyers,If First Fiddler’s sells the house by means

of a second-bidder sealed-bid auction (Vickrey auction),what will be the

bank’s expected revenue from the sale? (Choose the closest answer.)

(a) $500,000

(b) $474,074

(c) $466,667

(d) $666,667

(e) $266,667

17.2 An antique cabinet is being sold by means of an English auction.

There are four bidders,Natalie,Heidi,Linda,and Eva,These bidders

are unacquainted with each other and do not collude,Natalie values the

cabinet at $1,200,Heidi values it at $950,Linda values it at $1,700,and

Eva values it at $700,If the bidders bid in their rational self-interest,the

cabinet will be sold to

(a) Linda for about $1,700.

(b) Natalie for about $1,200.

(c) either Linda or Natalie for about $1,200,Which of these two buyers

gets it is randomly determined.

(d) Linda for slightly more than $1,200.

478 AUCTIONS (Ch,17)

(e) either Linda or Natalie for about $950,Which of these two buyers

gets it is randomly determined.

17.3 A dealer decides to sell an antique automobile by means of an English

auction with a reservation price of $900,There are two bidders,The

dealer believes that there are only three possible values that each bidder’s

willingness to pay might take,$6,300,$2,700,and $900,Each bidder has

a probability of 1/3 of having each of these willingnesses to pay,and the

probabilities of the two bidders are independent of the other’s valuation.

Assuming that the two bidders bid rationally and do not collude,the

dealer’s expected revenue from selling the automobile is

(a) $4,500.

(b) $3,300.

(c) $2,700.

(d) $2,100.

(e) $6,300.

17.4 A dealer decides to sell an oil painting by means of an English

auction with a reservation price of slightly below $81,000,If he fails to

get a bid as high as his reservation price,he will burn the painting,There

are two bidders,The dealer believes that each bidder’s willingness to

pay will take one of the three values,$90,000,$81,000,and $45,000,The

dealer believes that each bidder has a probability of 1/3 of having each

of these three values,The probability distribution of each buyer’s value

is independent of that of the other’s,Assuming that the two bidders bid

rationally and do not collude,the dealer’s expected revenue from selling

the painting is slightly less than

(a) $73,000.

(b) $81,000.

(c) $45,000.

(d) $63,000.

NAME 479

(e) $72,000.

17.5 Jerry’s Auction House in Purloined Hubcap,Oregon,holds sealed-

bid used car auctions every Wednesday,Each car is sold to the highest

bidder at the second-highest bidder’s bid,On average,2/3 of the cars that

are auctioned are lemons and 1/3 are good used cars,A good used car is

worth $1,500 to any buyer,A lemon is worth $150 to any buyer,Most

buyers can do no better than picking at random from among these used

cars,The only exception is Al Crankcase,Recall that Al can sometimes

detect lemons by tasting the oil on the car’s dipstick,A good car never

fails Al’s test,but half of the lemons fail his test,Al attends every auction,

licks every dipstick,and bids his expected value of every car given the

results of his test,Al will bid:

(a) $825 for cars that pass his test and $150 for cars that fail his test.

Normal bidders will get only lemons.

(b) $750 for cars that pass his test and $500 for cars that fail his test.

Normal bidders will get only lemons.

(c) $500 for cars that pass his test and $150 for cars that fail his test.

Normal bidders will get good cars only 1/6 of the time.

(d) $600 for cars that pass his test and $250 for cars that fail his test.

Normal bidders will get good cars only 1/6 of the time.

(e) $300 for cars that pass his test and $150 for cars that fail his test.

Normal bidders will get good cars only 1/12 of the time.

480 AUCTIONS (Ch,17)

Quiz 18 NAME

Technology

18.1 This problem will be easier if you have done Problem 18.1,A rm

has the production functionf(x

1;x

2

)=x

0:90

1

x

0:30

2

,The isoquant on which

output is 40

3=10

has the equation

(a) x

2

=40x

3

1

.

(b) x

2

=40x

3:33

1

.

(c) x

1

=x

2

=3.

(d) x

2

=40x

0:30

1

.

(e) x

1

=0:30x

0:70

2

.

18.2 A rm has the production function f(x;y)=x

0:70

y

0:30

.This rm

has

(a) decreasing returns to scale and dimininishing marginal product for

factor x.

(b) increasing returns to scale and decreasing marginal product of factor

x.

(c) decreasing returns to scale and increasing marginal product for factor

x.

(d) constant returns to scale.

(e) None of the other options are correct.

18.3 A rm uses 3 factors of production,Its production function is

f(x;y;z)=minfx

5

=y;y

4;(z

6

x

6

)=y

2

g,If the amount of each input is

multiplied by 6,its output will be multiplied by

(a) 7,776.

(b) 1,296.

(c) 216.

(d) 0.

482 TECHNOLOGY (Ch,18)

(e) The answer depends on the original choice of x,y,andz.

18.4 A rm has a production function f(x;y)=1:20(x

0:10

+y

0:10

)

1

when-

ever x>0andy>0,When the amounts of both inputs are positive,

this rm has

(a) increasing returns to scale.

(b) decreasing returns to scale.

(c) constant returns to scale.

(d) increasing returns to scale if x+y>1 and decreasing returns to scale

otherwise.

(e) increasing returns to scale if output is less than 1 and decreasing

returns to scale if output is greater than 1.

Quiz 19 NAME

Profit Maximization

19.1 In Problem 19.1,the production function is F(L)=6L

2=3

,Suppose

that the cost per unit of labor is 8 and the price of output is 8,how many

units of labor will the rm hire?

(a) 128

(b) 64

(c) 32

(d) 192

(e) None of the other options are correct.

19.2 In Problem 19.2,the production function is given by f(x)=4x

1=2

.

If the price of the commodity produced is 70 per unit and the cost of the

input is 35 per unit,how much pro ts will the rm make if it maximizes

pro ts?

(a) 560

(b) 278

(c) 1,124

(d) 545

(e) 283

19.3 In Problem 19.11,the production function is f(x

1;x

2

)=x

1=2

1

x

1=2

2

.If

the price of factor 1 is 8 and the price of factor 2 is 16,in what proportions

should the rm use factors 1 and 2 if it wants to maximize pro ts?

(a) x

1

= x

2

.

(b) x

1

=0:50x

2

.

(c) x

1

=2x

2

.

(d) We can’t tell without knowing the price of output.

484 PROFIT MAXIMIZATION (Ch,19)

(e) x

1

=16x

2

.

19.4 In Problem 19.9,when Farmer Hoglund applies N pounds of fer-

tilizer per acre,the marginal product of fertilizer is 1?(N=200) bushels

of corn,If the price of corn is $4 per bushel and the price of fertilizer

is $1.20 per pound,then how many pounds of fertilizer per acre should

Farmer Hoglund use in order to maximize his pro ts?

(a) 140

(b) 280

(c) 74

(d) 288

(e) 200

Quiz 20 NAME

Cost Minimization

20.1 Suppose that Nadine in Problem 20.1 has a production function

3x

1

+ x

2

,If the factor prices are 9 for factor 1 and 4 for factor 2,how

much will it cost her to produce 50 units of output?

(a) 1,550

(b) 150

(c) 200

(d) 875

(e) 175

20.2 In Problem 20.2,suppose that a new alloy is invented which uses

copper and zinc in xed proportions,where one unit of output requires 3

units of copper and 3 units of zinc for each unit of alloy produced,If no

other inputs are needed,if the price of copper is 2 and the price of zinc

is 2,what is the average cost per unit when 4,000 units of the alloy are

produced?

(a) 6.33

(b) 666.67

(c) 0.67

(d) 12

(e) 6,333.33

20.3 In Problem 20.3,the production function is f(L;M)=4L

1=2

M

1=2

,

where L is the number of units of labor and M is the number of machines

used,If the cost of labor is $25 per unit and the cost of machines is $64

per unit,then the total cost of producing 6 units of output will be

(a) $120.

(b) $267.

(c) $150.

486 COST MINIMIZATION (Ch,20)

(d) $240.

(e) None of the other options are correct.

20.4 Suppose that in the short run,the rm in Problem 20.3 which has

production function F(L;M)=4L

1=2

M

1=2

must use 25 machines,If the

cost of labor is 8 per unit and the cost of machines is 7 per unit,the

short-run total cost of producing 200 units of output is

(a) 1,500.

(b) 1,400.

(c) 1,600.

(d) 1,950.

(e) 975.

20.5 In Problem 20.12,Al’s production function for deer is f(x

1;x

2

)=

(2x

1

+ x

2

)

1=2

where x

1

is the amount of plastic and x

2

is the amount of

wood used,If the cost of plastic is $2 per unit and the cost of wood is $4

per unit,then the cost of producing 8 deer is

(a) $64.

(b) $70.

(c) $256.

(d) $8.

(e) $32.

20.6 Two rms,Wickedly E cient Widgets and Wildly Nepotistic Wid-

gets,produce widgets with the same production function y = K

1=2

L

1=2

where K is the input of capital and L is the input of labor,Each company

can hire labor at $1 per unit and capital at $1 per unit,WEW produces

10 widgets per week,choosing its input combination so as to produce

these 10 widgets in the cheapest way possible,WNW also produces 10

widgets per week,but its dotty CEO requires it to use twice as much

labor as WEW uses,Given that it must use twice as many laborers as

WEW does,and must produce the same output,how much more larger

are WNW’s total costs than WEW’s?

(a) $10 per week

(b) $20 per week

(c) $15 per week

(d) $5 per week

(e) $2 per week

Quiz 21 NAME

Cost Curves

21.1 In Problem 21.2,if Mr,Dent Carr’s total costs are 4s

2

+75s + 60,

then if he repairs 15 cars,his average variable costs will be

(a) 135.

(b) 139.

(c) 195.

(d) 270.

(e) 97.50.

21.2 In Problem 21.3,Rex Carr could pay $10 for a shovel that lasts one

year and pay $5 a car to his brother Scoop to bury the cars,or he could

buy a low-quality car smasher that costs $200 a year to own and that

smashes cars at a marginal cost of $1 per car,If it is also possible for

Rex to buy a high-quality hydraulic car smasher that cost $300 per year

to own and if with this smasher he could dispose of cars at a cost of $0.80

per car,it would be worthwhile for him to buy this high-quality smasher

smasher if

(a) he plans to dispose of at least 500 cars per year.

(b) he plans to dispose of no more than 250 cars per year.

(c) he plans to dispose of at least 510 cars per year.

(d) he plans to dispose of no more than 500 cars per year.

(e) he plans to dispose of at least 250 cars per year.

21.3 Mary Magnolia in Problem 21.4 has variable costs equal to y

2

=F

where y is the number of bouquets she sells per month and where F is the

number of square feet of space in her shop,If Mary has signed a lease for

a shop with 1,600 square feet and if she is not able to get out of the lease

or to expand her store in the short run,and if the price of a bouquet is

$3 per unit,how many bouquets per month should she sell in the short

run?

(a) 1,600

488 COST CURVES (Ch,21)

(b) 800

(c) 2,400

(d) 3,600

(e) 2,640

21.4 Touchie MacFeelie’s production function is,1J

1=2

L

3=4

,whereJ is

the number of old jokes used and L is the number of hours of cartoonists’

labor,Touchie is stuck with 900 old jokes for which he paid 6 dollars each.

If the wage rate for cartoonists is 5,then the total cost of producing 24

comics books is

(a) 5,480.

(b) 2,740.

(c) 8,220.

(d) 5,504.

(e) 1,370.

21.5 Recall that Touchie McFeelie’s production function for comic books

is,1J

1=2

L

3=4

,Suppose that Touchie can vary both jokes and cartoonists’

labor,If old jokes cost $2 each and cartoonists’ labor costs $18 per hour,

then the cheapest way to produce comics books requires using jokes and

labor in the ratio J=L =

(a) 9.

(b) 12.

(c) 3.

(d) 2/3.

(e) 6.

Quiz 22 NAME

Firm Supply

22.1 Suppose that Dent Carr’s long-run total cost of repairing s cars per

week is c(s)=3s

2

+ 192,If the price he receives for repairing a car is 36,

then in the long run,how many cars will he x per week if he maximizes

pro ts?

(a) 6

(b) 0

(c) 12

(d) 9

(e) 18

22.2 In Problem 22.9,suppose that Irma’s production function is

f(x

1;x

2

)=(minfx

1;2x

2

g)

1=2

,If the price of factor 1 is w

1

=6and

the price of factor 2 is w

2

= 4,then her supply function is given by the

equation:

(a) S(p)=p=16.

(b) S(p)=pmaxfw

1;2w

2

g

2

.

(c) S(p)=pminfw

1;2w

2

g

2

.

(d) S(p)=8p.

(e) S(p)=minf6p;8pg.

22.3 A rm has the long-run cost function C(q)=2q

2

+ 8,In the long

run,it will supply a positive amount of output,so long as the price is

greater than

(a) 16.

(b) 24.

(c) 4.

(d) 8.

(e) 13.

490 FIRM SUPPLY (Ch,22)

Quiz 23 NAME

Industry Supply

23.1 In Problem 23.1,if the cost of plaster and labor is $9 per gnome and

everything else is as in the problem,what is the lowest price of gnomes

at which there would be a positive supply in the long run?

(a) $9

(b) $18

(c) $11.20

(d) $9.90

(e) $10.80

23.2 Suppose that the garden gnome industry was in long-run equilib-

rium given the circumstances described in Problem 23.1,Suppose,as in

Problem 23.2,that it was discovered to everyone’s surprise,on January 1,

1993 after it was to late to change orders for gnome molds,that the cost

of the plaster and labor needed to make a gnome had changed to 8,If

the demand curve does not change,what will happen to the equilibrium

price of gnomes?

(a) It rises by 1.

(b) It falls by 1.

(c) It stays constant.

(d) It rises by 8.

(e) It falls by 4.

23.3 Suppose that the garden gnome industry was in long run equilib-

rium as described in Problem 23.1 and that on January 1,1993,the cost

of plaster and labor remained at $7 per gnome,and the government in-

troduced a tax of $10 on every garden gnome sold,Then the equilibrium

price of garden gnomes in 1993 would be

(a) $17.

(b) $9.20.

492 INDUSTRY SUPPLY (Ch,23)

(c) $7.

(d) $10.

(e) $27.

23.4 Suppose that the cost of capturing a cockatoo and transporting him

to the U,S,is about $40 per bird,Cockatoos are drugged and smuggled

in suitcases to the U,S,Half of the smuggled cockatoos die in transit.

Each smuggled cockatoo has a 10% probability of being discovered,in

which case the smuggler is ned,If the ne imposed for each smuggled

cockatoo is increased to $900,then the equilibrium price of cockatoos in

theU.S.willbe

(a) $288.89.

(b) $130.

(c) $85.

(d) $67.

(e) $200.

23.5 In Problem 23.13,in the absence of government interference,there

is a constant marginal cost of $5 per ounce for growing marijuana and

delivering it to buyers,If the probability that any shipment of marijuana

is seized is 0.20 and the ne if a shipper is caught is $20 per ounce,then

the equilibrium price of marijuana per ounce is

(a) $11.25.

(b) $9.

(c) $25.

(d) $4.

(e) $6.

23.6 In Problem 23.8,the supply curve of any rm is S

i

(p)=p=2,If a

rm produces 3 units of output,what are its total variable costs?

(a) $18

(b) $7

(c) $13.50

(d) $9

(e) There is not enough information given to determine total variable

costs.

Quiz 24 NAME

Monopoly

24.1 In Problem 24.1,if the demand schedule for Bong’s book is Q =

3;000? 100p,the cost of having the book typeset is 10,000,and the

marginal cost of printing an extra book is $4,he would maximize his

pro ts by

(a) having it typeset and selling 1,300 copies.

(b) having it typeset and selling 1,500 copies.

(c) not having it typeset and not selling any copies.

(d) having it typeset and selling 2,600 copies.

(e) having it typeset and selling 650 copies.

24.2 In Problem 24.2,if the demand for pigeon pies is p(y)=70?y=2,

then what level of output will maximize Peter’s pro ts?

(a) 74

(b) 14

(c) 140

(d) 210

(e) None of the above

24.3 A pro t-maximizing monopoly faces an inverse demand function

described by the equation p(y)=70?y and its total costs are c(y)=5y,

where prices and costs are measured in dollars,In the past it was not

taxed,but now it must pay a tax of 8 dollars per unit of output,After

the tax,the monopoly will

(a) increase its price by $8.

(b) increase its price by $12.

(c) increase its price by $4.

(d) leave its price constant.

494 MONOPOLY (Ch,24)

(e) None of the other options are correct.

24.4 A rm has invented a new beverage called Slops,It doesn’t taste

very good,but it gives people a craving for Lawrence Welk’s music and

Professor Johnson’s jokes,Some people are willing to pay money for this

e ect,so the demand for Slops is given by the equation q =14?p.Slops

can be made at zero marginal cost from old-fashioned macroeconomics

books dissolved in bathwater,But before any Slops can be produced,the

rm must undertake a xed cost of 54,Since the inventor has a patent

on Slops,it can be a monopolist in this new industry.

(a) The rm will produce 7 units of Slops.

(b) A Pareto improvement could be achieved by having the government

pay the rm a subsidy of 59 and insisting that the rm o er Slops at zero

price.

(c) From the point of view of social e ciency,it is best that no Slops be

produced.

(d) The rm will produce 14 units of Slops.

(e) None of the other options are correct.

Quiz 25 NAME

Monopoly Behavior

25.1 (See Problem 25.1.) If demand in the U.S,is given by Q

1

=23;400?

900p

1

,wherep

1

is the price in the U.S,and if the demand in England

is given by 2;800? 200p

2

where p

2

is the price in England,then the

di erence between the price charged in England and the price charged in

the U.S,will be

(a) 6.

(b) 12.

(c) 0.

(d) 14.

(e) 18.

25.2 (See Problem 25.2.) A monopolist faces a demand curve described

by p(y) = 100?2y and has constant marginal costs of 16 and zero xed

costs,If this monopolist is able to practice perfect price discrimination,

its total pro ts will be

(a) 1,764.

(b) 21.

(c) 882.

(d) 2,646.

(e) 441.

25.3 A price-discriminating monopolist sells in two separate markets such

that goods sold in one market are never resold in the other,It charges 4 in

one market and 8 in the other market,At these prices,the price elasticity

inthe rstmarketis?1:50 and the price elasticity in the second market

is?0:10,Which of the following actions is sure to raise the monopolists

pro ts?

(a) Lower p

2

.

(b) Raise p

2

.

496 MONOPOLY BEHAVIOR (Ch,25)

(c) Raise p

1

and lower p

2

.

(d) Raise both p

1

and p

2

.

(e) Raise p

2

and lower p

1

.

25.4 The demand for Professor Bongmore’s new book is given by the

function Q =2;000?100p,If the cost of having the book typeset is

8,000,if the marginal cost of printing an extra copy is 4,and if he has no

other costs,then he would maximize his pro ts by

(a) having it typeset and selling 800 copies.

(b) having it typeset and selling 1,000 copies.

(c) not having it typeset and not selling any copies.

(d) having it typeset and selling 1,600 copies.

(e) having it typeset and selling 400 copies.

Quiz 26 NAME

Factor Markets

26.1 Suppose that in Problem 26.2,the demand curve for mineral water

is given by p =30?12q,wherep is the price per bottle paid by consumers

and q is the number of bottles purchased by consumers,Mineral water

is supplied to consumers by a monopolistic distributor,who buys from a

monopolist producer who is able to produce mineral water at zero cost.

The producer charges the distributor a price of c per bottle,where the

price c maximizes the producer’s total revenue,Given his marginal cost

of c,the distributor chooses an output to maximize pro ts,The price

paid by consumers under this arrangement is

(a) 15.

(b) 22.50.

(c) 2.50.

(d) 1.25.

(e) 7.50.

26.2 Suppose that the labor supply curve for a large university in a small

town is given by w =60+0:08L where L is number of units of labor per

week and w is the weekly wage paid per unit of labor,If the university

is currently hiring 1,000 units of labor per week,the marginal cost of an

additional unit of labor

(a) equals the wage rate.

(b) is twice the wage rate.

(c) equals the wage rate plus 160.

(d) equals the wage rate plus 80.

(e) equals the wage rate plus 240

26.3 Rabelaisian Restaurants has a monopoly in the town of Upper Duo-

denum,Its production function is Q =40L,whereL istheamountof

labor it uses and Q is the number of meals produced,Rabelaisian Restau-

rants nds that in order to hire L units of labor,it must pay a wage of

40 +,1L per unit of labor,The demand curve for meals at Rabelaisian

Restaurants is given by P =30:75?Q=1;000,The pro t-maximizing

output for Rabelasian Restaurants is

498 FACTOR MARKETS (Ch,26)

(a) 14,000.

(b) 28,000.

(c) 3,500.

(d) 3,000.

(e) 1,750.

Quiz 27 NAME

Oligopoly

27.1 Suppose that the duopolists Carl and Simon in Problem 27.1 face

a demand function for pumpkins of Q =13;200?800P,whereQ is the

total number of pumpkins that reach the market and P is the price of

pumpkins,Suppose further that each farmer has a constant marginal

cost of $0.50 for each pumpkin produced,If Carl believes that Simon is

going to produce Q

s

pumpkins this year,then the reaction function tells

us how many pumpkins Carl should produce in order to maximize his

pro ts,Carl’s reaction function is R

C

(Q

s

)=

(a) 6;400?Q

s

=2.

(b) 13;200?800Q

s

.

(c) 13;200?1;600Q

s

.

(d) 3;200?Q

s

=2.

(e) 9;600?Q

s

.

27.2 If in Problem 27.4,the inverse demand for bean sprouts were given

by P(Y ) = 290?4Y and the total cost of producing y units for any

rm were TC(Y)=50Y,and if the industry consisted of two Cournot

duopolists,then in equilibrium each rm’s production would be

(a) 30 units.

(b) 15 units.

(c) 10 units.

(d) 20 units.

(e) 18.13 units.

27.3 In Problem 27.5,suppose that Grinch and Grubb go into the wine

business in a small country where wine is di cult to grow,The demand

for wine is given by p = $360?:2Q where p is the price and Q is the total

quantity sold,The industry consists of just the two Cournot duopolists,

Grinch and Grubb,Imports are prohibited,Grinch has constant marginal

costs of $15 and Grubb has marginal costs of $75,How much is Grinch’s

output in equilibrium?

500 OLIGOPOLY (Ch,27)

(a) 675

(b) 1,350

(c) 337.50

(d) 1,012.50

(e) 2,025

27.4 In Problem 27.6,suppose that two Cournot duopolists serve the

Peoria-Dubuque route,and the demand curve for tickets per day is Q =

200?2p (so p = 100?Q=2),Total costs of running a flight on this route

are 700+40q whereq is the number of passengers on the flight,Each flight

has a capacity of 80 passengers,In Cournot equilibrium,each duopolist

will run one flight per day and will make a daily pro t of

(a) 100.

(b) 350.

(c) 200.

(d) 200.

(e) 2,400.

27.5 In Problem 27.4,suppose that the market demand curve for bean

sproutsisgivenbyP = 880?2Q,whereP is the price and Q is total

industry output,Suppose that the industry has two rms,a Stackleberg

leader,and a follower,Each rm has a constant marginal cost of $80 per

unit of output,In equilibrium,total output by the two rms will be

(a) 200.

(b) 100.

(c) 300.

(d) 400.

(e) 50.

27.6 There are two rms in the blastopheme industry,The demand

curve for blastophemes is given by p =2;100?3q,Each rm has one

manufacturing plant and each rm i has a cost function C(q

i

)=q

2

i

where

q

i

is the output of rm i,The two rms form a cartel and arrange to

split total industry pro ts equally,Under this cartel arrangement,they

will maximize joint pro ts if

NAME 501

(a) and only if each rm produces 150 units in its plant.

(b) they produce a total of 300 units,no matter which rm produces

them.

(c) and only if they each produce a total of 350 units.

(d) they produce a total of 233.33 units,no matter which rm produces

them.

(e) they shut down one of the two plants,having the other operate as a

monopoly and splitting the pro ts.

502 OLIGOPOLY (Ch,27)

Quiz 28 NAME

Game Theory

28.1 (See Problem 28.1.) Big Pig and Little Pig have two possible strate-

gies,Press the Button,and Wait at the Trough,If both pigs choose Wait,

both get 4,If both pigs press the button then Big Pig gets 5 and Little

Pig gets 5,If Little Pig presses the button and Big Pig waits,then Big

Pig gets 10 and Little Pig gets 0,Finally,if Big Pig presses and Little

Pig waits,then Big Pig gets 4 and Little Pig gets 2,In Nash equilibrium,

(a) Little Pig will get a payo of 2 and Big Pig will get a payo of 4.

(b) Little Pig will get a payo of 5 and Big Pig will get a payo of 5.

(c) both pigs will wait at the trough.

(d) Little Pig will get a payo of zero.

(e) the pigs must be using mixed strategies.

28.2 (See Problem 28.6.) Two players are engaged in a game of \chicken."

There are two possible strategies,Swerve and Drive Straight,A player

who chooses to Swerve is called \Chicken" and gets a payo of zero,

regardless of what the other player does,A player who chooses to Drive

Straight gets a payo of 32 if the other player swerves and a payo of

48 if the other player also chooses to Drive Straight,This game has two

pure strategy equilibria and

(a) a mixed strategy equilibrium in which each player swerves with prob-

ability 0.60 and drives straight with probability 0.40.

(b) two mixed strategies in which players alternate between swerving and

driving straight.

(c) a mixed strategy equilibrium in which one player swerves with prob-

ability 0.60 and the other swerves with probability 0.40.

(d) a mixed strategy in which each player swerves with probability 0.30

and drives straight with probability 0.70.

504 GAME THEORY (Ch,28)

(e) no mixed strategies.

28.3 The old Michigan football coach had only two strategies,run the ball

to the left side of the line,and run the ball to the right side,The defense

can concentrate either on the left side or the right side of Michigan’s

line,If the opponent concentrates on the wrong side,Michigan is sure to

gain at least 5 yards,If the defense defended the left side and Michigan

ran left,Michigan would be stopped for no gain,But if the opponent

defended the right side when Michigan ran right,Michigan would still

gain at least 5 yards with probability 0.40,It is the last play of the

game and Michigan needs to gain 5 yards to win,Both sides choose Nash

equilibrium strategies,In Nash equilibrium,Michigan would

(a) be sure to run to the right side.

(b) run to the right side with probability 0.63.

(c) run to the right side with probability 0.77.

(d) run with equal probability to one side or the other.

(e) run to the right side with probability 0.60.

28.4 Suppose that in the Hawk-Dove game discussed in Problem 28.3,

the payo to each player is?4 if both play Hawk,If both play Dove,

the payo to each player is 1 and if one plays Hawk and the other plays

Dove,the one that plays Hawk gets a payo of 3 and the one that plays

Dove gets 0,In equilibrium,we would expect Hawks and Doves to do

equally well,This happens when the proportion of the total population

that plays Hawk is

(a) 0.33.

(b) 0.17.

(c) 0.08.

(d) 0.67.

(e) 1.

28.5 (See Problem 28.11.) If the number of persons who attend the club

meeting this week is X,then the number of people who will attend next

week is 27 + 0:70X,What is a long-run equilibrium attendance for this

club?

(a) 27

(b) 38.57

(c) 54

(d) 90

(e) 63

Quiz 29 NAME

Exchange

29.1 An economy has two people Charlie and Doris,There are two goods,

apples and bananas,Charlie has an initial endowment of 3 apples and

12 bananas,Doris has an initial endowment of 6 apples and 6 bananas.

Charlie’s utility function is U(A

C;B

C

)=A

C

B

C

where A

C

is his apple

consumption and B

C

is his banana consumption,Doris’s utility function

is U(A

D;B

D

)=A

D

B

D

where A

D

and B

D

are her apple and banana

consumptions,At every Pareto optimal allocation,

(a) Charlie consumes the same number of apples as Doris.

(b) Charlie consumes 9 apples for every 18 bananas that he consumes.

(c) Doris consumes equal numbers of apples and bananas.

(d) Charlie consumes more bananas per apple than Doris does.

(e) Doris consumes apples and bananas in the ratio of 6 apples for every

6 bananas that she consumes.

29.2 In Problem 29.4,Ken’s utility function is U(Q

K;W

K

)=Q

K

W

K

and Barbie’s utility function is U(Q

B;W

B

)=Q

B

W

B

,If Ken’s initial

endowment were 3 units of quiche and 10 units of wine and Barbie’s

endowment were 6 units of quiche and 10 units of wine,then at any Pareto

optimal allocation where both persons consume some of each good,

(a) Ken would consume 3 units of quiche for every 10 units of wine.

(b) Barbie would consume twice as much quiche as Ken.

(c) Ken would consume 9 units of quiche for every 20 units of wine that

he consumed.

(d) Barbie would consume 6 units of quiche for every 10 units of wine

that she consumed.

(e) None of the other options are correct.

29.3 In Problem 29.1,suppose that Morris has the utility function

U(b;w)=6b +24w and Philip has the utility function U(b;w)=bw.

If we draw an Edgeworth box with books on the horizontal axis and wine

on the vertical axis and if we measure Morris’s consumptions from the

lower left corner of the box,then the contract curve contains

506 EXCHANGE (Ch,29)

(a) a straight line running from the upper right corner of the box to the

lower left.

(b) a curve that gets steeper as you move from left to right.

(c) a straight line with slope 1=4 passing through the lower left corner of

the box.

(d) a straight line with slope 1=4 passing through the upper right corner

of the box.

(e) a curve that gets flatter as you move from left to right.

29.4 In Problem 29.2,Astrid’s utility function is U(H

a;C

A

)=H

A

C

A

.

Birger’s utility function is minfH

B;C

B

g,Astrid’s initial endowment is no

cheese and 4 units of herring,and Birger’s initial endowments are 6 units

of cheese and no herring,Where p is a competitive equilibrium price of

herring and cheese is the numeraire,it must be that demand equals supply

in the herring market,This implies that

(a) 6=(p +1)+2=4.

(b) 6=4=p.

(c) 4=6=p.

(d) 6=p +4=2p =6.

(e) minf4;6g= p.

29.5 Suppose that in Problem 29.8,Mutt’s utility function is U(m;j)=

maxf3m;jg and Je ’s utility function is U(m;j)=2m + j.Mutis

initially endowed with 4 units of milk and 2 units of juice,Je is initially

endowed with 4 units of milk and 6 units of juice,If we draw an Edgeworth

box with milk on the horizontal axis and juice on the vertical axis and if

we measure goods for Mutt by the distance from the lower left corner of

the box,then the set of Pareto optimal allocations includes the

(a) left edge of the Edgeworth box but no other edges.

(b) bottom edge of the Edgeworth box but no other edges.

(c) left edge and bottom edge of the Edgeworth box.

(d) right edge of the Edgeworth box but no other edges.

NAME 507

(e) right edge and top edge of the Edgeworth box.

29.6 In Problem 29.3,Professor Nightsoil’s utility function,U

N

(B

N;P

N

),

is B

N

+4P

1=2

N

and Dean Interface’s utility function is U

I

(B

I;P

I

)=B

I

+

2P

1=2

I

,If Nightsoil’s initial endowment is 7 bromides and 15 platitudes

and if Interface’s initial endowment is 7 bromides and 25 platitudes,then

at any Pareto e cient allocation where both persons consume positive

amounts of both goods,it must be that

(a) Nightsoil consumes the same ratio of bromides to platitudes as Inter-

face.

(b) Interface consumes 8 platitudes.

(c) Interface consumes 7 bromides.

(d) Interface consumes 3 bromides.

(e) Interface consumes 5 platitudes.

508 EXCHANGE (Ch,29)

Quiz 30 NAME

Production

30.1 Suppose that in Problem 30.1,Tip can write 5 pages of term papers

or solve 20 workbook problems in an hour,while Spot can write 2 pages

of term papers or solve 6 workbook problems in an hour,If they each

decide to work a total of 7 hours,and to share their output then if they

produce as many pages of term paper as possible given that they produce

30 workbook problems,

(a) Spot will spend all of his time writing term papers and Tip will spend

some time at each task.

(b) Tip will spend all of his time writing term papers and Spot will spend

some time at each task.

(c) bothstudentswillspendsometimeateachtask.

(d) Spot will write term papers only and Tip will do workbook problems

only.

(e) Tip will write term papers only and Spot will do workbook problems

only.

30.2 Al and Bill are the only workers in a small factory which makes

geegaws and doodads,Al can make 3 geegaws per hour or 15 doodads per

hour,Bill can make 2 geegaws per hour or 6 doodads per hour,Assuming

that neither of them nds one task more odious than the other,

(a) Al has comparative advantage in producing geegaws,and Bill has

comparative advantage in producing doodads.

(b) Bill has comparative advantage in producing geegaws,and Al has

comparative advantage in producing doodads.

(c) Al has comparative advantage in producing both geegaws and doo-

dads.

(d) Bill has comparative advantage in producing both geegaws and doo-

dads.

510 PRODUCTION (Ch,30)

(e) both persons have comparative advantage in producing doodads.

30.3 (See Problem 30.5.) Every consumer has a red-money income and

a blue-money income and each commodity has a red price and a blue

price,You can buy a good by paying for it either with blue money at

the blue price,or with red money at the red price,Harold has 10 units

of red money to spend and 18 units of blue money to spend,The red

price of ambrosia is 1 and the blue price of ambrosia is 2,The red price

of bubblegum is 1 and the blue price of bubblegum is 1,If ambrosia is on

the horizontal axis,and bubblegum on the vertical,axis,then Harold’s

budget set is bounded

(a) by two line segments,one running from (0,28) to (10,18) and another

running from (10,18) to (19,0).

(b) by two line segments one running from (0,28) to (9,10) and the other

running from (9,10) to (19,0).

(c) by two line segments,one running from (0,27)to (10,18) and the other

running from (10,18) to (20,0).

(d) a vertical line segment and a horizontal line segement,intersecting at

(10,18).

(e) a vertical line segment and a horizontal line segment,intersecting at

(9,10).

30.4 (See Problem 30.2.) Robinson Crusoe has exactly 12 hours per day

to spend gathering coconuts or catching sh,He can catch 4 sh per hour

or he can pick 16 coconuts per hour,His utility function is U(F;C)=FC

where F is his consumption of sh and C is his consumption of coconuts.

If he allocates his time in the best possible way between catching sh and

picking coconuts,his consumption will be the same as it would be if he

could buy sh and coconuts in a competitive market where the price of

coconuts is 1,and where

(a) his income is 192 and the price of sh is 4.

(b) his income is 48 and the price of sh is 4.

(c) his income is 240 and the price of sh is 4.

(d) his income is 192 and the price of sh is 0.25.

(e) his income is 120 and the price of sh is 0.25.

30.5 On a certain island there are only two goods,wheat and milk,The

only scarce resource is land,There are 1,000 acres of land,An acre of land

will produce either 16 units of milk or 37 units of wheat,Some citizens

have lots of land,some have just a little bit,The citizens of the island

all have utility functions of the form U(M;W)=MW,At every Pareto

optimal allocation,

NAME 511

(a) the number of units of milk produced equals the number of units of

wheat produced.

(b) total milk production is 8,000.

(c) all citizens consume the same commodity bundle.

(d) every consumer’s marginal rate of substitution between milk and

wheat is?1.

(e) None of the above is true at every Pareto optimal allocation.

512 PRODUCTION (Ch,30)

Quiz 31 NAME

Welfare

31.1 A Borda count is used to decide an election between 3 candidates,

x,y,and z where a score of 1 is awarded to a rst choice,2 to a second

choice and 3 to a third choice,There are 25 voters,7 voters rank the

candidates x rst,y second,z third; 4 voters rank the candidates x rst,z

second,y third; 6 rank the candidates,z rst,y second,x third; 8 voters

rank the candidates,y rst,z second,x third,Which candidate wins?

(a) Candidate x.

(b) Candidate y.

(c) Candidate z.

(d) There is a tie between x and y,with z coming in third.

(e) There is a tie between y and z,with x coming in third.

31.2 A parent has two children living in cities with di erent costs of

living,The cost of living in city B is 3 times the cost of living in city A.

The child in city A has an income of 3,000 and the child in city B has an

income of $9,000,The parent wants to give a total of $4,000 to her two

children,Her utility function is U(C

A;C

B

)=C

A

C

B

,whereC

A

and C

B

are the consumptions of the children living in cities A and B respectively.

She will choose to

(a) give each child $2,000,even though this will buy less goods for the

child in city B.

(b) give the child in city B 3 times as much money as the child in city A.

(c) give the child in city A 3 times as much money as the child in city B.

(d) give the child in city B 1.50 times as much money as the child in

city A.

(e) give the child in city A 1.50 times as much money as the child in

city B.

31.3 Suppose that Paul and David from Problem 31.7 have utility func-

tions U =5A

P

+ O

P

and U = A

D

+5O

D

,respectively,where A

P

and

O

P

are Paul’s consumptions of apples and oranges and A

D

and O

D

are

David’s consumptions of apples and oranges,The total supply of apples

and oranges to be divided between them is 8 apples and 8 oranges,The

\fair" allocations consist of all allocations satisfying the following condi-

tions.

514 WELFARE (Ch,31)

(a) A

D

= A

P

and O

D

= O

P

.

(b) 10A

P

+2O

P

is at least 48,and 2A

D

+10O

D

is at least 48.

(c) 5A

P

+O

P

is at least 48,and 2A

D

+5O

D

is at least 48.

(d) A

D

+O

D

is at least 8,and A

S

+O

S

is at least 8.

(e) 5A

P

+O

P

is at least A

D

+5O

D

,andA

D

+5O

D

is at least 5A

P

+O

P

.

31.4 Suppose that Romeo in Problem 31.8 has the utility function U =

S

8

R

S

4

J

and Juliet has the utility function U = S

4

R

S

8

J

,whereS

R

is Romeo’s

spaghetti consumption and S

J

is Juliet’s,They have 96 units of spaghetti

to divide between them.

(a) Romeo would want to give Juliet some spaghetti if he had more than

48 units of spaghetti.

(b) Juliet would want to give Romeo some spaghetti if she has more than

62 units.

(c) Romeo and Juliet would never disagree about how to divide the

spaghetti.

(d) Romeo would want to give Juliet some spaghetti if he has more than

60 units of spaghetti.

(e) Juliet would want to give Romeo some spaghetti if she has more than

64 units of spaghetti.

31.5 Hat eld and McCoy burn with hatred for each other,They both

consume corn whisky,Hat eld’s utility function is U = W

H

W

2=8

M

and

McCoy’s utility is U = W

M

W

2=8

H

,whereW

H

is Hat eld’s whisky con-

sumption and W

M

is McCoy’s whisky consumption,measured in gallons.

The sheri has a total of 28 units of con scated whisky that he could

give back to them,For some reason,the sheri wants them both to be as

happy as possible,and he wants to treat them equally,The sheri should

give them each

(a) 14 gallons.

(b) 4 gallons and spill 20 gallons in the creek.

(c) 2 gallons and spill 24 gallons in the creek.

(d) 8 gallons and spill the rest in the creek.

(e) 1 gallon and spill the rest in the creek.

Quiz 32 NAME

Externalities

32.1 Suppose that in Horsehead,Massachusetts,the cost of operating

a lobster boat is $3,000 per month,Suppose that if X lobster boats

operate in the bay,the total monthly revenue from lobster boats in the

bay is $1;000(23x?x

2

),If there are no restrictions on entry and new

boats come into the bay until there is no pro t to be made by a new

entrant,then the number of boats that enter will be X1,If the number

of boats that operate in the bay is regulated to maximize total pro ts,

the number of boats in the bay will be X2.

(a) X1 = 20 and X2 = 20.

(b) X1 = 10 and X2=8.

(c) X1 = 20 and X2 = 10.

(d) X1 = 24 and X2 = 14.

(e) None of the other options are correct.

32.2 An apiary is located next to an apple orchard,The apiary produces

honey and the apple orchard produces apples,The cost function of the

apiary is C

H

(H;A)=H

2

=100?1A and the cost function of the apple

orchard is C

A

(H;A)=A

2

=100,where H and A are the number of units

of honey and apples produced respectively,The price of honey is 8 and

the price of apples is 7 per unit,Let A1 be the output of apples if the

rms operate independently,and let A2 be the output of apples if the

rms are operated by a single owner,It follows that

(a) A1 = 175 and A2 = 350.

(b) A1=A2 = 350.

(c) A1 = 200 and A2 = 350.

(d) A1 = 350 and A2 = 400.

516 EXTERNALITIES (Ch,32)

(e) A1 = 400 and A2 = 350.

32.3 Martin’s utility is U(c;d;h)=2c +5d?d

2

2h,whered is the

number of hours per day that he spends driving around,h is the number

of hours per day spent driving around by other people in his home town

and c is the amount of money he has left to spend on other stu besides

gasoline and auto repairs,Gas and auto repairs cost $.50 per hour of

driving,All the people in Martin’s home town have the same tastes,If

each citizen believes that his own driving will not a ect the amount of

driving done by others,they will all drive D1hoursperday,Iftheyall

drive the same amount,they would all be best o if each drove D2hours

per day,where

(a) D1=2andD2=1.

(b) D1=D2=2.

(c) D1=4andD2=2.

(d) D1=5andD2=0.

(e) D1 = 24 and D2=0.

32.4 (See Problems 32.8,32.9.) An airport is located next to a housing

development,Where X is the number of planes that land per day and Y

is the number of houses in the housing development,pro ts of the airport

are 22X?X

2

and pro ts of the developer are 32Y?Y

2

XY.Let

H1 be the number of houses built if a single pro t-maximizing company

owns the airport and the housing development,Let H2bethenumberof

houses built if the airport and the housing development are operated in-

dependently and the airport has to pay the developer the total \damages"

XY done by the planes to developer’s pro ts,Then

(a) H1=H2 = 14.

(b) H1 = 14 and H2 = 16.

(c) H1 = 16 and H2 = 14.

(d) H1 = 16 and H2 = 15.

(e) H1 = 15 and H2 = 19.

32.5 (See Problem 32.5.) A clothing store and a jeweler are located

side by side in a shopping mall,If the clothing store spends C dollars

on advertising and the jeweler spends J dollars on advertising,then the

pro ts of the clothing store will be (48 + J)C?2C

2

and the pro ts of

the jeweler will be (42 +C)J?2J

2

,The clothing store gets to choose his

amount of advertising rst,knowing that the jeweler will nd out how

much the clothing store advertised before deciding how much to spend.

The amount spent by the clothing store will be

NAME 517

(a) 16.71.

(b) 46.

(c) 69.

(d) 11.50.

(e) 34.50.

518 EXTERNALITIES (Ch,32)

Quiz 33 NAME

Law

33.1 Consider Madame Norrell,in Problem 33.1,She gets 5 logx if she

delivers x buttons to her fence,She has to pay a ne Fxif she is caught,

and she has a 10 percent chance of getting caught,If she is caught,she

cannot collect anything from her fence,How big should the ne be if we

want to limit Madam Norrell to taking 5 buttons?

(a) 4.5

(b) 5.5

(c) 9

(d) 11

(e) 12

33.2 Consider Jim and Dick,described in Problem 33.2,Jim rides at

speed s and has money m; his utility function is 10s +m,Dick walks at

speed w and has money m; his utility function is 10w + m.Thecostof

an accident to Jim is c

J

(s;w)=s

2

+ w

2

,and the cost of an accident to

Dick is also c

D

(s;w)=s

2

+w

2

,If there is no liability,how fast will Dick

and Jim move?

(a) s =10andw = 10.

(b) s =5andw =5.

(c) s =5andw = 10.

(d) s =10andw =5.

(e) s =15andw = 15.

520 LAW (Ch,33)

Quiz 34 NAME

Information Technology

34.1 If the demand function for the DoorKnobs operating system is re-

lated to perceived market share s and actual market share t by the equa-

tion p = 512s(1?x),then in the long run,the highest price at which

DoorKnobs could sustain a market share of 3/4 is

(a) $156.

(b) $64.

(c) $96.

(d) $128.

(e) $256.

34.2 Eleven consumers are trying to decide whether to connect to a new

communications network,Consumer 1 is of type 1,consumer 2 is of type

2,consumer 3 is of type 3,and so on,Where k is the number of consumers

connected to the network (including oneself),a consumer of type n has

willingness to pay to belong to this network equal to k times n.Whatis

the highest price at which 7 consumers could all connect to the network

and either make a pro t or at least break even?

(a) $40

(b) $33

(c) $25

(d) $40

(e) $35

34.3 Professor Kremepu ’s new,user-friendly textbook has just been

published,This book will be used in classes for two years,after which it

will be replaced by a new edition,The publisher charges a price of p

1

in

the rst year and p

2

in the second year,After the rst year,bookstores

buy back used copies for p

2

=2 and resell them to students in the second

year for p

2

,(Students are indi erent between new and used copies.) The

cost to a student of owning the book during the rst year is therefore

p

1

(p

2

=2),In the rst year of publication,the number of students

willing to pay $v to own a copy of the book for a year is 60;000?1;000v.

The number of students taking the course in the rst year who are willing

522 INFORMATION TECHNOLOGY (Ch,34)

to pay $w to keep the book for reference rather than sell it at the end of

the year is 60;000?5;000w,The number of persons who are taking the

course in the second year and are willing to pay at least $p for a copy of

the book is 50;000?1;000p,If the publisher sets a price of p

1

in the rst

year and p

2

p

1

in the second year,then the total number of copies of

the book that the publisher sells over the two years will be

(a) 120;000?1;000p

1

1;000p

2

.

(b) 120;000?1;000(p

1

p

2

=2).

(c) 120;000?3;000p

2

.

(d) 110;000?1;000(p

1

+p

2

=2).

(e) 110;000?1;500p

2

.

Quiz 35 NAME

Public Goods

35.1 Just north of the town of Muskrat,Ontario,is the town of Brass

Monkey,population 500,Brass Monkey,like Muskrat,has a single pub-

lic good,the town skating rink and a single private good,Labatt’s ale.

Everyone’s utility function is U

i

(X

i;Y)=X

i

64=Y,whereX

i

is the

number of bottles of ale consumed by i and Y is the size of the skating

rink in square meters,The price of ale is $1 per bottle,The cost of the

skating rink to the city is $5 per square meter,Everyone has an income

of at least $5,000,What is the Pareto e cient size for the town skating

rink?

(a) 80 square meters

(b) 200 square meters

(c) 100 square meters

(d) 165 square meters

(e) None of the other options are correct.

35.2 Recall Bob and Ray in Problem 35.4,They are thinking of buying a

sofa,Bob’s utility function is U

B

(S;M

B

)=(1+S)M

B

,and Ray’s utility

function is U

R

(S;M

R

)=(4+S)M

R

,whereS = 0 if they don’t get the sofa

and S = 1 if they do and where M

B

and M

R

are the amounts of money

they have respectively to spend on their private consumptions,Bob has

a total of $800 to spend on the sofa and other stu,Ray has a total of

$2,000 to spend on the sofa and other stu,The maximum amount that

they could pay for the sofa and still arrange to both be better o than

without it is

(a) $1,200.

(b) $500.

(c) $450.

(d) $800.

524 PUBLIC GOODS (Ch,35)

(e) $1,600.

35.3 Recall Bonnie and Clyde from Problem 35.5,Suppose that their

total pro ts are 48H,whereH is the number of hours they work per

year,Their utility functions are,respectively,U

B

(C

B;H)=C

B

0:01H

2

and U

C

(C

C;H)=C

C

0:01H

2

,whereC

B

and C

C

are their private goods

consumptions and H is the number of hours they work per year,If they

nd a Pareto optimal choice of hours of work and income distribution,it

must be that the number of hours they work per year is

(a) 1,300.

(b) 1,800.

(c) 1,200.

(d) 550.

(e) 650.

35.4 Recall Lucy and Melvin from Problem 35.6,Lucy’s utility function

is 2X

L

+ G,and Melvin’s utility function is X

M

G,where G is their ex-

penditures on the public goods they share in their apartment and where

X

L

and X

M

are their respective private consumption expenditures,The

total amount they have to spend on private goods and public goods is

32,000,They agree on a Pareto optimal pattern of expenditures in which

the amount that is spent on Lucy’s private consumption is 8,000,How

much do they spent on public goods?

(a) 8,000

(b) 16,000

(c) 8,050

(d) 4,000

(e) There is not enough information here to be able to determine the

answer.

Quiz 36 NAME

Information

36.1 As in Problem 36.2,suppose that low-productivity workers have

marginal products of 10 and high-productivity workers have marginal

products of 16,The community has equal numbers of each type of worker.

The local community college o ers a course in microeconomics,High-

productivity workers think taking this course is as bad as a wage cut of

4,and low-productivity workers think it is as bad as a wage cut of 7.

(a) There is a separating equilibrium in which high-productivity workers

take the course and are paid 16 and low-productivity workers do not take

the course and are paid 10.

(b) There is no separating equilibrium and no pooling equilibrium.

(c) There is no separating equilibrium,but there is a pooling equilibrium

in which everybody is paid 13.

(d) There is a separating equilibrium in which high-productivity workers

take the course and are paid 20 and low-productivity workers do not take

the course and are paid 10.

(e) There is a separating equilibrium in which high-productivity workers

take the course and are paid 16 and low-productivity workers are paid 13.

36.2 Suppose that in Enigma,Ohio,Klutzes have productivity of $1,000

and Kandos have productivity of $5,000 per month,You can’t tell Klutzes

from Kandos by looking at them or asking them,and it is too expensive

to monitor individual productivity,Kandos,however,have more patience

than Klutzes,Listening to an hour of dull lectures is as bad as losing $200

for a Klutz and $100 for a Kando,There will be a separating equilibrium

in which anybody who attends a course of H hours of lectures is paid

$5,000 per month and anybody who does not is paid $1,000 per month

(a) if H<40 and H>20.

(b) if H<80 and H>20.

(c) for all positive values of H.

(d) only in the limit as H approaches in nity.

526 INFORMATION (Ch,36)

(e) if H<35 and H>17:50.

36.3 In Rustbucket,Michigan,there are 200 used cars for sale,Half of

them are good,and half of them are lemons,Owners of lemons are willing

to sell them for $300,Owners of good used cars are willing to sell them for

prices above $1,100 but will keep them if the price is lower than $1,100.

There is a large number of potential buyers who are willing to pay $400

for a lemon and $2,100 for a good car,Buyers can’t tell good cars from

bad,but original owners know.

(a) There will be an equilibrium in which all used cars sell for $1,250.

(b) The only equilibrium is one in which all used cars on the market are

lemons and they sell for 400.

(c) There will be an equilibrium in which lemons sell for 300 and good

used cars sell for 1,100.

(d) There will be an equilibrium in which all used cars sell for 700.

(e) There will be an equilibrium in which lemons sell for 400 and good

used cars sell for 2,100.

36.4 Suppose that in Burnt Clutch,Pa.,the quality distribution of the

1000 used cars on the market is such that the number of used cars of value

less than V is V=2,Original owners must sell their used cars,Original

owners know what their cars are worth,but buyers can’t determine a car’s

quality until they buy it,An owner can either take his car to an appraiser

and pay the appraiser $100 to appraise the car (accurately and credibly),

or he can sell the car unappraised,In equilibrium,car owners will have

their cars appraised if and only if their value is at least

(a) $100.

(b) $500.

(c) $300.

(d) $200.

(e) $400.

The Market

Introduction,The problems in this chapter examine some variations on

the apartment market described in the text,In most of the problems we

work with the true demand curve constructed from the reservation prices

of the consumers rather than the \smoothed" demand curve that we used

in the text.

Remember that the reservation price of a consumer is that price

where he is just indi erent between renting or not renting the apartment.

At any price below the reservation price the consumer will demand one

apartment,at any price above the reservation price the consumer will de-

mand zero apartments,and exactly at the reservation price the consumer

will be indi erent between having zero or one apartment.

You should also observe that when demand curves have the \stair-

case" shape used here,there will typically be a range of prices where

supply equals demand,Thus we will ask for the the highest and lowest

price in the range.

1.1 (3) Suppose that we have 8 people who want to rent an apartment.

Their reservation prices are given below,(To keep the numbers small,

think of these numbers as being daily rent payments.)

Person=ABCDEFGH

Price = 40 25 30 35 10 18 15 5

(a) Plot the market demand curve in the following graph,(Hint,When

the market price is equal to some consumer i’s reservation price,there

will be two di erent quantities of apartments demanded,since consumer

i will be indi erent between having or not having an apartment.)

2 THE MARKET (Ch,1)

012345678

10

20

30

40

60

50

Price

Apartments

(b) Suppose the supply of apartments is xed at 5 units,In this case

there is a whole range of prices that will be equilibrium prices,What is

the highest price that would make the demand for apartments equal to 5

units? $18.

(c) What is the lowest price that would make the market demand equal

to 5 units? $15.

(d) With a supply of 4 apartments,which of the people A{H end up

getting apartments? A,B,C,D.

(e) What if the supply of apartments increases to 6 units,What is the

range of equilibrium prices? $10 to $15.

1.2 (3) Suppose that there are originally 5 units in the market and that

1 of them is turned into a condominium.

(a) Suppose that person A decides to buy the condominium,What will

be the highest price at which the demand for apartments will equal the

supply of apartments? What will be the lowest price? Enter your an-

swers in column A,in the table,Then calculate the equilibrium prices of

apartments if B,C,:::,decide to buy the condominium.

NAME 3

Person A B C D E F G H

High price 18 18 18 18 25 25 25 25

Low price 15 15 15 15 18 15 18 18

(b) Suppose that there were two people at each reservation price and 10

apartments,What is the highest price at which demand equals supply?

18,Suppose that one of the apartments was turned into a condo-

minium,Is that price still an equilibrium price? Yes.

1.3 (2) Suppose now that a monopolist owns all the apartments and that

he is trying to determine which price and quantity maximize his revenues.

(a) Fill in the box with the maximum price and revenue that the monop-

olist can make if he rents 1,2,:::,8 apartments,(Assume that he must

charge one price for all apartments.)

Number 1 2 3 4 5 6 7 8

Price 40 35 30 25 18 15 10 5

Revenue 40 70 90 100 90 90 70 40

(b) Which of the people A{F would get apartments? A,B,C,D.

(c) If the monopolist were required by law to rent exactly 5 apartments,

what price would he charge to maximize his revenue? $18.

(d) Who would get apartments? A,B,C,D,F.

(e) If this landlord could charge each individual a di erent price,and he

knew the reservation prices of all the individuals,what is the maximum

revenue he could make if he rented all 5 apartments? $148.

(f) If 5 apartments were rented,which individuals would get the apart-

ments? A,B,C,D,F.

1.4 (2) Suppose that there are 5 apartments to be rented and that the

city rent-control board sets a maximum rent of $9,Further suppose that

people A,B,C,D,and E manage to get an apartment,while F,G,and

H are frozen out.

4 THE MARKET (Ch,1)

(a) If subletting is legal|or,at least,practiced|who will sublet to whom

in equilibrium? (Assume that people who sublet can evade the city rent-

control restrictions.) E,who is willing to pay only

$10 for an apartment would sublet to F,

who is willing to pay $18.

(b) What will be the maximum amount that can be charged for the sublet

payment? $18.

(c) If you have rent control with unlimited subletting allowed,which of

the consumers described above will end up in the 5 apartments? A,

B,C,D,F.

(d) How does this compare to the market outcome? It’s the

same.

1.5 (2) In the text we argued that a tax on landlords would not get

passed along to the renters,What would happen if instead the tax was

imposed on renters?

(a) To answer this question,consider the group of people in Problem 1.1.

What is the maximum that they would be willing to pay to the landlord

if they each had to pay a $5 tax on apartments to the city? Fill in the

box below with these reservation prices.

Person A B C D E F G H

Reservation Price 35 20 25 30 5 13 10 0

(b) Using this information determine the maximum equilibrium price if

there are 5 apartments to be rented,$13.

(c) Of course,the total price a renter pays consists of his or her rent plus

the tax,This amount is $18.

(d) How does this compare to what happens if the tax is levied on the

landlords? It’s the same.

Chapter 2 NAME

Budget Constraint

Introduction,These workouts are designed to build your skills in de-

scribing economic situations with graphs and algebra,Budget sets are a

good place to start,because both the algebra and the graphing are very

easy,Where there are just two goods,a consumer who consumes x

1

units

of good 1 and x

2

units of good 2 is said to consume the consumption bun-

dle,(x

1;x

2

),Any consumption bundle can be represented by a point on

a two-dimensional graph with quantities of good 1 on the horizontal axis

and quantities of good 2 on the vertical axis,If the prices are p

1

for good 1

and p

2

for good 2,and if the consumer has income m,then she can a ord

any consumption bundle,(x

1;x

2

),such thatp

1

x

1

+p

2

x

2

m,On a graph,

the budget line is just the line segment with equation p

1

x

1

+ p

2

x

2

= m

and with x

1

and x

2

both nonnegative,The budget line is the boundary

of the budget set,All of the points that the consumer can a ord lie on

one side of the line and all of the points that the consumer cannot a ord

lie on the other.

If you know prices and income,you can construct a consumer’s bud-

get line by nding two commodity bundles that she can \just a ord" and

drawing the straight line that runs through both points.

Example,Myrtle has 50 dollars to spend,She consumes only apples and

bananas,Apples cost 2 dollars each and bananas cost 1 dollar each,You

are to graph her budget line,where apples are measured on the horizontal

axis and bananas on the vertical axis,Notice that if she spends all of her

income on apples,she can a ord 25 apples and no bananas,Therefore

her budget line goes through the point (25;0) on the horizontal axis,If

she spends all of her income on bananas,she can a ord 50 bananas and

no apples,Therfore her budget line also passes throught the point (0;50)

on the vertical axis,Mark these two points on your graph,Then draw a

straight line between them,This is Myrtle’s budget line.

What if you are not told prices or income,but you know two com-

modity bundles that the consumer can just a ord? Then,if there are just

two commodities,you know that a unique line can be drawn through two

points,so you have enough information to draw the budget line.

Example,Laurel consumes only ale and bread,If she spends all of her

income,she can just a ord 20 bottles of ale and 5 loaves of bread,Another

commodity bundle that she can a ord if she spends her entire income is

10 bottles of ale and 10 loaves of bread,If the price of ale is 1 dollar per

bottle,how much money does she have to spend? You could solve this

problem graphically,Measure ale on the horizontal axis and bread on the

vertical axis,Plot the two points,(20;5) and (10;10),that you know to

be on the budget line,Draw the straight line between these points and

extend the line to the horizontal axis,This point denotes the amount of

6 BUDGET CONSTRAINT (Ch,2)

ale Laurel can a ord if she spends all of her money on ale,Since ale costs

1 dollar a bottle,her income in dollars is equal to the largest number of

bottles she can a ord,Alternatively,you can reason as follows,Since

the bundles (20;5) and (10;10) cost the same,it must be that giving up

10 bottles of ale makes her able to a ord an extra 5 loaves of bread,So

bread costs twice as much as ale,The price of ale is 1 dollar,so the price

of bread is 2 dollars,The bundle (20;5)costsasmuchasherincome.

Therefore her income must be 20 1+5 2 = 30.

When you have completed this workout,we hope that you will be

able to do the following:

Write an equation for the budget line and draw the budget set on a

graph when you are given prices and income or when you are given

two points on the budget line.

Graph the e ects of changes in prices and income on budget sets.

Understand the concept of numeraire and know what happens to the

budget set when income and all prices are multiplied by the same

positive amount.

Know what the budget set looks like if one or more of the prices is

negative.

See that the idea of a \budget set" can be applied to constrained

choices where there are other constraints on what you can have,in

addition to a constraint on money expenditure.

NAME 7

2.1 (0) You have an income of $40 to spend on two commodities,Com-

modity 1 costs $10 per unit,and commodity 2 costs $5 per unit.

(a) Write down your budget equation,10x

1

+5x

2

=40.

(b) If you spent all your income on commodity 1,how much could you

buy? 4.

(c) If you spent all of your income on commodity 2,how much could

you buy? 8,Use blue ink to draw your budget line in the graph

below.

02468

2

4

6

x1

x2

8

Blue Line

Red Line

Black Line

Black Shading

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

,,,,,,

Blue

Shading

(d) Suppose that the price of commodity 1 falls to $5 while everything else

stays the same,Write down your new budget equation,5x

1

+5x

2

=

40,On the graph above,use red ink to draw your new budget line.

(e) Suppose that the amount you are allowed to spend falls to $30,while

the prices of both commodities remain at $5,Write down your budget

equation,5x

1

+5x

2

=30,Use black ink to draw this budget

line.

(f) On your diagram,use blue ink to shade in the area representing com-

modity bundles that you can a ord with the budget in Part (e) but could

not a ord to buy with the budget in Part (a),Use black ink or pencil to

shade in the area representing commodity bundles that you could a ord

with the budget in Part (a) but cannot a ord with the budget in Part

(e).

2.2 (0) On the graph below,draw a budget line for each case.

8 BUDGET CONSTRAINT (Ch,2)

(a) p

1

=1,p

2

=1,m = 15,(Use blue ink.)

(b) p

1

=1,p

2

=2,m = 20,(Use red ink.)

(c) p

1

=0,p

2

=1,m = 10,(Use black ink.)

(d) p

1

= p

2

,m =15p

1

,(Use pencil or black ink,Hint,How much of

good 1 could you a ord if you spend your entire budget on good 1?)

0 5 10 15 20

5

10

15

x1

x2

20

Blue Line

Red Line

Black Line

2.3 (0) Your budget is such that if you spend your entire income,you

can a ord either 4 units of good x and 6 units of good y or 12 units of x

and 2 units of y.

(a) Mark these two consumption bundles and draw the budget line in the

graph below.

0481216

4

8

12

x

y

16

NAME 9

(b) What is the ratio of the price of x to the price of y? 1/2.

(c) If you spent all of your income on x,howmuchx could you buy?

16.

(d) If you spent all of your income on y,howmuchy could you buy?

8.

(e) Write a budget equation that gives you this budget line,where the

price of x is 1,x+2y =16.

(f) Write another budget equation that gives you the same budget line,

but where the price of x is 3,3x+6y =48.

2.4 (1) Murphy was consuming 100 units of X and 50 units of Y.The

price of X rose from 2 to 3,The price of Y remained at 4.

(a) How much would Murphy’s income have to rise so that he can still

exactly a ord 100 units of X and 50 units of Y? $100.

2.5 (1) If Amy spent her entire allowance,she could a ord 8 candy bars

and 8 comic books a week,She could also just a ord 10 candy bars and

4 comic books a week,The price of a candy bar is 50 cents,Draw her

budget line in the box below,What is Amy’s weekly allowance? $6.

0 8 16 24 32

8

16

24

Candy bars

Comic books

32

12

10 BUDGET CONSTRAINT (Ch,2)

2.6 (0) In a small country near the Baltic Sea,there are only three

commodities,potatoes,meatballs,and jam,Prices have been remark-

ably stable for the last 50 years or so,Potatoes cost 2 crowns per sack,

meatballs cost 4 crowns per crock,and jam costs 6 crowns per jar.

(a) Write down a budget equation for a citizen named Gunnar who has

an income of 360 crowns per year,Let P stand for the number of sacks of

potatoes,M for the number of crocks of meatballs,and J for the number

of jars of jam consumed by Gunnar in a year,2P +4M +6J =

360.

(b) The citizens of this country are in general very clever people,but they

are not good at multiplying by 2,This made shopping for potatoes excru-

ciatingly di cult for many citizens,Therefore it was decided to introduce

a new unit of currency,such that potatoes would be the numeraire,A

sack of potatoes costs one unit of the new currency while the same rel-

ative prices apply as in the past,In terms of the new currency,what is

the price of meatballs? 2 crowns.

(c) In terms of the new currency,what is the price of jam? 3

crowns.

(d) What would Gunnar’s income in the new currency have to be for him

to be exactly able to a ord the same commodity bundles that he could

a ord before the change? 180 crowns.

(e) Write down Gunnar’s new budget equation,P +2M +3J =

180,Is Gunnar’s budget set any di erent than it was before the change?

No.

2.7 (0) Edmund Stench consumes two commodities,namely garbage and

punk rock video cassettes,He doesn’t actually eat the former but keeps

it in his backyard where it is eaten by billy goats and assorted vermin.

The reason that he accepts the garbage is that people pay him $2 per

sack for taking it,Edmund can accept as much garbage as he wishes at

that price,He has no other source of income,Video cassettes cost him

$6 each.

(a) If Edmund accepts zero sacks of garbage,how many video cassettes

can he buy? 0.

NAME 11

(b) If he accepts 15 sacks of garbage,how many video cassettes can he

buy? 5.

(c) Write down an equation for his budget line,6C?2G =0.

(d) Draw Edmund’s budget line and shade in his budget set.

0 5 10 15 20

5

10

15

Video cassettes

Garbage

20

Budget Line

Budget Set

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

2.8 (0) If you think Edmund is odd,consider his brother Emmett.

Emmett consumes speeches by politicians and university administrators.

He is paid $1 per hour for listening to politicians and $2 per hour for

listening to university administrators,(Emmett is in great demand to help

ll empty chairs at public lectures because of his distinguished appearance

and his ability to refrain from making rude noises.) Emmett consumes

one good for which he must pay,We have agreed not to disclose what

that good is,but we can tell you that it costs $15 per unit and we shall

call it Good X,In addition to what he is paid for consuming speeches,

Emmett receives a pension of $50 per week.

0255075100

25

50

75

Politician speeches

Administrator speeches

100

12 BUDGET CONSTRAINT (Ch,2)

(a) Write down a budget equation stating those combinations of the three

commodities,Good X,hours of speeches by politicians (P),and hours of

speeches by university administrators (A) that Emmett could a ord to

consume per week,15X?1P?2A =50.

(b) On the graph above,draw a two-dimensional diagram showing the

locus of consumptions of the two kinds of speeches that would be possible

for Emmett if he consumed 10 units of Good X per week.

2.9 (0) Jonathan Livingstone Yuppie is a prosperous lawyer,He

has,in his own words,\outgrown those con ning two-commodity lim-

its." Jonathan consumes three goods,unblended Scotch whiskey,de-

signer tennis shoes,and meals in French gourmet restaurants,The price

of Jonathan’s brand of whiskey is $20 per bottle,the price of designer

tennis shoes is $80 per pair,and the price of gourmet restaurant meals

is $50 per meal,After he has paid his taxes and alimony,Jonathan has

$400 a week to spend.

(a) Write down a budget equation for Jonathan,where W stands for

the number of bottles of whiskey,T stands for the number of pairs of

tennis shoes,and M for the number of gourmet restaurant meals that he

consumes,20W +80T +50M = 400.

(b) Draw a three-dimensional diagram to show his budget set,Label the

intersections of the budget set with each axis.

M

TW

8

5

20

(c) Suppose that he determines that he will buy one pair of designer tennis

shoes per week,What equation must be satis ed by the combinations of

restaurant meals and whiskey that he could a ord? 20W+50M =

320.

2.10 (0) Martha is preparing for exams in economics and sociology,She

has time to read 40 pages of economics and 30 pages of sociology,In the

same amount of time she could also read 30 pages of economics and 60

pages of sociology.

NAME 13

(a) Assuming that the number of pages per hour that she can read of

either subject does not depend on how she allocates her time,how many

pages of sociology could she read if she decided to spend all of her time

on sociology and none on economics? 150 pages,(Hint,You

have two points on her budget line,so you should be able to determine

the entire line.)

(b) How many pages of economics could she read if she decided to spend

all of her time reading economics? 50 pages.

2.11 (1) Harry Hype has $5,000 to spend on advertising a new kind of

dehydrated sushi,Market research shows that the people most likely to

buy this new product are recent recipients of M.B.A,degrees and lawyers

who own hot tubs,Harry is considering advertising in two publications,

a boring business magazine and a trendy consumer publication for people

who wish they lived in California.

Fact 1,Ads in the boring business magazine cost $500 each and ads in

the consumer magazine cost $250 each.

Fact 2,Each ad in the business magazine will be read by 1,000 recent

M.B.A.’s and 300 lawyers with hot tubs.

Fact 3,Each ad in the consumer publication will be read by 300 recent

M.B.A.’s and 250 lawyers who own hot tubs.

Fact 4,Nobody reads more than one ad,and nobody who reads one

magazine reads the other.

(a) If Harry spends his entire advertising budget on the business pub-

lication,his ad will be read by 10,000 recent M.B.A.’s and by

3,000 lawyers with hot tubs.

(b) If he spends his entire advertising budget on the consumer publication,

his ad will be read by 6,000 recent M.B.A.’s and by 5,000

lawyers with hot tubs.

(c) Suppose he spent half of his advertising budget on each publication.

His ad would be read by 8,000 recent M.B.A.’s and by 4,000

lawyers with hot tubs.

(d) Draw a \budget line" showing the combinations of number of readings

by recent M.B.A.’s and by lawyers with hot tubs that he can obtain if he

spends his entire advertising budget,Does this line extend all the way

to the axes? No,Sketch,shade in,and label the budget set,which

includes all the combinations of MBA’s and lawyers he can reach if he

spends no more than his budget.

14 BUDGET CONSTRAINT (Ch,2)

(e) Let M stand for the number of instances of an ad being read by an

M.B.A,and L stand for the number of instances of an ad being read by

a lawyer,This budget line is a line segment that lies on the line with

equation M +2L =16,With a xed advertising budget,how

many readings by M.B.A.’s must he sacri ce to get an additional reading

by a lawyer with a hot tub? 2.

0481216

4

8

12

Lawyers x 1000

MBA's x 1000

16

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

,,,,,,,,

c

a

b

5

10

Budget

Set

Budget line

2

3

6

2.12 (0) On the planet Mungo,they have two kinds of money,blue

money and red money,Every commodity has two prices|a red-money

price and a blue-money price,Every Mungoan has two incomes|a red

income and a blue income.

In order to buy an object,a Mungoan has to pay that object’s red-

money price in red money and its blue-money price in blue money,(The

shops simply have two cash registers,and you have to pay at both registers

to buy an object.) It is forbidden to trade one kind of money for the other,

and this prohibition is strictly enforced by Mungo’s ruthless and e cient

monetary police.

There are just two consumer goods on Mungo,ambrosia and bubble

gum,All Mungoans prefer more of each good to less.

The blue prices are 1 bcu (bcu stands for blue currency unit) per

unit of ambrosia and 1 bcu per unit of bubble gum.

The red prices are 2 rcus (red currency units) per unit of ambrosia

and 6 rcus per unit of bubble gum.

(a) On the graph below,draw the red budget (with red ink) and the

blue budget (with blue ink) for a Mungoan named Harold whose blue

income is 10 and whose red income is 30,Shade in the \budget set"

containing all of the commodity bundles that Harold can a ord,given

NAME 15

its

two budget constraints,Remember,Harold has to have enough blue

money and enough red money to pay both the blue-money cost and the

red-money cost of a bundle of goods.

0 5 10 15 20

5

10

15

Ambrosia

Gum

20

Blue Lines

Red Line

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

(b) Another Mungoan,Gladys,faces the same prices that Harold faces

and has the same red income as Harold,but Gladys has a blue income of

20,Explain how it is that Gladys will not spend its entire blue income

no matter what its tastes may be,(Hint,Draw Gladys’s budget lines.)

The blue budget line lies strictly outside

the red budget line,so to satisfy both

budgets,one must be strictly inside the

red budget line.

(c) A group of radical economic reformers on Mungo believe that the

currency rules are unfair,\Why should everyone have to pay two prices

for everything?" they ask,They propose the following scheme,Mungo

will continue to have two currencies,every good will have a blue price and

a red price,and every Mungoan will have a blue income and a red income.

But nobody has to pay both prices,Instead,everyone on Mungo must

declare itself to be either a Blue-Money Purchaser (a \Blue") or a Red-

Money Purchaser (a \Red") before it buys anything at all,Blues must

make all of their purchases in blue money at the blue prices,spending

only their blue incomes,Reds must make all of their purchases in red

money,spending only their red incomes.

Suppose that Harold has the same income after this reform,and that

prices do not change,Before declaring which kind of purchaser it will be,

We refer to all Mungoans by the gender-neutral pronoun,\it." Al-

though Mungo has two sexes,neither of them is remotely like either of

ours.

16 BUDGET CONSTRAINT (Ch,2)

Harold contemplates the set of commodity bundles that it could a ord

by making one declaration or the other,Let us call a commodity bundle

\attainable" if Harold can a ord it by declaring itself to be a \Blue" and

buying the bundle with blue money or if Harold can a ord the bundle

by declaring itself to be a \Red" and buying it with red money,On the

diagram below,shade in all of the attainable bundles.

0 5 10 15 20

5

10

15

Ambrosia

Gum

20

Blue Line

Red Line

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

2.13 (0) Are Mungoan budgets really so fanciful? Can you think of sit-

uations on earth where people must simultaneously satisfy more than one

budget constraint? Is money the only scarce resource that people use up

when consuming? Consumption of many commodities

takes time as well as money,People have

to simultaneously satisfy a time budget

and a money budget,Other examples--people

may have a calorie budget or a cholesterol

budget or an alcohol-intake budget.

Chapter 3 NAME

Preferences

Introduction,In the previous section you learned how to use graphs to

show the set of commodity bundles that a consumer can a ord,In this

section,you learn to put information about the consumer’s preferences on

the same kind of graph,Most of the problems ask you to draw indi erence

curves.

Sometimes we give you a formula for the indi erence curve,Then

all you have to do is graph a known equation,But in some problems,we

give you only \qualitative" information about the consumer’s preferences

and ask you to sketch indi erence curves that are consistent with this

information,This requires a little more thought,Don’t be surprised or

disappointed if you cannot immediately see the answer when you look

at a problem,and don’t expect that you will nd the answers hiding

somewhere in your textbook,The best way we know to nd answers is to

\think and doodle." Draw some axes on scratch paper and label them,

then mark a point on your graph and ask yourself,\What other points on

the graph would the consumer nd indi erent to this point?" If possible,

draw a curve connecting such points,making sure that the shape of the

line you draw reflects the features required by the problem,This gives

you one indi erence curve,Now pick another point that is preferred to

the rst one you drew and draw an indi erence curve through it.

Example,Jocasta loves to dance and hates housecleaning,She has strictly

convex preferences,She prefers dancing to any other activity and never

gets tired of dancing,but the more time she spends cleaning house,the less

happy she is,Let us try to draw an indi erence curve that is consistent

with her preferences,There is not enough information here to tell us

exactly where her indi erence curves go,but there is enough information

to determine some things about their shape,Take a piece of scratch

paper and draw a pair of axes,Label the horizontal axis \Hours per day of

housecleaning." Label the vertical axis \Hours per day of dancing." Mark

a point a little ways up the vertical axis and write a 4 next to it,At this

point,she spends 4 hours a day dancing and no time housecleaning,Other

points that would be indi erent to this point would have to be points

where she did more dancing and more housecleaning,The pain of the

extra housekeeping should just compensate for the pleasure of the extra

dancing,So an indi erence curve for Jocasta must be upward sloping.

Because she loves dancing and hates housecleaning,it must be that she

prefers all the points above this indi erence curve to all of the points on

or below it,If Jocasta has strictly convex preferences,then it must be

that if you draw a line between any two points on the same indi erence

curve,all the points on the line (except the endpoints) are preferred to

the endpoints,For this to be the case,it must be that the indi erence

curve slopes upward ever more steeply as you move to the right along it.

You should convince yourself of this by making some drawings on scratch

18 PREFERENCES (Ch,3)

paper,Draw an upward-sloping curve passing through the point (0;4)

and getting steeper as one moves to the right.

When you have completed this workout,we hope that you will be

able to do the following:

Given the formula for an indi erence curve,draw this curve,and nd

its slope at any point on the curve.

Determine whether a consumer prefers one bundle to another or is

indi erent between them,given speci c indi erence curves.

Draw indi erence curves for the special cases of perfect substitutes

and perfect complements.

Draw indi erence curves for someone who dislikes one or both com-

modities.

Draw indi erence curves for someone who likes goods up to a point

but who can get \too much" of one or more goods.

Identify weakly preferred sets and determine whether these are con-

vex sets and whether preferences are convex.

Know what the marginal rate of substitution is and be able to deter-

mine whether an indi erence curve exhibits \diminishing marginal

rate of substitution."

Determine whether a preference relation or any other relation be-

tween pairs of things is transitive,whether it is reflexive,and whether

it is complete.

3.1 (0) Charlie likes both apples and bananas,He consumes nothing else.

The consumption bundle where Charlie consumes x

A

bushels of apples

per year and x

B

bushels of bananas per year is written as (x

A;x

B

),Last

year,Charlie consumed 20 bushels of apples and 5 bushels of bananas,It

happens that the set of consumption bundles (x

A;x

B

) such that Charlie

is indi erent between (x

A;x

B

)and(20;5) is the set of all bundles such

that x

B

= 100=x

A

,The set of bundles (x

A;x

B

) such that Charlie is just

indi erent between (x

A;x

B

) and the bundle (10;15) is the set of bundles

such that x

B

= 150=x

A

.

(a) On the graph below,plot several points that lie on the indi erence

curve that passes through the point (20;5),and sketch this curve,using

blue ink,Do the same,using red ink,for the indi erence curve passing

through the point (10;15).

(b) Use pencil to shade in the set of commodity bundles that Charlie

weakly prefers to the bundle (10;15),Use blue ink to shade in the set

of commodity bundles such that Charlie weakly prefers (20;5) to these

bundles.

NAME 19

010203040

10

20

30

Apples

Bananas

40

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Blue Curve

Pencil Shading

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,,,,,,,,,,,,,,,,,,,,,,,,

Red Curve

Blue Shading

For each of the following statements about Charlie’s preferences,write

\true" or \false."

(c) (30;5) (10;15),True.

(d) (10;15) (20;5),True.

(e) (20;5) (10;10),True.

(f) (24;4) (11;9:1),False.

(g) (11;14) (2;49),True.

(h) A set is convex if for any two points in the set,the line segment

between them is also in the set,Is the set of bundles that Charlie weakly

prefers to (20;5) a convex set? Yes.

(i) Is the set of bundles that Charlie considers inferior to (20;5) a convex

set? No.

(j) The slope of Charlie’s indi erence curve through a point,(x

A;x

B

),is

known as his marginal rate of substitution at that point.

20 PREFERENCES (Ch,3)

(k) Remember that Charlie’s indi erence curve through the point (10;10)

has the equation x

B

= 100=x

A

,Those of you who know calculus will

remember that the slope of a curve is just its derivative,which in this

case is?100=x

2

A

,(If you don’t know calculus,you will have to take our

word for this.) Find Charlie’s marginal rate of substitution at the point,

(10;10),?1.

(l) What is his marginal rate of substitution at the point (5;20)4.

(m) What is his marginal rate of substitution at the point (20;5)?

(?:25).

(n) Do the indi erence curves you have drawn for Charlie exhibit dimin-

ishing marginal rate of substitution? Yes.

3.2 (0) Ambrose consumes only nuts and berries,Fortunately,he likes

both goods,The consumption bundle where Ambrose consumes x

1

units

of nuts per week and x

2

units of berries per week is written as (x

1;x

2

).

The set of consumption bundles (x

1;x

2

) such that Ambrose is indi erent

between (x

1;x

2

)and(1;16) is the set of bundles such that x

1

0,x

2

0,

and x

2

=20?4

p

x

1

,The set of bundles (x

1;x

2

) such that (x

1;x

2

)

(36;0) is the set of bundles such that x

1

0,x

2

0andx

2

=24?4

p

x

1

.

(a) On the graph below,plot several points that lie on the indi erence

curve that passes through the point (1;16),and sketch this curve,using

blue ink,Do the same,using red ink,for the indi erence curve passing

through the point (36;0).

(b) Use pencil to shade in the set of commodity bundles that Ambrose

weakly prefers to the bundle (1;16),Use red ink to shade in the set of

all commodity bundles (x

1;x

2

) such that Ambrose weakly prefers (36;0)

to these bundles,Is the set of bundles that Ambrose prefers to (1;16) a

convex set? Yes.

(c) What is the slope of Ambrose’s indi erence curve at the point (9;8)?

(Hint,Recall from calculus the way to calculate the slope of a curve,If

you don’t know calculus,you will have to draw your diagram carefully

and estimate the slope.)?2=3.

NAME 21

(d) What is the slope of his indi erence curve at the point (4;12)1.

,,,,,,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,

010203040

10

20

30

Nuts

Berries

40

Pencil Shading

Red Curve

,

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,,,,,,,,,,,,,,,,,,,,,

Red

Shading

Blue Curve

(e) What is the slope of his indi erence curve at the point (9;12)2=3

at the point (4;16)1.

(f) Do the indi erence curves you have drawn for Ambrose exhibit dimin-

ishing marginal rate of substitution? Yes.

(g) Does Ambrose have convex preferences? Yes.

3.3 (0) Shirley Sixpack is in the habit of drinking beer each evening

while watching \The Best of Bowlerama" on TV,She has a strong thumb

and a big refrigerator,so she doesn’t care about the size of the cans that

beer comes in,she only cares about how much beer she has.

(a) On the graph below,draw some of Shirley’s indi erence curves be-

tween 16-ounce cans and 8-ounce cans of beer,Use blue ink to draw these

indi erence curves.

22 PREFERENCES (Ch,3)

02468

2

4

6

16-ounce

8-ounce

8

Blue Lines

Red Lines

(b) Lorraine Quiche likes to have a beer while she watches \Masterpiece

Theatre." She only allows herself an 8-ounce glass of beer at any one

time,Since her cat doesn’t like beer and she hates stale beer,if there is

more than 8 ounces in the can she pours the excess into the sink,(She

has no moral scruples about wasting beer.) On the graph above,use red

ink to draw some of Lorraine’s indi erence curves.

3.4 (0) Elmo nds himself at a Coke machine on a hot and dusty Sunday.

The Coke machine requires exact change|two quarters and a dime,No

other combination of coins will make anything come out of the machine.

No stores are open; no one is in sight,Elmo is so thirsty that the only

thing he cares about is how many soft drinks he will be able to buy with

the change in his pocket; the more he can buy,the better,While Elmo

searches his pockets,your task is to draw some indi erence curves that

describe Elmo’s preferences about what he nds.

NAME 23

02468

2

4

6

Quarters

Dimes

8

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,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,,,,,,,,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

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,,,,,,,,,,,,

,

,

,

,,,,,,,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,,,,,,

Red

shading

Blue

shading

Black

lines

(a) If Elmo has 2 quarters and a dime in his pockets,he can buy 1 soft

drink,How many soft drinks can he buy if he has 4 quarters and 2 dimes?

2.

(b) Use red ink to shade in the area on the graph consisting of all com-

binations of quarters and dimes that Elmo thinks are just indi erent to

having 2 quarters and 1 dime,(Imagine that it is possible for Elmo to

have fractions of quarters or of dimes,but,of course,they would be use-

less in the machine.) Now use blue ink to shade in the area consisting of

all combinations that Elmo thinks are just indi erent to having 4 quarters

and 2 dimes,Notice that Elmo has indi erence \bands," not indi erence

curves.

(c) Does Elmo have convex preferences between dimes and quarters?

Yes.

(d) Does Elmo always prefer more of both kinds of money to less? No.

(e) Does Elmo have a bliss point? No.

(f) If Elmo had arrived at the Coke machine on a Saturday,the drugstore

across the street would have been open,This drugstore has a soda foun-

tain that will sell you as much Coke as you want at a price of 4 cents an

ounce,The salesperson will take any combination of dimes and quarters

in payment,Suppose that Elmo plans to spend all of the money in his

pocket on Coke at the drugstore on Saturday,On the graph above,use

pencil or black ink to draw one or two of Elmo’s indi erence curves be-

tween quarters and dimes in his pocket,(For simplicity,draw your graph

24 PREFERENCES (Ch,3)

as if Elmo’s fractional quarters and fractional dimes are accepted at the

corresponding fraction of their value.) Describe these new indi erence

curves in words,Line segments with slope?2:5.

3.5 (0) Randy Ratpack hates studying both economics and history,The

more time he spends studying either subject,the less happy he is,But

Randy has strictly convex preferences.

(a) Sketch an indi erence curve for Randy where the two commodities

are hours per week spent studying economics and hours per week spent

studying history,Will the slope of an indi erence curve be positive or

negative? Negative.

(b) Do Randy’s indi erence curves get steeper or flatter as you move from

left to right along one of them? Steeper.

02468

2

4

6

Hours studying economics

Hours studying history

8

Preference

direction

3.6 (0) Flossy Toothsome likes to spend some time studying and some

time dating,In fact her indi erence curves between hours per week spent

studying and hours per week spent dating are concentric circles around

her favorite combination,which is 20 hours of studying and 15 hours of

dating per week,The closer she is to her favorite combination,the happier

she is.

NAME 25

(a) Suppose that Flossy is currently studying 25 hours a week and dating

3hoursaweek,Wouldsheprefertobestudying30hoursaweekand

dating8hoursaweek? Yes,(Hint,Remember the formula for the

distance between two points in the plane?)

(b) On the axes below,draw a few of Flossy’s indi erence curves and

use your diagram to illustrate which of the two time allocations discussed

above Flossy would prefer.

010203040

10

20

30

Hours studying

Hours dating

40

,

(25,3)

(30,8)

(20,15)

Preference

direction

3.7 (0) Joan likes chocolate cake and ice cream,but after 10 slices of

cake,she gets tired of cake,and eating more cake makes her less happy.

Joan always prefers more ice cream to less,Joan’s parents require her to

eat everything put on her plate,In the axes below,use blue ink to draw a

set of indi erence curves that depict her preferences between plates with

di erent amounts of cake and ice cream,Be sure to label the axes.

(a) Suppose that Joan’s preferences are as before,but that her parents

allow her to leave anything on her plate that she doesn’t want,On the

graph below,use red ink to draw some indi erence curves depicting her

preferences between plates with di erent amounts of cake and ice cream.

Blue curves

Red curves

Ice cream

Chocolate cake

10

Preference

direction

26 PREFERENCES (Ch,3)

3.8 (0) Professor Goodheart always gives two midterms in his commu-

nications class,He only uses the higher of the two scores that a student

gets on the midterms when he calculates the course grade.

(a) Nancy Lerner wants to maximize her grade in this course,Let x

1

be

her score on the rst midterm and x

2

be her score on the second midterm.

Which combination of scores would Nancy prefer,x

1

=20andx

2

=70

or x

1

=60andx

2

= 60? (20,70).

(b) On the graph below,use red ink to draw an indi erence curve showing

all of the combinations of scores that Nancy likes exactly as much as

x

1

=20andx

2

= 70,Also use red ink to draw an indi erence curve

showing the combinations that Nancy likes exactly as much as x

1

=60

and x

2

= 60.

(c) Does Nancy have convex preferences over these combinations? No.

020406080

20

40

60

Grade on first midterm

Grade on second midterm

80

,

Preference

direction

Blue curves

Red

curves

(d) Nancy is also taking a course in economics from Professor Stern.

Professor Stern gives two midterms,Instead of discarding the lower grade,

Professor Stern discards the higher one,Let x

1

be her score on the rst

midterm and x

2

be her score on the second midterm,Which combination

of scores would Nancy prefer,x

1

=20andx

2

=70orx

1

=60and

x

2

= 50? (60,50).

(e) On the graph above,use blue ink to draw an indi erence curve showing

all of the combinations of scores on her econ exams that Nancy likes

exactly as well as x

1

=20andx

2

= 70,Also use blue ink to draw an

indi erence curve showing the combinations that Nancy likes exactly as

well as x

1

=60andx

2

= 50,Does Nancy have convex preferences over

these combinations? Yes.

NAME 27

3.9 (0) Mary Granola loves to consume two goods,grapefruits and

avocados.

(a) On the graph below,the slope of an indi erence curve through any

point where she has more grapefruits than avocados is?2,This means

that when she has more grapefruits than avocados,she is willing to give

up 2 grapefruit(s) to get one avocado.

(b) On the same graph,the slope of an indi erence curve at points where

she has fewer grapefruits than avocados is?1=2,This means that when

she has fewer grapefruits than avocados,she is just willing to give up

1/2 grapefruit(s) to get one avocado.

(c) On this graph,draw an indi erence curve for Mary through bundle

(10A;10G),Draw another indi erence curve through (20A;20G).

010203040

10

20

30

Avocados

Grapefruits

40

45

Slope -2

Slope -1/2

(d) Does Mary have convex preferences? Yes.

3.10 (2) Ralph Rigid likes to eat lunch at 12 noon,However,he also

likes to save money so he can buy other consumption goods by attending

the \early bird specials" and \late lunchers" promoted by his local diner.

Ralph has 15 dollars a day to spend on lunch and other stu,Lunch at

noon costs $5,If he delays his lunch until t hours after noon,he is able

to buy his lunch for a price of $5?t,Similarly if he eats his lunch t hours

before noon,he can buy it for a price of $5?t,(This is true for fractions

of hours as well as integer numbers of hours.)

(a) If Ralph eats lunch at noon,how much money does he have per day

to spend on other stu? $10.

28 PREFERENCES (Ch,3)

(b) How much money per day would he have left for other stu if he ate

at 2 P.M.? $12.

(c) On the graph below,use blue ink to draw the broken line that shows

combinations of meal time and money for other stu that Ralph can just

a ord,On this same graph,draw some indi erence curves that would be

consistent with Ralph choosing to eat his lunch at 11 A.M.

0

11 12 1 2

5

10

15

Time

Money

20

10

3.11 (0) Henry Hanover is currently consuming 20 cheeseburgers and 20

Cherry Cokes a week,A typical indi erence curve for Henry is depicted

below.

Cheeseburgers

Cherry Coke

10 20 30 400

40

30

20

10

NAME 29

(a) If someone o ered to trade Henry one extra cheeseburger for every

Coke he gave up,would Henry want to do this? No.

(b) What if it were the other way around,for every cheeseburger Henry

gave up,he would get an extra Coke,Would he accept this o er? Yes.

(c) At what rate of exchange would Henry be willing to stay put at his

current consumption level? 2 cheeseburgers for 1

Coke.

3.12 (1) Tommy Twit is happiest when he has 8 cookies and 4 glasses of

milk per day,Whenever he has more than his favorite amount of either

food,giving him still more makes him worse o,Whenever he has less

than his favorite amount of either food,giving him more makes him better

o,His mother makes him drink 7 glasses of milk and only allows him 2

cookies per day,One day when his mother was gone,Tommy’s sadistic

sister made him eat 13 cookies and only gave him 1 glass of milk,despite

the fact that Tommy complained bitterly about the last 5 cookies that she

made him eat and begged for more milk,Although Tommy complained

later to his mother,he had to admit that he liked the diet that his sister

forced on him better than what his mother demanded.

(a) Use black ink to draw some indi erence curves for Tommy that are

consistent with this story.

0

(8,4)

(13,1)

(2,7)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Cookies

1

2

3

4

5

6

7

8

9

10

11

12

Milk

30 PREFERENCES (Ch,3)

(b) Tommy’s mother believes that the optimal amount for him to consume

is 7 glasses of milk and 2 cookies,She measures deviations by absolute

values,If Tommy consumes some other bundle,say,(c;m),she measures

his departure from the optimal bundle by D = j7?mj+j2?cj.The

larger D is,the worse o she thinks Tommy is,Use blue ink in the graph

above to sketch a few of Mrs,Twit’s indi erence curves for Tommy’s

consumption,(Hint,Before you try to draw Mrs,Twit’s indi erence

curves,we suggest that you take a piece of scrap paper and draw a graph

of the locus of points (x

1;x

2

) such that jx

1

j+jx

2

j=1.)

3.13 (0) Coach Steroid likes his players to be big,fast,and obedient,If

player A is better than player B in two of these three characteristics,then

Coach Steroid prefers A to B,but if B is better than A in two of these

three characteristics,then Steroid prefers B to A,Otherwise,Steroid is

indi erent between them,Wilbur Westinghouse weighs 340 pounds,runs

very slowly,and is fairly obedient,Harold Hotpoint weighs 240 pounds,

runs very fast,and is very disobedient,Jerry Jacuzzi weighs 150 pounds,

runs at average speed,and is extremely obedient.

(a) Does Steroid prefer Westinghouse to Hotpoint or vice versa? He

prefers Westinghouse to Hotpoint.

(b) Does Steroid prefer Hotpoint to Jacuzzi or vice versa? He

prefers Hotpoint to Jacuzzi.

(c) Does Steroid prefer Westinghouse to Jacuzzi or vice versa? He

prefers Jacuzzi to Westinghouse.

(d) Does Coach Steroid have transitive preferences? No.

(e) After several losing seasons,Coach Steroid decides to change his way of

judging players,According to his new preferences,Steroid prefers player

A to player B if player A is better in all three of the characteristics that

Steroid values,and he prefers B to A if player B is better at all three

things,He is indi erent between A and B if they weigh the same,are

equally fast,and are equally obedient,In all other cases,Coach Steroid

simply says \A and B are not comparable."

(f) Are Coach Steroid’s new preferences complete? No.

(g) Are Coach Steroid’s new preferences transitive? Yes.

NAME 31

(h) Are Coach Steroid’s new preferences reflexive? Yes.

3.14 (0) The Bear family is trying to decide what to have for din-

ner,Baby Bear says that his ranking of the possibilities is (honey,grubs,

Goldilocks),Mama Bear ranks the choices (grubs,Goldilocks,honey),

while Papa Bear’s ranking is (Goldilocks,honey,grubs),They decide to

take each pair of alternatives and let a majority vote determine the family

rankings.

(a) Papa suggests that they rst consider honey vs,grubs,and then the

winner of that contest vs,Goldilocks,Which alternative will be chosen?

Goldilocks.

(b) Mama suggests instead that they consider honey vs,Goldilocks and

then the winner vs,grubs,Which gets chosen? Grubs.

(c) What order should Baby Bear suggest if he wants to get his favorite

food for dinner? Grubs versus Goldilocks,then

Honey versus the winner.

(d) Are the Bear family’s \collective preferences," as determined by vot-

ing,transitive? No.

3.15 (0) Olson likes strong co ee,the stronger the better,But he can’t

distinguish small di erences,Over the years,Mrs,Olson has discovered

that if she changes the amount of co ee by more than one teaspoon in

her six-cup pot,Olson can tell that she did it,But he cannot distinguish

di erences smaller than one teaspoon per pot,Where A and B are two

di erent cups of co ee,let us write A B if Olson prefers cup A to

cup B,Let us write A B if Olson either prefers A to B,or can’t tell

the di erence between them,Let us write A B if Olson can’t tell the

di erence between cups A and B,Suppose that Olson is o ered cups A,

B,andC all brewed in the Olsons’ six-cup pot,Cup A was brewed using

14 teaspoons of co ee in the pot,CupB was brewed using 14.75 teaspoons

of co ee in the pot and cup C was brewed using 15.5 teaspoons of co ee

in the pot,For each of the following expressions determine whether it is

true of false.

(a) A B,True.

(b) B A,True.

32 PREFERENCES (Ch,3)

(c) B C,True.

(d) A C,False.

(e) C A,False.

(f) A B,True.

(g) B A,True.

(h) B C,True.

(i) A C,False.

(j) C A,True.

(k) A B,False.

(l) B A,False.

(m) B C,False.

(n) A C,False.

(o) C A,True.

(p) Is Olson’s \at-least-as-good-as" relation,,transitive? No.

(q) Is Olson’s \can’t-tell-the-di erence" relation,,transitive? No.

(r) is Olson’s \better-than" relation,,transitive,Yes.

Chapter 4 NAME

Utility

Introduction,In the previous chapter,you learned about preferences

and indi erence curves,Here we study another way of describing prefer-

ences,the utility function,A utility function that represents a person’s

preferences is a function that assigns a utility number to each commodity

bundle,The numbers are assigned in such a way that commodity bundle

(x;y) gets a higher utility number than bundle (x

0;y

0

) if and only if the

consumer prefers (x;y)to(x

0;y

0

),If a consumer has the utility function

U(x

1;x

2

),then she will be indi erent between two bundles if they are

assigned the same utility.

If you know a consumer’s utility function,then you can nd the

indi erence curve passing through any commodity bundle,Recall from

the previous chapter that when good 1 is graphed on the horizontal axis

and good 2 on the vertical axis,the slope of the indi erence curve passing

through a point (x

1;x

2

)isknownasthemarginal rate of substitution.An

important and convenient fact is that the slope of an indi erence curve is

minus the ratio of the marginal utility of good 1 to the marginal utility of

good 2,For those of you who know even a tiny bit of calculus,calculating

marginal utilities is easy,To nd the marginal utility of either good,

you just take the derivative of utility with respect to the amount of that

good,treating the amount of the other good as a constant,(If you don’t

know any calculus at all,you can calculate an approximation to marginal

utility by the method described in your textbook,Also,at the beginning

of this section of the workbook,we list the marginal utility functions for

commonly encountered utility functions,Even if you can’t compute these

yourself,you can refer to this list when later problems require you to use

marginal utilities.)

Example,Arthur’s utility function is U(x

1;x

2

)=x

1

x

2

,Let us nd the

indi erence curve for Arthur that passes through the point (3;4),First,

calculate U(3;4) = 3 4 = 12,The indi erence curve through this

point consists of all (x

1;x

2

) such that x

1

x

2

= 12,This last equation

is equivalent to x

2

=12=x

1

,Therefore to draw Arthur’s indi erence

curve through (3;4),just draw the curve with equation x

2

=12=x

1

.At

the point (x

1;x

2

),the marginal utility of good 1 is x

2

and the marginal

utility of good 2 is x

1

,Therefore Arthur’s marginal rate of substitution

at the point (3;4) is?x

2

=x

1

=?4=3.

Example,Arthur’s uncle,Basil,has the utility function U

(x

1;x

2

)=

3x

1

x

2

10,Notice that U

(x

1;x

2

)=3U(x

1;x

2

)?10,where U(x

1;x

2

)is

Arthur’s utility function,Since U

is a positive multiple of U minus a con-

stant,it must be that any change in consumption that increases U will also

increase U

(and vice versa),Therefore we say that Basil’s utility function

is a monotonic increasing transformation of Arthur’s utility function,Let

34 UTILITY (Ch,4)

us nd Basil’s indi erence curve through the point (3;4),First we nd

that U

(3;4) = 3 3 4?10 = 26,The indi erence curve passing through

this point consists of all (x

1;x

2

) such that 3x

1

x

2

10 = 26,Simplify this

last expression by adding 10 to both sides of the equation and dividing

both sides by 3,You nd x

1

x

2

= 12,or equivalently,x

2

=12=x

1

.This

is exactly the same curve as Arthur’s indi erence curve through (3;4).

We could have known in advance that this would happen,because if two

consumers’ utility functions are monotonic increasing transformations of

each other,then these consumers must have the same preference relation

between any pair of commodity bundles.

When you have nished this workout,we hope that you will be able

to do the following:

Draw an indi erence curve through a speci ed commodity bundle

when you know the utility function.

Calculate marginal utilities and marginal rates of substitution when

you know the utility function.

Determine whether one utility function is just a \monotonic transfor-

mation" of another and know what that implies about preferences.

Find utility functions that represent preferences when goods are per-

fect substitutes and when goods are perfect complements.

Recognize utility functions for commonly studied preferences such as

perfect substitutes,perfect complements,and other kinked indi er-

ence curves,quasilinear utility,and Cobb-Douglas utility.

4.0 WarmUpExercise,This is the rst of several \warm up ex-

ercises" that you will nd in Workouts,These are here to help you see

how to do calculations that are needed in later problems,The answers to

all warm up exercises are in your answer pages,If you nd the warm up

exercises easy and boring,go ahead|skip them and get on to the main

problems,You can come back and look at them if you get stuck later.

This exercise asks you to calculate marginal utilities and marginal

rates of substitution for some common utility functions,These utility

functions will reappear in several chapters,so it is a good idea to get to

know them now,If you know calculus,you will nd this to be a breeze.

Even if your calculus is shaky or nonexistent,you can handle the rst three

utility functions just by using the de nitions in the textbook,These three

are easy because the utility functions are linear,If you do not know any

calculus,ll in the rest of the answers from the back of the workbook and

keep a copy of this exercise for reference when you encounter these utility

functions in later problems.

NAME 35

u(x

1;x

2

) MU

1

(x

1;x

2

) MU

2

(x

1;x

2

) MRS(x

1;x

2

)

2x

1

+3x

2

2 3?2=3

4x

1

+6x

2

4 6?2=3

ax

1

+bx

2

a b?a=b

2

p

x

1

+x

2

1

p

x

1

1?

1

p

x

1

lnx

1

+x

2

1=x

1

1?1=x

1

v(x

1

)+x

2

v

0

(x

1

) 1?v

0

(x

1

)

x

1

x

2

x

2

x

1

x

2

=x

1

x

a

1

x

b

2

ax

a?1

1

x

b

2

bx

a

1

x

b?1

2

ax

2

bx

1

(x

1

+2)(x

2

+1) x

2

+1 x

1

+2?

x

2

+1

x

1

+2

(x

1

+a)(x

2

+b) x

2

+b x

1

+a?

x

2

+b

x

1

+a

x

a

1

+x

a

2

ax

a?1

1

ax

a?1

2

x

1

x

2

a?1

36 UTILITY (Ch,4)

4.1 (0) Remember Charlie from Chapter 3? Charlie consumes apples and

bananas,We had a look at two of his indi erence curves,In this problem

we give you enough information so you can nd all of Charlie’s indi erence

curves,We do this by telling you that Charlie’s utility function happens

to be U(x

A;x

B

)=x

A

x

B

.

(a) Charlie has 40 apples and 5 bananas,Charlie’s utility for the bun-

dle (40;5) is U(40;5) = 200,The indi erence curve through (40;5)

includes all commodity bundles (x

A;x

B

) such that x

A

x

B

= 200,So

the indi erence curve through (40;5) has the equation x

B

=

200

x

A

,On

the graph below,draw the indi erence curve showing all of the bundles

that Charlie likes exactly as well as the bundle (40;5).

010203040

10

20

30

Apples

Bananas

40

(b) Donna o ers to give Charlie 15 bananas if he will give her 25 apples.

Would Charlie have a bundle that he likes better than (40;5) if he makes

this trade? Yes,What is the largest number of apples that Donna

could demand from Charlie in return for 15 bananas if she expects him to

be willing to trade or at least indi erent about trading? 30,(Hint,If

Donna gives Charlie 15 bananas,he will have a total of 20 bananas,If he

has 20 bananas,how many apples does he need in order to be as well-o

as he would be without trade?)

4.2 (0) Ambrose,whom you met in the last chapter,continues to thrive

on nuts and berries,You saw two of his indi erence curves,One indif-

ference curve had the equation x

2

=20?4

p

x

1

,and another indi erence

curve had the equation x

2

=24?4

p

x

1

,wherex

1

is his consumption of

NAME 37

nuts and x

2

is his consumption of berries,Now it can be told that Am-

brose has quasilinear utility,In fact,his preferences can be represented

by the utility function U(x

1;x

2

)=4

p

x

1

+x

2

.

(a) Ambrose originally consumed 9 units of nuts and 10 units of berries.

His consumption of nuts is reduced to 4 units,but he is given enough

berries so that he is just as well-o as he was before,After the change,

how many units of berries does Ambrose consume? 14.

(b) On the graph below,indicate Ambrose’s original consumption and

sketch an indi erence curve passing through this point,As you can verify,

Ambrose is indi erent between the bundle (9,10) and the bundle (25,2).

If you doubled the amount of each good in each bundle,you would have

bundles (18,20) and (50,4),Are these two bundles on the same indi er-

ence curve? No,(Hint,How do you check whether two bundles are

indi erent when you know the utility function?)

0 5 10 15 20

5

10

15

Nuts

Berries

20

(9,10)

(c) What is Ambrose’s marginal rate of substitution,MRS(x

1;x

2

),when

he is consuming the bundle (9;10)? (Give a numerical answer.)?2=3.

What is Ambrose’s marginal rate of substitution when he is consuming

the bundle (9;20)2=3.

(d) We can write a general expression for Ambrose’s marginal rate of

substitution when he is consuming commodity bundle (x

1;x

2

),This is

MRS(x

1;x

2

)=?2=

p

x

1

,Although we always write MRS(x

1;x

2

)

as a function of the two variables,x

1

and x

2

,we see that Ambrose’s utility

function has the special property that his marginal rate of substitution

does not change when the variable x

2

changes.

38 UTILITY (Ch,4)

4.3 (0) Burt’s utility function is U(x

1;x

2

)=(x

1

+2)(x

2

+ 6),where x

1

is the number of cookies and x

2

is the number of glasses of milk that he

consumes.

(a) What is the slope of Burt’s indi erence curve at the point where he

is consuming the bundle (4;6)2,Use pencil or black ink to draw

a line with this slope through the point (4;6),(Try to make this graph

fairly neat and precise,since details will matter.) The line you just drew

is the tangent line to the consumer’s indi erence curve at the point (4;6).

(b) The indi erence curve through the point (4;6) passes through the

points ( 10,0),(7,2 ),and (2,12 ),Use blue ink

to sketch in this indi erence curve,Incidentally,the equation for Burt’s

indi erence curve through the point (4;6) is x

2

= 72=(x

1

+2)?6.

0481216

4

8

12

Cookies

Glasses of milk

16

a

b

Red Line

Black Line

Blue curve

(c) Burt currently has the bundle (4;6),Ernie o ers to give Burt 9

glasses of milk if Burt will give Ernie 3 cookies,If Burt makes this trade,

he would have the bundle (1;15),Burt refuses to trade,Was this

a wise decision? Yes,U(1;15) = 63 <U(4;6) = 72.

Mark the bundle (1;15) on your graph.

(d) Ernie says to Burt,\Burt,your marginal rate of substitution is?2.

That means that an extra cookie is worth only twice as much to you as

an extra glass of milk,I o ered to give you 3 glasses of milk for every

cookie you give me,If I o er to give you more than your marginal rate

of substitution,then you should want to trade with me." Burt replies,

NAME 39

\Ernie,you are right that my marginal rate of substitution is?2,That

means that I am willing to make small trades where I get more than 2

glasses of milk for every cookie I give you,but 9 glasses of milk for 3

cookies is too big a trade,My indi erence curves are not straight lines,

you see." Would Burt be willing to give up 1 cookie for 3 glasses of milk?

Yes,U(3;9) = 75 >U(4;6) = 72,Would Burt object to

giving up 2 cookies for 6 glasses of milk? No,U(2;12) = 72 =

U(4;6).

(e) On your graph,use red ink to draw a line with slope?3 through the

point (4;6),This line shows all of the bundles that Burt can achieve by

trading cookies for milk (or milk for cookies) at the rate of 1 cookie for

every 3 glasses of milk,Only a segment of this line represents trades that

make Burt better o than he was without trade,Label this line segment

on your graph AB.

4.4 (0) Phil Rupp’s utility function is U(x;y)=maxfx;2yg.

(a) On the graph below,use blue ink to draw and label the line whose

equation is x = 10,Also use blue ink to draw and label the line whose

equation is 2y = 10.

(b) If x =10and2y<10,then U(x;y)= 10,If x<10 and 2y = 10,

then U(x;y)= 10.

(c) Now use red ink to sketch in the indi erence curve along which

U(x;y) = 10,Does Phil have convex preferences? No.

0 5 10 15 20

5

10

15

x

y

20

Red

indifference

curve

Blue

lines

x=10

2y=10

40 UTILITY (Ch,4)

4.5 (0) As you may recall,Nancy Lerner is taking Professor Stern’s

economics course,She will take two examinations in the course,and her

score for the course is the minimum of the scores that she gets on the two

exams,Nancy wants to get the highest possible score for the course.

(a) Write a utility function that represents Nancy’s preferences over al-

ternative combinations of test scores x

1

and x

2

on tests 1 and 2 re-

spectively,U(x

1;x

2

)= minfx

1;x

2

g,or any monotonic

transformation.

4.6 (0) Remember Shirley Sixpack and Lorraine Quiche from the last

chapter? Shirley thinks a 16-ounce can of beer is just as good as two

8-ounce cans,Lorraine only drinks 8 ounces at a time and hates stale

beer,so she thinks a 16-ounce can is no better or worse than an 8-ounce

can.

(a) Write a utility function that represents Shirley’s preferences between

commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer.

Let X stand for the number of 8-ounce cans and Y stand for the number

of 16-ounce cans,u(X;Y)=X +2Y.

(b) Now write a utility function that represents Lorraine’s preferences.

u(X;Y)=X +Y.

(c) Would the function utility U(X;Y) = 100X+200Y represent Shirley’s

preferences? Yes,Would the utility function U(x;y)=(5X +10Y )

2

represent her preferences? Yes,Would the utility function U(x;y)=

X +3Y represent her preferences? No.

(d) Give an example of two commodity bundles such that Shirley likes

the rst bundle better than the second bundle,while Lorraine likes the

second bundle better than the rst bundle,Shirley prefers

(0,2) to (3,0),Lorraine disagrees.

4.7 (0) Harry Mazzola has the utility function u(x

1;x

2

)=minfx

1

+

2x

2;2x

1

+ x

2

g,wherex

1

is his consumption of corn chips and x

2

is his

consumption of french fries.

(a) On the graph below,use a pencil to draw the locus of points along

which x

1

+2x

2

=2x

1

+x

2

,Use blue ink to show the locus of points for

which x

1

+2x

2

= 12,and also use blue ink to draw the locus of points

for which 2x

1

+x

2

= 12.

NAME 41

(b) On the graph you have drawn,shade in the region where both of the

following inequalities are satis ed,x

1

+2x

2

12 and 2x

1

+ x

2

12.

At the bundle (x

1;x

2

)=(8;2),one sees that 2x

1

+ x

2

= 18 and

x

1

+2x

2

= 12,Therefore u(8;2) = 12.

(c) Use black ink to sketch in the indi erence curve along which Harry’s

utility is 12,Use red ink to sketch in the indi erence curve along which

Harry’s utility is 6,(Hint,Is there anything about Harry Mazzola that

reminds you of Mary Granola?)

02468

2

4

6

Corn chips

French fries

8

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

Pencil line

Red

line

Blue

lines

Black line

Blue

lines

(d) At the point where Harry is consuming 5 units of corn chips and 2

units of french fries,how many units of corn chips would he be willing to

trade for one unit of french fries? 2.

4.8 (1) Vanna Boogie likes to have large parties,She also has a strong

preference for having exactly as many men as women at her parties,In

fact,Vanna’s preferences among parties can be represented by the utility

function U(x;y)=minf2x?y;2y?xg where x is the number of women

and y is the number of men at the party,On the graph below,let us try

to draw the indi erence curve along which Vanna’s utility is 10.

(a) Use pencil to draw the locus of points at which x = y.Whatpoint

on this gives Vanna a utility of 10? (10;10),Use blue ink to draw

the line along which 2y?x = 10,When minf2x?y;2y?xg =2y?x,

42 UTILITY (Ch,4)

there are (more men than women,more women than men)? More

women,Draw a squiggly red line over the part of the blue line for which

U(x;y)=minf2x?y;2y?xg=2y?x,This shows all the combinations

that Vanna thinks are just as good as (10;10) but where there are (more

men than women,more women than men)? More women,Now

draw a blue line along which 2x?y = 10,Draw a squiggly red line over

the part of this new blue line for which minf2x?y;2y?xg=2x?y.Use

pencil to shade in the area on the graph that represents all combinations

that Vanna likes at least as well as (10;10).

(b) Suppose that there are 9 men and 10 women at Vanna’s party,Would

Vanna think it was a better party or a worse party if 5 more men came

to her party? Worse.

(c) If Vanna has 16 women at her party and more men than women,and

if she thinks the party is exactly as good as having 10 men and 10 women,

how many men does she have at the party? 22,If Vanna has 16 women

at her party and more women than men,and if she thinks the party is

exactly as good as having 10 men and 10 women,how many men does

she have at her party? 13.

(d) Vanna’s indi erence curves are shaped like what letter of the alpha-

bet? V.

0 5 10 15 20

5

10

15

x

y

20

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

Pencil

line

Blue

lines

Squiggly

red

lines

4.9 (0) Suppose that the utility functions u(x;y)andv(x;y) are related

by v(x;y)=f(u(x;y)),In each case below,write \Yes" if the function

f is a positive monotonic transformation and \No" if it is not,(Hint for

NAME 43

calculus users,A di erentiable function f(u) is an increasing function of

u if its derivative is positive.)

(a) f(u)=3:141592u,Yes.

(b) f(u)=5;000?23u,No.

(c) f(u)=u?100;000,Yes.

(d) f(u)=log

10

u,Yes.

(e) f(u)=?e

u

,Yes.

(f) f(u)=1=u,No.

(g) f(u)=?1=u,Yes.

4.10 (0) Martha Modest has preferences represented by the utility func-

tion U(a;b)=ab=100,where a is the number of ounces of animal crackers

that she consumes and b is the number of ounces of beans that she con-

sumes.

(a) On the graph below,sketch the locus of points that Martha nds

indi erent to having 8 ounces of animal crackers and 2 ounces of beans.

Also sketch the locus of points that she nds indi erent to having 6 ounces

of animal crackers and 4 ounces of beans.

02468

2

4

6

Animal crackers

Beans

8

(8,2)

(6,4)

44 UTILITY (Ch,4)

(b) Bertha Brassy has preferences represented by the utility function

V(a;b)=1;000a

2

b

2

,wherea is the number of ounces of animal crack-

ers that she consumes and b is the number of ounces of beans that she

consumes,On the graph below,sketch the locus of points that Bertha

nds indi erent to having 8 ounces of animal crackers and 2 ounces of

beans,Also sketch the locus of points that she nds indi erent to having

6 ounces of animal crackers and 4 ounces of beans.

02468

2

4

6

Animal crackers

Beans

8

(8,2)

(6,4)

(c) Are Martha’s preferences convex? Yes,Are Bertha’s? Yes.

(d) What can you say about the di erence between the indi erence curves

you drew for Bertha and those you drew for Martha? There is no

difference.

(e) How could you tell this was going to happen without having to draw

the curves? Their utility functions only differ

by a monotonic transformation.

4.11 (0) Willy Wheeler’s preferences over bundles that contain non-

negative amounts of x

1

and x

2

are represented by the utility function

U(x

1;x

2

)=x

2

1

+x

2

2

.

(a) Draw a few of his indi erence curves,What kind of geometric g-

ure are they? Quarter circles centered at the

origin,Does Willy have convex preferences? No.

NAME 45

02468

2

4

6

x1

x2

8

Calculus 4.12 (0) Joe Bob has a utility function given by u(x

1;x

2

)=x

2

1

+2x

1

x

2

+

x

2

2

.

(a) Compute Joe Bob’s marginal rate of substitution,MRS(x

1;x

2

)=

1.

(b) Joe Bob’s straight cousin,Al,has a utility function v(x

1;x

2

)=x

2

+x

1

.

Compute Al’s marginal rate of substitution,MRS(x

1;x

2

)=?1.

(c) Do u(x

1;x

2

)andv(x

1;x

2

) represent the same preferences? Yes.

Can you show that Joe Bob’s utility function is a monotonic transforma-

tion of Al’s? (Hint,Some have said that Joe Bob is square.) Notice

that u(x

1;x

2

)=[v(x

1;x

2

)]

2

.

4.13 (0) The idea of assigning numerical values to determine a preference

ordering over a set of objects is not limited in application to commodity

bundles,The Bill James Baseball Abstract argues that a baseball player’s

batting average is not an adequate measure of his o ensive productivity.

Batting averages treat singles just the same as extra base hits,Further-

more they do not give credit for \walks," although a walk is almost as

good as a single,James argues that a double in two at-bats is better than

a single,but not as good as two singles,To reflect these considerations,

James proposes the following index,which he calls \runs created." Let A

be the number of hits plus the number of walks that a batter gets in a sea-

son,Let B be the number of total bases that the batter gets in the season.

(Thus,if a batter has S singles,W walks,D doubles,T triples,and H

46 UTILITY (Ch,4)

home runs,then A = S+D+T+H+W and B = S+W+2D+3T+4H.)

Let N be the number of times the batter bats,Then his index of runs

created in the season is de ned to be AB=N and will be called his RC.

(a) In 1987,George Bell batted 649 times,He had 39 walks,105 singles,

32 doubles,4 triples,and 47 home runs,In 1987,Wade Boggs batted 656

times,He had 105 walks,130 singles,40 doubles,6 triples,and 24 home

runs,In 1987,Alan Trammell batted 657 times,He had 60 walks,140

singles,34 doubles,3 triples,and 28 home runs,In 1987,Tony Gwynn

batted 671 times,He had 82 walks,162 singles,36 doubles,13 triples,and

7 home runs,We can calculate A,the number of hits plus walks,B the

number of total bases,and RC,the runs created index for each of these

players,For Bell,A = 227,B = 408,RC = 143,For Boggs,A = 305,

B = 429,RC = 199,For Trammell,A = 265,B = 389,RC = 157,For

Gwynn,A = 300,B = 383,RC = 171.

(b) If somebody has a preference ordering among these players,based only

on the runs-created index,which player(s) would she prefer to Trammell?

Boggs and Gwynn.

(c) The di erences in the number of times at bat for these players are

small,and we will ignore them for simplicity of calculation,On the graph

below,plot the combinations of A and B achieved by each of the players.

Draw four \indi erence curves," one through each of the four points you

have plotted,These indi erence curves should represent combinations of

A and B that lead to the same number of runs-created.

0 120 180 240 300 360

Number of hits plus walks

80

160

240

320

400

Number of total bases

480

60

Bell

Trammell

Gwynn

Boggs

NAME 47

4.14 (0) This problem concerns the runs-created index discussed in the

preceding problem,Consider a batter who bats 100 times and always

either makes an out,hits for a single,or hits a home run.

(a) Let x be the number of singles and y be the number of home runs

in 100 at-bats,Suppose that the utility function U(x;y)bywhichwe

evaluate alternative combinations of singles and home runs is the runs-

created index,Then the formula for the utility function is U(x;y)=

(x+y)(x+4y)=100.

(b) Let’s try to nd out about the shape of an indi erence curve between

singles and home runs,Hitting 10 home runs and no singles would give

him the same runs-created index as hitting 20 singles and no

home runs,Mark the points (0;10) and (x;0),where U(x;0) = U(0;10).

(c) Where x is the number of singles you solved for in the previous part,

mark the point (x=2;5) on your graph,Is U(x=2;5) greater than or less

than or equal to U(0;10)? Greater than,Is this consistent with

the batter having convex preferences between singles and home runs?

Yes.

0 5 10 15 20

5

10

15

Singles

Home runs

20

(0,10)

(20,0)

(10,5)

Preference

direction

48 UTILITY (Ch,4)

Chapter 5 NAME

Choice

Introduction,You have studied budgets,and you have studied prefer-

ences,Now is the time to put these two ideas together and do something

with them,In this chapter you study the commodity bundle chosen by a

utility-maximizing consumer from a given budget.

Given prices and income,you know how to graph a consumer’s bud-

get,If you also know the consumer’s preferences,you can graph some of

his indi erence curves,The consumer will choose the \best" indi erence

curve that he can reach given his budget,But when you try to do this,you

have to ask yourself,\How do I nd the most desirable indi erence curve

that the consumer can reach?" The answer to this question is \look in the

likely places." Where are the likely places? As your textbook tells you,

there are three kinds of likely places,These are,(i) a tangency between

an indi erence curve and the budget line; (ii) a kink in an indi erence

curve; (iii) a \corner" where the consumer specializes in consuming just

one good.

Here is how you nd a point of tangency if we are told the consumer’s

utility function,the prices of both goods,and the consumer’s income,The

budget line and an indi erence curve are tangent at a point (x

1;x

2

)ifthey

have the same slope at that point,Now the slope of an indi erence curve

at (x

1;x

2

)istheratio?MU

1

(x

1;x

2

)=MU

2

(x

1;x

2

),(This slope is also

known as the marginal rate of substitution.) The slope of the budget line

is?p

1

=p

2

,Therefore an indi erence curve is tangent to the budget line

at the point (x

1;x

2

)whenMU

1

(x

1;x

2

)=MU

2

(x

1;x

2

)=p

1

=p

2

.Thisgives

us one equation in the two unknowns,x

1

and x

2

,If we hope to solve

for the x’s,we need another equation,That other equation is the budget

equation p

1

x

1

+ p

2

x

2

= m,With these two equations you can solve for

(x

1;x

2

).

Example,A consumer has the utility function U(x

1;x

2

)=x

2

1

x

2

.The

price of good 1 is p

1

= 1,the price of good 2 is p

2

= 3,and his income

is 180,Then,MU

1

(x

1;x

2

)=2x

1

x

2

and MU

2

(x

1;x

2

)=x

2

1

.There-

fore his marginal rate of substitution is?MU

1

(x

1;x

2

)=MU

2

(x

1;x

2

)=

2x

1

x

2

=x

2

1

=?2x

2

=x

1

,This implies that his indi erence curve will be

tangent to his budget line when?2x

2

=x

1

=?p

1

=p

2

=?1=3,Simplifying

this expression,we have 6x

2

= x

1

,This is one of the two equations we

need to solve for the two unknowns,x

1

and x

2

,The other equation is

the budget equation,In this case the budget equation is x

1

+3x

2

= 180.

Solving these two equations in two unknowns,we nd x

1

= 120 and

Some people have trouble remembering whether the marginal rate

of substitution is?MU

1

=MU

2

or?MU

2

=MU

1

,It isn’t really crucial to

remember which way this goes as long as you remember that a tangency

happens when the marginal utilities of any two goods are in the same

proportion as their prices.

50 CHOICE (Ch,5)

x

2

= 20,Therefore we know that the consumer chooses the bundle

(x

1;x

2

) = (120;20).

For equilibrium at kinks or at corners,we don’t need the slope of

the indi erence curves to equal the slope of the budget line,So we don’t

have the tangency equation to work with,But we still have the budget

equation,The second equation that you can use is an equation that tells

you that you are at one of the kinky points or at a corner,You will see

exactly how this works when you work a few exercises.

Example,A consumer has the utility function U(x

1;x

2

)=minfx

1;3x

2

g.

The price of x

1

is 2,the price of x

2

is 1,and her income is 140,Her

indi erence curves are L-shaped,The corners of the L’s all lie along the

line x

1

=3x

2

,She will choose a combination at one of the corners,so this

gives us one of the two equations we need for nding the unknowns x

1

and

x

2

,The second equation is her budget equation,which is 2x

1

+x

2

= 140.

Solve these two equations to nd that x

1

=60andx

2

= 20,So we know

that the consumer chooses the bundle (x

1;x

2

)=(60;20).

When you have nished these exercises,we hope that you will be

able to do the following:

Calculate the best bundle a consumer can a ord at given prices and

income in the case of simple utility functions where the best a ord-

able bundle happens at a point of tangency.

Find the best a ordable bundle,given prices and income for a con-

sumer with kinked indi erence curves.

Recognize standard examples where the best bundle a consumer can

a ord happens at a corner of the budget set.

Draw a diagram illustrating each of the above types of equilibrium.

Apply the methods you have learned to choices made with some kinds

of nonlinear budgets that arise in real-world situations.

5.1 (0) We begin again with Charlie of the apples and bananas,Recall

that Charlie’s utility function is U(x

A;x

B

)=x

A

x

B

,Suppose that the

price of apples is 1,the price of bananas is 2,and Charlie’s income is 40.

(a) On the graph below,use blue ink to draw Charlie’s budget line,(Use

a ruler and try to make this line accurate.) Plot a few points on the

indi erence curve that gives Charlie a utility of 150 and sketch this curve

with red ink,Now plot a few points on the indi erence curve that gives

Charlie a utility of 300 and sketch this curve with black ink or pencil.

NAME 51

010203040

10

20

30

Apples

Bananas

40

a

e

Blue

budget

line

Red

curves

Black curve

Pencil line

(b) Can Charlie a ord any bundles that give him a utility of 150? Yes.

(c) Can Charlie a ord any bundles that give him a utility of 300? No.

(d) On your graph,mark a point that Charlie can a ord and that gives

him a higher utility than 150,Label that point A.

(e) Neither of the indi erence curves that you drew is tangent to Charlie’s

budget line,Let’s try to nd one that is,At any point,(x

A;x

B

),Charlie’s

marginal rate of substitution is a function of x

A

and x

B

,In fact,if you

calculate the ratio of marginal utilities for Charlie’s utility function,you

will nd that Charlie’s marginal rate of substitution is MRS(x

A;x

B

)=

x

B

=x

A

,This is the slope of his indi erence curve at (x

A;x

B

),The

slope of Charlie’s budget line is?1=2 (give a numerical answer).

(f) Write an equation that implies that the budget line is tangent to an

indi erence curve at (x

A;x

B

),?x

B

=x

A

=?1=2,There are

many solutions to this equation,Each of these solutions corresponds to

a point on a di erent indi erence curve,Use pencil to draw a line that

passes through all of these points.

52 CHOICE (Ch,5)

(g) The best bundle that Charlie can a ord must lie somewhere on the

line you just penciled in,It must also lie on his budget line,If the point

is outside of his budget line,he can’t a ord it,If the point lies inside

of his budget line,he can a ord to do better by buying more of both

goods,On your graph,label this best a ordable bundle with an E.This

happens where x

A

= 20 and x

B

= 10,Verify your answer by

solving the two simultaneous equations given by his budget equation and

the tangency condition.

(h) What is Charlie’s utility if he consumes the bundle (20;10)? 200.

(i) On the graph above,use red ink to draw his indi erence curve through

(20,10),Does this indi erence curve cross Charlie’s budget line,just touch

it,or never touch it? Just touch it.

5.2 (0) Clara’s utility function is U(X;Y)=(X +2)(Y + 1),where X

is her consumption of good X and Y is her consumption of good Y.

(a) Write an equation for Clara’s indi erence curve that goes through the

point (X;Y)=(2;8),Y =

36

X+2

1,On the axes below,sketch

Clara’s indi erence curve for U = 36.

0481216

4

8

12

Y

16

11

11

U=36

X

(b) Suppose that the price of each good is 1 and that Clara has an income

of 11,Draw in her budget line,Can Clara achieve a utility of 36 with

this budget? Yes.

NAME 53

(c) At the commodity bundle,(X;Y),Clara’s marginal rate of substitu-

tion is?

Y+1

X+2

:

(d) If we set the absolute value of the MRS equal to the price ratio,we

have the equation

Y+1

X+2

=1:

(e) The budget equation is X +Y =11.

(f) Solving these two equations for the two unknowns,X and Y,we nd

X = 5 and Y = 6.

5.3 (0) Ambrose,the nut and berry consumer,has a utility function

U(x

1;x

2

)=4

p

x

1

+x

2

,wherex

1

is his consumption of nuts and x

2

is his

consumption of berries.

(a) The commodity bundle (25;0) gives Ambrose a utility of 20,Other

points that give him the same utility are (16;4),(9,8 ),(4,

12 ),(1,16 ),and (0,20 ),Plot these points on

the axes below and draw a red indi erence curve through them.

(b) Suppose that the price of a unit of nuts is 1,the price of a unit of

berries is 2,and Ambrose’s income is 24,Draw Ambrose’s budget line

with blue ink,How many units of nuts does he choose to buy? 16

units.

(c) How many units of berries? 4 units.

(d) Find some points on the indi erence curve that gives him a utility of

25 and sketch this indi erence curve (in red).

(e) Now suppose that the prices are as before,but Ambrose’s income is

34,Draw his new budget line (with pencil),How many units of nuts will

he choose? 16 units,How many units of berries? 9 units.

54 CHOICE (Ch,5)

010152025

Nuts

5

10

15

20

Berries

5 30

Red

curve

Blue line

Red curve

Pencil line

Blue

line

(f) Now let us explore a case where there is a \boundary solution." Sup-

pose that the price of nuts is still 1 and the price of berries is 2,but

Ambrose’s income is only 9,Draw his budget line (in blue),Sketch the

indi erence curve that passes through the point (9;0),What is the slope

of his indi erence curve at the point (9;0)2=3.

(g) What is the slope of his budget line at this point1=2.

(h) Which is steeper at this point,the budget line or the indi erence

curve? Indifference curve.

(i) Can Ambrose a ord any bundles that he likes better than the point

(9;0)? No.

5.4 (1) Nancy Lerner is trying to decide how to allocate her time in

studying for her economics course,There are two examinations in this

course,Her overall score for the course will be the minimum of her scores

on the two examinations,She has decided to devote a total of 1,200

minutes to studying for these two exams,and she wants to get as high an

overall score as possible,She knows that on the rst examination if she

doesn’t study at all,she will get a score of zero on it,For every 10 minutes

that she spends studying for the rst examination,she will increase her

score by one point,If she doesn’t study at all for the second examination

she will get a zero on it,For every 20 minutes she spends studying for

the second examination,she will increase her score by one point.

NAME 55

(a) On the graph below,draw a \budget line" showing the various com-

binations of scores on the two exams that she can achieve with a total of

1,200 minutes of studying,On the same graph,draw two or three \indif-

ference curves" for Nancy,On your graph,draw a straight line that goes

through the kinks in Nancy’s indi erence curves,Label the point where

this line hits Nancy’s budget with the letter A,Draw Nancy’s indi erence

curve through this point.

0 40 60 80 100

Score on test 1

20

40

60

80

Score on test 2

20 120

a

Budget line

"L" shaped

indifference

curves

(b) Write an equation for the line passing through the kinks of Nancy’s

indi erence curves,x

1

= x

2

.

(c) Write an equation for Nancy’s budget line,10x

1

+20x

2

=

1;200.

(d) Solve these two equations in two unknowns to determine the intersec-

tion of these lines,This happens at the point (x

1;x

2

)= (40;40).

(e) Given that she spends a total of 1,200 minutes studying,Nancy will

maximize her overall score by spending 400 minutes studying for the

rst examination and 800 minutes studying for the second examina-

tion.

5.5 (1) In her communications course,Nancy also takes two examina-

tions,Her overall grade for the course will be the maximum of her scores

on the two examinations,Nancy decides to spend a total of 400 minutes

studying for these two examinations,If she spends m

1

minutes studying

56 CHOICE (Ch,5)

for the rst examination,her score on this exam will be x

1

= m

1

=5,If

she spends m

2

minutes studying for the second examination,her score on

this exam will be x

2

= m

2

=10.

(a) On the graph below,draw a \budget line" showing the various combi-

nations of scores on the two exams that she can achieve with a total of 400

minutes of studying,On the same graph,draw two or three \indi erence

curves" for Nancy,On your graph,nd the point on Nancy’s budget line

that gives her the best overall score in the course.

(b) Given that she spends a total of 400 minutes studying,Nancy will

maximize her overall score by achieving a score of 80 on the rst

examination and 0 on the second examination.

(c) Her overall score for the course will then be 80.

020406080

20

40

60

Score on test 1

Score on test 2

80

,

Preference

direction

Max,

overall

score

5.6 (0) Elmer’s utility function is U(x;y)=minfx;y

2

g.

(a) If Elmer consumes 4 units of x and 3 units of y,his utility is 4.

(b) If Elmer consumes 4 units of x and 2 units of y,his utility is 4.

(c) If Elmer consumes 5 units of x and 2 units of y,his utility is 4.

(d) On the graph below,use blue ink to draw the indi erence curve for

Elmer that contains the bundles that he likes exactly as well as the bundle

(4;2).

NAME 57

(e) On the same graph,use blue ink to draw the indi erence curve for

Elmer that contains bundles that he likes exactly as well as the bundle

(1;1) and the indi erence curve that passes through the point (16;5).

(f) On your graph,use black ink to show the locus of points at which

Elmer’s indi erence curves have kinks,What is the equation for this

curve? x = y

2

.

(g) On the same graph,use black ink to draw Elmer’s budget line when

the price of x is 1,the price of y is 2,and his income is 8,What bundle

does Elmer choose in this situation? (4,2).

0 8 12 16 20

x

4

8

12

16

y

4 24

Black

line

(16,5)

Blue

curves

Blue curve

Black curve

Chosen

bundle

(h) Suppose that the price of x is 10 and the price of y is 15 and Elmer

buys 100 units of x,What is Elmer’s income? 1,150,(Hint,At rst

you might think there is too little information to answer this question.

But think about how much y he must be demanding if he chooses 100

units of x.)

5.7 (0) Linus has the utility function U(x;y)=x+3y.

(a) On the graph below,use blue ink to draw the indi erence curve passing

through the point (x;y)=(3;3),Use black ink to sketch the indi erence

curve connecting bundles that give Linus a utility of 6.

58 CHOICE (Ch,5)

0481216

4

8

12

Y

16

X

(3,3)

Blue

curve

Black

curve

Red line

(b) On the same graph,use red ink to draw Linus’s budget line if the

price of x is 1 and the price of y is 2 and his income is 8,What bundle

does Linus choose in this situation? (0,4).

(c) What bundle would Linus choose if the price of x is 1,the price of y

is 4,and his income is 8? (8,0).

5.8 (2) Remember our friend Ralph Rigid from Chapter 3? His favorite

diner,Food for Thought,has adopted the following policy to reduce the

crowds at lunch time,if you show up for lunch t hours before or after

12 noon,you get to deduct t dollars from your bill,(This holds for any

fraction of an hour as well.)

NAME 59

0

11 12 1 2

5

10

15

Time

Money

20

10

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyy

Red curves

Blue budget set

(a) Use blue ink to show Ralph’s budget set,On this graph,the horizontal

axis measures the time of day that he eats lunch,and the vertical axis

measures the amount of money that he will have to spend on things other

than lunch,Assume that he has $20 total to spend and that lunch at

noon costs $10,(Hint,How much money would he have left if he ate at

noon?at1P.M.?at11A.M.?)

(b) Recall that Ralph’s preferred lunch time is 12 noon,but that he is

willing to eat at another time if the food is su ciently cheap,Draw

some red indi erence curves for Ralph that would be consistent with his

choosing to eat at 11 A.M.

5.9 (0) Joe Grad has just arrived at the big U,He has a fellowship that

covers his tuition and the rent on an apartment,In order to get by,Joe

has become a grader in intermediate price theory,earning $100 a month.

Out of this $100 he must pay for his food and utilities in his apartment.

His utilities expenses consist of heating costs when he heats his apartment

and air-conditioning costs when he cools it,To raise the temperature of

his apartment by one degree,it costs $2 per month (or $20 per month

to raise it ten degrees),To use air-conditioning to cool his apartment by

a degree,it costs $3 per month,Whatever is left over after paying the

utilities,he uses to buy food at $1 per unit.

60 CHOICE (Ch,5)

02030405060

Temperature

20

40

60

80

100

Food

120

10 70 80 90 100

December September August

Black budget constraint

Blue budget constraint

Red budget constraint

(a) When Joe rst arrives in September,the temperature of his apartment

is 60 degrees,If he spends nothing on heating or cooling,the temperature

in his room will be 60 degrees and he will have $100 left to spend on food.

If he heated the room to 70 degrees,he would have $80 left to spend

on food,If he cooled the room to 50 degrees,he would have $70 left

to spend on food,On the graph below,show Joe’s September budget

constraint (with black ink),(Hint,You have just found three points that

Joe can a ord,Apparently,his budget set is not bounded by a single

straight line.)

(b) In December,the outside temperature is 30 degrees and in August

poor Joe is trying to understand macroeconomics while the temperature

outside is 85 degrees,On the same graph you used above,draw Joe’s

budget constraints for the months of December (in blue ink) and August

(in red ink).

(c) Draw a few smooth (unkinky) indi erence curves for Joe in such a way

that the following are true,(i) His favorite temperature for his apartment

would be 65 degrees if it cost him nothing to heat it or cool it,(ii)Joe

chooses to use the furnace in December,air-conditioning in August,and

neither in September,(iii) Joe is better o in December than in August.

(d) In what months is the slope of Joe’s budget constraint equal to the

slope of his indi erence curve? August and December.

NAME 61

(e) In December Joe’s marginal rate of substitution between food and

degrees Fahrenheit is -2,In August,his MRS is 3.

(f) Since Joe neither heats nor cools his apartment in September,we

cannot determine his marginal rate of substitution exactly,but we do

know that it must be no smaller than -2 and no larger than

3,(Hint,Look carefully at your graph.)

5.10 (0) Central High School has $60,000 to spend on computers and

other stu,so its budget equation is C + X =60;000,where C is ex-

penditure on computers and X is expenditures on other things,C.H.S.

currently plans to spend $20,000 on computers.

The State Education Commission wants to encourage \computer lit-

eracy" in the high schools under its jurisdiction,The following plans have

been proposed.

Plan A,This plan would give a grant of $10,000 to each high school in

the state that the school could spend as it wished.

Plan B,This plan would give a $10,000 grant to any high school,so

long as the school spent at least $10,000 more than it currently spends on

computers,Any high school can choose not to participate,in which case it

does not receive the grant,but it doesn’t have to increase its expenditure

on computers.

Plan C,Plan C is a \matching grant." For every dollar’s worth of

computers that a high school orders,the state will give the school 50

cents.

Plan D,This plan is like plan C,except that the maximum amount of

matching funds that any high school could get from the state would be

limited to $10,000.

(a) Write an equation for Central High School’s budget if plan A is

adopted,C + X =70;000,Use black ink to draw the bud-

get line for Central High School if plan A is adopted.

(b) If plan B is adopted,the boundary of Central High School’s budget set

has two separate downward-sloping line segments,One of these segments

describes the cases where C.H.S,spends at least $30,000 on computers.

This line segment runs from the point (C;X)=(70;000;0) to the point

(C;X)= (30,000,40,000).

(c) Another line segment corresponds to the cases where C.H.S,spends

less than $30,000 on computers,This line segment runs from (C;X)=

(30,000,30,000) to the point (C;X)=(0;60;000),Use red

ink to draw these two line segments.

62 CHOICE (Ch,5)

(d) If plan C is adopted and Central High School spendsC dollars on com-

puters,then it will have X =60;000?:5C dollars left to spend on other

things,Therefore its budget line has the equation,5C+X=60,000.

Use blue ink to draw this budget line.

(e) If plan D is adopted,the school district’s budget consists of two

line segments that intersect at the point where expenditure on comput-

ers is 20,000 and expenditure on other instructional materials is

50,000.

(f) The slope of the flatter line segment is?:5,The slope of the

steeper segment is?1,Use pencil to draw this budget line.

0 20 30 40 50 60

Thousands of dollars worth of computers

10

20

30

40

50

Thousands of dollars worth of other things

60

10

Red budget line

(plan B)

Black budget line (plan A)

Pencil budget line

(plan D)

Blue

budget

line

(plan C)

5.11 (0) Suppose that Central High School has preferences that can

be represented by the utility function U(C;X)=CX

2

,Let us try to

determine how the various plans described in the last problem will a ect

the amount that C.H.S,spends on computers.

NAME 63

(a) If the state adopts none of the new plans,nd the expenditure on

computers that maximizes the district’s utility subject to its budget con-

straint,20,000.

(b) If plan A is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,23,333.

(c) On your graph,sketch the indi erence curve that passes through the

point (30,000,40,000) if plan B is adopted,At this point,which is steeper,

the indi erence curve or the budget line? The budget line.

(d) If plan B is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,(Hint,Look

at your graph.) 30,000.

(e) If plan C is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,40,000.

(f) If plan D is adopted,nd the expenditure on computers that maxi-

mizes the district’s utility subject to its budget constraint,23,333.

5.12 (0) The telephone company allows one to choose between two

di erent pricing plans,For a fee of $12 per month you can make as

many local phone calls as you want,at no additional charge per call.

Alternatively,you can pay $8 per month and be charged 5 cents for each

local phone call that you make,Suppose that you have a total of $20 per

month to spend.

(a) On the graph below,use black ink to sketch a budget line for someone

who chooses the rst plan,Use red ink to draw a budget line for someone

who chooses the second plan,Where do the two budget lines cross?

(80;8).

64 CHOICE (Ch,5)

0 40 60 80 100

Local phone calls

4

8

12

16

Other goods

20 120

Black line

Red line

Pencil curve

Blue curve

(b) On the graph above,use pencil to draw indi erence curves for some-

one who prefers the second plan to the rst,Use blue ink to draw an

indi erence curve for someone who prefers the rst plan to the second.

5.13 (1) This is a puzzle|just for fun,Lewis Carroll (1832-1898),

author of Alice in Wonderland and Through the Looking Glass,was a

mathematician,logician,and political scientist,Carroll loved careful rea-

soning about puzzling things,Here Carroll’s Alice presents a nice bit

of economic analysis,At rst glance,it may seem that Alice is talking

nonsense,but,indeed,her reasoning is impeccable.

\I should like to buy an egg,please." she said timidly,\How do you

sell them?"

\Fivepence farthing for one|twopence for two," the Sheep replied.

\Then two are cheaper than one?" Alice said,taking out her purse.

\Only you must eat them both if you buy two," said the Sheep.

\Then I’ll have one please," said Alice,as she put the money down

on the counter,For she thought to herself,\They mightn’t be at all nice,

you know."

(a) Let us try to draw a budget set and indi erence curves that are

consistent with this story,Suppose that Alice has a total of 8 pence to

spend and that she can buy either 0,1,or 2 eggs from the Sheep,but no

fractional eggs,Then her budget set consists of just three points,The

point where she buys no eggs is (0;8),Plot this point and label it A.On

your graph,the point where she buys 1 egg is (1;2

3

4

),(A farthing is 1/4

of a penny.) Plot this point and label it B.

(b) The point where she buys 2 eggs is (2;6),Plot this point and

label it C,If Alice chooses to buy 1 egg,she must like the bundleB better

than either the bundle A or the bundle C,Draw indi erence curves for

Alice that are consistent with this behavior.

NAME 65

01234

2

4

6

Eggs

Other goods

8

b

a

c

66 CHOICE (Ch,5)

Chapter 6 NAME

Demand

Introduction,In the previous chapter,you found the commodity bundle

that a consumer with a given utility function would choose in a speci c

price-income situation,In this chapter,we take this idea a step further.

We nd demand functions,which tell us for any prices and income you

might want to name,how much of each good a consumer would want,In

general,the amount of each good demanded may depend not only on its

own price,but also on the price of other goods and on income,Where

there are two goods,we write demand functions for Goods 1 and 2 as

x

1

(p

1;p

2;m)andx

2

(p

1;p

2;m).

When the consumer is choosing positive amounts of all commodities

and indi erence curves have no kinks,the consumer chooses a point of

tangency between her budget line and the highest indi erence curve that

it touches.

Example,Consider a consumer with utility function U(x

1;x

2

)=(x

1

+

2)(x

2

+ 10),To nd x

1

(p

1;p

2;m)andx

2

(p

1;p

2;m),we need to nd a

commodity bundle (x

1;x

2

) on her budget line at which her indi erence

curve is tangent to her budget line,The budget line will be tangent to

the indi erence curve at (x

1;x

2

) if the price ratio equals the marginal

rate of substitution,For this utility function,MU

1

(x

1;x

2

)=x

2

+10 and

MU

2

(x

1;x

2

)=x

1

+ 2,Therefore the \tangency equation" is p

1

=p

2

=

(x

2

+ 10)=(x

1

+ 2),Cross-multiplying the tangency equation,one nds

p

1

x

1

+2p

1

= p

2

x

2

+10p

2

.

The bundle chosen must also satisfy the budget equation,p

1

x

1

+

p

2

x

2

= m,This gives us two linear equations in the two unknowns,x

1

and x

2

,You can solve these equations yourself,using high school algebra.

You will nd that the solution for the two \demand functions" is

x

1

=

m?2p

1

+10p

2

2p

1

x

2

=

m+2p

1

10p

2

2p

2

:

There is one thing left to worry about with the \demand functions" we

just found,Notice that these expressions will be positive only if m?2p

1

+

10p

2

> 0andm+2p

1

10p

2

> 0,If either of these expressions is negative,

then it doesn’t make sense as a demand function,What happens in this

For some utility functions,demand for a good may not be a ected by

all of these variables,For example,with Cobb-Douglas utility,demand

for a good depends on the good’s own price and on income but not on the

other good’s price,Still,there is no harm in writing demand for Good

1 as a function of p

1

,p

2

,andm,It just happens that the derivative of

x

1

(p

1;p

2;m) with respect to p

2

is zero.

68 DEMAND (Ch,6)

case is that the consumer will choose a \boundary solution" where she

consumes only one good,At this point,her indi erence curve will not be

tangent to her budget line.

When a consumer has kinks in her indi erence curves,she may choose

a bundle that is located at a kink,In the problems with kinks,you

will be able to solve for the demand functions quite easily by looking

at diagrams and doing a little algebra,Typically,instead of nding a

tangency equation,you will nd an equation that tells you \where the

kinks are." With this equation and the budget equation,you can then

solve for demand.

You might wonder why we pay so much attention to kinky indi er-

ence curves,straight line indi erence curves,and other \funny cases."

Our reason is this,In the funny cases,computations are usually pretty

easy,But often you may have to draw a graph and think about what

you are doing,That is what we want you to do,Think and ddle with

graphs,Don’t just memorize formulas,Formulas you will forget,but the

habit of thinking will stick with you.

When you have nished this workout,we hope that you will be able

to do the following:

Find demand functions for consumers with Cobb-Douglas and other

similar utility functions.

Find demand functions for consumers with quasilinear utility func-

tions.

Find demand functions for consumers with kinked indi erence curves

and for consumers with straight-line indi erence curves.

Recognize complements and substitutes from looking at a demand

curve.

Recognize normal goods,inferior goods,luxuries,and necessities from

looking at information about demand.

Calculate the equation of an inverse demand curve,given a simple

demand equation.

6.1 (0) Charlie is back|still consuming apples and bananas,His util-

ity function is U(x

A;x

B

)=x

A

x

B

,We want to nd his demand func-

tion for apples,x

A

(p

A;p

B;m),and his demand function for bananas,

x

B

(p

A;p

B;m).

(a) When the prices arep

A

andp

B

and Charlie’s income ism,the equation

for Charlie’s budget line isp

A

x

A

+p

B

x

B

= m,The slope of Charlie’s indif-

ference curve at the bundle (x

A;x

B

)is?MU

1

(x

A;x

B

)=MU

2

(x

A;x

B

)=

x

B

=x

A

,The slope of Charlie’s budget line is?p

A

=p

B

,Char-

lie’s indi erence curve will be tangent to his budget line at the point

(x

A;x

B

) if the following equation is satis ed,p

A

=p

B

= x

B

=x

A

.

NAME 69

(b) You now have two equations,the budget equation and the tan-

gency equation,that must be satis ed by the bundle demanded,Solve

these two equations for x

A

and x

B

,Charlie’s demand function for ap-

ples is x

A

(p

A;p

B;m)=

m

2p

A

,and his demand function for bananas is

x

B

(p

A;p

B;m)=

m

2p

B

.

(c) In general,the demand for both commodities will depend on the price

of both commodities and on income,But for Charlie’s utility function,

the demand function for apples depends only on income and the price

of apples,Similarly,the demand for bananas depends only on income

and the price of bananas,Charlie always spends the same fraction of his

income on bananas,What fraction is this? 1=2.

6.2 (0) Douglas Corn eld’s preferences are represented by the utility

function u(x

1;x

2

)=x

2

1

x

3

2

,The prices of x

1

and x

2

are p

1

and p

2

.

(a) The slope of Corn eld’s indi erence curve at the point (x

1;x

2

)is

2x

2

=3x

1

.

(b) If Corn eld’s budget line is tangent to his indi erence curve at (x

1;x

2

),

then

p

1

x

1

p

2

x

2

= 2/3,(Hint,Look at the equation that equates the slope

of his indi erence curve with the slope of his budget line.) When he is

consuming the best bundle he can a ord,what fraction of his income does

Douglas spend on x

1

2/5.

(c) Other members of Doug’s family have similar utility functions,but

the exponents may be di erent,or their utilities may be multiplied by a

positive constant,If a family member has a utility function U(x;y)=

cx

a

1

x

b

2

where a,b,andc are positive numbers,what fraction of his or her

income will that family member spend on x

1

a/(a+b).

6.3 (0) Our thoughts return to Ambrose and his nuts and berries,Am-

brose’s utility function is U(x

1;x

2

)=4

p

x

1

+ x

2

,wherex

1

is his con-

sumption of nuts and x

2

is his consumption of berries.

(a) Let us nd his demand function for nuts,The slope of Ambrose’s

indi erence curve at (x

1;x

2

)is?

2

p

x

1

,Setting this slope equal to

the slope of the budget line,you can solve for x

1

without even using the

budget equation,The solution is x

1

=

2p

2

p

1

2

.

70 DEMAND (Ch,6)

(b) Let us nd his demand for berries,Now we need the budget equation.

In Part (a),you solved for the amount of x

1

that he will demand,The

budget equation tells us that p

1

x

1

+ p

2

x

2

= M,Plug the solution that

you found for x

1

into the budget equation and solve for x

2

as a function

of income and prices,The answer is x

2

=

M

p

2

4

p

2

p

1

.

(c) When we visited Ambrose in Chapter 5,we looked at a \boundary

solution," where Ambrose consumed only nuts and no berries,In that

example,p

1

=1,p

2

=2,andM = 9,If you plug these numbers into the

formulas we found in Parts (a) and (b),you nd x

1

= 16,and

x

2

=?3:5,Since we get a negative solution for x

2

,it must be that

the budget line x

1

+2x

2

= 9 is not tangent to an indi erence curve when

x

2

0,The best that Ambrose can do with this budget is to spend all

of his income on nuts,Looking at the formulas,we see that at the prices

p

1

=1andp

2

= 2,Ambrose will demand a positive amount of both goods

if and only if M> 16.

6.4 (0) Donald Fribble is a stamp collector,The only things other

than stamps that Fribble consumes are Hostess Twinkies,It turns out

that Fribble’s preferences are represented by the utility function u(s;t)=

s +lnt where s is the number of stamps he collects and t is the number

of Twinkies he consumes,The price of stamps is p

s

and the price of

Twinkies is p

t

,Donald’s income is m.

(a) Write an expression that says that the ratio of Fribble’s marginal

utility for Twinkies to his marginal utility for stamps is equal to the ratio

of the price of Twinkies to the price of stamps,1=t = p

t

=p

s

,(Hint:

The derivative of lnt with respect to t is 1=t,and the derivative of s with

respect to s is 1.)

(b) You can use the equation you found in the last part to show that if he

buys both goods,Donald’s demand function for Twinkies depends only

on the price ratio and not on his income,Donald’s demand function for

Twinkies is t(p

s;p

t;m)=p

s

=p

t

.

(c) Notice that for this special utility function,if Fribble buys both goods,

then the total amount of money that he spends on Twinkies has the

peculiar property that it depends on only one of the three variables m,

p

t

,andp

s

,namely the variable p

s

,(Hint,The amount of money that

he spends on Twinkies is p

t

t(p

s;p

t;m).)

NAME 71

(d) Since there are only two goods,any money that is not spent on

Twinkies must be spent on stamps,Use the budget equation and Don-

ald’s demand function for Twinkies to nd an expression for the number

of stamps he will buy if his income is m,the price of stamps is p

s

and the

price of Twinkies is p

t

,s =

m

p

s

1.

(e) The expression you just wrote down is negative if m<p

s

,Surely

it makes no sense for him to be demanding negative amounts of postage

stamps,If m<p

s

,what would Fribble’s demand for postage stamps be?

s =0 What would his demand for Twinkies be? t = m=p

t

.

(Hint,Recall the discussion of boundary optimum.)

(f) Donald’s wife complains that whenever Donald gets an extra dollar,

he always spends it all on stamps,Is she right? (Assume that m>p

s

.)

Yes.

(g) Suppose that the price of Twinkies is $2 and the price of stamps is $1.

On the graph below,draw Fribble’s Engel curve for Twinkies in red ink

and his Engel curve for stamps in blue ink,(Hint,First draw the Engel

curves for incomes greater than $1,then draw them for incomes less than

$1.)

02468

2

4

6

Quantities

Income

8

Blue

line

Red line

1

0.5

6.5 (0) Shirley Sixpack,as you will recall,thinks that two 8-ounce cans

of beer are exactly as good as one 16-ounce can of beer,Suppose that

these are the only sizes of beer available to her and that she has $30 to

spend on beer,Suppose that an 8-ounce beer costs $.75 and a 16-ounce

beer costs $1,On the graph below,draw Shirley’s budget line in blue ink,

and draw some of her indi erence curves in red.

72 DEMAND (Ch,6)

010203040

10

20

30

16-ounce cans

8-ounce cans

40

Blue

budget

line

Red

curves

Red curve

(a) At these prices,which size can will she buy,or will she buy some of

each? 16-ounce cans.

(b) Suppose that the price of 16-ounce beers remains $1 and the price of

8-ounce beers falls to $.55,Will she buy more 8-ounce beers? No.

(c) What if the price of 8-ounce beers falls to $.40? How many 8-ounce

beers will she buy then? 75 cans.

(d) If the price of 16-ounce beers is $1 each and if Shirley chooses some

8-ounce beers and some 16-ounce beers,what must be the price of 8-ounce

beers? $.50.

(e) Now let us try to describe Shirley’s demand function for 16-ounce beers

as a function of general prices and income,Let the prices of 8-ounce and

16-ounce beers be p

8

and p

16

,and let her income be m.Ifp

16

< 2p

8

,then

the number of 16-ounce beers she will demand is m=p

16

,If p

16

> 2p

8

,

then the number of 16-ounce beers she will demand is 0,If p

16

=

2 p

8

,she will be indi erent between any a ordable combinations.

6.6 (0) Miss Mu et always likes to have things \just so." In fact the

only way she will consume her curds and whey is in the ratio of 2 units of

whey per unit of curds,She has an income of $20,Whey costs $.75 per

unit,Curds cost $1 per unit,On the graph below,draw Miss Mu et’s

budget line,and plot some of her indi erence curves,(Hint,Have you

noticed something kinky about Miss Mu et?)

NAME 73

(a) How many units of curds will Miss Mu et demand in this situation?

8 units,How many units of whey? 16 units.

0 8 16 24 32

8

16

24

Curds

Whey

32

w = 2c

Budget

line

Indifference

curves

(b) Write down Miss Mu et’s demand function for whey as a function

of the prices of curds and whey and of her income,where p

c

is the price

of curds,p

w

is the price of whey,and m is her income,D(p

c;p

w;m)=

m

p

w

+p

c

=2

,(Hint,You can solve for her demands by solving two equa-

tions in two unknowns,One equation tells you that she consumes twice

as much whey as curds,The second equation is her budget equation.)

6.7 (1) Mary’s utility function is U(b;c)=b+ 100c?c

2

,whereb is the

number of silver bells in her garden and c is the number of cockle shells.

She has 500 square feet in her garden to allocate between silver bells and

cockle shells,Silver bells each take up 1 square foot and cockle shells each

take up 4 square feet,She gets both kinds of seeds for free.

(a) To maximize her utility,given the size of her garden,Mary should

plant 308 silver bells and 48 cockle shells,(Hint,Write down

her \budget constraint" for space,Solve the problem as if it were an

ordinary demand problem.)

(b) If she suddenly acquires an extra 100 square feet for her garden,how

much should she increase her planting of silver bells? 100 extra

silver bells,How much should she increase her planting of

cockle shells? Not at all.

74 DEMAND (Ch,6)

(c) If Mary had only 144 square feet in her garden,how many cockle

shells would she grow? 36.

(d) If Mary grows both silver bells and cockle shells,then we know that

the number of square feet in her garden must be greater than 192.

6.8 (0) Casper consumes cocoa and cheese,He has an income of $16.

Cocoa is sold in an unusual way,There is only one supplier and the more

cocoa one buys from him,the higher the price one has to pay per unit.

In fact,x units of cocoa will cost Casper a total of x

2

dollars,Cheese is

sold in the usual way at a price of $2 per unit,Casper’s budget equation,

therefore,is x

2

+2y =16wherex is his consumption of cocoa and y is

his consumption of cheese,Casper’s utility function is U(x;y)=3x+y.

(a) On the graph below,draw the boundary of Casper’s budget set in

blue ink,Use red ink to sketch two or three of his indi erence curves.

0481216

4

8

12

Cheese

16

Cocoa

Red

indifference

curves

Blue budget line

(b) Write an equation that says that at the point (x;y),the slope

of Casper’s budget \line" equals the slope of his indi erence \curve."

2x=2=3=1,Casper demands 3 units of cocoa and 3.5

units of cheese.

6.9 (0) Perhaps after all of the problems with imaginary people and

places,you would like to try a problem based on actual fact,The U.S.

government’s Bureau of Labor Statistics periodically makes studies of

family budgets and uses the results to compile the consumer price index.

These budget studies and a wealth of other interesting economic data can

be found in the annually published Handbook of Labor Statistics,The

NAME 75

tables below report total current consumption expenditures and expendi-

tures on certain major categories of goods for 5 di erent income groups

in the United States in 1961,People within each of these groups all had

similar incomes,Group A is the lowest income group and Group E is the

highest.

Table 6.1

Expenditures by Category for Various Income Groups in 1961

Income Group A B C D E

Food Prepared at Home 465 783 1078 1382 1848

Food Away from Home 68 171 213 384 872

Housing 626 1090 1508 2043 4205

Clothing 119 328 508 830 1745

Transportation 139 519 826 1222 2048

Other 364 745 1039 1554 3490

Total Expenditures 1781 3636 5172 7415 14208

Table 6.2

Percentage Allocation of Family Budget

Income Group A B C D E

Food Prepared at Home 26 22 21 19 13

Food Away from Home 3.8 4.7 4.1 5.2 6.1

Housing 35 30 29 28 30

Clothing 6.7 9.0 9.8 11 12

Transportation 7.8 14 16 17 14

(a) Complete Table 6.2.

(b) Which of these goods are normal goods? All of them.

(c) Which of these goods satisfy your textbook’s de nition of luxury

goods at most income levels? Food away from home,

clothing,transportation.

76 DEMAND (Ch,6)

(d) Which of these goods satisfy your textbook’s de nition of necessity

goods at most income levels? Food prepared at home,

housing.

(e) On the graph below,use the information from Table 6.1 to draw

\Engel curves." (Use total expenditure on current consumption as income

for purposes of drawing this curve.) Use red ink to draw the Engel curve

for food prepared at home,Use blue ink to draw an Engel curve for food

away from home,Use pencil to draw an Engel curve for clothing,How

does the shape of an Engel curve for a luxury di er from the shape of

an Engel curve for a necessity? The curve for a luxury

gets flatter as income rises,the curve for

a necessity gets steeper.

0 750 1500 2250 3000

3

6

9

Total expenditures (thousands of dollars)

12

Expenditure on specific goods

Red lineBlue

line

Pencil

line

6.10 (0) Percy consumes cakes and ale,His demand function for cakes

is q

c

= m?30p

c

+20p

a

,wherem is his income,p

a

is the price of ale,p

c

is the price of cakes,and q

c

is his consumption of cakes,Percy’s income

is $100,and the price of ale is $1 per unit.

(a) Is ale a substitute for cakes or a complement? Explain,A

substitute,An increase in the price of

ale increases demand for cakes.

NAME 77

(b) Write an equation for Percy’s demand function for cakes where income

and the price of ale are held xed at $100 and $1,q

c

= 120?30p

c

.

(c) Write an equation for Percy’s inverse demand function for cakes where

income is $100 and the price of ale remains at $1,p

c

=4?q

c

=30.

At what price would Percy buy 30 cakes? $3,Use blue ink to draw

Percy’s inverse demand curve for cakes.

(d) Suppose that the price of ale rises to $2.50 per unit and remains

there,Write an equation for Percy’s inverse demand for cakes,p

c

=

5?q

c

=30,Use red ink to draw in Percy’s new inverse demand curve

for cakes.

0306090120

1

2

3

Number of cakes

Price

4

Blue Line

Red Line

6.11 (0) Richard and Mary Stout have fallen on hard times,but remain

rational consumers,They are making do on $80 a week,spending $40 on

food and $40 on all other goods,Food costs $1 per unit,On the graph

below,use black ink to draw a budget line,Label their consumption

bundle with the letter A.

(a) The Stouts suddenly become eligible for food stamps,This means

that they can go to the agency and buy coupons that can be exchanged

for $2 worth of food,Each coupon costs the Stouts $1,However,the

maximum number of coupons they can buy per week is 10,On the graph,

draw their new budget line with red ink.

78 DEMAND (Ch,6)

(b) If the Stouts have homothetic preferences,how much more food will

they buy once they enter the food stamp program? 5 units.

0 40 60 80 100 120

20

40

60

80

100

Dollars worth of other things

120

20

a

New consumption point

45

Red budget line

Black budget line

Food

Calculus 6.12 (2) As you may remember,Nancy Lerner is taking an economics

course in which her overall score is the minimum of the number of correct

answers she gets on two examinations,For the rst exam,each correct

answer costs Nancy 10 minutes of study time,For the second exam,each

correct answer costs her 20 minutes of study time,In the last chapter,

you found the best way for her to allocate 1200 minutes between the two

exams,Some people in Nancy’s class learn faster and some learn slower

than Nancy,Some people will choose to study more than she does,and

some will choose to study less than she does,In this section,we will nd

a general solution for a person’s choice of study times and exam scores as

a function of the time costs of improving one’s score.

(a) Suppose that if a student does not study for an examination,he or

she gets no correct answers,Every answer that the student gets right

on the rst examination costs P

1

minutes of studying for the rst exam.

Every answer that he or she gets right on the second examination costs

P

2

minutes of studying for the second exam,Suppose that this student

spends a total of M minutes studying for the two exams and allocates

the time between the two exams in the most e cient possible way,Will

the student have the same number of correct answers on both exams?

NAME 79

Yes,Write a general formula for this student’s overall score for the

course as a function of the three variables,P

1

,P

2

,andM,S =

M

P

1

+P

2

.

If this student wants to get an overall score of S,with the smallest pos-

sible total amount of studying,this student must spend P

1

S minutes

studying for the rst exam and P

2

S studying for the second exam.

(b) Suppose that a student has the utility function

U(S;M)=S?

A

2

M

2;

where S is the student’s overall score for the course,M is the number

of minutes the student spends studying,and A is a variable that reflects

how much the student dislikes studying,In Part (a) of this problem,you

found that a student who studies for M minutes and allocates this time

wisely between the two exams will get an overall score of S =

M

P

1

+P

2

.

Substitute

M

P

1

+P

2

for S in the utility function and then di erentiate with

respect to M to nd the amount of study time,M,that maximizes the

student’s utility,M =

1

A(P

1

+P

2

)

,Your answer will be a function of

the variables P

1

,P

2

,andA,If the student chooses the utility-maximizing

amount of study time and allocates it wisely between the two exams,he

or she will have an overall score for the course of S =

1

A(P

1

+P

2

)

2

.

(c) Nancy Lerner has a utility function like the one presented above,She

chose the utility-maximizing amount of study time for herself,For Nancy,

P

1

=10andP

2

= 20,She spent a total of M =1;200 minutes studying

for the two exams,This gives us enough information to solve for the

variable A in Nancy’s utility function,In fact,for Nancy,A =

1

36;000

.

(d) Ed Fungus is a student in Nancy’s class,Ed’s utility function is just

like Nancy’s,with the same value of A,But Ed learns more slowly than

Nancy,In fact it takes Ed exactly twice as long to learn anything as it

takes Nancy,so that for him,P

1

=20andP

2

= 40,Ed also chooses his

amount of study time so as to maximize his utility,Find the ratio of the

amount of time Ed spends studying to the amount of time Nancy spends

studying,1/2,Will his score for the course be greater than half,

equal to half,or less than half of Nancy’s? Less than half.

6.13 (1) Here is a puzzle for you,At rst glance,it would appear that

there is not nearly enough information to answer this question,But when

you graph the indi erence curve and think about it a little,you will see

that there is a neat,easily calculated solution.

80 DEMAND (Ch,6)

Kinko spends all his money on whips and leather jackets,Kinko’s

utility function is U(x;y)=minf4x;2x+yg,wherex is his consumption

of whips and y is his consumption of leather jackets,Kinko is consuming

15 whips and 10 leather jackets,The price of whips is $10,You are to

nd Kinko’s income.

(a) Graph the indi erence curve for Kinko that passes through the point

(15;10),What is the slope of this indi erence curve at (15;10)2.

What must be the price of leather jackets if Kinko chooses this point?

$5,Now,what is Kinko’s income? 15 10 + 10 5 = 200.

010203040

10

20

30

Whips

Leather jackets

40

(15,10)

Indifference

curve

2x + y = 40

4x = 40

Chapter 7 NAME

Revealed Preference

Introduction,In the last section,you were given a consumer’s pref-

erences and then you solved for his or her demand behavior,In this

chapter we turn this process around,you are given information about a

consumer’s demand behavior and you must deduce something about the

consumer’s preferences,The main tool is the weak axiom of revealed pref-

erence,This axiom says the following,If a consumer chooses commodity

bundle A when she can a ord bundleB,then she will never choose bundle

B from any budget in which she can also a ord A,The idea behind this

axiomisthatifyouchooseA when you could have had B,you must like

A better than B,But if you like A better than B,then you will never

choose B when you can have A,If somebody chooses A when she can

a ord B,we say that for her,A is directly revealed preferred to B.The

weak axiom says that if A is directly revealed preferred to B,thenB is

not directly revealed preferred to A.

Example,Let us look at an example of how you check whether one bundle

is revealed preferred to another,Suppose that a consumer buys the bundle

(x

A

1;x

A

2

)=(2;3) at prices (p

A

1;p

A

2

)=(1;4),The cost of bundle (x

A

1;x

A

2

)

at these prices is (2 1) + (3 4) = 14,Bundle (2;3) is directly revealed

preferred to all the other bundles that she can a ord at prices (1;4),when

she has an income of 14,For example,the bundle (5;2) costs only 13 at

prices (1;4),so we can say that for this consumer (2;3) is directly revealed

preferred to (1;4).

You will also have some problems about price and quantity indexes.

A price index is a comparison of average price levels between two di erent

times or two di erent places,If there is more than one commodity,it is not

necessarily the case that all prices changed in the same proportion,Let us

suppose that we want to compare the price level in the \current year" with

the price level in some \base year." One way to make this comparison

is to compare the costs in the two years of some \reference" commodity

bundle,Two reasonable choices for the reference bundle come to mind.

One possibility is to use the current year’s consumption bundle for the

reference bundle,The other possibility is to use the bundle consumed

in the base year,Typically these will be di erent bundles,If the base-

year bundle is the reference bundle,the resulting price index is called the

Laspeyres price index,If the current year’s consumption bundle is the

reference bundle,then the index is called the Paasche price index.

Example,Suppose that there are just two goods,In 1980,the prices

were (1;3) and a consumer consumed the bundle (4;2),In 1990,the

prices were (2;4) and the consumer consumed the bundle (3;3),The cost

of the 1980 bundle at 1980 prices is (1 4)+ (3 2) = 10,The cost of this

same bundle at 1990 prices is (2 4) + (4 2) = 16,If 1980 is treated

as the base year and 1990 as the current year,the Laspeyres price ratio

82 REVEALED PREFERENCE (Ch,7)

is 16=10,To calculate the Paasche price ratio,you nd the ratio of the

cost of the 1990 bundle at 1990 prices to the cost of the same bundle at

1980 prices,The 1990 bundle costs (2 3) + (4 3) = 18 at 1990 prices.

The same bundle cost (1 3) + (3 3) = 12 at 1980 prices,Therefore

the Paasche price index is 18=12,Notice that both price indexes indicate

that prices rose,but because the price changes are weighted di erently,

the two approaches give di erent price ratios.

Making an index of the \quantity" of stu consumed in the two

periods presents a similar problem,How do you weight changes in the

amount of good 1 relative to changes in the amount of good 2? This time

we could compare the cost of the two periods’ bundles evaluated at some

reference prices,Again there are at least two reasonable possibilities,the

Laspeyres quantity index and the Paasche quantity index,The Laspeyres

quantity index uses the base-year prices as the reference prices,and the

Paasche quantity index uses current prices as reference prices.

Example,In the example above,the Laspeyres quantity index is the ratio

of the cost of the 1990 bundle at 1980 prices to the cost of the 1980 bundle

at 1980 prices,The cost of the 1990 bundle at 1980 prices is 12 and the

cost of the 1980 bundle at 1980 prices is 10,so the Laspeyres quantity

index is 12/10,The cost of the 1990 bundle at 1990 prices is 18 and

the cost of the 1980 bundle at 1990 prices is 16,Therefore the Paasche

quantity index is 18/16.

When you have completed this section,we hope that you will be able

to do the following:

Decide from given data about prices and consumption whether one

commodity bundle is preferred to another.

Given price and consumption data,calculate Paasche and Laspeyres

price and quantity indexes.

Use the weak axiom of revealed preferences to make logical deduc-

tions about behavior.

Use the idea of revealed preference to make comparisons of well-being

across time and across countries.

7.1 (0) When prices are (4;6),Goldie chooses the bundle (6;6),and

when prices are (6;3),she chooses the bundle (10;0).

(a) On the graph below,show Goldie’s rst budget line in red ink and

her second budget line in blue ink,Mark her choice from the rst budget

with the label A,and her choice from the second budget with the label

B.

(b) Is Goldie’s behavior consistent with the weak axiom of revealed pref-

erence? No.

NAME 83

0 5 10 15 20

5

10

15

Good 1

Good 2

20

a

b

Blue line

Red line

7.2 (0) Freddy Frolic consumes only asparagus and tomatoes,which are

highly seasonal crops in Freddy’s part of the world,He sells umbrellas for

a living,which provides a fluctuating income depending on the weather.

But Freddy doesn’t mind; he never thinks of tomorrow,so each week he

spends as much as he earns,One week,when the prices of asparagus and

tomatoes were each $1 a pound,Freddy consumed 15 pounds of each,Use

blue ink to show the budget line in the diagram below,Label Freddy’s

consumption bundle with the letter A.

(a) What is Freddy’s income? $30.

(b) The next week the price of tomatoes rose to $2 a pound,but the price

of asparagus remained at $1 a pound,By chance,Freddy’s income had

changed so that his old consumption bundle of (15,15) was just a ordable

at the new prices,Use red ink to draw this new budget line on the graph

below,Does your new budget line go through the point A? Yes.

What is the slope of this line1=2.

(c) How much asparagus can he a ord now if he spent all of his income

on asparagus? 45 pounds.

(d) What is Freddy’s income now? $45.

84 REVEALED PREFERENCE (Ch,7)

(e) Use pencil to shade the bundles of goods on Freddy’s new red budget

line that he de nitely will not purchase with this budget,Is it possible

that he would increase his consumption of tomatoes when his budget

changes from the blue line to the red one? No.

010203040

10

20

30

Asparagus

Tomatoes

40

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

a

Blue line

Red line

Pencil

shading

7.3 (0) Pierre consumes bread and wine,For Pierre,the price of bread

is 4 francs per loaf,and the price of wine is 4 francs per glass,Pierre has

an income of 40 francs per day,Pierre consumes 6 glasses of wine and 4

loaves of bread per day.

Bob also consumes bread and wine,For Bob,the price of bread is

1/2 dollar per loaf and the price of wine is 2 dollars per glass,Bob has

an income of $15 per day.

(a) If Bob and Pierre have the same tastes,can you tell whether Bob is

better o than Pierre or vice versa? Explain,Bob is better

off,He can afford Pierre’s bundle and

still have income left.

(b) Suppose prices and incomes for Pierre and Bob are as above and that

Pierre’s consumption is as before,Suppose that Bob spends all of his

income,Give an example of a consumption bundle of wine and bread such

that,if Bob bought this bundle,we would know that Bob’s tastes are not

the same as Pierre’s tastes,7.5 wine and 0 bread,for

example,If they had the same preferences,

NAME 85

this violates WARP,since each can afford

but rejects the other’s bundle.

7.4 (0) Here is a table of prices and the demands of a consumer named

Ronald whose behavior was observed in 5 di erent price-income situa-

tions.

Situation p

1

p

2

x

1

x

2

A 1 1 5 35

B 1 2 35 10

C 1 1 10 15

D 3 1 5 15

E 1 2 10 10

(a) Sketch each of his budget lines and label the point chosen in each case

by the letters A,B,C,D,and E.

(b) Is Ronald’s behavior consistent with the Weak Axiom of Revealed

Preference? Yes.

(c) Shade lightly in red ink all of the points that you are certain are worse

for Ronald than the bundle C.

(d) Suppose that you are told that Ronald has convex and monotonic

preferences and that he obeys the strong axiom of revealed preference.

Shade lightly in blue ink all of the points that you are certain are at least

as good as the bundle C.

010203040

10

20

30

x1

x2

40

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

yyyyyyyyyyyyyyyyyyyyy

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,

,

,

,,

,,

e

d c

b

a

Red shading

Blue shading

86 REVEALED PREFERENCE (Ch,7)

7.5 (0) Horst and Nigel live in di erent countries,Possibly they have

di erent preferences,and certainly they face di erent prices,They each

consume only two goods,x and y,Horst has to pay 14 marks per unit of

x and 5 marks per unit of y,Horst spends his entire income of 167 marks

on 8 units of x and 11 units of y,Good x costs Nigel 9 quid per unit and

good y costs him 7 quid per unit,Nigel buys 10 units of x and 9 units of

y.

(a) Which prices and income would Horst prefer,Nigel’s income and prices

or his own,or is there too little information to tell? Explain your answer.

Horst prefers Nigel’s budget to his own.

With Nigel’s budget,he can afford his own

bundle with money left over.

(b) Would Nigel prefer to have Horst’s income and prices or his own,or

is there too little information to tell? There is too little

information to tell.

7.6 (0) Here is a table that illustrates some observed prices and choices

for three di erent goods at three di erent prices in three di erent situa-

tions.

Situation p

1

p

2

p

3

x

1

x

2

x

3

A 1 2 8 2 1 3

B 4 1 8 3 4 2

C 3 1 2 2 6 2

(a) We will ll in the table below as follows,Where i and j stand for any

of the letters A,B,and C in Row i and Column j of the matrix,write

the value of the Situation-j bundle at the Situation-i prices,For example,

in Row A and Column A,we put the value of the bundle purchased in

Situation A at Situation A prices,From the table above,we see that in

Situation A,the consumer bought bundle (2;1;3) at prices (1;2;8),The

cost of this bundle A at prices A is therefore (1 2)+(2 1)+(8 3) = 28,

so we put 28 in Row A,Column A,In Situation B the consumer bought

bundle (3;4;2),The value of the Situation-B bundle,evaluated at the

situation-A prices is (1 3) + (2 4) + (8 2) = 27,so put 27 in Row

A,Column B,We have lled in some of the boxes,but we leave a few for

you to do.

NAME 87

Prices=Quantities A B C

A 28 27 30

B 33 32 30

C 13 17 16

(b) Fill in the entry in Row i and Column j of the table below with a D if

the Situation-i bundle is directly revealed preferred to the Situation-j bun-

dle,For example,in Situation A the consumer’s expenditure is $28,We

see that at Situation-A prices,he could also a ord the Situation-B bun-

dle,which cost 27,Therefore the Situation-A bundle is directly revealed

preferred to the Situation-B bundle,so we put a D in Row A,Column

B,Now let us consider Row B,Column A,The cost of the Situation-B

bundle at Situation-B prices is 32,The cost of the Situation-A bundle

at Situation-B prices is 33,So,in Situation B,the consumer could not

a ord the Situation-A bundle,Therefore Situation B is not directly re-

vealed preferred to Situation A,So we leave the entry in Row B,Column

A blank,Generally,there is a D in Row i Column j if the number in the

ij entry of the table in part (a) is less than or equal to the entry in Row

i,Columni,There will be a violation of WARP if for some i and j,there

is a D in Row i Column j and also a D in Row j,Columni.Dothese

observations violate WARP? No.

Situation A B C

A | D I

B I | D

C D I |

(c) Now ll in Row i,Columnj with an I if observation i is indirectly

revealed preferred to j,Do these observations violate the Strong Axiom

of Revealed Preference? Yes.

7.7 (0) It is January,and Joe Grad,whom we met in Chapter 5,is

shivering in his apartment when the phone rings,It is Mandy Manana,

one of the students whose price theory problems he graded last term.

Mandy asks if Joe would be interested in spending the month of February

in her apartment,Mandy,who has switched majors from economics to

political science,plans to go to Aspen for the month and so her apartment

will be empty (alas),All Mandy asks is that Joe pay the monthly service

charge of $40 charged by her landlord and the heating bill for the month

of February,Since her apartment is much better insulated than Joe’s,

it only costs $1 per month to raise the temperature by 1 degree,Joe

88 REVEALED PREFERENCE (Ch,7)

thanks her and says he will let her know tomorrow,Joe puts his earmu s

back on and muses,If he accepts Mandy’s o er,he will still have to pay

rent on his current apartment but he won’t have to heat it,If he moved,

heating would be cheaper,but he would have the $40 service charge,The

outdoor temperature averages 20 degrees Fahrenheit in February,and it

costs him $2 per month to raise his apartment temperature by 1 degree.

Joe is still grading homework and has $100 a month left to spend on food

and utilities after he has paid the rent on his apartment,The price of

food is still $1 per unit.

(a) Draw Joe’s budget line for February if he moves to Mandy’s apartment

and on the same graph,draw his budget line if he doesn’t move.

(b) After drawing these lines himself,Joe decides that he would be better

o not moving,From this,we can tell,using the principle of revealed

preference that Joe must plan to keep his apartment at a temperature of

less than 60 degrees.

(c) Joe calls Mandy and tells her his decision,Mandy o ers to pay half

the service charge,Draw Joe’s budget line if he accepts Mandy’s new

o er,Joe now accepts Mandy’s o er,From the fact that Joe accepted

this o er we can tell that he plans to keep the temperature in Mandy’s

apartment above 40 degrees.

0 10 20 30 40 50 60 70 80

20

40

60

80

100

120

Food

Don't move budget line

Move budget line

'New offer'

budget line

Temperature

7.8 (0) Lord Peter Pommy is a distinguished criminologist,schooled

in the latest techniques of forensic revealed preference,Lord Peter is in-

vestigating the disappearance of Sir Cedric Pinchbottom who abandoned

his aging mother on a street corner in Liverpool and has not been seen

NAME 89

since,Lord Peter has learned that Sir Cedric left England and is living

under an assumed name somewhere in the Empire,There are three sus-

pects,R,Preston McAfee of Brass Monkey,Ontario,Canada,Richard

Manning of North Shag,New Zealand,and Richard Stevenson of Gooey

Shoes,Falkland Islands,Lord Peter has obtained Sir Cedric’s diary,which

recorded his consumption habits in minute detail,By careful observation,

he has also discovered the consumption behavior of McAfee,Manning,and

Stevenson,All three of these gentlemen,like Sir Cedric,spend their entire

incomes on beer and sausage,Their dossiers reveal the following:

Sir Cedric Pinchbottom | In the year before his departure,Sir

Cedric consumed 10 kilograms of sausage and 20 liters of beer per

week,At that time,beer cost 1 English pound per liter and sausage

cost 1 English pound per kilogram.

R,Preston McAfee | McAfee is known to consume 5 liters of beer

and 20 kilograms of sausage,In Brass Monkey,Ontario beer costs 1

Canadian dollar per liter and sausage costs 2 Canadian dollars per

kilogram.

Richard Manning | Manning consumes 5 kilograms of sausage

and 10 liters of beer per week,In North Shag,a liter of beer costs

2 New Zealand dollars and sausage costs 2 New Zealand dollars per

kilogram.

Richard Stevenson | Stevenson consumes 5 kilograms of sausage

and 30 liters of beer per week,In Gooey Shoes,a liter of beer costs 10

Falkland Island pounds and sausage costs 20 Falkland Island pounds

per kilogram.

(a) Draw the budget line for each of the three fugitives,using a di erent

color of ink for each one,Label the consumption bundle that each chooses.

On this graph,superimpose Sir Cedric’s budget line and the bundle he

chose.

90 REVEALED PREFERENCE (Ch,7)

010203040

10

20

30

Beer

Sausage

40

McAfee

Manning

Pinchbottom

Stevenson

(b) After pondering the dossiers for a few moments,Lord Peter an-

nounced,\Unless Sir Cedric has changed his tastes,I can eliminate one

of the suspects,Revealed preference tells me that one of the suspects is

innocent." Which one? McAfee.

(c) After thinking a bit longer,Lord Peter announced,\If Sir Cedric

left voluntarily,then he would have to be better o than he was before.

Therefore if Sir Cedric left voluntarily and if he has not changed his tastes,

he must be living in Falklands.

7.9 (1) The McCawber family is having a tough time making ends meet.

They spend $100 a week on food and $50 on other things,A new welfare

program has been introduced that gives them a choice between receiving

a grant of $50 per week that they can spend any way they want,and

buying any number of $2 food coupons for $1 apiece,(They naturally

are not allowed to resell these coupons.) Food is a normal good for the

McCawbers,As a family friend,you have been asked to help them decide

on which option to choose,Drawing on your growing fund of economic

knowledge,you proceed as follows.

(a) On the graph below,draw their old budget line in red ink and label

their current choice C,Now use black ink to draw the budget line that

they would have with the grant,If they chose the coupon option,how

much food could they buy if they spent all their money on food coupons?

$300,How much could they spend on other things if they bought

NAME 91

no food? $150,Use blue ink to draw their budget line if they choose

the coupon option.

0 30 60 90 120 150 180 210 240

30

60

90

120

150

180

Other things

Food

c

a

b

Black budget line

Blue budget line

Red budget line

(b) Using the fact that food is a normal good for the McCawbers,and

knowing what they purchased before,darken the portion of the black

budget line where their consumption bundle could possibly be if they

chose the lump-sum grant option,Label the ends of this line segment A

and B.

(c) After studying the graph you have drawn,you report to the McCaw-

bers,\I have enough information to be able to tell you which choice to

make,You should choose the coupon because you can

get more food even when other expenditure

is constant.

(d) Mr,McCawber thanks you for your help and then asks,\Would you

have been able to tell me what to do if you hadn’t known whether food

was a normal good for us?" On the axes below,draw the same budget

lines you drew on the diagram above,but draw indi erence curves for

which food is not a normal good and for which the McCawbers would be

better o with the program you advised them not to take.

92 REVEALED PREFERENCE (Ch,7)

0 30 60 90 120 150 180 210 240

30

60

90

120

150

180

Other things

Food

c

a

b

Black budget

line

Blue budget line

Red budget line

7.10 (0) In 1933,the Swedish economist Gunnar Myrdal (who later won

a Nobel prize in economics) and a group of his associates at Stockholm

University collected a fantastically detailed historical series of prices and

price indexes in Sweden from 1830 until 1930,This was published in a

book called The Cost of Living in Sweden,In this book you can nd

100 years of prices for goods such as oat groats,hard rye bread,salted

cod sh,beef,reindeer meat,birchwood,tallow candles,eggs,sugar,and

co ee,There are also estimates of the quantities of each good consumed

by an average working-class family in 1850 and again in 1890.

The table below gives prices in 1830,1850,1890,and 1913,for flour,

meat,milk,and potatoes,In this time period,these four staple foods

accounted for about 2/3 of the Swedish food budget.

Prices of Staple Foods in Sweden

Prices are in Swedish kronor per kilogram,except for milk,which is in

Swedish kronor per liter.

1830 1850 1890 1913

Grain Flour,14,14,16,19

Meat,28,34,66,85

Milk,07,08,10,13

Potatoes,032,044,051,064

Based on the tables published in Myrdal’s book,typical consump-

tion bundles for a working-class Swedish family in 1850 and 1890 are

listed below,(The reader should be warned that we have made some

NAME 93

approximations and simpli cations to draw these simple tables from the

much more detailed information in the original study.)

Quantities Consumed by a Typical Swedish Family

Quantities are measured in kilograms per year,except for milk,which is

measured in liters per year.

1850 1890

Grain Flour 165 220

Meat 22 42

Milk 120 180

Potatoes 200 200

(a) Complete the table below,which reports the annual cost of the 1850

and 1890 bundles of staple foods at various years’ prices.

Cost of 1850 and 1890 Bundles at Various Years’ Prices

Cost 1850 bundle 1890 bundle

Cost at 1830 Prices 44.1 61.6

Cost at 1850 Prices 49.0 68.3

Cost at 1890 Prices 63.1 91.1

Cost at 1913 Prices 78.5 113.7

(b) Is the 1890 bundle revealed preferred to the 1850 bundle? Yes.

(c) The Laspeyres quantity index for 1890 with base year 1850 is the ratio

of the value of the 1890 bundle at 1850 prices to the value of the 1850

bundle at 1850 prices,Calculate the Laspeyres quantity index of staple

food consumption for 1890 with base year 1850,1.39.

(d) The Paasche quantity index for 1890 with base year 1850 is the ratio

of the value of the 1890 bundle at 1890 prices to the value of the 1850

bundle at 1890 prices,Calculate the Paasche quantity index for 1890 with

base year 1850,1.44.

(e) The Laspeyres price index for 1890 with base year 1850 is calculated

using 1850 quantities for weights,Calculate the Laspeyres price index for

1890 with base year 1850 for this group of four staple foods,1.29.

94 REVEALED PREFERENCE (Ch,7)

(f) If a Swede were rich enough in 1850 to a ord the 1890 bundle of staple

foods in 1850,he would have to spend 1.39 timesasmuchonthese

foods as does the typical Swedish worker of 1850.

(g) If a Swede in 1890 decided to purchase the same bundle of food staples

that was consumed by typical 1850 workers,he would spend the fraction

.69 of the amount that the typical Swedish worker of 1890 spends on

these goods.

7.11 (0) This question draws from the tables in the previous question.

Let us try to get an idea of what it would cost an American family at

today’s prices to purchase the bundle consumed by an average Swedish

family in 1850,In the United States today,the price of flour is about $.40

per kilogram,the price of meat is about $3.75 per kilogram,the price of

milk is about $.50 per liter,and the price of potatoes is about $1 per

kilogram,We can also compute a Laspeyres price index across time and

across countries and use it to estimate the value of a current US dollar

relative to the value of an 1850 Swedish kronor.

(a) How much would it cost an American at today’s prices to buy the bun-

dle of staple food commodities purchased by an average Swedish working-

class family in 1850? $408.

(b) Myrdal estimates that in 1850,about 2=3 of the average family’s

budget was spent on food,In turn,the four staples discussed in the last

question constitute about 2=3 of the average family’s food budget,If the

prices of other goods relative to the price of the food staples are similar

in the United States today to what they were in Sweden in 1850,about

how much would it cost an American at current prices to consume the

same overall consumption bundle consumed by a Swedish working-class

family in 1850? $919.

(c) Using the Swedish consumption bundle of staple foods in 1850 as

weights,calculate a Laspeyres price index to compare prices in current

American dollars relative to prices in 1850 Swedish kronor,8.35,If

we use this to estimate the value of current dollars relative to 1850 Swedish

kronor,we would say that a U.S,dollar today is worth about,12 1850

Swedish kronor.

7.12 (0) Suppose that between 1960 and 1985,the price of all goods

exactly doubled while every consumer’s income tripled.

NAME 95

(a) Would the Laspeyres price index for 1985,with base year 1960 be less

than 2,greater than 2,or exactly equal to 2? Exactly 2,What

about the Paasche price index? Exactly 2.

(b) If bananas are a normal good,will total banana consumption in-

crease? Yes,If everybody has homothetic preferences,can you de-

termine by what percentage total banana consumption must have in-

creased? Explain,Yes,by 50%,Everybody’s budget

line shifted out by 50%,With homothetic

preferences,the consumption of each good

increases in the same proportion.

7.13 (1) Norm and Sheila consume only meat pies and beer,Meat pies

used to cost $2 each and beer was $1 per can,Their gross income used

to be $60 per week,but they had to pay an income tax of $10,Use red

ink to sketch their old budget line for meat pies and beer.

02030405060

10

20

30

40

50

60

10

Pies

Beer

Black

budget

line

Red budget line

Blue budget line

96 REVEALED PREFERENCE (Ch,7)

(a) They used to buy 30 cans of beer per week and spent the rest of their

income on meat pies,How many meat pies did they buy? 10.

(b) The government decided to eliminate the income tax and to put a

sales tax of $1 per can on beer,raising its price to $2 per can,Assuming

that Norm and Sheila’s pre-tax income and the price of meat pies did not

change,draw their new budget line in blue ink.

(c) The sales tax on beer induced Norm and Sheila to reduce their beer

consumption to 20 cans per week,What happened to their consumption

of meat pies? Stayed the same--10,How much revenue

did this tax raise from Norm and Sheila? $20.

(d) This part of the problem will require some careful thinking,Suppose

that instead of just taxing beer,the government decided to tax both beer

andmeatpiesatthesame percentage rate,and suppose that the price

of beer and the price of meat pies each went up by the full amount of

the tax,The new tax rate for both goods was set high enough to raise

exactly the same amount of money from Norm and Sheila as the tax on

beer used to raise,This new tax collects $,50 for every bottle of beer

sold and $ 1 for every meat pie sold,(Hint,If both goods are

taxed at the same rate,the e ect is the same as an income tax.) How

large an income tax would it take to raise the same revenue as the $1 tax

on beer? $20,Now you can gure out how big a tax on each good

is equivalent to an income tax of the amount you just found.

(e) Use black ink to draw the budget line for Norm and Sheila that cor-

responds to the tax in the last section,Are Norm and Sheila better o

having just beer taxed or having both beer and meat pies taxed if both

sets of taxes raise the same revenue? Both,(Hint,Try to use the

principle of revealed preference.)

Chapter 8 NAME

Slutsky Equation

Introduction,It is useful to think of a price change as having two dis-

tinct e ects,a substitution e ect and an income e ect,The substitution

e ect of a price change is the change that would have happened if in-

come changed at the same time in such a way that the consumer could

exactly a ord her old consumption bundle,The rest of the change in the

consumer’s demand is called the income e ect,Why do we bother with

breaking a real change into the sum of two hypothetical changes? Because

we know things about the pieces that we wouldn’t know about the whole

without taking it apart,In particular,we know that the substitution ef-

fect of increasing the price of a good must reduce the demand for it,We

also know that the income e ect of an increase in the price of a good is

equivalent to the e ect of a loss of income,Therefore if the good whose

price has risen is a normal good,then both the income and substitution

e ect operate to reduce demand,But if the good is an inferior good,

income and substitution e ects act in opposite directions.

Example,A consumer has the utility function U(x

1;x

2

)=x

1

x

2

and an

income of $24,Initially the price of good 1 was $1 and the price of good 2

was $2,Then the price of good 2 rose to $3 and the price of good 1 stayed

at $1,Using the methods you learned in Chapters 5 and 6,you will nd

that this consumer’s demand function for good 1 is D

1

(p

1;p

2;m)=m=2p

1

and her demand function for good 2 is D

2

(p

1;p

2;m)=m=2p

2

,Therefore

initially she will demand 12 units of good 1 and 6 units of good 2,If,

when the price of good 2 rose to $3,her income had changed enough so

that she could exactly a ord her old bundle,her new income would have

to be (1 12)+ (3 6) = $30,At an income of $30,at the new prices,she

would demand D

2

(1;3;30) = 5 units of good 2,Before the change she

bought 6 units of 2,so the substitution e ect of the price change on her

demand for good 2 is 5?6=?1 units,Our consumer’s income didn’t

really change,Her income stayed at $24,Her actual demand for good 2

after the price change was D

2

(1;3;24) = 4,The di erence between what

she actually demanded after the price change and what she would have

demanded if her income had changed to let her just a ord the old bundle

is the income e ect,In this case the income e ect is 4?5=?1 units

of good 2,Notice that in this example,both the income e ect and the

substitution e ect of the price increase worked to reduce the demand for

good 2.

When you have completed this workout,we hope that you will be

able to do the following:

Find Slutsky income e ect and substitution e ect of a speci c price

change if you know the demand function for a good.

Show the Slutsky income and substitution e ects of a price change

98 SLUTSKY EQUATION (Ch,8)

on an indi erence curve diagram.

Show the Hicks income and substitution e ects of a price change on

an indi erence curve diagram.

Find the Slutsky income and substitution e ects for special util-

ity functions such as perfect substitutes,perfect complements,and

Cobb-Douglas.

Use an indi erence-curve diagram to show how the case of a Gi en

good might arise.

Show that the substitution e ect of a price increase unambiguously

decreases demand for the good whose price rose.

Apply income and substitution e ects to draw some inferences about

behavior.

8.1 (0) Gentle Charlie,vegetarian that he is,continues to consume

apples and bananas,His utility function is U(x

A;x

B

)=x

A

x

B

,The price

of apples is $1,the price of bananas is $2,and Charlie’s income is $40 a

day,The price of bananas suddenly falls to $1.

(a) Before the price change,Charlie consumed 20 apples and

10 bananas per day,On the graph below,use black ink to draw

Charlie’s original budget line and put the label A on his chosen consump-

tion bundle.

(b) If,after the price change,Charlie’s income had changed so that he

could exactly a ord his old consumption bundle,his new income would

have been 30,With this income and the new prices,Charlie would

consume 15 apples and 15 bananas,Use red ink to draw

the budget line corresponding to this income and these prices,Label the

bundle that Charlie would choose at this income and the new prices with

the letter B.

(c) Does the substitution e ect of the fall in the price of bananas make

him buy more bananas or fewer bananas? More bananas,How

many more or fewer? 5 more.

(d) After the price change,Charlie actually buys 20 apples and

20 bananas,Use blue ink to draw Charlie’s actual budget line

after the price change,Put the label C on the bundle that he actually

chooses after the price change,Draw 3 horizontal lines on your graph,one

from A to the vertical axis,one from B to the vertical axis,and one from

C to the vertical axis,Along the vertical axis,label the income e ect,the

substitution e ect,and the total e ect on the demand for bananas,Is the

NAME 99

blue line parallel to the red line or the black line that you drew before?

Red line.

010203040

10

20

30

Apples

Bananas

40

a

b

c

Total

Substitution

Income

Red line

Blue line

Black line

(e) The income e ect of the fall in the price of bananas on Charlie’s

demand for bananas is the same as the e ect of an (increase,decrease)

increase in his income of $ 10 per day,Does the income

e ect make him consume more bananas or fewer? More,How many

more or how many fewer? 5 more.

(f) Does the substitution e ect of the fall in the price of bananas make

Charlie consume more apples or fewer? Fewer,How many more or

fewer? 5 fewer,Does the income e ect of the fall in the price of

bananas make Charlie consume more apples or fewer? More,What

is the total e ect of the change in the price of bananas on the demand for

apples? Zero.

8.2 (0) Neville’s passion is ne wine,When the prices of all other

goods are xed at current levels,Neville’s demand function for high-

quality claret is q =,02m?2p,wherem is his income,p is the price

of claret (in British pounds),and q is the number of bottles of claret that

he demands,Neville’s income is 7,500 pounds,and the price of a bottle

of suitable claret is 30 pounds.

100 SLUTSKY EQUATION (Ch,8)

(a) How many bottles of claret will Neville buy? 90.

(b) If the price of claret rose to 40 pounds,how much income would Neville

have to have in order to be exactly able to a ord the amount of claret

and the amount of other goods that he bought before the price change?

8,400 pounds,At this income,and a price of 40 pounds,how

many bottles would Neville buy? 88 bottles.

(c) At his original income of 7,500 and a price of 40,how much claret

would Neville demand? 70 bottles.

(d) When the price of claret rose from 30 to 40,the number of bottles

that Neville demanded decreased by 20,The substitution e ect (in-

creased,reduced) reduced his demand by 2 bottles and

the income e ect (increased,reduced) reduced his demand by 18

bottles.

8.3 (0) Note,Do this problem only if you have read the section entitled

\Another Substitution E ect" that describes the \Hicks substitution ef-

fect",Consider the gure below,which shows the budget constraint and

the indi erence curves of good King Zog,Zog is in equilibrium with an

income of $300,facing prices p

X

=$4andp

Y

= $10.

C

E

F

Y

X

30

22.5

30 35 43 9075 120

NAME 101

(a) How much X does Zog consume? 30.

(b) If the price of X falls to $2.50,while income and the price of Y stay

constant,how much X will Zog consume? 35.

(c) How much income must be taken away from Zog to isolate the Hicksian

income and substitution e ects (i.e.,to make him just able to a ord to

reach his old indi erence curve at the new prices)? $75.

(d) The total e ect of the price change is to change consumption from

the point E to the point C.

(e) The income e ect corresponds to the movement from the point

F to the point C while the substitution e ect corre-

sponds to the movement from the point E to the point F.

(f) Is X a normal good or an inferior good? An inferior

good.

(g) On the axes below,sketch an Engel curve and a demand curve for

Good X that would be reasonable given the information in the graph

above,Be sure to label the axes on both your graphs.

Income

x

300

225

4330

102 SLUTSKY EQUATION (Ch,8)

Price

x

1

2

3

4

30

2.5

35

8.4 (0) Maude spends all of her income on delphiniums and hollyhocks.

She thinks that delphiniums and hollyhocks are perfect substitutes; one

delphinium is just as good as one hollyhock,Delphiniums cost $4 a unit

and hollyhocks cost $5 a unit.

(a) If the price of delphiniums decreases to $3 a unit,will Maude buy

more of them? Yes,What part of the change in consumption is due

to the income e ect and what part is due to the substitution e ect?

All due to income effect.

(b) If the prices of delphiniums and hollyhocks are respectively p

d

=$4

and p

h

= $5 and if Maude has $120 to spend,draw her budget line in

blue ink,Draw the highest indi erence curve that she can attain in red

ink,and label the point that she chooses as A.

NAME 103

010203040

10

20

30

Hollyhocks

Delphiniums

40

a

b

Red

curves

Black line

Blue line

(c) Now let the price of hollyhocks fall to $3 a unit,while the price of

delphiniums does not change,Draw her new budget line in black ink.

Draw the highest indi erence curve that she can now reach with red ink.

Label the point she chooses now as B.

(d) How much would Maude’s income have to be after the price of holly-

hocks fell,so that she could just exactly a ord her old commodity bundle

A? $120.

(e) When the price of hollyhocks fell to $3,what part of the change in

Maude’s demand was due to the income e ect and what part was due to

the substitution e ect? All substitution effect.

8.5 (1) Suppose that two goods are perfect complements,If the price

of one good changes,what part of the change in demand is due to the

substitution e ect,and what part is due to the income e ect? All

income effect.

8.6 (0) Douglas Corn eld’s demand function for good x is x(p

x;p

y;m)=

2m=5p

x

,His income is $1,000,the price of x is $5,and the price of y is

$20,If the price of x falls to $4,then his demand for x will change from

80 to 100.

(a) If his income were to change at the same time so that he could exactly

a ord his old commodity bundle at p

x

=4andp

y

= 20,what would his

new income be? 920,What would be his demand for x at this new

level of income,at prices p

x

=4andp

y

= 20? 92.

104 SLUTSKY EQUATION (Ch,8)

(b) The substitution e ect is a change in demand from 80 to

92,The income e ect of the price change is a change in demand from

92 to 100.

(c) On the axes below,use blue ink to draw Douglas Corn eld’s budget

line before the price change,Locate the bundle he chooses at these prices

on your graph and label this point A,Use black ink to draw Douglas

Corn eld’s budget line after the price change,Label his consumption

bundle after the change by B.

0 40 80 120 160 200 240 280 320

20

40

60

80

y

x

a

b

c

Blue line

Black line

Black line

(d) On the graph above,use black ink to draw a budget line with the new

prices but with an income that just allows Douglas to buy his old bundle,

A,Find the bundle that he would choose with this budget line and label

this bundle C.

8.7 (1) Mr,Consumer allows himself to spend $100 per month on

cigarettes and ice cream,Mr,C’s preferences for cigarettes and ice cream

are una ected by the season of the year.

(a) In January,the price of cigarettes was $1 per pack,while ice cream

cost $2 per pint,Faced with these prices,Mr,C bought 30 pints of ice

cream and 40 packs of cigarettes,Draw Mr,C’s January budget line with

blue ink and label his January consumption bundle with the letter J.

NAME 105

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

J

F

Blue

budget

line

Red budget line

Black budget line

A

Pencil

budget

line

Cigarettes

Ice cream

0

(b) In February,Mr,C again had $100 to spend and ice cream still cost

$2 per pint,but the price of cigarettes rose to $1.25 per pack,Mr,C

consumed 30 pints of ice cream and 32 packs of cigarettes,Draw Mr,C’s

February budget line with red ink and mark his February bundle with

the letter F,The substitution e ect of this price change would make him

buy (less,more,the same amount of) less cigarettes and (less,more,

thesameamountof) more ice cream,Since this is true and the total

change in his ice cream consumption was zero,it must be that the income

e ect of this price change on his consumption of ice cream makes him buy

(more,less,the same amount of) less ice cream,The income

e ect of this price change is like the e ect of an (increase,decrease)

106 SLUTSKY EQUATION (Ch,8)

decrease in his income,Therefore the information we have suggests

that ice cream is a(n) (normal,inferior,neutral) normal good.

(c) In March,Mr,C again had $100 to spend,Ice cream was on sale for $1

per pint,Cigarette prices,meanwhile,increased to $1.50 per pack,Draw

his March budget line with black ink,Is he better o than in January,

worse o,or can you not make such a comparison? Better off.

How does your answer to the last question change if the price of cigarettes

had increased to $2 per pack? Now you can’t tell.

8.8 (1) This problem continues with the adventures of Mr,Consumer

from the previous problem.

(a) In April,cigarette prices rose to $2 per pack and ice cream was still

on sale for $1 per pint,Mr,Consumer bought 34 packs of cigarettes and

32 pints of ice cream,Draw his April budget line with pencil and label

his April bundle with the letter A,Was he better o or worse o than

in January? Worse off,Was he better o or worse o than in

February,or can’t one tell? Better off.

(b) In May,cigarettes stayed at $2 per pack and as the sale on ice cream

ended,the price returned to $2 per pint,On the way to the store,how-

ever,Mr,C found $30 lying in the street,He then had $130 to spend on

cigarettes and ice cream,Draw his May budget with a dashed line,With-

out knowing what he purchased,one can determine whether he is better

o than he was in at least one previous month,Which month or months?

He is better off in May than in February.

(c) In fact,Mr,C buys 40 packs of cigarettes and 25 pints of ice cream

in May,Does he satisfy WARP? No.

8.9 (2) In the last chapter,we studied a problem involving food prices

and consumption in Sweden in 1850 and 1890.

(a) Potato consumption was the same in both years,Real income must

have gone up between 1850 and 1890,since the amount of food staples

purchased,as measured by either the Laspeyres or the Paasche quantity

index,rose,The price of potatoes rose less rapidly than the price of either

meat or milk,and at about the same rate as the price of grain flour,So

real income went up and the price of potatoes went down relative to

other goods,From this information,determine whether potatoes were

NAME 107

most likely a normal or an inferior good,Explain your answer.

If potatoes were a normal good,both the

fall in potato price and the rise in income

would increase the demand for potatoes,But

potato consumption did not increase,So

potatoes must be an inferior good.

(b) Can one also tell from these data whether it is likely that pota-

toes were a Gi en good? If potatoes were a

Giffen good,then the fall in the price

of potatoes would decrease demand and the

rise in income would also decrease demand

for potatoes,But potato demand stayed

constant,So potatoes were probably not a

Giffen good.

8.10 (1) Agatha must travel on the Orient Express from Istanbul to

Paris,The distance is 1,500 miles,A traveler can choose to make any

fraction of the journey in a rst-class carriage and travel the rest of the

way in a second-class carriage,The price is 10 cents a mile for a second-

class carriage and 20 cents a mile for a rst-class carriage,Agatha much

prefers rst-class to second-class travel,but because of a misadventure in

an Istanbul bazaar,she has only $200 left with which to buy her tickets.

Luckily,she still has her toothbrush and a suitcase full of cucumber sand-

wiches to eat on the way,Agatha plans to spend her entire $200 on her

tickets for her trip,She will travel rst class as much as she can a ord

to,but she must get all the way to Paris,and $200 is not enough money

to get her all the way to Paris in rst class.

(a) On the graph below,use red ink to show the locus of combinations

of rst- and second-class tickets that Agatha can just a ord to purchase

with her $200,Use blue ink to show the locus of combinations of rst-

and second-class tickets that are su cient to carry her the entire distance

from Istanbul to Paris,Locate the combination of rst- and second-class

miles that Agatha will choose on your graph and label it A.

108 SLUTSKY EQUATION (Ch,8)

0 400 800 1200 1600

400

800

1200

Second-class miles

First-class miles

1600

Red

line

Blue line

Black line

a

Pencil line

b

c

(b) Let m

1

be the number of miles she travels by rst-class coach and m

2

be the number of miles she travels by second-class coach,Write down two

equations that you can solve to nd the number of miles she chooses to

travel by rst-class coach and the number of miles she chooses to travel

by second-class coach.,2m

1

+,1m

2

= 200,m

1

+ m

2

=

1;500.

(c) The number of miles that she travels by second-class coach is

1,000.

(d) Just before she was ready to buy her tickets,the price of second-class

tickets fell to $.05 while the price of rst-class tickets remained at $.20.

On the graph that you drew above,use pencil to show the combinations

of rst-class and second-class tickets that she can a ord with her $200

at these prices,On your graph,locate the combination of rst-class and

second-class tickets that she would now choose,(Remember,she is going

to travel as much rst-class as she can a ord to and still make the 1,500

mile trip on $200.) Label this point B.Howmanymilesdoesshetravel

by second class now? 666.66,(Hint,For an exact solution you

will have to solve two linear equations in two unknowns.) Is second-class

travel a normal good for Agatha? No,Is it a Gi en good for her?

Yes.

NAME 109

8.11 (0) We continue with the adventures of Agatha,from the previous

problem,Just after the price change from $.10 per mile to $.05 per mile

for second-class travel,and just before she had bought any tickets,Agatha

misplaced her handbag,Although she kept most of her money in her sock,

the money she lost was just enough so that at the new prices,she could

exactly a ord the combination of rst- and second-class tickets that she

would have purchased at the old prices,How much money did she lose?

$50,On the graph you started in the previous problem,use black ink

to draw the locus of combinations of rst- and second-class tickets that

she can just a ord after discovering her loss,Label the point that she

chooses with a C,How many miles will she travel by second class now?

1,000.

(a) Finally,poor Agatha nds her handbag again,How many miles will

she travel by second class now (assuming she didn’t buy any tickets before

she found her lost handbag)? 666.66,When the price of second-

class tickets fell from $.10 to $.05,how much of a change in Agatha’s de-

mand for second-class tickets was due to a substitution e ect? None.

How much of a change was due to an income e ect333:33.

110 SLUTSKY EQUATION (Ch,8)

Chapter 9 NAME

Buying and Selling

Introduction,In previous chapters,we studied the behavior of con-

sumers who start out without owning any goods,but who had some money

with which to buy goods,In this chapter,the consumer has an initial en-

dowment,which is the bundle of goods the consumer owns before any

trades are made,A consumer can trade away from his initial endowment

by selling one good and buying the other.

The techniques that you have already learned will serve you well here.

To nd out how much a consumer demands at given prices,you nd his

budget line and then nd a point of tangency between his budget line and

an indi erence curve,To determine a budget line for a consumer who

is trading from an initial endowment and who has no source of income

other than his initial endowment,notice two things,First,the initial

endowment must lie on the consumer’s budget line,This is true because,

no matter what the prices are,the consumer can always a ord his initial

endowment,Second,if the prices are p

1

and p

2

,the slope of the budget

line must be?p

1

=p

2

,This is true,since for every unit of good 1 the

consumer gives up,he can get exactly p

1

=p

2

units of good 2,Therefore

if you know the prices and you know the consumer’s initial endowment,

then you can always write an equation for the consumer’s budget line.

After all,if you know one point on a line and you know its slope,you

can either draw the line or write down its equation,Once you have the

budget equation,you can nd the bundle the consumer chooses,using the

same methods you learned in Chapter 5.

Example,A peasant consumes only rice and sh,He grows some rice and

some sh,but not necessarily in the same proportion in which he wants

to consume them,Suppose that if he makes no trades,he will have 20

units of rice and 5 units of sh,The price of rice is 1 yuan per unit,and

the price of sh is 2 yuan per unit,The value of the peasant’s endowment

is (1 20) + (2 5) = 30,Therefore the peasant can consume any bundle

(R;F) such that (1 R)+(2 F) = 30.

Perhaps the most interesting application of trading from an initial

endowment is the theory of labor supply,To study labor supply,we

consider the behavior of a consumer who is choosing between leisure and

other goods,The only thing that is at all new or \tricky" is nding

the appropriate budget constraint for the problem at hand,To study

labor supply,we think of the consumer as having an initial endowment of

leisure,some of which he may trade away for goods.

In most applications we set the price of \other goods" at 1,The

wage rate is the price of leisure,The role that is played by income in

the ordinary consumer-good model is now played by \full income." A

worker’s full income is the income she would have if she chose to take no

leisure.

112 BUYING AND SELLING (Ch,9)

Example,Sherwin has 18 hours a day which he divides between labor and

leisure,He can work as many hours a day as he wishes for a wage of $5

per hour,He also receives a pension that gives him $10 a day whether he

works or not,The price of other goods is $1 per unit,If Sherwin makes no

trades at all,he will have 18 hours of leisure and 10 units of other goods.

Therefore Sherwin’s initial endowment is 18 hours of leisure a day and

$10 a day for other goods,Let R be the amount of leisure that he has per

day,and let C be the number of dollars he has to spend per day on other

goods,If his wage is $5 an hour,he can a ord to consume bundle (R;C)

if it costs no more per day than the value of his initial endowment,The

value of his initial endowment (his full income) is $10 + ($5 18) = $100

per day,Therefore Sherwin’s budget equation is 5R +C = 100.

9.1 (0) Abishag Appleby owns 20 quinces and 5 kumquats,She has no

income from any other source,but she can buy or sell either quinces or

kumquats at their market prices,The price of kumquats is four times the

price of quinces,There are no other commodities of interest.

(a) How many quinces could she have if she was willing to do without

kumquats? 40,How many kumquats could she have if she was willing

to do without quinces? 10.

010203040

10

20

30

Quinces

Kumquats

40

Red line

Blue line

e

c

Squiggly

line

(b) Draw Abishag’s budget set,using blue ink,and label the endowment

bundle with the letter E,If the price of quinces is 1 and the price of

kumquats is 4,write Abishag’s budget equation,Q +4K =40.

If the price of quinces is 2 and the price of kumquats is 8,write Abishag’s

budget equation,2Q+8K =80,What e ect does doubling both

NAME 113

prices have on the set of commodity bundles that Abishag can a ord?

No effect.

(c) Suppose that Abishag decides to sell 10 quinces,Label her nal

consumption bundle in your graph with the letter C.

(d) Now,after she has sold 10 quinces and owns the bundle labelled C,

suppose that the price of kumquats falls so that kumquats cost the same

as quinces,On the diagram above,draw Abishag’s new budget line,using

red ink.

(e) If Abishag obeys the weak axiom of revealed preference,then there are

some points on her red budget line that we can be sure Abishag will not

choose,On the graph,make a squiggly line over the portion of Abishag’s

red budget line that we can be sure she will not choose.

9.2 (0) Mario has a small garden where he raises eggplant and tomatoes.

He consumes some of these vegetables,and he sells some in the market.

Eggplants and tomatoes are perfect complements for Mario,since the only

recipes he knows use them together in a 1:1 ratio,One week his garden

yielded 30 pounds of eggplant and 10 pounds of tomatoes,At that time

the price of each vegetable was $5 per pound.

(a) What is the monetary value of Mario’s endowment of vegetables?

$200.

(b) On the graph below,use blue ink to draw Mario’s budget line,Mario

ends up consuming 20 pounds of tomatoes and 20 pounds

of eggplant,Draw the indi erence curve through the consumption bundle

that Mario chooses and label this bundle A.

(c) Suppose that before Mario makes any trades,the price of tomatoes

rises to $15 a pound,while the price of eggplant stays at $5 a pound.

What is the value of Mario’s endowment now? $300,Draw his new

budget line,using red ink,He will now choose a consumption bundle

consisting of 15 tomatoes and 15 eggplants.

(d) Suppose that Mario had sold his entire crop at the market for a total

of $200,intending to buy back some tomatoes and eggplant for his own

consumption,Before he had a chance to buy anything back,the price of

tomatoes rose to $15,while the price of eggplant stayed at $5,Draw his

budget line,using pencil or black ink,Mario will now consume 10

pounds of tomatoes and 10 pounds of eggplant.

114 BUYING AND SELLING (Ch,9)

(e) Assuming that the price of tomatoes rose to $15 from $5 before Mario

made any transactions,the change in the demand for tomatoes due to

the substitution e ect was 0,The change in the demand for

tomatoes due to the ordinary income e ect was?10,The change

in the demand for tomatoes due to the endowment income e ect was

+5,The total change in the demand for tomatoes was?5.

010203040

10

20

30

40

Red line

Blue line

a

Black line

Tomatoes

Eggplant

9.3 (0) Lucetta consumes only two goods,A and B,Her only source of

income is gifts of these commodities from her many admirers,She doesn’t

always get these goods in the proportions in which she wants to consume

them in,but she can always buy or sell A at the price p

A

=1andB at

the price p

B

= 2,Lucetta’s utility function is U(a;b)=ab,wherea is the

amount of A she consumes and b istheamountofB she consumes.

(a) Suppose that Lucetta’s admirers give her 100 units of A and 200 units

of B,In the graph below,use red ink to draw her budget line,Label her

initial endowment E.

(b) What are Lucetta’s gross demands for A? 250 units,And for

B? 125 units.

(c) What are Lucetta’s net demands? 150 of A and?75 of

B.

NAME 115

(d) Suppose that before Lucetta has made any trades,the price of good

B falls to 1,and the price of good A stays at 1,Draw Lucetta’s budget

line at these prices on your graph,using blue ink.

(e) Does Lucetta’s consumption of good B rise or fall? It rises.

By how much? 25 units,What happens to Lucetta’s consumption

of good A? It decreases by 100 units.

0 225 300

100

200

300

400

500

600

75

Good A

Good B

Red budget line

Blue budget line

150

e

(f) Suppose that before the price of goodB fell,Lucetta had exchanged all

of her gifts for money,planning to use the money to buy her consumption

bundle later,How much good B will she choose to consume? 250

units,How much good A? 250 units.

(g) Explain why her consumption is di erent depending on whether she

was holding goods or money at the time of the price change,In

the former case,the fall in p

B

makes her

poorer because she is a net seller of good

B,In the latter case,her income doesn’t

116 BUYING AND SELLING (Ch,9)

change.

9.4 (0) Priscilla nds it optimal not to engage in trade at the going

prices and just consumes her endowment,Priscilla has no kinks in her

indi erence curves,and she is endowed with positive amounts of both

goods,Use pencil or black ink to draw a budget line and an indi erence

curve for Priscilla that would be consistent with these facts,Suppose that

the price of good 2 stays the same,but the price of good 1 falls below the

level at which Priscilla made no trade,Use blue ink to show her new bud-

get line,Priscilla satis es the weak axiom of revealed preference,Could

it happen that Priscilla will consume less of good 1 than before? Explain.

No,If p

1

falls,then with the new budget,

she can still afford her old bundle,She

could afford the bundles with less of good

1 than her endowment at the old prices,By

WARP she won’t choose them now.

X2

X1

e

Black

budget

line

Blue

budget

line

9.5 (0) Potatoes are a Gi en good for Paddy,who has a small potato

farm,The price of potatoes fell,but Paddy increased his potato consump-

tion,At rst this astonished the village economist,who thought that a

decrease in the price of a Gi en good was supposed to reduce demand.

But then he remembered that Paddy was a net supplier of potatoes,With

the help of a graph,he was able to explain Paddy’s behavior,In the axes

below,show how this could have happened,Put \potatoes" on the hor-

izontal axis and \all other goods" on the vertical axis,Label the old

equilibrium A and the new equilibrium B.DrawapointC so that the

Slutsky substitution e ect is the movement from A to C and the Slutsky

NAME 117

income e ect is the movement from C to B,On this same graph,you are

also going to have to show that potatoes are a Gi en good,To do this,

draw a budget line showing the e ect of a fall in the price of potatoes if

Paddy didn’t own any potatoes,but only had money income,Label the

new consumption point under these circumstances by D.(Warning:You

probably will need to make a few dry runs on some scratch paper to get

the whole story straight.)

All other goods

Potatoes

b

e

c

a

d

9.6 (0) Recall the travails of Agatha,from the previous chapter,She

had to travel 1,500 miles from Istanbul to Paris,She had only $200 with

which to buy rst-class and second-class tickets on the Orient Express

when the price of rst-class tickets was $.20 and the price of second-class

tickets was $.10,She bought tickets that enabled her to travel all the

way to Paris,with as many miles of rst class as she could a ord,After

she boarded the train,she discovered to her amazement that the price of

second-class tickets had fallen to $.05 while the price of rst-class tickets

remained at $.20,She also discovered that on the train it was possible to

buy or sell rst-class tickets for $.20 a mile and to buy or sell second-class

tickets for $.05 a mile,Agatha had no money left to buy either kind of

ticket,but she did have the tickets that she had already bought.

(a) On the graph below,use pencil to show the combinations of tickets

that she could a ord at the old prices,Use blue ink to show the combi-

nations of tickets that will take her exactly 1,500 miles,Mark the point

that she chooses with the letter A.

118 BUYING AND SELLING (Ch,9)

0 400 800 1200 1600

400

800

1200

Second-class miles

First-class miles

1600

Red line

Blue line

a

Pencil line

(b) Use red ink to draw a line showing all of the combinations of rst-class

and second-class travel that she can a ord when she is on the train,by

trading her endowment of tickets at the new prices that apply on board

the train.

(c) On your graph,show the point that she chooses after nding out

about the price change,Does she choose more,less,or the same amount

of second-class tickets? The same.

9.7 (0) Mr,Cog works in a machine factory,He can work as many

hours per day as he wishes at a wage rate of w.LetC be the number of

dollars he has to spend on consumer goods and let R be the number of

hours of leisure that he chooses.

(a) Suppose that Mr,Cog earns $8 an hour and has 18 hours per day

to devote to labor or leisure,and suppose that he has $16 of nonlabor

income per day,Write an equation for his budget between consumption

and leisure,C+8R = 160,Use blue ink to draw his budget line

in the graph below,His initial endowment is the point where he doesn’t

work,but keeps all of his leisure,Mark this point on the graph below with

the letter A,(When your draw your graph,remember that although Cog

can choose to work and thereby \sell" some of his endowment of leisure,

he cannot \buy leisure" by paying somebody else to loaf for him.) If Mr.

Cog has the utility function U(R;C)=CR,how many hours of leisure

per day will he choose? 10,How many hours per day will he work?

8.

NAME 119

01216204

40

80

120

160

200

240

4

Leisure

Consumption

Black budget line

Red budget line

Blue budget line

8

a

(b) Suppose that Mr,Cog’s wage rate rose to $12 an hour,Use red ink

to draw his new budget line,(He still has $16 a day in nonlabor income.)

If Mr,Cog continued to work exactly as many hours as he did before the

wage increase,how much more money would he have each day to spend

on consumption? $32,But with his new budget line,he chooses to

work 8

1

3

hours,and so his consumption actually increases by

$36.

(c) Suppose that Mr,Cog still receives $8 an hour but that his nonlabor

income rises to $48 per day,Use black ink to draw his budget line,How

many hours does he choose to work? 6.

(d) Suppose that Mr,Cog has a wage of $w perhourandanonlabor

income of $m,As before,assume that he has 18 hours to divide between

labor and leisure,Cog’s budget line has the equation C+wR = m+18w.

Using the same methods you used in the chapter on demand functions,

nd the amount of leisure that Mr,Cog will demand as a function of

wages and of nonlabor income,(Hint,Notice that this is just the same

as nding the demand for R when the price of R is w,the price of C is

120 BUYING AND SELLING (Ch,9)

1,and income is m +18w.) Mr,Cog’s demand function for leisure is

R(w;m)= 9+(m=2w),Mr,Cog’s supply function for labor is

therefore 18?R(w;m)= 9?m=2w.

9.8 (0) Fred has just arrived at college and is trying to gure out how to

supplement the meager checks that he gets from home,\How can anyone

live on $50 a week for spending money?" he asks,But he asks to no

avail,\If you want more money,get a job," say his parents,So Fred

glumly investigates the possibilities,The amount of leisure time that he

has left after allowing for necessary activities like sleeping,brushing teeth,

and studying for economics classes is 50 hours a week,He can work as

many hours per week at a nearby Taco Bell for $5 an hour,Fred’s utility

function for leisure and money to spend on consumption is U(C;L)=CL.

(a) Fred has an endowment that consists of $50 of money to spend on

consumption and 50 hours of leisure,some of which he might \sell"

for money,The money value of Fred’s endowment bundle,including both

his money allowance and the market value of his leisure time is therefore

$300,Fred’s \budget line" for leisure and consumption is like a budget

lineforsomeonewhocanbuythesetwogoodsatapriceof$1perunit

of consumption and a price of $5 per unit of leisure,The only

di erence is that this budget line doesn’t run all the way to the horizontal

axis.

(b) On the graph below,use black ink to show Fred’s budget line,(Hint:

Find the combination of leisure and consumption expenditures that he

could have if he didn’t work at all,Find the combination he would have

if he chose to have no leisure at all,What other points are on your graph?)

On the same graph,use blue ink to sketch the indi erence curves that

give Fred utility levels of 3,000,4,500,and 7,500.

(c) If you maximized Fred’s utility subject to the above budget,how

much consumption would he choose? $150,(Hint,Remember how

to solve for the demand function of someone with a Cobb-Douglas utility

function?)

(d) The amount of leisure that Fred will choose to consume is 30

hours,This means that his optimal labor supply will be 20 hours.

NAME 121

0304506

50

100

150

200

250

300

10

Leisure

Consumption

Black

budget

line

Blue

indifference

curve (3000)

20

Blue indifference curve

(4500)

Blue

indifference

curve

(7500)

9.9 (0) George Johnson earns $5 per hour in his job as a tru e snif-

fer,After allowing time for all of the activities necessary for bodily up-

keep,George has 80 hours per week to allocate between leisure and labor.

Sketch the budget constraints for George resulting from the following

government programs.

(a) There is no government subsidy or taxation of labor income,(Use

blue ink on the graph below.)

020406080

100

200

300

Leisure

Consumption

400

Blue budget line

Red budget line

122 BUYING AND SELLING (Ch,9)

(b) All individuals receive a lump-sum payment of $100 per week from the

government,There is no tax on the rst $100 per week of labor income.

But all labor income above $100 per week is subject to a 50% income tax.

(Use red ink on the graph above.)

(c) If an individual is not working,he receives a payment of $100,If he

works he does not receive the $100,and all wages are subject to a 50%

income tax,(Use blue ink on the graph below.)

020406080

100

200

300

Leisure

Consumption

400

Red budget line

Blue budget line

(d) The same conditions as in Part (c) apply,with the exception that the

rst 20 hours of labor are exempt from the tax,(Use red ink on the graph

above.)

(e) All wages are taxed at 50%,but as an incentive to encourage work,

the government gives a payment of $100 to anyone who works more than

20 hours a week,(Use blue ink on the graph below.)

NAME 123

020406080

100

200

300

Leisure

Consumption

400

Blue budget line

9.10 (0) In the United States,real wage rates in manufacturing have

risen steadily from 1890 to the present,In the period from 1890 to 1930,

the length of the work week was reduced dramatically,But after 1930,

despite continuing growth of real wage rates,the length of the work week

has stayed remarkably constant at about 40 hours per week.

Hourly Wages and Length of Work Week

in U.S,Manufacturing,1890-1983

Sources,Handbook of Labor Statistics,1983 and U.S,Economic History,

by Albert Niemi (p,274),Wages are in 1983 dollars.

Year Wage Hours Worked

1890 1.89 59.0

1909 2.63 51.0

1920 3.11 47.4

1930 3.69 42.1

1940 5.27 38.1

1950 6.86 40.5

1960 8.56 39.7

1970 9.66 39.8

1983 10.74 40.1

124 BUYING AND SELLING (Ch,9)

(a) Use these data to plot a \labor supply curve" on the graph below.

0304506

2

4

6

8

10

12

10

Hourly wage rate (in 1983 dollars)

20

Hours of work per week

(b) At wage rates below $4 an hour,does the workweek get longer or

shorter as the wage rate rises? Shorter.

(c) The data in this table could be consistent with workers choosing var-

ious hours a week to work,given the wage rate,An increase in wages

has both an endowment income e ect and a substitution e ect,The

substitution e ect alone would make for a (longer,shorter) longer

workweek,If leisure is a normal good,the endowment income e ect tends

to make people choose (more,less) more leisure and a (longer,shorter)

shorter workweek,At wage rates below $4 an hour,the (substi-

tution,endowment income) endowment income e ect appears

to dominate,How would you explain what happens at wages above $4

an hour? Substitution and endowment income

effects cancel each other out,so the

work week stays roughly constant.

(d) Between 1890 and 1909,wage rates rose by 39 percent,but

weekly earnings rose by only 20 percent,For this period,the

NAME 125

gain in earnings (overstates,understates) understates the gain

in worker’s wealth,since they chose to take (more,less) more leisure

in 1909 than they took in 1890.

9.11 (0) Professor Mohamed El Hodiri of the University of Kansas,in

a classic tongue-in-cheek article \The Economics of Sleeping," Manifold,

vol,17,1975,o ered the following analysis,\Assume there are 24 hours

in a day,Daily consumption being x and hours of sleep s,the consumer

maximizes a utility function of the form u = x

2

s,wherex = w(24?s),

with w being the wage rate."

(a) In El Hodiri’s model,does the optimal amount of sleeping increase,

decrease,or stay the same as wages increase? Stays the same.

(b) How many hours of sleep per day is best in El Hodiri’s model? 8.

9.12 (0) Wendy and Mac work in fast food restaurants,Wendy gets $4

an hour for the rst 40 hours that she works and $6 an hour for every

hour beyond 40 hours a week,Mac gets $5 an hour no matter how many

hours he works,Each has 80 hours a week to allocate between work and

leisure and neither has any income from sources other than labor,Each

has a utility function U = cr,wherec is consumption and r is leisure.

Each can choose the number of hours to work.

(a) How many hours will Mac choose to work? 40.

(b) Wendy’s budget \line" has a kink in it at the point wherer = 40

and c = 160,Use blue ink for the part of her budget line where she

would be if she does not work overtime,Use red ink for the part where

she would be if she worked overtime.

126 BUYING AND SELLING (Ch,9)

020406080

100

200

300

Leisure

Consumption

400

Blue part of line

Red part of line

(c) The blue line segment that you drew lies on a line with equation

c +4r = 320,The red line that you drew lies on a line with

equation c +6r = 400,(Hint,For the red line,you know one

point on the line and you know its slope.)

(d) If Wendy was paid $4 an hour no matter how many hours she worked,

she would work 40 hours and earn a total of $160 a week.

On your graph,use black ink to draw her indi erence curve through this

point.

(e) Will Wendy choose to work overtime? Yes,What is the best

choice for Wendy from the red budget line? (c;r)= (200;33:3).

How many hours a week will she work? 46.6.

(f) Suppose that the jobs are equally agreeable in all other respects,Since

Wendy and Mac have the same preferences,they will be able to agree

about who has the better job,Who has the better job? Mac,(Hint:

Calculate Wendy’s utility when she makes her best choice,Calculate what

her utility would be if she had Mac’s job and chose the best amount of

time to work.)

NAME 127

9.13 (1) Wally Piper is a plumber,He charges $10 per hour for his work

and he can work as many hours as he likes,Wally has no source of income

other than his labor,He has 168 hours per week to allocate between labor

and leisure,On the graph below,draw Wally’s budget set,showing the

various combinations of weekly leisure and income that Wally can a ord.

0 120 160 200 240

400

800

1200

1600

2000

2400

40

Income

80

Leisure

128

Red

budget

line

(a) Write down Wally’s budget equation,I +10R =1;680.

(b) While self-employed,Wally chose to work 40 hours per week,The

construction rm,Glitz and Drywall,had a rush job to complete,They

o ered Wally $20 an hour and said that he could work as many hours as

he liked,Wally still chose to work only 40 hours per week,On the graph

you drew above,draw in Wally’s new budget line.

(c) Wally has convex preferences and no kinks in his indi erence curves.

On the graph,draw indi erence curves that are consistent with his choice

of working hours when he was self-employed and when he worked for Glitz

and Drywall.

(d) Glitz and Drywall were in a great hurry to complete their project and

wanted Wally to work more than 40 hours,They decided that instead of

paying him $20 per hour,they would pay him only $10 an hour for the

rst 40 hours that he worked per week and $20 an hour for every hour of

128 BUYING AND SELLING (Ch,9)

\overtime" that he worked beyond 40 hours per week,On the graph that

you drew above,use red ink to sketch in Wally’s budget line with this

pay schedule,Draw the indi erence curve through the point that Wally

chooses with this pay schedule,Will Wally work more than 40 hours or

less than 40 hours per week with this pay schedule? More.

9.14 (1) Felicity loves her job,She is paid $10 an hour and can work

as many hours a day as she wishes,She chooses to work only 5 hours

a day,She says the job is so interesting that she is happier working at

this job than she would be if she made the same income without working

at all,A skeptic asks,\If you like the job better than not working at

all,why don’t you work more hours and earn more money?" Felicity,

who is entirely rational,patiently explains that work may be desirable on

average but undesirable on the margin,The skeptic insists that she show

him her indi erence curves and her budget line.

(a) On the axes below,draw a budget line and indi erence curves that are

consistent with Felicity’s behavior and her remarks,Put leisure on the

horizontal axis and income on the vertical axis,(Hint,Where does the

indi erence curve through her actual choice hit the vertical line l = 24?)

Income

Leisure

50

240

24

9.15 (2) Dudley’s utility function is U(C;R)=C?(12?R)

2

,whereR

is the amount of leisure he has per day,He has 16 hours a day to divide

between work and leisure,He has an income of $20 a day from nonlabor

sources,The price of consumption goods is $1 per unit.

(a) If Dudley can work as many hours a day as he likes but gets zero

wages for his labor,how many hours of leisure will he choose? 12.

NAME 129

(b) If Dudley can work as many hours a day as he wishes for a wage rate

of $10 an hour,how many hours will he choose to work? (Hint,Write

down Dudley’s budget constraint,Solve for his labor supply,Remember

that the amount of labor he wishes to supply is 16 minus his demand for

leisure.) 9.

(c) If Dudley’s nonlabor income decreased to $5 a day,how many hours

would he choose to work? 7.

(d) Suppose that Dudley has to pay an income tax of 20 percent on all

of his income,and suppose that his before-tax wage remained at $10 an

hour and his before-tax nonlabor income was $20 per day; how many

hours would he choose to work? 8.

130 BUYING AND SELLING (Ch,9)

Chapter 10 NAME

Intertemporal Choice

Introduction,The theory of consumer saving uses techniques that you

have already learned,In order to focus attention on consumption over

time,we will usually consider examples where there is only one consumer

good,but this good can be consumed in either of two time periods,We

will be using two \tricks." One trick is to treat consumption in period 1

and consumption in period 2 as two distinct commodities,If you make

period-1 consumption the numeraire,then the \price" of period-2 con-

sumption is the amount of period-1 consumption that you have to give

up to get an extra unit of period-2 consumption,This price turns out to

be 1=(1 +r),where r is the interest rate.

The second trick is in the way you treat income in the two di erent

periods,Suppose that a consumer has an income of m

1

in period 1 and

m

2

in period 2 and that there is no inflation,The total amount of period-

1 consumption that this consumer could buy,if he borrowed as much

money as he could possibly repay in period 2,is m

1

+

m

2

1+r

.Asyou

work the exercises and study the text,it should become clear that the

consumer’s budget equation for choosing consumption in the two periods

is always

c

1

+

c

2

1+r

= m

1

+

m

2

1+r

:

This budget constraint looks just like the standard budget constraint that

you studied in previous chapters,where the price of \good 1" is 1,the

price of \good 2" is 1=(1 + r),and \income" is m

1

+

m

2

(1+r)

,Therefore

if you are given a consumer’s utility function,the interest rate,and the

consumer’s income in each period,you can nd his demand for consump-

tion in periods 1 and 2 using the methods you already know,Having

solved for consumption in each period,you can also nd saving,since the

consumer’s saving is just the di erence between his period-1 income and

his period-1 consumption.

Example,A consumer has the utility function U(c

1;c

2

)=c

1

c

2

.Thereis

no inflation,the interest rate is 10%,and the consumer has income 100

in period 1 and 121 in period 2,Then the consumer’s budget constraint

c

1

+c

2

=1:1 = 100 + 121=1:1 = 210,The ratio of the price of good 1 to the

price of good 2 is 1 +r =1:1,The consumer will choose a consumption

bundle so that MU

1

=MU

2

=1:1,But MU

1

= c

2

and MU

2

= c

1

,sothe

consumer must choose a bundle such that c

2

=c

1

=1:1,Take this equation

together with the budget equation to solve for c

1

and c

2

,The solution is

c

1

= 105 and c

2

= 115:50,Since the consumer’s period-1 income is only

100,he must borrow 5 in order to consume 105 in period 1,To pay back

principal and interest in period 2,he must pay 5.50 out of his period-2

income of 121,This leaves him with 115.50 to consume.

You will also be asked to determine the e ects of inflation on con-

132 INTERTEMPORAL CHOICE (Ch,10)

sumer behavior,The key to understanding the e ects of inflation is to

see what happens to the budget constraint.

Example,Suppose that in the previous example,there happened to be

an inflation rate of 6%,and suppose that the price of period-1 goods is

1,Then if you save $1 in period 1 and get it back with 10% interest,

you will get back $1.10 in period 2,But because of the inflation,goods

in period 2 cost 1.06 dollars per unit,Therefore the amount of period-1

consumption that you have to give up to get a unit of period-2 consump-

tion is 1:06=1:10 =,964 units of period-2 consumption,If the consumer’s

money income in each period is unchanged,then his budget equation is

c

1

+,964c

2

= 210,This budget constraint is the same as the budget

constraint would be if there were no inflation and the interest rate were

r,where:964 = 1=(1 + r),The value of r that solves this equation is

known as the real rate of interest,In this case the real rate of interest

is about,038,When the interest rate and inflation rate are both small,

the real rate of interest is closely approximated by the di erence between

the nominal interest rate,(10% in this case) and the inflation rate (6%

in this case),that is,:038,10?:06,As you will see,this is not such a

good approximation if inflation rates and interest rates are large.

10.1 (0) Peregrine Pickle consumes (c

1;c

2

)andearns(m

1;m

2

)inperiods

1 and 2 respectively,Suppose the interest rate is r.

(a) Write down Peregrine’s intertemporal budget constraint in present

value terms,c

1

+

c

2

(1+r)

= m

1

+

m

2

(1+r)

.

(b) If Peregrine does not consume anything in period 1,what is the most

he can consume in period 2? m

1

(1+r)+m

2

,This is the (future

value,present value) of his endowment,Future value.

(c) If Peregrine does not consume anything in period 2,what is the most

he can consume in period 1? m

1

+

m

2

(1+r)

,This is the (future value,

present value) of his endowment,Present value,What is the

slope of Peregrine’s budget line(1 +r).

10.2 (0) Molly has a Cobb-Douglas utility function U(c

1;c

2

)=c

a

1

c

1?a

2

,

where 0 <a<1andwherec

1

and c

2

are her consumptions in periods 1

and 2 respectively,We saw earlier that if utility has the form u(x

1;x

2

)=

x

a

1

x

1?a

2

and the budget constraint is of the \standard" form p

1

x

1

+p

2

x

2

=

m,then the demand functions for the goods are x

1

= am=p

1

and x

2

=

(1?a)m=p

2

.

NAME 133

(a) Suppose that Molly’s income is m

1

in period 1 and m

2

in period 2.

Write down her budget constraint in terms of present values,c

1

+

c

2

=(1 +r)=m

1

+m

2

=(1 +r).

(b) We want to compare this budget constraint to one of the standard

form,In terms of Molly’s budget constraint,what is p

1

1,What

is p

2

1=(1 +r),What is m? m

1

+m

2

=(1 +r).

(c) If a =,2,solve for Molly’s demand functions for consumption in

each period as a function of m

1

,m

2

,andr,Her demand function for

consumption in period 1 is c

1

=,2m

1

+,2m

2

=(1 + r),Her

demand function for consumption in period 2 is c

2

=,8(1+r)m

1

+

:8m

2

.

(d) An increase in the interest rate will decrease her period-1

consumption,It will increase her period-2 consumption and

increase her savings in period 1.

10.3 (0) Nickleby has an income of $2,000 this year,and he expects an

income of $1,100 next year,He can borrow and lend money at an interest

rate of 10%,Consumption goods cost $1 per unit this year and there is

no inflation.

134 INTERTEMPORAL CHOICE (Ch,10)

01234

1

2

3

Consumption this year in 1,000s

Consumption next year in 1,000s

4

e

a

Squiggly

line

Red line

Blue

line

(a) What is the present value of Nickleby’s endowment? $3,000.

What is the future value of his endowment? $3,300,With blue

ink,show the combinations of consumption this year and consumption

next year that he can a ord,Label Nickelby’s endowment with the letter

E.

(b) Suppose that Nickleby has the utility function U(C

1;C

2

)=C

1

C

2

.

Write an expression for Nickleby’s marginal rate of substitution between

consumption this year and consumption next year,(Your answer will be

a function of the variables C

1;C

2

.) MRS =?C

2

=C

1

.

(c) What is the slope of Nickleby’s budget line1:1,Write an

equation that states that the slope of Nickleby’s indi erence curve is equal

to the slope of his budget line when the interest rate is 10%,1:1=

C

2

=C

1

,Also write down Nickleby’s budget equation,C

1

+

C

2

=1:1=3;000.

(d) Solve these two equations,Nickleby will consume 1,500 units

in period 1 and 1,650 units in period 2,Label this point A on your

diagram.

NAME 135

(e) Will he borrow or save in the rst period? Save,How much?

500.

(f) On your graph use red ink to show what Nickleby’s budget line would

be if the interest rate rose to 20%,Knowing that Nickleby chose the

point A at a 10% interest rate,even without knowing his utility function,

you can determine that his new choice cannot be on certain parts of his

new budget line,Draw a squiggly mark over the part of his new budget

line where that choice can not be,(Hint,Close your eyes and think of

WARP.)

(g) Solve for Nickleby’s optimal choice when the interest rate is 20%.

Nickleby will consume 1,458.3 units in period 1 and 1,750

units in period 2.

(h) Will he borrow or save in the rst period? Save,How much?

541.7.

10.4 (0) Decide whether each of the following statements is true or

false,Then explain why your answer is correct,based on the Slutsky

decomposition into income and substitution e ects.

(a) \If both current and future consumption are normal goods,an increase

in the interest rate will necessarily make a saver save more." False.

Substitution effect makes him consume less

in period 1 and save more,For a saver,

income effect works in opposite direction.

Either effect could dominate.

(b) \If both current and future consumption are normal goods,an in-

crease in the interest rate will necessarily make a saver choose more

consumption in the second period." True,The income

and substitution effects both lead to more

consumption in the second period.

10.5 (1) Laertes has an endowment of $20 each period,He can borrow

money at an interest rate of 200%,and he can lend money at a rate of

0%,(Note,If the interest rate is 0%,for every dollar that you save,you

get back $1 in the next period,If the interest rate is 200%,then for every

dollar you borrow,you have to pay back $3 in the next period.)

136 INTERTEMPORAL CHOICE (Ch,10)

(a) Use blue ink to illustrate his budget set in the graph below,(Hint:

The boundary of the budget set is not a single straight line.)

010203040

10

20

30

C1

C2

40

Red line

Blue line

Black line

(b) Laertes could invest in a project that would leave him with m

1

=30

and m

2

= 15,Besides investing in the project,he can still borrow at 200%

interest or lend at 0% interest,Use red ink to draw the new budget set

in the graph above,Would Laertes be better o or worse o by investing

in this project given his possibilities for borrowing or lending? Or can’t

one tell without knowing something about his preferences? Explain.

Better off,If he invests in the project,

he can borrow or lend to get any bundle he

could afford without investing.

(c) Consider an alternative project that would leave Laertes with the

endowment m

1

= 15,m

2

= 30,Again suppose he can borrow and lend

as above,But if he chooses this project,he can’t do the rst project.

Use pencil or black ink to draw the budget set available to Laertes if he

chooses this project,Is Laertes better o or worse o by choosing this

project than if he didn’t choose either project? Or can’t one tell without

knowing more about his preferences? Explain,Can’t tell,He

can afford some things he couldn’t afford

originally,But some things he could afford

before,he can’t afford if he invests in

this project.

10.6 (0) The table below reports the inflation rate and the annual rate

of return on treasury bills in several countries for the years 1984 and 1985.

NAME 137

Inflation Rate and Interest Rate for Selected Countries

% Inflation % Inflation % Interest % Interest

Country Rate,1984 Rate,1985 Rate,1984 Rate,1985

United States 3.6 1.9 9.6 7.5

Israel 304.6 48.1 217.3 210.1

Switzerland 3.1 0.8 3.6 4.1

W,Germany 2.2?0:2 5.3 4.2

Italy 9.2 5.8 15.3 13.9

Argentina 90.0 672.2 NA NA

Japan 0.6 2.0 NA NA

(a) In the table below,use the formula that your textbook gives for the

exact real rate of interest to compute the exact real rates of interest.

(b) What would the nominal rate of return on a bond in Argentina have

to be to give a real rate of return of 5% in 1985? 710:8%,What

would the nominal rate of return on a bond in Japan have to be to give

a real rate of return of 5% in 1985? 7.1%.

(c) Subtracting the inflation rate from the nominal rate of return gives

a good approximation to the real rate for countries with a low rate of

inflation,For the United States in 1984,the approximation gives you

6% while the more exact method suggested by the text gives you

5.79%,But for countries with very high inflation this is a poor

approximation,The approximation gives you?87:3% for Israel

in 1984,while the more exact formula gives you?21:57%,For

Argentina in 1985,the approximation would tell us that a bond yielding

a nominal rate of 677.7% would yield a real interest rate of 5%,This

contrasts with the answer 710.8% that you found above.

138 INTERTEMPORAL CHOICE (Ch,10)

Real Rates of Interest in 1984 and 1985

Country 1984 1985

United States 5.7 5.5

Israel?21:57 109.4

Switzerland 0.5 3.33

W,Germany 3.0 4.4

Italy 5.6 7.6

10.7 (0) We return to the planet Mungo,On Mungo,macroeconomists

and bankers are jolly,clever creatures,and there are two kinds of money,

red money and blue money,Recall that to buy something in Mungo you

have to pay for it twice,once with blue money and once with red money.

Everything has a blue-money price and a red-money price,and nobody

is ever allowed to trade one kind of money for the other,There is a blue-

money bank where you can borrow and lend blue money at a 50% annual

interest rate,There is a red-money bank where you can borrow and lend

red money at a 25% annual interest rate.

A Mungoan named Jane consumes only one commodity,ambrosia,

but it must decide how to allocate its consumption between this year and

next year,Jane’s income this year is 100 blue currency units and no red

currency units,Next year,its income will be 100 red currency units and

no blue currency units,The blue currency price of ambrosia is one b.c.u.

per flagon this year and will be two b.c.u.’s per flagon next year,The red

currency price of ambrosia is one r.c.u,per flagon this year and will be

the same next year.

(a) If Jane spent all of its blue income in the rst period,it would be

enough to pay the blue price for 100 flagons of ambrosia,If Jane

saved all of this year’s blue income at the blue-money bank,it would

have 150 b.c.u.’s next year,This would give it enough blue currency

to pay the blue price for 75 flagons of ambrosia,On the graph

below,draw Jane’s blue budget line,depicting all of those combinations

of current and next period’s consumption that it has enough blue income

to buy.

NAME 139

025 75100

25

50

75

Ambrosia this period

Ambrosia next period

100

50

(b) If Jane planned to spend no red income in the next period and to

borrow as much red currency as it can pay back with interest with next

period’s red income,how much red currency could it borrow? 80.

(c) The (exact) real rate of interest on blue money is?25%,The

real rate of interest on red money is 25%.

(d) On the axes below,draw Jane’s blue budget line and its red budget

line,Shade in all of those combinations of current and future ambrosia

consumption that Jane can a ord given that it has to pay with both

currencies.

140 INTERTEMPORAL CHOICE (Ch,10)

025 75100

25

50

75

Ambrosia this period

Ambrosia next period

100

50

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

Blue

line

Red line

c

(e) It turns out that Jane nds it optimal to operate on its blue budget

line and beneath its red budget line,Find such a point on your graph and

mark it with a C.

(f) On the following graph,show what happens to Jane’s original budget

set if the blue interest rate rises and the red interest rate does not change.

On your graph,shade in the part of the new budget line where Jane’s

new demand could possibly be,(Hint,Apply the principle of revealed

preference,Think about what bundles were available but rejected when

Jane chose to consume at C before the change in blue interest rates.)

NAME 141

025 75100

25

50

75

Ambrosia this period

Ambrosia next period

100

50

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,

Blue

line

Shaded region

c

New blue

line

Red

line

10.8 (0) Mr,O,B,Kandle will only live for two periods,In the rst

period he will earn $50,000,In the second period he will retire and live

on his savings,His utility function is U(c

1;c

2

)=c

1

c

2

,wherec

1

is con-

sumption in period 1 and c

2

is consumption in period 2,He can borrow

and lend at the interest rate r =,10.

(a) If the interest rate rises,will his period-1 consumption increase,de-

crease,or stay the same? Stay the same,His demand

for c

1

is,5(m

1

+m

2

=(1 +r)) and m

2

=0.

(b) Would an increase in the interest rate make him consume more or

less in the second period? More,He saves the same

amount,but with higher interest rates,he

gets more back next period.

(c) If Mr,Kandle’s income is zero in period 1,and $ 55,000 in period 2,

would an increase in the interest rate make him consume more,less,or

thesameamountinperiod1? Less.

10.9 (1) Harvey Habit’s utility function is U(c

1;c

2

)=minfc

1;c

2

g,where

c

1

is his consumption of bread in period 1 and c

2

is his consumption of

bread in period 2,The price of bread is $1 per loaf in period 1,The

interest rate is 21%,Harvey earns $2,000 in period 1 and he will earn

$1,100 in period 2.

142 INTERTEMPORAL CHOICE (Ch,10)

(a) Write Harvey’s budget constraint in terms of future value,assuming

no inflation,1:21c

1

+c

2

=3;520.

(b) How much bread does Harvey consume in the rst period and how

much money does he save? (The answer is not necessarily an integer.)

He picks c

1

= c

2

,Substitute into the budget

to find c

1

=3;520=2:21 = 1;592:8,He saves

2;000?3;520=2:21 = 407:2.

(c) Suppose that Harvey’s money income in both periods is the same as

before,the interest rate is still 21%,but there is a 10% inflation rate.

Then in period 2,a loaf of bread will cost $ 1.10,Write down Har-

vey’s budget equation for period-1 and period-2 bread,given this new

information,1:21c

1

+1:1c

2

=3;520.

10.10 (2) In an isolated mountain village,the only crop is corn,Good

harvests alternate with bad harvests,This year the harvest will be 1,000

bushels,Next year it will be 150 bushels,There is no trade with the

outside world,Corn can be stored from one year to the next,but rats

will eat 25% of what is stored in a year,The villagers have Cobb-Douglas

utility functions,U(c

1;c

2

)=c

1

c

2

where c

1

is consumption this year,and

c

2

is consumption next year.

(a) Use red ink to draw a \budget line," showing consumption possibilities

for the village,with this year’s consumption on the horizontal axis and

next year’s consumption on the vertical axis,Put numbers on your graph

to show where the budget line hits the axes.

Next year's consumption

This year's consumption

150

1250

1000

900

1150

1136

1111

Red line

Black line

Blue line

NAME 143

(b) How much corn will the villagers consume this year? 600

bushels,How much will the rats eat? 100 bushels,How

much corn will the villagers consume next year? 450 bushels.

(c) Suppose that a road is built to the village so that now the village is

able to trade with the rest of the world,Now the villagers are able to buy

and sell corn at the world price,which is $1 per bushel,They are also

able to borrow and lend money at an interest rate of 10%,On your graph,

use blue ink to draw the new budget line for the villagers,Solve for the

amount they would now consume in the rst period 568 bushels

and the second period 624 bushels.

(d) Suppose that all is as in the last part of the question except that there

is a transportation cost of $.10 per bushel for every bushel of grain hauled

into or out of the village,On your graph,use black ink or pencil to draw

the budget line for the village under these circumstances.

10.11 (0) The table below records percentage interest rates and inflation

rates for the United States in some recent years,Complete this table.

Inflation and Interest in the United States,1965-1985

Year 1965 1970 1975 1978 1980 1985

CPI,Start of Year 38.3 47.1 66.3 79.2 100.0 130.0

CPI,End of Year 39.4 49.2 69.1 88.1 110.4 133

% Inflation Rate 2.9 4.3 4.2 11.3 10.4 2.3

Nominal Int,Rate 4.0 6.4 5.8 7.2 11.6 7.5

Real Int,Rate 1.1 2.1 1.6?3.7 1.09 5.07

(a) People complained a great deal about the high interest rates in

the late 70s,In fact,interest rates had never reached such heights

in modern times,Explain why such complaints are misleading.

Nominal interest rates were high,but so

was inflation,Real interest rates were

not high,(They were negative in 1978.)

144 INTERTEMPORAL CHOICE (Ch,10)

(b) If you gave up a unit of consumption goods at the beginning of 1985

and saved your money at interest,you could use the proceeds of your

saving to buy 1.05 units of consumption goods at the beginning of

1986,If you gave up a unit of consumption goods at the beginning of

1978 and saved your money at interest,you would be able to use the

proceeds of your saving to buy,96 units of consumption goods at the

beginning of 1979.

10.12 (1) Marsha Mellow doesn’t care whether she consumes in period

1 or in period 2,Her utility function is simply U(c

1;c

2

)=c

1

+ c

2

.Her

initial endowment is $20 in period 1 and $40 in period 2,In an antique

shop,she discovers a cookie jar that is for sale for $12 in period 1 and that

she is certain she can sell for $20 in period 2,She derives no consumption

bene ts from the cookie jar,and it costs her nothing to store it for one

period.

(a) On the graph below,label her initial endowment,E,and use blue ink

to draw the budget line showing combinations of period-1 and period-2

consumption that she can a ord if she doesn’t buy the cookie jar,On the

same graph,label the consumption bundle,A,that she would have if she

did not borrow or lend any money but bought the cookie jar in period 1,

sold it in period 2,and used the proceeds to buy period-2 consumption.

If she cannot borrow or lend,should Marsha invest in the cookie jar?

Yes.

(b) Suppose that Marsha can borrow and lend at an interest rate of 50%.

On the graph where you labelled her initial endowment,draw the budget

line showing all of the bundles she can a ord if she invests in the cookie

jar and borrows or lends at the interest rate of 50%,On the same graph

use red ink to draw one or two of Marsha’s indi erence curves.

020406080

20

40

60

Period-1 consumption

Period-2 consumption

80

,

e

a

Blue

line

Red

curves

NAME 145

(c) Suppose that instead of consumption in the two periods being per-

fect substitutes,they are perfect complements,so that Marsha’s utility

function is minfc

1;c

2

g,If she cannot borrow or lend,should she buy the

cookie jar? No,If she can borrow and lend at an interest rate of 50%,

should she invest in the cookie jar? Yes,If she can borrow or lend as

much at an interest rate of 100%,should she invest in the cookie jar?

No.

146 INTERTEMPORAL CHOICE (Ch,10)

Chapter 11 NAME

Asset Markets

Introduction,The fundamental equilibrium condition for asset markets

is that in equilibrium the rate of return on all assets must be the same.

Thus if you know the rate of interest and the cash flow generated by an

asset,you can predict what its market equilibrium price will be,This

condition has many interesting implications for the pricing of durable

assets,Here you will explore several of these implications.

Example,A drug manufacturing rm owns the patent for a new medicine.

The patent will expire on January 1,1996,at which time anyone can pro-

duce the drug,Whoever owns the patent will make a pro t of $1,000,000

per year until the patent expires,For simplicity,let us suppose that prof-

its for any year are all collected on December 31,The interest rate is

5%,Let us gure out what the selling price of the patent rights will be

on January 1,1993,On January 1,1993,potential buyers realize that

owning the patent will give them $1,000,000 every year starting 1 year

from now and continuing for 3 years,The present value of this cash flow

is

$

1;000;000

(1:05)

+

1;000;000

(1:05)

2

+

1;000;000

(1:05)

3

$2;723;248:

Nobody would pay more than this amount for the patent since if you put

$2,723,248 at 5% interest,you could collect $1,000,000 a year from the

bank for 3 years,starting 1 year from now,The patent wouldn’t sell for

less than $2,723,248,since if it sold for less,one would get a higher rate

of return by investing in this patent than one could get from investing in

anything else,What will the price of the patent be on January 1,1994?

At that time,the patent is equivalent to a cash flow of $1,000,000 in 1

year and another $1,000,000 in 2 years,The present value of this flow,

viewed from the standpoint of January 1,1994,will be

$

1;000;000

(1:05)

+

1;000;000

(1:05)

2

$1;859;310:

A slightly more di cult problem is one where the cash flow from an

asset depends on how the asset is used,To nd the price of such an asset,

one must ask what will be the present value of the cash flow that the asset

yields if it is managed in such a way as to maximize its present value.

Example,People will be willing to pay $15 a bottle to drink a certain wine

this year,Next year they would be willing to pay $25,and the year after

that they would be willing to pay $26,After that,it starts to deteriorate

and the amount people are willing to pay to drink it falls,The interest

rate is 5%,We can determine not only what the wine will sell for but

also when it will be drunk,If the wine is drunk in the rst year,it would

have to sell for $15,But no rational investor is going to sell the wine for

148 ASSET MARKETS (Ch,11)

$15 in the rst year,because it will sell for $25 one year later,This is a

66:66% rate of return,which is better than the rate of interest,When the

interest rate is 5%,investors are willing to pay at least $25=1:05 = $23:81

for the wine,So investors must outbid drinkers,and none will be drunk

this year,Will investors want to hold onto the wine for 2 years? In 2

years,the wine will be worth $26,so the present value of buying the wine

and storing it for 2 years is $26=(1:05)

2

= $23:58,This is less than the

present value of holding the wine for 1 year and selling it for $25,So,we

conclude that the wine will be drunk after 1 year,Its current selling price

will be $23:81,and 1 year from now,it will sell for $25.

11.0 Warm Up Exercise,Here are a few problems on present val-

ues,In all of the following examples,assume that you can both borrow

and lend at an annual interest rate of r andthattheinterestratewill

remain the same forever.

(a) You would be indi erent between getting $1 now and 1+r dollars,

one year from now,because if you put the dollar in the bank,then one

year from now you could get back 1+r dollars from the bank.

(b) You would be indi erent between getting 1 dollar(s) one

year from now and getting $1=(1 +r) dollars now,because 1=(1 +r)

deposited in the bank right now would enable you to withdraw principal

and interest worth $1.

(c) For any X>0,you would be indi erent between getting X=(1 +

r) dollars right now and $X one year from now,The present value of

$X received one year from now is X=(1 +r) dollars.

(d) The present value of an obligation to pay $X one year from now is

X=(1 +r) dollars.

(e) The present value of $X,to be received 2 years from now,is

X=(1 +r)

2

dollars.

(f) The present value of an asset that pays X

t

dollars t years from now

is X

t

=(1 +r)

t

dollars.

NAME 149

(g) The present value of an asset that pays $X

1

one year from now,$X

2

in

two years,and $X

10

ten years from now is X

1

=(1+r)+X

2

=(1+

r)

2

+X

10

=(1 +r)

10

dollars.

(h) Thepresentvalueofanassetthatpaysaconstantamount,$X per

year forever can be computed in two di erent ways,One way is to gure

out the amount of money you need in the bank so that the bank would

give you $X per year,forever,without ever exhausting your principal.

The annual interest received on a bank account of X=r dollars will

be $X,Therefore having X=r dollars right now is just as good as

getting $X a year forever.

(i) Another way to calculate the present value of $X a year forever is to

evaluate the in nite series

P

1

i=1

X=(1+r)

i

,This series is known

as a geometric series,Whenever r>0,this sum is well de ned

and is equal to X=r.

(j) If the interest rate is 10%,the present value of receiving $1,000 one

year from now will be,to the nearest dollar,$909,The present

value of receiving $1,000 a year forever,will be,to the nearest dollar,

$10,000.

(k) If the interest rate is 10%,what is the present value of an asset that

requires you to pay out $550 one year from now and will pay you back

$1,210 two years from now550=(1:1) + 1;210=(1:1)

2

=

500 dollars.

11.1 (0) An area of land has been planted with Christmas trees,On

December 1,ten years from now,the trees will be ready for harvest,At

that time,the standing Christmas trees can be sold for $1,000 per acre.

The land,after the trees have been removed,will be worth $200 per acre.

There are no taxes or operating expenses,but also no revenue from this

land until the trees are harvested,The interest rate is 10%.

(a) What can we expect the market price of the land to be?

$1;200=(1:1)

10

$463 per acre.

150 ASSET MARKETS (Ch,11)

(b) Suppose that the Christmas trees do not have to be sold after 10

years,but could be sold in any year,Their value if they are cut before

they are 10 years old is zero,After the trees are 10 years old,an acre of

trees is worth $1,000 and its value will increase by $100 per year for the

next 20 years,After the trees are cut,the land on which the trees stood

can always be sold for $200 an acre,When should the trees be cut to

maximize the present value of the payments received for trees and land?

After 10 years,What will be the market price of an acre of

land? Still $463.

11.2 (0) Publicity agents for the Detroit Felines announce the signing

of a phenomenal new quarterback,Archie Parabola,They say that the

contract is worth $1,000,000 and will be paid in 20 installments of $50,000

per year starting one year from now and with one new installment each

year for next 20 years,The contract contains a clause that guarantees he

will get all of the money even if he is injured and cannot play a single game.

Sports writers declare that Archie has become an \instant millionaire."

(a) Archie’s brother,Fenwick,who majored in economics,explains to

Archie that he is not a millionaire,In fact,his contract is worth less than

half a million dollars,Explain in words why this is so.

The present value of $50,000 a year for 20

years is less than $1,000,000,since the

present value of a dollar received in the

future is less than $1.

Archie’s college course on \Sports Management" didn’t cover present

values,So his brother tried to reason out the calculation for him,Here

is how it goes:

(b) Suppose that the interest rate is 10% and is expected to remain at

10% forever,How much would it cost the team to buy Archie a perpetuity

that would pay him and his heirs $1 per year forever,startingin1year?

$10.

(c) How much would it cost to buy a perpetuity that paid $50,000 a year

forever,starting in one year? $500,000.

In the last part,you found the present value of Archie’s contract

if he were going to get $50,000 a year forever,But Archie is not going

to get $50;000 a year forever,The payments stop after 20 years,The

present value of Archie’s actual contract is the same as the present value

of a contract that pays him $50,000 a year forever,but makes him pay

back $50,000 each year,forever,starting 21 years from now,Therefore

you can nd the present value of Archie’s contract by subtracting the

NAME 151

present value of $50,000 a year forever,starting 21 years from now from

the present value of $50,000 a year forever.

(d) If the interest rate is and will remain at 10%,a stream of payments

of $50,000 a year,starting 21 years from now has the same present value

as a lump sum of $ 500,000 to be received all at once,exactly 20

years from now.

(e) If the interest rate is and will remain at 10%,what is the present value

of $50,000 per year forever,starting 21 years from now? $75,000.

(Hint,The present value of $1 to be paid in 20 years is 1=(1+r)

20

=,15.)

(f) Now calculate the present value of Archie’s contract,8:50

50;000 = $425;000.

11.3 (0) Professor Thesis is puzzling over the formula for the present

value of a stream of payments of $1 a year,starting 1 year from now and

continuing forever,He knows that the value of this stream is expressed

by the in nite series

S =

1

1+r

+

1

(1 +r)

2

+

1

(1 +r)

3

+:::;

but he can’t remember the simpli ed formula for this sum,All he knows

is that if the rst payment were to arrive today,rather than a year from

now,the present value of the sum would be $1 higher,So he knows that

S +1=1+

1

(1 +r)

+

1

(1 +r)

2

+

1

(1 +r)

3

+::::

Professor Antithesis su ers from a similar memory lapse,He can’t

remember the formula for S either,But,he knows that the present value

of $1 a year forever,starting right now has to be 1 + r times as large as

the present value of $1 a year,starting a year from now,(This is true

because if you advance any income stream by a year,you multiply its

present value by 1+r.) That is,

1+

1

(1 +r)

+

1

(1 +r)

2

+

1

(1 +r)

3

+:::=(1+r)S:

(a) If Professor Thesis and Professor Antithesis put their knowledge to-

gether,they can express a simple equation involving only the variable S.

This equation is S +1= (1 +r)S,Solving this equation,they nd

that S = 1=r.

152 ASSET MARKETS (Ch,11)

(b) The two professors have also forgotten the formula for the present

value of a stream of $1 per year starting next year and continuing for K

years,They agree to call this number S(K) and they see that

S(K)=

1

(1 +r)

+

1

(1 +r)

2

+:::+

1

(1 +r)

K

:

Professor Thesis notices that if each of the payments came 1 year earlier,

the present value of the resulting stream of payments would be

1+

1

(1 +r)

+

1

(1 +r)

2

+:::+

1

(1 +r)

K?1

= S(K)+1?

1

(1 +r)

K

:

Professor Antithesis points out that speeding up any stream of payments

by a year is also equivalent to multiplying its present value by (1 + r).

Putting their two observations together,the two professors noticed an

equation that could be solved for S(K),This equation is S(K)+1?

1

(1+r)

K

= S(K)(1 +r),Solving this equation for S(K),they nd

that the formula for S(K)is S(K)=(1?

1

(1+r)

K

)=r.

Calculus 11.4 (0) You are the business manager of P,Bunyan Forests,Inc.,and

are trying to decide when you should cut your trees,The market value of

the lumber that you will get if you let your trees reach the age of t years

is given by the function W(t)=e

:20t?:001t

2

,Mr,Bunyan can earn an

interest rate of 5% per year on money in the bank.

The rate of growth of the market value of the trees will be greater

than 5% until the trees reach 75 years of age,(Hint,It follows

from elementary calculus that if F(t)=e

g(t)

,thenF

0

(t)=F(t)=g

0

(t).)

(a) If he is only interested in the trees as an investment,how old should

Mr,Bunyan let the trees get? 75 years.

(b) At what age do the trees have the greatest market value? 100

years.

11.5 (0) You expect the price of a certain painting to rise by 8% per

year forever,The market interest rate for borrowing and lending is 10%.

Assume there are no brokerage costs in purchasing or selling.

(a) If you pay a price of $x for the painting now and sell it in a year,how

much has it cost you to hold the painting rather than to have loaned the

$x at the market interest rate? It has cost,02x.

NAME 153

(b) You would be willing to pay $100 a year to have the painting on your

walls,Write an equation that you can solve for the price x at which you

would be just willing to buy the painting,02x = 100.

(c) How much should you be willing to pay to buy the painting?

$5,000.

11.6 (2) Ashley is thinking of buying a truckload of wine for investment

purposes,He can borrow and lend as much as he likes at an annual

interest rate of 10%,He is looking at three kinds of wine,To keep our

calculations simple,let us assume that handling and storage costs are

negligible.

Wine drinkers would pay exactly $175 a case to drink Wine A today.

But if Wine A is allowed to mature for one year,it will improve,In fact

wine drinkers will be willing to pay $220 a case to drink this wine one

year from today,After that,the wine gradually deteriorates and becomes

less valuable every year.

From now until one year from now,Wine B is indistinguishable

from Wine A,But instead of deteriorating after one year,Wine B will

improve,In fact the amount that wine drinkers would be willing to pay

to drink Wine B will be $220 a case in one year and will rise by $10 per

case per year for the next 30 years.

Wine drinkers would be willing to pay $100 per case to drink Wine

C right now,But one year from now,they will be willing to pay $250

per case to drink it and the amount they will be willing to pay to drink

it will rise by $50 per case per year for the next 20 years.

(a) What is the most Ashley would be willing to pay per case for Wine

A? $200.

(b) What is the most Ashley would be willing to pay per case for Wine

B? $200,(Hint,When will Wine B be drunk?)

(c) How old will Wine C be when it rst becomes worthwhile for investors

to sell o their holdings and for drinkers to drink it? 6 years.

(Hint,When does the rate of return on holding wine get to 10%?)

(d) What will the price of Wine C beatthetimeitis rstdrunk?

$500 per case.

154 ASSET MARKETS (Ch,11)

(e) What is the most that Ashley would be willing to pay today for a case

of Wine C? (Hint,What is the present value of his investment if he sells

it to a drinker at the optimal time?) Express your answer in exponential

notation without calculating it out,$500=1:1

6

.

11.7 (0) Fisher Brown is taxed at 40% on his income from ordinary

bonds,Ordinary bonds pay 10% interest,Interest on municipal bonds is

not taxed at all.

(a) If the interest rate on municipal bonds is 7%,should he buy municipal

bonds or ordinary bonds? Brown should buy municipal

bonds.

(b) Hunter Black makes less money than Fisher Brown and is taxed at

only 25% on his income from ordinary bonds,Which kind of bonds should

he buy? Black should buy ordinary bonds.

(c) If Fisher has $1,000,000 in bonds and Hunter has $10,000 in bonds,

how much tax does Fisher pay on his interest from bonds? 0.

How much tax does Hunter pay on his interest from bonds? $250.

(d) The government is considering a new tax plan under which no interest

income will be taxed,If the interest rates on the two types of bonds do

not change,and Fisher and Hunter are allowed to adjust their portfolios,

how much will Fisher’s after-tax income be increased? $30,000.

How much will Hunter’s after-tax income be increased? $250.

(e) What would the change in the tax law do to the demand for municipal

bonds if the interest rates did not change? It would reduce

it to zero.

(f) What interest rate will new issues of municipal bonds have to pay

in order to attract purchasers? They will have to pay

10%.

NAME 155

(g) What do you think will happen to the market price of the old mu-

nicipal bonds,which had a 7% yield originally? The price of

the old bonds will fall until their yield

equals 10%.

11.8 (0) In the text we discussed the market for oil assuming zero

production costs,but now suppose that it is costly to get the oil out of

the ground,Suppose that it costs $5 dollars per barrel to extract oil from

the ground,Let the price in period t be denoted by p

t

and let r be the

interest rate.

(a) If a rm extracts a barrel of oil in period t,howmuchpro tdoesit

make in period t? p

t

5.

(b) If a rm extracts a barrel of oil in period t+1,how muchpro tdoes

it make in period t+1? p

t+1

5.

(c) What is the present value of the pro ts from extracting a barrel of oil

in period t+1? (p

t+1

5)=(1+r)

t+1

,What is the present value of

pro t from extracting a barrel of oil in periodt? (p

t

5)=(1+r)

t

.

(d) If the rm is willing to supply oil in each of the two periods,what

must be true about the relation between the present value of pro ts

from sale of a barrel of oil in the two periods? The present

values must be equal,Express this relation as an equation.

p

t+1

5

(1+r)

t+1

=

p

t

5

(1+r)

t

.

(e) Solve the equation in the above part for p

t+1

as a function of p

t

and

r,p

t+1

=(1+r)p

t

5r.

(f) Is the percentage rate of price increase between periods larger or

smaller than the interest rate? The percent change in

price is smaller.

11.9 (0) Dr,No owns a bond,serial number 007,issued by the James

Company,The bond pays $200 for each of the next three years,at which

time the bond is retired and pays its face value of $2,000.

156 ASSET MARKETS (Ch,11)

(a) How much is the James bond 007 worth to Dr,No at an interest rate

of 10%? 200=1:1+200=1:1

2

+200=1:1

3

+2;000=1:1

3

=

2;000.

(b) How valuable is James bond 007 at an interest rate of 5%?

200=1:05 + 200=1:05

2

+ 200=1:05

3

+2;000=1:05

3

=

2;272:32.

(c) Ms,Yes o ers Dr,No $2,200 for the James bond 007,Should Dr,No

say yes or no to Ms,Yes if the interest rate is 10%? Yes,What if

the interest rate is 5%? No.

(d) In order to destroy the world,Dr,No hires Professor Know to develop

a nasty zap beam,In order to lure Professor Know from his university

position,Dr,No will have to pay the professor $200 a year,The nasty

zap beam will take three years to develop,at the end of which it can be

built for $2,000,If the interest rate is 5%,how much money will Dr,No

need today to nance this dastardly program? $2,272.32,

which is the present value calculated in

the first part of the problem,If the interest rate

were 10%,would the world be in more or less danger from Dr,No?

More danger,since the dastardly plan is

now cheaper.

11.10 (0) Chillingsworth owns a large,poorly insulated home,His

annual fuel bill for home heating averages $300 per year,An insulation

contractor suggests to him the following options.

Plan A,Insulate just the attic,If he does this,he will permanently

reduce his fuel consumption by 15%,Total cost of insulating the attic is

$300.

Plan B,Insulate the attic and the walls,If he does this,he will perma-

nently reduce his fuel consumption by 20%,Total cost of insulating the

attic and the walls is $500.

Plan C,Insulate the attic and the walls,and install a solar heating unit.

If he does this,he will permanently reduce his fuel costs to zero,Total cost

of this option is $7,000 for the solar heater and $500 for the insulating.

NAME 157

(a) Assume for simplicity of calculations that the house and the insulation

will last forever,Calculate the present value of the dollars saved on fuel

from each of the three options if the interest rate is 10%,The present

values are,Plan A? $450,Plan B? $600,Plan C? $3,000.

(b) Each plan requires an expenditure of money to undertake,The di er-

ence between the present value and the present cost of each plan is,Plan

A? 450?300 = 150,Plan B? 600?500 = 100,Plan

C? 3;000?7;500 =?4;500.

(c) If the price of fuel is expected to remain constant,which option should

he choose if he can borrow and lend at an annual interest rate of 10%?

A.

(d) Which option should he choose if he can borrow and lend at an annual

rate of 5%? B.

(e) Suppose that the government o ers to pay half of the cost of any

insulation or solar heating device,Which option would he now choose at

interest rates 10%? B,5%? C.

(f) Suppose that there is no government subsidy but that fuel prices are

expected to rise by 5% per year,What is the present value of fuel savings

from each of the three proposals if interest rates are 10%? (Hint,If

a stream of income is growing at x% and being discounted at y%,its

present value should be the same as that of a constant stream of income

discounted at (y?x)%.) Plan A? $900,Plan B? $1,200.

Plan C? $6,000,Which proposal should Chillingsworth choose if

interest rates are 10%? B,5%? C.

11.11 (1) Have you ever wondered if a college education is nancially

worthwhile? The U.S,Census Bureau collects data on income and educa-

tion that throws some light on this question,A recent census publication

(Current Population Reports,Series P-70,No,11) reports the average an-

nual wage income in 1984 of persons aged 35{44 by the level of schooling

achieved,The average wage income of high school graduates was $13,000

per year,The average wage income of persons with bachelor’s degrees

was $24,000 per year,The average wage income of persons with master’s

degrees was $28,000 per year,The average wage income of persons with

Ph.D.’s was $40,000 per year,These income di erences probably over-

state the return to education itself,because it is likely that those people

who get more education tend to be more able than those who get less.

158 ASSET MARKETS (Ch,11)

Some of the income di erence is,therefore,a return to ability rather than

to education,But just to get a rough idea of returns to education,let us

see what would be the return if the reported wage di erences are all due

to education.

(a) Suppose that you have just graduated from high school at age 18,You

want to estimate the present value of your lifetime earnings if you do not

go to college but take a job immediately,To do this,you have to make

some assumptions,Assume that you would work for 47 years,until you

are 65 and then retire,Assume also that you would make $13,000 a year

for the rest of your life,(If you were going to do this more carefully,you

would want to take into account that people’s wages vary with their age,

but let’s keep things simple for this problem.) Assume that the interest

rate is 5%,Find the present value of your lifetime earnings,(Hint,First

nd out the present value of $13,000 a year forever,Subtract from this

the present value of $13,000 a year forever,starting 47 years from now.)

$233,753.

(b) Again,supposing you have just graduated from high school at age 18,

and you want to estimate the present value of your life time earnings if

you go to college for 4 years and do not earn any wages until you graduate

from college,Assume that after graduating from college,you would work

for 43 years at $24,000 per year,What would be the present value of your

lifetime earnings? $346,441.

(c) Now calculate the present value of your lifetime earnings if you get a

master’s degree,Assume that if you get a master’s,you have no earnings

for 6 years and then you work for 41 years at $28,000 per year,What

would be the present value of your lifetime income? $361,349.

(d) Finally calculate the present value of your lifetime earnings if you get

a Ph.D,Assume that if you get a Ph.D.,you will have no earnings for 8

years and then you work for 39 years at $40,000 per year,What would

be the present value of your lifetime income? $460,712.

(e) Consider the case of someone who married right after nishing high

school and stopped her education at that point,Suppose that she is now

45 years old,Her children are nearly adults,and she is thinking about

going back to work or going to college,Assuming she would earn the

average wage for her educational level and would retire at age 65,what

would be the present value of her lifetime earnings if she does not go to

college? $162,000.

NAME 159

(f) What would be the present value of her lifetime earnings if she goes to

college for 4 years and then takes a job until she is 65? $213,990.

(g) If college tuition is $5,000 per year,is it nancially worthwhile

for her to go to college? Explain,Yes,the gain in

the present value of her income exceeds the

present value of tuition.

11.12 (0) As you may have noticed,economics is a di cult major,Are

their any rewards for all this e ort? The U.S,census publication discussed

in the last problem suggests that there might be,There are tables report-

ing wage income by the eld in which one gets a degree,For bachelor’s

degrees,the most lucrative majors are economics and engineering,The

average wage incomes for economists are about $28,000 per year and for

engineers are about $27,000,Psychology majors average about $15,000 a

year and English majors about $14,000 per year.

(a) Can you think of any explanation for these di erences? Some

might say that economics,like accounting

or mortuary science,is so boring the

pay has to be high to get you to do

it,Others would suggest that the ability

to do well in economics is scarce and

is valued by the marketplace,Perhaps

the English majors and psychology majors

include disproportionately many persons

who are not full-time participants in the

labor force,No doubt there are several

other good partial explanations.

(b) The same table shows that the average person with an advanced degree

in business earns $38,000 per year and the average person with a degree

in medicine earns $45,000 per year,Suppose that an advanced degree

in business takes 2 years after one spends 4 years getting a bachelor’s

degree and that a medical degree takes 4 years after getting a bachelor’s

160 ASSET MARKETS (Ch,11)

degree,Suppose that you are 22 years old and have just nished college,If

r =,05,nd the present value of lifetime earnings for a graduating senior

who will get an advanced degree in business and earn the average wage

rate for someone with this degree until retiring at 65,$596,000.

Make a similar calculation for medicine,$630,000.

11.13 (0) On the planet Stinko,the principal industry is turnip growing.

For centuries the turnip elds have been fertilized by guano which was

deposited by the now-extinct giant scissor-billed kiki-bird,It costs $5 per

ton to mine kiki-bird guano and deliver it to the elds,Unfortunately,the

country’s stock of kiki-bird guano is about to be exhausted,Fortunately

the scientists on Stinko have devised a way of synthesizing kiki-guano from

political science textbooks and swamp water,This method of production

makes it possible to produce a product indistinguishable from kiki-guano

and to deliver it to the turnip elds at a cost of $30 per ton,The interest

rate on Stinko is 10%,There are perfectly competitive markets for all

commodities.

(a) Given the current price and the demand function for kiki-guano,the

last of the deposits on Stinko will be exhausted exactly one year from

now,Next year,the price of kiki-guano delivered to the elds will have

to be $30,so that the synthetic kiki-guano industry will just break even.

The owners of the guano deposits know that next year,they would get a

net return of $25 a ton for any guano they have left to sell,In equilibrium,

what must be the current price of kiki-guano delivered to the turnip elds?

The price of guano delivered to the field

must be the $5 + the present value of $25.

This is 5+25=1:1=27:73,(Hint,In equilibrium,sellers

must be indi erent between selling their kiki-guano right now or at any

other time before the total supply is exhausted,But we know that they

must be willing to sell it right up until the day,one year from now,when

the supply will be exhausted and the price will be $30,the cost of synthetic

guano.)

(b) Suppose that everything is as we have said previously except that the

deposits of kiki-guano will be exhausted 10 years from now,What must

be the current price of kiki-guano? (Hint,1:1

10

=2:59.)

5+25=(1:1)

10

=14:65.

Chapter 12 NAME

Uncertainty

Introduction,In Chapter 11,you learned some tricks that allow you to

use techniques you already know for studying intertemporal choice,Here

you will learn some similar tricks,so that you can use the same methods

to study risk taking,insurance,and gambling.

One of these new tricks is similar to the trick of treating commodi-

ties at di erent dates as di erent commodities,This time,we invent

new commodities,which we call contingent commodities.Ifeitheroftwo

events A or B could happen,then we de ne one contingent commodity

as consumption if A happens and another contingent commodity as con-

sumption if B happens,The second trick is to nd a budget constraint

that correctly speci es the set of contingent commodity bundles that a

consumer can a ord.

This chapter presents one other new idea,and that is the notion

of von Neumann-Morgenstern utility,A consumer’s willingness to take

various gambles and his willingness to buy insurance will be determined

by how he feels about various combinations of contingent commodities.

Often it is reasonable to assume that these preferences can be expressed

by a utility function that takes the special form known as von Neumann-

Morgenstern utility,The assumption that utility takes this form is called

the expected utility hypothesis,If there are two events,1 and 2 with

probabilities

1

and

2

,and if the contingent consumptions are c

1

and

c

2

,then the von Neumann-Morgenstern utility function has the special

functional form,U(c

1;c

2

)=

1

u(c

1

)+

2

u(c

2

),The consumer’s behavior

is determined by maximizing this utility function subject to his budget

constraint.

Example,You are thinking of betting on whether the Cincinnati Reds

will make it to the World Series this year,A local gambler will bet with

you at odds of 10 to 1 against the Reds,You think the probability that

the Reds will make it to the World Series is =,2,If you don’t bet,

you are certain to have $1,000 to spend on consumption goods,Your

behavior satis es the expected utility hypothesis and your von Neumann-

Morgenstern utility function is

1

p

c

1

+

2

p

c

2

.

The contingent commodities are dollars if the Reds make the World

Series and dollars if the Reds don’t make the World Series.Letc

W

be

your consumption contingent on the Reds making the World Series and

c

NW

be your consumption contingent on their not making the Series.

Betting on the Reds at odds of 10 to 1 means that if you bet $x on the

Reds,then if the Reds make it to the Series,you make a net gain of $10x,

but if they don’t,you have a net loss of $x,Since you had $1,000 before

betting,if you bet $x on the Reds and they made it to the Series,you

would have c

W

=1;000 + 10x to spend on consumption,If you bet $x

on the Reds and they didn’t make it to the Series,you would lose $x,

162 UNCERTAINTY (Ch,12)

and you would have c

NW

=1;000?x,By increasing the amount $x that

you bet,you can make c

W

larger and c

NW

smaller,(You could also bet

against the Reds at the same odds,If you bet $x against the Reds and

they fail to make it to the Series,you make a net gain of,1x and if they

make it to the Series,you lose $x,If you work through the rest of this

discussion for the case where you bet against the Reds,you will see that

the same equations apply,with x being a negative number.) We can use

the above two equations to solve for a budget equation,From the second

equation,we have x =1;000?c

NW

,Substitute this expression for x into

the rst equation and rearrange terms to nd c

W

+10c

NW

=11;000,or

equivalently,:1c

W

+ c

NW

=1;100,(The same budget equation can be

written in many equivalent ways by multiplying both sides by a positive

constant.)

Then you will choose your contingent consumption bundle (c

W;c

NW

)

to maximize U(c

W;c

NW

)=:2

p

c

W

+,8

p

c

NW

subject to the budget

constraint,:1c

W

+c

NW

=1;100,Using techniques that are now familiar,

you can solve this consumer problem,From the budget constraint,you

see that consumption contingent on the Reds making the World Series

costs 1=10 as much as consumption contingent on their not making it,If

you set the marginal rate of substitution between c

W

and c

NW

equal to

the price ratio and simplify the resulting expression,you will nd that

c

NW

=,16c

W

,This equation,together with the budget equation implies

that c

W

=$4;230:77 and c

NW

= $676:92,You achieve this bundle by

betting $323:08 on the Reds,If the Reds make it to the Series,you will

have $1;000 + 10 323:08 = $4;230:80,If not,you will have $676:92.

(We rounded the solutions to the nearest penny.)

12.1 (0) In the next few weeks,Congress is going to decide whether

or not to develop an expensive new weapons system,If the system is

approved,it will be very pro table for the defense contractor,General

Statics,Indeed,if the new system is approved,the value of stock in

General Statics will rise from $10 per share to $15 a share,and if the

project is not approved,the value of the stock will fall to $5 a share,In

his capacity as a messenger for Congressman Kickback,Buzz Condor has

discovered that the weapons system is much more likely to be approved

than is generally thought,On the basis of what he knows,Condor has

decided that the probability that the system will be approved is 3/4 and

the probability that it will not be approved is 1/4,Let c

A

be Condor’s

consumption if the system is approved and c

NA

be his consumption if

the system is not approved,Condor’s von Neumann-Morgenstern utility

function is U(c

A;c

NA

)=:75 lnc

A

+,25 lnc

NA

,Condor’s total wealth is

$50,000,all of which is invested in perfectly safe assets,Condor is about

to buy stock in General Statics.

(a) If Condor buys x shares of stock,and if the weapons system is ap-

proved,he will make a pro t of $5 per share,Thus the amount he can

consume,contingent on the system being approved,is c

A

= $50;000+5x.

If Condor buys x shares of stock,and if the weapons system is not ap-

proved,then he will make a loss of $ 5 per share,Thus the

NAME 163

amount he can consume,contingent on the system not being approved,is

c

NA

= 50;000?5x.

(b) You can solve for Condor’s budget constraint on contingent commod-

ity bundles (c

A;c

NA

) by eliminating x from these two equations,His bud-

get constraint can be written as,5 c

A

+,5 c

NA

=50;000.

(c) Buzz Condor has no moral qualms about trading on inside informa-

tion,nor does he have any concern that he will be caught and punished.

To decide how much stock to buy,he simply maximizes his von Neumann-

Morgenstern utility function subject to his budget,If he sets his marginal

rate of substitution between the two contingent commodities equal to

their relative prices and simpli es the equation,he nds that c

A

=c

NA

=

3,(Reminder,Where a is any constant,the derivative of alnx

with respect to x is a=x.)

(d) Condor nds that his optimal contingent commodity bundle is

(c

A;c

NA

)= (75,000,25,000),To acquire this contingent com-

modity bundle,he must buy 5,000 shares of stock in General Statics.

12.2 (0) Willy owns a small chocolate factory,located close to a river

that occasionally floods in the spring,with disastrous consequences,Next

summer,Willy plans to sell the factory and retire,The only income he

will have is the proceeds of the sale of his factory,If there is no flood,

the factory will be worth $500,000,If there is a flood,then what is left

of the factory will be worth only $50,000,Willy can buy flood insurance

at a cost of $.10 for each $1 worth of coverage,Willy thinks that the

probability that there will be a flood this spring is 1/10,Let c

F

denote the

contingent commodity dollars if there is a flood and c

NF

denote dollars

if there is no flood,Willy’s von Neumann-Morgenstern utility function is

U(c

F;c

NF

)=:1

p

c

F

+:9

p

c

NF

.

(a) If he buys no insurance,then in each contingency,Willy’s consumption

will equal the value of his factory,so Willy’s contingent commodity bundle

will be (c

F;c

NF

)= (50;000;500;000).

(b) To buy insurance that pays him $x in case of a flood,Willy must

pay an insurance premium of,1x,(The insurance premium must be

paid whether or not there is a flood.) If Willy insures for $x,thenif

there is a flood,he gets $x in insurance bene ts,Suppose that Willy has

contracted for insurance that pays him $x in the event of a flood,Then

after paying his insurance premium,he will be able to consume c

F

=

50;000 +:9x,If Willy has this amount of insurance and there is

no flood,then he will be able to consume c

NF

= 500;000?:1x.

164 UNCERTAINTY (Ch,12)

(c) You can eliminate x from the two equations for c

F

and c

NF

that

you found above,This gives you a budget equation for Willy,Of course

there are many equivalent ways of writing the same budget equation,

since multiplying both sides of a budget equation by a positive constant

yields an equivalent budget equation,The form of the budget equation in

which the \price" of c

NF

is 1 can be written as,9c

NF

+,1 c

F

=

455,000.

(d) Willy’s marginal rate of substitution between the two contingent com-

modities,dollars if there is no flood and dollars if there is a flood,is

MRS(c

NF;c

F

)=?

:9

p

c

F

:1

p

c

NF

,To nd his optimal bundle of contingent

commodities,you must set this marginal rate of substitution equal to

the number =?9,Solving this equation,you nd that Willy will

choose to consume the two contingent commodities in the ratio

c

NF

=c

F

=1.

(e) Since you know the ratio in which he will consume c

NF

and c

F

,and

you know his budget equation,you can solve for his optimal consumption

bundle,which is (c

NF;c

F

)= (455;000; 455;000),Willy will

buy an insurance policy that will pay him $450,000 if there is a

flood,The amount of insurance premium that he will have to pay is

$45,000.

12.3 (0) Clarence Bunsen is an expected utility maximizer,His pref-

erences among contingent commodity bundles are represented by the ex-

pected utility function

u(c

1;c

2;

1;

2

)=

1

p

c

1

+

2

p

c

2

:

Clarence’s friend,Hjalmer Ingqvist,has o ered to bet him $1,000 on the

outcome of the toss of a coin,That is,if the coin comes up heads,Clarence

must pay Hjalmer $1,000 and if the coin comes up tails,Hjalmer must

pay Clarence $1,000,The coin is a fair coin,so that the probability of

heads and the probability of tails are both 1/2,If he doesn’t accept the

bet,Clarence will have $10,000 with certainty,In the privacy of his car

dealership o ce over at Bunsen Motors,Clarence is making his decision.

(Clarence uses the pocket calculator that his son,Elmer,gave him last

Christmas,You will nd that it will be helpful for you to use a calculator

too.) Let Event 1 be \coin comes up heads" and let Event 2 be \coin

comes up tails."

NAME 165

(a) If Clarence accepts the bet,then in Event 1,he will have 9,000

dollars and in Event 2,he will have 11,000 dollars.

(b) Since the probability of each event is 1/2,Clarence’s expected utility

for a gamble in which he gets c

1

in Event 1 and c

2

in Event 2 can be

described by the formula

1

2

p

c

1

+

1

2

p

c

2

,Therefore Clarence’s

expected utility if he accepts the bet with Hjalmer will be 99.8746.

(Use that calculator.)

(c) If Clarence decides not to bet,then in Event 1,he will have

10,000 dollars and in Event 2,he will have 10,000 dollars.

Therefore if he doesn’t bet,his expected utility will be 100.

(d) Having calculated his expected utility if he bets and if he does not bet,

Clarence determines which is higher and makes his decision accordingly.

Does Clarence take the bet? No.

12.4 (0) It is a slow day at Bunsen Motors,so since he has his calcu-

lator warmed up,Clarence Bunsen (whose preferences toward risk were

described in the last problem) decides to study his expected utility func-

tion more closely.

(a) Clarence rst thinks about really big gambles,What if he bet his

entire $10,000 on the toss of a coin,where he loses if heads and wins if

tails? Then if the coin came up heads,he would have 0 dollars and if it

came up tails,he would have $20,000,His expected utility if he took the

bet would be 70.71,while his expected utility if he didn’t take the

bet would be 100,Thereforeheconcludesthathewouldnottake

such a bet.

(b) Clarence then thinks,\Well,of course,I wouldn’t want to take a

chance on losing all of my money on just an ordinary bet,But,what

if somebody o ered me a really good deal,Suppose I had a chance to

bet where if a fair coin came up heads,I lost my $10,000,but if it came

up tails,I would win $50,000,Would I take the bet? If I took the bet,

my expected utility would be 122.5,If I didn’t take the bet,my

expected utility would be 100,Therefore I should take the bet."

166 UNCERTAINTY (Ch,12)

(c) Clarence later asks himself,\If I make a bet where I lose my $10,000

if the coin comes up heads,what is the smallest amount that I would have

to win in the event of tails in order to make the bet a good one for me

to take?" After some trial and error,Clarence found the answer,You,

too,might want to nd the answer by trial and error,but it is easier to

nd the answer by solving an equation,On the left side of your equation,

you would write down Clarence’s utility if he doesn’t bet,On the right

side of the equation,you write down an expression for Clarence’s utility

if he makes a bet such that he is left with zero consumption in Event 1

and x in Event 2,Solve this equation for x,The answer to Clarence’s

question is where x =10;000,The equation that you should write is

100 =

1

2

p

x,The solution is x = 40;000.

(d) Your answer to the last part gives you two points on Clarence’s in-

di erence curve between the contingent commodities,money in Event 1

and money in Event 2,(Poor Clarence has never heard of indi erence

curves or contingent commodities,so you will have to work this part for

him,while he heads over to the Chatterbox Cafe for morning co ee.) One

of these points is where money in both events is $10,000,On the graph

below,label this point A,The other is where money in Event 1 is zero

and money in Event 2 is 40,000,On the graph below,label this

point B.

010203040

10

20

30

Money in Event 1 (x 1,000)

Money in Event 2 (x 1,000)

40

a

b

c

d

(e) You can quickly nd a third point on this indi erence curve,The

coin is a fair coin,and Clarence cares whether heads or tails turn up only

because that determines his prize,Therefore Clarence will be indi erent

between two gambles that are the same except that the assignment of

prizes to outcomes are reversed,In this example,Clarence will be indif-

ferent between point B on the graph and a point in which he gets zero if

Event 2 happens and 40,000 if Event 1 happens,Find this point

on the Figure above and label it C.

NAME 167

(f) Another gamble that is on the same indi erence curve for Clarence

as not gambling at all is the gamble where he loses $5,000 if heads turn

up and where he wins 6,715.73 dollars if tails turn up,(Hint,To

solve this problem,put the utility of not betting on the left side of an

equation and on the right side of the equation,put the utility of having

$10;000?$5;000 in Event 1 and $10;000 +x in Event 2,Then solve the

resulting equation for x.) On the axes above,plot this point and label it

D,Now sketch in the entire indi erence curve through the points that

you have labeled.

12.5 (0) Hjalmer Ingqvist’s son-in-law,Earl,has not worked out very

well,It turns out that Earl likes to gamble,His preferences over contin-

gent commodity bundles are represented by the expected utility function

u(c

1;c

2;

1;

2

)=

1

c

2

1

+

2

c

2

2

:

(a) Just the other day,some of the boys were down at Skoog’s tavern

when Earl stopped in,They got to talking about just how bad a bet they

could get him to take,At the time,Earl had $100,Kenny Olson shu ed

a deck of cards and o ered to bet Earl $20 that Earl would not cut a spade

from the deck,Assuming that Earl believed that Kenny wouldn’t cheat,

the probability that Earl would win the bet was 1/4 and the probability

that Earl would lose the bet was 3/4,If he won the bet,Earl would

have 120 dollars and if he lost the bet,he would have 80

dollars,Earl’s expected utility if he took the bet would be 8,400,

and his expected utility if he did not take the bet would be 10,000.

Therefore he refused the bet.

(b) Just when they started to think Earl might have changed his ways,

Kenny o ered to make the same bet with Earl except that they would

bet $100 instead of $20,What is Earl’s expected utility if he takes that

bet? 10,000,Would Earl be willing to take this bet? He is

just indifferent about taking it or not.

(c) Let Event 1 be the event that a card drawn from a fair deck of cards is

a spade,Let Event 2 be the event that the card is not a spade,Earl’s pref-

erences between income contingent on Event 1,c

1

,and income contingent

on Event 2,c

2

,can be represented by the equation u =

1

4

c

2

1

+

3

4

c

2

2

.

Use blue ink on the graph below to sketch Earl’s indi erence curve passing

through the point (100;100).

168 UNCERTAINTY (Ch,12)

0 50 100 150 200

50

100

150

Money in Event 1

Money in Event 2

200

Blue curve

Red curves

(d) On the same graph,let us draw Hjalmer’s son-in-law Earl’s indif-

ference curves between contingent commodities where the probabilities

are di erent,Suppose that a card is drawn from a fair deck of cards.

Let Event 1 be the event that the card is black,Let event 2 be the event

that the card drawn is red,Suppose each event has probability 1/2,Then

Earl’s preferences between income contingent on Event 1 and income con-

tingent on Event 2 are represented by the formula u =

1

2

c

2

1

+

1

2

c

2

2

.

On the graph,use red ink to show two of Earl’s indi erence curves,in-

cluding the one that passes through (100;100).

12.6 (1) Sidewalk Sam makes his living selling sunglasses at the board-

walk in Atlantic City,If the sun shines Sam makes $30,and if it rains

Sam only makes $10,For simplicity,we will suppose that there are only

two kinds of days,sunny ones and rainy ones.

(a) One of the casinos in Atlantic City has a new gimmick,It is accepting

bets on whether it will be sunny or rainy the next day,The casino sells

dated \rain coupons" for $1 each,If it rains the next day,the casino will

give you $2 for every rain coupon you bought on the previous day,If it

doesn’t rain,your rain coupon is worthless,In the graph below,mark

Sam’s \endowment" of contingent consumption if he makes no bets with

the casino,and label it E.

NAME 169

010203040

10

20

30

Cs

Cr

40

e

a

Blue

line

Red

line

(b) On the same graph,mark the combination of consumption contingent

on rain and consumption contingent on sun that he could achieve by

buying 10 rain coupons from the casino,Label it A.

(c) On the same graph,use blue ink to draw the budget line representing

all of the other patterns of consumption that Sam can achieve by buying

rain coupons,(Assume that he can buy fractional coupons,but not neg-

ative amounts of them.) What is the slope of Sam’s budget line at points

above and to the left of his initial endowment? The slope is

1.

(d) Suppose that the casino also sells sunshine coupons,These tickets

also cost $1,With these tickets,the casino gives you $2 if it doesn’t rain

and nothing if it does,On the graph above,use red ink to sketch in the

budget line of contingent consumption bundles that Sam can achieve by

buying sunshine tickets.

(e) If the price of a dollar’s worth of consumption when it rains is set equal

to 1,what is the price of a dollar’s worth of consumption if it shines?

The price is 1.

12.7 (0) Sidewalk Sam,from the previous problem,has the utility func-

tion for consumption in the two states of nature

u(c

s;c

r; )=c

1?

s

c

r;

where c

s

is the dollar value of his consumption if it shines,c

r

is the dollar

value of his consumption if it rains,and is the probability that it will

rain,The probability that it will rain is =,5.

170 UNCERTAINTY (Ch,12)

(a) How many units of consumption is it optimal for Sam to consume

conditional on rain? 20 units.

(b) How many rain coupons is it optimal for Sam to buy? 10.

12.8 (0) Sidewalk Sam’s brother Morgan von Neumanstern is an ex-

pected utility maximizer,His von Neumann-Morgenstern utility function

for wealth is u(c)=lnc,Sam’s brother also sells sunglasses on another

beach in Atlantic City and makes exactly the same income as Sam does.

He can make exactly the same deal with the casino as Sam can.

(a) If Morgan believes that there is a 50% chance of rain and a 50% chance

of sun every day,what would his expected utility of consuming (c

s;c

r

)

be? u =

1

2

lnc

s

+

1

2

lnc

r

.

(b) How does Morgan’s utility function compare to Sam’s? Is one a

monotonic transformation of the other? Morgan’s utility

function is just the natural log of Sam’s,

so the answer is yes.

(c) What will Morgan’s optimal pattern of consumption be? Answer:

Morgan will consume 20 on the sunny days and 20 on

the rainy days,How does this compare to Sam’s consumption? This

is the same as Sam’s consumption.

12.9 (0) Billy John Pigskin of Mule Shoe,Texas,has a von Neumann-

Morgenstern utility function of the form u(c)=

p

c:Billy John also weighs

about 300 pounds and can outrun jackrabbits and pizza delivery trucks.

Billy John is beginning his senior year of college football,If he is not

seriously injured,he will receive a $1,000,000 contract for playing pro-

fessional football,If an injury ends his football career,he will receive a

$10,000 contract as a refuse removal facilitator in his home town,There

is a 10% chance that Billy John will be injured badly enough to end his

career.

(a) What is Billy John’s expected utility? We calculate

:1

p

10;000 +:9

p

1;000;000 = 910.

NAME 171

(b) If Billy John pays $p for an insurance policy that would give him

$1,000,000 if he su ered a career-ending injury while in college,then he

would be sure to have an income of $1;000;000?p no matter what hap-

pened to him,Write an equation that can be solved to nd the largest

price that Billy John would be willing to pay for such an insurance policy.

The equation is 910 =

q

1;000;000?p.

(c) Solve this equation for p,p = 171;900.

12.10 (1) You have $200 and are thinking about betting on the Big

Game next Saturday,Your team,the Golden Boars,are scheduled to

play their traditional rivals the Robber Barons,It appears that the going

odds are 2 to 1 against the Golden Boars,That is to say if you want

to bet $10 on the Boars,you can nd someone who will agree to pay

you $20 if the Boars win in return for your promise to pay him $10 if

the Robber Barons win,Similarly if you want to bet $10 on the Robber

Barons,you can nd someone who will pay you $10 if the Robber Barons

win,in return for your promise to pay him $20 if the Robber Barons lose.

Suppose that you are able to make as large a bet as you like,either on

the Boars or on the Robber Barons so long as your gambling losses do

not exceed $200,(To avoid tedium,let us ignore the possibility of ties.)

(a) If you do not bet at all,you will have $200 whether or not the Boars

win,If you bet $50 on the Boars,then after all gambling obligations are

settled,you will have a total of 300 dollars if the Boars win and

150 dollars if they lose,On the graph below,use blue ink to draw a

line that represents all of the combinations of \money if the Boars win"

and \money if the Robber Barons win" that you could have by betting

from your initial $200 at these odds.

172 UNCERTAINTY (Ch,12)

0 100 200 300 400

100

200

300

Money if the Boars win

Money if the Boars lose

400

e

c

d

Red line

Blue line

(b) Label the point on this graph where you would be if you did not bet

at all with an E.

(c) After careful thought you decide to bet $50 on the Boars,Label the

point you have chosen on the graph with a C,Suppose that after you have

made this bet,it is announced that the star Robber Baron quarterback

su ered a sprained thumb during a tough economics midterm examination

and will miss the game,The market odds shift from 2 to 1 against the

Boars to \even money" or 1 to 1,That is,you can now bet on either

team and the amount you would win if you bet on the winning team is

the same as the amount that you would lose if you bet on the losing team.

You cannot cancel your original bet,but you can make new bets at the

new odds,Suppose that you keep your rst bet,but you now also bet

$50 on the Robber Barons at the new odds,If the Boars win,then after

you collect your winnings from one bet and your losses from the other,

how much money will you have left? $250,If the Robber Barons

win,how much money will you have left after collecting your winnings

and paying o your losses? $200.

(d) Use red ink to draw a line on the diagram you made above,showing

the combinations of \money if the Boars win" and \money if the Robber

Barons win" that you could arrange for yourself by adding possible bets

at the new odds to the bet you made before the news of the quarterback’s

misfortune,On this graph,label the point D that you reached by making

the two bets discussed above.

12.11 (2) The certainty equivalent of a lottery is the amount of money

you would have to be given with certainty to be just as well-o with that

lottery,Suppose that your von Neumann-Morgenstern utility function

NAME 173

over lotteries that give you an amount x if Event 1 happens and y if

Event 1 does not happen is U(x;y; )=

p

x+(1? )

p

y,where is the

probability that Event 1 happens and 1? is the probability that Event

1 does not happen.

(a) If =,5,calculate the utility of a lottery that gives you $10,000

if Event 1 happens and $100 if Event 1 does not happen,55 =

:5 100 +:5 10:

(b) If you were sure to receive $4,900,what would your utility be? 70.

(Hint,If you receive $4,900 with certainty,then you receive $4,900 in

both events.)

(c) Given this utility function and =,5,write a general formula for the

certainty equivalent of a lottery that gives you $x if Event 1 happens and

$y if Event 1 does not happen,(:5x

1=2

+:5y

1=2

)

2

.

(d) Calculate the certainty equivalent of receiving $10,000 if Event 1 hap-

pens and $100 if Event 1 does not happen,$3,025.

12.12 (0) Dan Partridge is a risk averter who tries to maximize the

expected value of

p

c,wherec is his wealth,Dan has $50,000 in safe

assets and he also owns a house that is located in an area where there

are lots of forest res,If his house burns down,the remains of his house

and the lot it is built on would be worth only $40,000,giving him a total

wealth of $90,000,If his home doesn’t burn,it will be worth $200,000

and his total wealth will be $250,000,The probability that his home will

burn down is,01.

(a) Calculate his expected utility if he doesn’t buy re insurance.

$498.

(b) Calculate the certainty equivalent of the lottery he faces if he doesn’t

buy re insurance,$248,004.

(c) Suppose that he can buy insurance at a price of $1 per $100 of in-

surance,For example if he buys $100,000 worth of insurance,he will pay

$1,000 to the company no matter what happens,but if his house burns,

he will also receive $100,000 from the company,If Dan buys $160,000

worth of insurance,he will be fully insured in the sense that no matter

what happens his after-tax wealth will be $248,400.

174 UNCERTAINTY (Ch,12)

(d) Therefore if he buys full insurance,the certainty equivalent of his

wealth is $248,400,and his expected utility is

p

248;800.

12.13 (0) Portia has been waiting a long time for her ship to come in

and has concluded that there is a 25% chance that it will arrive today,If

it does come in today,she will receive $1,600,If it does not come in today,

it will never come and her wealth will be zero,Portia has a von Neumann-

Morgenstern utility such that she wants to maximize the expected value

of

p

c,wherec is total income,What is the minimum price at which she

will sell the rights to her ship? $100.

Chapter 13 NAME

Risky Assets

Introduction,Here you will solve the problems of consumers who wish

to divide their wealth optimally between a risky asset and a safe asset.

The expected rate of return on a portfolio is just a weighted average of

the rate of return on the safe asset and the expected rate of return on

the risky asset,where the weights are the fractions of the consumer’s

wealth held in each,The standard deviation of the portfolio return is

just the standard deviation of the return on the risky asset times the

fraction of the consumer’s wealth held in the risky asset,Sometimes

you will look at the problem of a consumer who has preferences over

the expected return and the risk of her portfolio and who faces a budget

constraint,Since a consumer can always put all of her wealth in the

safe asset,one point on this budget constraint will be the combination

of the safe rate of return and no risk (zero standard deviation),Now

as the consumer puts x percent of her wealth into the risky asset,she

gains on that amount the di erence between the expected rate of return

for the risky asset and the rate of return on the safe asset,But she also

absorbs some risk,So the slope of the budget line will be the di erence

between the two returns divided by the standard deviation of the portfolio

that has x percent of the consumer’s wealth invested in the risky asset.

You can then apply the usual indi erence curve{budget line analysis to

nd the consumer’s optimal choice of risk and expected return given her

preferences,(Remember that if the standard deviation is plotted on the

horizontal axis and if less risk is preferred to more,the better bundles will

lie to the northwest.) You will also be asked to apply the result from the

Capital Asset Pricing Model that the expected rate of return on any asset

is equal to the sum of the risk-free rate of return plus the risk adjustment.

Remember too that the expected rate of return on an asset is its expected

change in price divided by its current price.

13.1 (3) Ms,Lynch has a choice of two assets,The rst is a risk-free

assetthato ersarateofreturnofr

f

,and the second is a risky asset (a

china shop that caters to large mammals) that has an expected rate of

return of r

m

and a standard deviation of

m

.

(a) If x is the percent of wealth Ms,Lynch invests in the risky asset,

what is the equation for the expected rate of return on the portfolio?

r

x

= xr

m

+(1?x)r

f

,What is the equation for the standard

deviation of the portfolio?

x

= x

m

.

176 RISKY ASSETS (Ch,13)

(b) By solving the second equation above for x and substituting the result

into the rst equation,derive an expression for the rate of return on the

portfolio in terms of the portfolio’s riskiness,r

x

=

r

m

r

f

m

x

+r

f

.

(c) Suppose that Ms,Lynch can borrow money at the interest rate r

f

and invest it in the risky asset,If r

m

= 20,r

f

= 10,and

m

= 10,what

will be Ms,Lynch’s expected return if she borrows an amount equal to

100% of her initial wealth and invests it in the risky asset? (Hint,This

is just like investing 200% of her wealth in the risky asset.) Apply

the formula r

x

= xr

m

+(1?x)r

f

with x =2 to

get r

x

=2 20?1 10 = 30.

(d) Suppose that Ms,Lynch can borrow or lend at the risk-free rate,If

r

f

is 10%,r

m

is 20%,and

m

is 10%,what is the formula for the \budget

line" Ms,Lynch faces? r

x

=

x

+10,Plot this budget line in the

graph below.

010203040

10

20

30

Standard deviation

Expected return

40

Budget line

U=0

U=5

U=10

(e) Which of the following risky assets would Ms,Lynch prefer to her

present risky asset,assuming she can only invest in one risky asset at a

time and that she can invest a fraction of her wealth in whichever risky

asset she chooses? Write the word \better," \worse," or \same" after

each of the assets.

Asset A with r

a

=17% and

a

=5%,Better.

Asset B with r

b

=30% and

b

= 25%,Worse.

NAME 177

Asset C with r

c

=11% and

c

=1%,Same.

Asset D with r

d

=25% and

d

= 14%,Better.

(f) Suppose Ms,Lynch’s utility function has the formu(r

x;

x

)=r

x

2

x

.

How much of her portfolio will she invest in the original risky asset?

(You might want to graph a few of Ms,Lynch’s indi erence curves be-

fore answering; e.g.,graph the combinations of r

x

and

x

that imply

u(r

x;

x

)=0;1;:::),She will not invest anything

in the risky asset.

13.2 (3) Fenner Smith is contemplating dividing his portfolio between

two assets,a risky asset that has an expected return of 30% and a standard

deviation of 10%,and a safe asset that has an expected return of 10%

and a standard deviation of 0%.

(a) If Mr,Smith invests x percent of his wealth in the risky asset,what

will be his expected return? r

x

=30x+ 10(1?x).

(b) If Mr,Smith invests x percent of his wealth in the risky asset,what

will be the standard deviation of his wealth?

x

=10x.

(c) Solve the above two equations for the expected return on Mr,Smith’s

wealth as a function of the standard deviation he accepts,The

budget line is r

x

=2

x

+10.

(d) Plot this \budget line" on the graph below.

0 5 10 15 20

10

20

30

Standard deviation

Expected return

40

Budget line

Indifference

curves

Optimal choice

178 RISKY ASSETS (Ch,13)

(e) If Mr,Smith’s utility function is u(r

x;

x

)=minfr

x;30?2

x

g,then

Mr,Smith’s optimal value of r

x

is 20,and his optimal value of

x

is 5,(Hint,You will need to solve two equations in two unknowns.

One of the equations is the budget constraint.)

(f) Plot Mr,Smith’s optimal choice and an indi erence curve through it

in the graph.

(g) What fraction of his wealth should Mr,Smith invest in the risky asset?

Using the answer to Part (a),we find an x

that solves 20 = r

x

=30x + 10(1?x),The

answer is x =,5.

13.3 (2) Assuming that the Capital Asset Pricing Model is valid,com-

plete the following table,In this table p

0

is the current price of asset i

and Ep

1

is the expected price of asset i in the next period.

r

f

r

m

r

i

i

p

0

Ep

1

10 20 10 0 100 110

10 20 25 1.5 100 125

10 15 20 2 200 240

0 30 20 2=3 40 48

10 22 10 0 80 88

13.4 (2) Farmer Alf Alpha has a pasture located on a sandy hill,The

return to him from this pasture is a random variable depending on how

much rain there is,In rainy years the yield is good; in dry years the yield

is poor,The market value of this pasture is $5,000,The expected return

from this pasture is $500 with a standard deviation of $100,Every inch

of rain above average means an extra $100 in pro t and every inch of rain

below average means another $100 less pro t than average,Farmer Alf

has another $5,000 that he wants to invest in a second pasture,There are

two possible pastures that he can buy.

(a) One is located on low land that never floods,This pasture yields

an expected return of $500 per year no matter what the weather is like.

What is Alf Alpha’s expected rate of return on his total investment if he

buys this pasture for his second pasture? 10%,What is the standard

deviation of his rate of return in this case? 10%.

NAME 179

(b) Another pasture that he could buy is located on the very edge of the

river,This gives very good yields in dry years but in wet years it floods.

This pasture also costs $5,000,The expected return from this pasture is

$500 and the standard deviation is $100,Every inch of rain below average

means an extra $100 in pro t and every inch of rain above average means

another $100 less pro t than average,If Alf buys this pasture and keeps

his original pasture on the sandy hill,what is his expected rate of return

on his total investment? 10%,What is the standard deviation of the

rate of return on his total investment in this case? 0%.

(c) If Alf is a risk averter,which of these two pastures should he buy

and why? He should choose the second pasture

since it has the same expected return and

lower risk.

180 RISKY ASSETS (Ch,13)

Chapter 14 NAME

Consumer’s Surplus

Introduction,In this chapter you will study ways to measure a con-

sumer’s valuation of a good given the consumer’s demand curve for it.

The basic logic is as follows,The height of the demand curve measures

how much the consumer is willing to pay for the last unit of the good

purchased|the willingness to pay for the marginal unit,Therefore the

sum of the willingnesses-to-pay for each unit gives us the total willingness

to pay for the consumption of the good.

In geometric terms,the total willingness to pay to consume some

amount of the good is just the area under the demand curve up to that

amount,This area is called gross consumer’s surplus or total bene t

of the consumption of the good,If the consumer has to pay some amount

in order to purchase the good,then we must subtract this expenditure in

order to calculate the (net) consumer’s surplus.

When the utility function takes the quasilinear form,u(x)+m,the

area under the demand curve measures u(x),and the area under the

demand curve minus the expenditure on the other good measures u(x)+

m,Thus in this case,consumer’s surplus serves as an exact measure of

utility,and the change in consumer’s surplus is a monetary measure of a

change in utility.

If the utility function has a di erent form,consumer’s surplus will not

be an exact measure of utility,but it will often be a good approximation.

However,if we want more exact measures,we can use the ideas of the

compensating variation and the equivalent variation.

Recall that the compensating variation is the amount of extra income

that the consumer would need at the new prices to be as well o as she

was facing the old prices; the equivalent variation is the amount of money

that it would be necessary to take away from the consumer at the old

prices to make her as well o as she would be,facing the new prices.

Although di erent in general,the change in consumer’s surplus and the

compensating and equivalent variations will be the same if preferences are

quasilinear.

In this chapter you will practice:

Calculating consumer’s surplus and the change in consumer’s surplus

Calculating compensating and equivalent variations

Example,Suppose that the inverse demand curve is given by P(q)=

100?10q and that the consumer currently has 5 units of the good,How

much money would you have to pay him to compensate him for reducing

his consumption of the good to zero?

Answer,The inverse demand curve has a height of 100 when q =0

and a height of 50 when q = 5,The area under the demand curve is a

trapezoid with a base of 5 and heights of 100 and 50,We can calculate

182 CONSUMER’S SURPLUS (Ch,14)

the area of this trapezoid by applying the formula

Area of a trapezoid = base

1

2

(height

1

+height

2

):

In this case we have A =5

1

2

(100 + 50) = $375.

Example,Suppose now that the consumer is purchasing the 5 units at a

price of $50 per unit,If you require him to reduce his purchases to zero,

how much money would be necessary to compensate him?

In this case,we saw above that his gross bene ts decline by $375.

On the other hand,he has to spend 5 50 = $250 less,The decline in

net surplus is therefore $125.

Example,Suppose that a consumer has a utility function u(x

1;x

2

)=

x

1

+ x

2

,Initially the consumer faces prices (1;2) and has income 10.

If the prices change to (4;2),calculate the compensating and equivalent

variations.

Answer,Since the two goods are perfect substitutes,the consumer

will initially consume the bundle (10;0) and get a utility of 10,After the

prices change,she will consume the bundle (0;5) and get a utility of 5.

After the price change she would need $20 to get a utility of 10; therefore

the compensating variation is 20?10 = 10,Before the price change,she

would need an income of 5 to get a utility of 5,Therefore the equivalent

variation is 10?5=5.

14.1 (0) Sir Plus consumes mead,and his demand function for tankards

of mead is given by D(p) = 100?p,wherep is the price of mead in

shillings.

(a) If the price of mead is 50 shillings per tankard,how many tankards of

mead will he consume? 50.

(b) How much gross consumer’s surplus does he get from this consump-

tion? 3,750.

(c) How much money does he spend on mead? 2,500.

(d) What is his net consumer’s surplus from mead consumption?

1,250.

14.2 (0) Here is the table of reservation prices for apartments taken

from Chapter 1:

Person=ABCDEFGH

Price = 40 25 30 35 10 18 15 5

NAME 183

(a) If the equilibrium rent for an apartment turns out to be $20,which

consumers will get apartments? A,B,C,D.

(b) If the equilibrium rent for an apartment turns out to be $20,what

is the consumer’s (net) surplus generated in this market for person A?

20,For person B? 5.

(c) If the equilibrium rent is $20,what is the total net consumers’ surplus

generated in the market? 50.

(d) If the equilibrium rent is $20,what is the total gross consumers’

surplus in the market? 130.

(e) If the rent declines to $19,how much does the gross surplus increase?

0.

(f) If the rent declines to $19,how much does the net surplus increase?

4.

Calculus 14.3 (0) Quasimodo consumes earplugs and other things,His utility

function for earplugs x and money to spend on other goods y is given by

u(x;y) = 100x?

x

2

2

+y:

(a) What kind of utility function does Quasimodo have? Quasilinear.

(b) What is his inverse demand curve for earplugs? p = 100?x.

(c) If the price of earplugs is $50,how many earplugs will he consume?

50.

(d) If the price of earplugs is $80,how many earplugs will he consume?

20.

(e) Suppose that Quasimodo has $4,000 in total to spend a month,What

is his total utility for earplugs and money to spend on other things if the

price of earplugs is $50? $5,250.

184 CONSUMER’S SURPLUS (Ch,14)

(f) What is his total utility for earplugs and other things if the price of

earplugs is $80? $4,200.

(g) Utility decreases by 1,050 when the price changes from $50 to

$80.

(h) What is the change in (net) consumer’s surplus when the price changes

from $50 to $80? 1,050.

14.4 (2) In the graph below,you see a representation of Sarah Gamp’s

indi erence curves between cucumbers and other goods,Suppose that

the reference price of cucumbers and the reference price of \other goods"

are both 1.

Cucumbers

Other goods

0

40

30

20

10

10 20 30 40

B

A

(a) What is the minimum amount of money that Sarah would need in

order to purchase a bundle that is indi erent to A? 20.

(b) What is the minimum amount of money that Sarah would need in

order to purchase a bundle that is indi erent to B? 30.

(c) Suppose that the reference price for cucumbers is 2 and the reference

price for other goods is 1,How much money does she need in order to

purchase a bundle that is indi erent to bundle A? 30.

(d) What is the minimum amount of money that Sarah would need to

purchase a bundle that is indi erent to B using these new prices? 40.

NAME 185

(e) No matter what prices Sarah faces,the amount of money she needs

to purchase a bundle indi erent to A must be (higher,lower) than the

amount she needs to purchase a bundle indi erent to B,lower.

14.5 (2) Bernice’s preferences can be represented by u(x;y)=minfx;yg,

where x is pairs of earrings and y is dollars to spend on other things,She

faces prices (p

x;p

y

)=(2;1) and her income is 12.

(a) Draw in pencil on the graph below some of Bernice’s indi erence

curves and her budget constraint,Her optimal bundle is 4 pairs

of earrings and 4 dollars to spend on other things.

0481216

4

8

12

Pairs of earrings

Dollars for other things

16

Black line

Pencil lines

Red

line

Blue lines

(b) The price of a pair of earrings rises to $3 and Bernice’s income stays

the same,Using blue ink,draw her new budget constraint on the graph

above,Her new optimal bundle is 3 pairs of earrings and

3 dollars to spend on other things.

(c) What bundle would Bernice choose if she faced the original prices and

had just enough income to reach the new indi erence curve? (3;3).

Draw with red ink the budget line that passes through this bundle at

the original prices,How much income would Bernice need at the original

prices to have this (red) budget line? $9.

186 CONSUMER’S SURPLUS (Ch,14)

(d) The maximum amount that Bernice would pay to avoid the price

increase is $3,This is the (compensating,equivalent) variation in

income,Equivalent.

(e) What bundle would Bernice choose if she faced the new prices and had

just enough income to reach her original indi erence curve? (4;4).

Draw with black ink the budget line that passes through this bundle at

the new prices,How much income would Bernice have with this budget?

$16.

(f) In order to be as well-o as she was with her original bundle,Bernice’s

original income would have to rise by $4,This is the (compensating,

equivalent) variation in income,Compensating.

Calculus 14.6 (0) Ulrich likes video games and sausages,In fact,his preferences

can be represented by u(x;y)=ln(x +1)+y where x is the number of

video games he plays and y is the number of dollars that he spends on

sausages,Let p

x

be the price of a video game and m be his income.

(a) Write an expression that says that Ulrich’s marginal rate of substi-

tution equals the price ratio,( Hint,Remember Donald Fribble from

Chapter 6?) 1=(x+1)=p

x

.

(b) Since Ulrich has quasilinear preferences,you can solve this

equation alone to get his demand function for video games,which is

x =1=p

x

1,His demand function for the dollars to spend on

sausages is y = m?1+p.

(c) Video games cost $:25 and Ulrich’s income is $10,Then Ulrich de-

mands 3 video games and 9.25 dollars’ worth of sausages.

His utility from this bundle is 10.64,(Round o to two decimal

places.)

(d) If we took away all of Ulrich’s video games,how much money would

he need to have to spend on sausages to be just as well-o as before?

$10.64.

NAME 187

(e) Now an amusement tax of $.25 is put on video games and is passed

on in full to consumers,With the tax in place,Ulrich demands 1

video game and 9.5 dollars’ worth of sausages,His utility from this

bundle is 10.19,(Round o to two decimal places.)

(f) Now if we took away all of Ulrich’s video games,how much money

would he have to have to spend on sausages to be just as well-o as with

the bundle he purchased after the tax was in place? $10.19.

(g) What is the change in Ulrich’s consumer surplus due to the tax?

:45 How much money did the government collect from Ulrich by

means of the tax? $.25.

Calculus 14.7 (1) Lolita,an intelligent and charming Holstein cow,consumes

only two goods,cow feed (made of ground corn and oats) and hay,Her

preferences are represented by the utility function U(x;y)=x?x

2

=2+y,

where x is her consumption of cow feed and y is her consumption of hay.

Lolita has been instructed in the mysteries of budgets and optimization

and always maximizes her utility subject to her budget constraint,Lolita

has an income of $m that she is allowed to spend as she wishes on cow

feed and hay,The price of hay is always $1,and the price of cow feed will

be denoted by p,where0<p 1.

(a) Write Lolita’s inverse demand function for cow feed,(Hint,Lolita’s

utility function is quasilinear,When y is the numeraire and the price of

x is p,the inverse demand function for someone with quasilinear utility

f(x)+y is found by simply setting p = f

0

(x).) p =1?x.

(b) If the price of cow feed is p and her income is m,howmuchhaydoes

Lolita choose? (Hint,The money that she doesn’t spend on feed is used

to buy hay.) m?p(1?p).

(c) Plug these numbers into her utility function to nd out the utility level

that she enjoys at this price and this income,u = m+(1?p)

2

=2.

(d) Suppose that Lolita’s daily income is $3 and that the price of feed is

$:50,What bundle does she buy? (1=2;11=4),What bundle would

she buy if the price of cow feed rose to $1? (0;3).

188 CONSUMER’S SURPLUS (Ch,14)

(e) How much money would Lolita be willing to pay to avoid having the

price of cow feed rise to $1? 1=8,This amount is known as the

equivalent variation.

(f) Suppose that the price of cow feed rose to $1,How much extra money

would you have to pay Lolita to make her as well-o as she was at the

old prices? 1=8,This amount is known as the compensating

variation,Which is bigger,the compensating or the equivalent variation,

or are they the same? Same.

(g) At the price $.50 and income $3,how much (net) consumer’s surplus

is Lolita getting? 1=8.

14.8 (2) F,Flintstone has quasilinear preferences and his inverse demand

function for Brontosaurus Burgers is P(b)=30?2b,Mr,Flintstone is

currently consuming 10 burgers at a price of 10 dollars.

(a) How much money would he be willing to pay to have this amount

rather than no burgers at all? $200,What is his level of (net)

consumer’s surplus? $100.

(b) The town of Bedrock,the only supplier of Brontosaurus Burgers,

decides to raise the price from $10 a burger to $14 a burger,What

is Mr,Flintstone’s change in consumer’s surplus? At price

$10,consumer’s surplus is $100,At $14,

he demands 8 burgers,for net consumer’s

surplus of

1

2

(16 8) = 64,The change in

consumer’s surplus is?$36.

14.9 (1) Karl Kapitalist is willing to produce p=2?20 chairs at every

price,p>40,At prices below 40,he will produce nothing,If the price

of chairs is $100,Karl will produce 30 chairs,At this price,how

much is his producer’s surplus?

1

2

(60 30) = 900.

14.10 (2) Ms,Q,Moto loves to ring the church bells for up to 10

hours a day,Where m is expenditure on other goods,and x is hours of

bell ringing,her utility is u(m;x)=m +3x for x 10,If x>10,she

develops painful blisters and is worse o than if she didn’t ring the bells.

NAME 189

Her income is equal to $100 and the sexton allows her to ring the bell for

10 hours.

(a) Due to complaints from the villagers,the sexton has decided to restrict

Ms,Moto to 5 hours of bell ringing per day,This is bad news for Ms.

Moto,In fact she regards it as just as bad as losing $15 dollars of

income.

(b) The sexton relents and o ers to let her ring the bells as much as she

likes so long as she pays $2 per hour for the privilege,How much ringing

does she do now? 10 hours,This tax on her activities is as bad

as a loss of how much income? $20.

(c) The villagers continue to complain,The sexton raises the price of

bell ringing to $4 an hour,How much ringing does she do now? 0

hours,This tax,as compared to the situation in which she could

ring the bells for free,is as bad as a loss of how much income? $30.

190 CONSUMER’S SURPLUS (Ch,14)

Chapter 15 NAME

Market Demand

Introduction,Some problems in this chapter will ask you to construct

the market demand curve from individual demand curves,The market

demand at any given price is simply the sum of the individual demands at

that price,The key thing to remember in going from individual demands

to the market demand is to add quantities,Graphically,you sum the

individual demands horizontally to get the market demand,The market

demand curve will have a kink in it whenever the market price is high

enough that some individual demand becomes zero.

Sometimes you will need to nd a consumer’s reservation price for

a good,Recall that the reservation price is the price that makes the

consumer indi erent between having the good at that price and not hav-

ing the good at all,Mathematically,the reservation price p

satis es

u(0;m)=u(1;m?p

),where m is income and the quantity of the other

good is measured in dollars.

Finally,some of the problems ask you to calculate price and/or in-

come elasticities of demand,These problems are especially easy if you

know a little calculus,If the demand function is D(p),and you want to

calculate the price elasticity of demand when the price is p,you only need

to calculate dD(p)=dp and multiply it by p=q.

15.0 Warm Up Exercise,(Calculating elasticities.) Here are

some drills on price elasticities,For each demand function,nd an ex-

pression for the price elasticity of demand,The answer will typically be

a function of the price,p,As an example,consider the linear demand

curve,D(p)=30?6p.ThendD(p)=dp =?6andp=q = p=(30?6p),so

the price elasticity of demand is?6p=(30?6p).

(a) D(p)=60?p,?p=(60?p).

(b) D(p)=a?bp,?bp=(a?bp).

(c) D(p)=40p

2

,?2.

(d) D(p)=Ap

b

,?b.

(e) D(p)=(p+3)

2

,?2p=(p+3).

192 MARKET DEMAND (Ch,15)

(f) D(p)=(p+a)

b

,?bp=(p+a).

15.1 (0) In Gas Pump,South Dakota,there are two kinds of consumers,

Buick owners and Dodge owners,Every Buick owner has a demand func-

tion for gasoline D

B

(p)=20?5p for p 4andD

B

(p)=0ifp>4.

Every Dodge owner has a demand function D

D

(p)=15?3p for p 5

and D

D

(p)=0forp>5,(Quantities are measured in gallons per week

and price is measured in dollars.) Suppose that Gas Pump has 150 con-

sumers,100 Buick owners,and 50 Dodge owners.

(a) If the price is $3,what is the total amount demanded by each indi-

vidual Buick Owner? 5,And by each individual Dodge owner?

6.

(b) What is the total amount demanded by all Buick owners? 500.

What is the total amount demanded by all Dodge owners? 300.

(c) What is the total amount demanded by all consumers in Gas Pump

at a price of 3? 800.

(d) On the graph below,use blue ink to draw the demand curve repre-

senting the total demand by Buick owners,Use black ink to draw the

demand curve representing total demand by Dodge owners,Use red ink

to draw the market demand curve for the whole town.

(e) At what prices does the market demand curve have kinks? At

p =4 and p =5.

(f) When the price of gasoline is $1 per gallon,how much does weekly

demand fall when price rises by 10 cents? 65 gallons.

(g) When the price of gasoline is $4.50 per gallon,how much does weekly

demand fall when price rises by 10 cents? 15 gallons.

(h) When the price of gasoline is $10 per gallon,how much does weekly

demand fall when price rises by 10 cents? Remains at zero.

NAME 193

0 1500 2000 2500 3000

1

2

3

4

5

6

500

Dollars per gallon

1000

Gallons per week

Blue line

Black

line

Red line

15.2 (0) For each of the following demand curves,compute the inverse

demand curve.

(a) D(p)=maxf10?2p;0g,p(q)=5?q=2 if q<10.

(b) D(p) = 100=

p

p,p(q)=10;000=q

2

.

(c) lnD(p)=10?4p,p(q)=(10?lnq)=4.

(d) lnD(p)=ln20?2lnp,p(q)=

q

20=q.

15.3 (0) The demand function of dog breeders for electric dog polishers

is q

b

=maxf200?p;0g,and the demand function of pet owners for electric

dog polishers is q

o

=maxf90?4p;0g.

(a) At price p,what is the price elasticity of dog breeders’ demand for

electric dog polishersp=(200?p),What is the price elasticity

of pet owners’ demand4p=(90?4p).

194 MARKET DEMAND (Ch,15)

(b) At what price is the dog breeders’ elasticity equal to?1? $100.

At what price is the pet owners’ elasticity equal to?1? $11.25.

(c) On the graph below,draw the dog breeders’ demand curve in blue

ink,the pet owners’ demand curve in red ink,and the market demand

curve in pencil.

(d) Find a nonzero price at which there is positive total demand for dog

polishers and at which there is a kink in the demand curve,$22.50.

What is the market demand function for prices below the kink? 290?

5p,What is the market demand function for prices above the kink?

200?p.

(e) Where on the market demand curve is the price elasticity equal to

1? $100,At what price will the revenue from the sale of electric

dog polishers be maximized? $100,If the goal of the sellers is to

maximize revenue,will electric dog polishers be sold to breeders only,to

pet owners only,or to both? Breeders only.

NAME 195

0 150 200 250 300

50

100

150

200

250

300

50

Price

100

Quantity

Blue line

Red

line

Pencil line

22.5

90 290

Calculus 15.4 (0) The demand for kitty litter,in pounds,is lnD(p)=1;000?

p+lnm,wherep is the price of kitty litter and m is income.

(a) What is the price elasticity of demand for kitty litter when p =2and

m = 5002,When p =3andm = 5003,When p =4and

m =1;5004.

(b) What is the income elasticity of demand for kitty litter when p =2

and m = 500? 1,When p =2andm =1;000? 1.

When p =3andm =1;500? 1.

196 MARKET DEMAND (Ch,15)

(c) What is the price elasticity of demand when price is p and income is

mp,The income elasticity of demand? 1.

Calculus 15.5 (0) The demand function for drangles is q(p)=(p +1)

2

.

(a) What is the price elasticity of demand at price p2p=(p+1).

(b) At what price is the price elasticity of demand for drangles equal to

1? When the price equals 1.

(c) Write an expression for total revenue from the sale of drangles as

a function of their price,R(p)=pq = p=(p +1)

2

,Use

calculus to nd the revenue-maximizing price,Don’t forget to check the

second-order condition,Differentiating and solving

gives p =1.

(d) Suppose that the demand function for drangles takes the more general

form q(p)=(p+a)

b

where a>0andb>1,Calculate an expression for

the price elasticity of demand at price p,?bp=(p+a),At what

price is the price elasticity of demand equal to?1? p = a=(b?1).

15.6 (0) Ken’s utility function is u

K

(x

1;x

2

)=x

1

+ x

2

and Barbie’s

utility function is u

B

(x

1;x

2

)=(x

1

+1)(x

2

+ 1),A person can buy 1

unit of good 1 or 0 units of good 1,It is impossible for anybody to buy

fractional units or to buy more than 1 unit,Either person can buy any

quantity of good 2 that he or she can a ord at a price of $1 per unit.

(a) Where m is Barbie’s wealth and p

1

is the price of good 1,write an

equation that can be solved to nd Barbie’s reservation price for good 1.

(m?p

1

+1)2=m +1,What is Barbie’s reservation price

for good 1? p =(m+1)=2,What is Ken’s reservation price for

good 1? $1.

(b) If Ken and Barbie each have a wealth of 3,plot the market demand

curve for good 1.

NAME 197

01234

1

2

3

4

Price

Quantity

15.7 (0) The demand function for yo-yos is D(p;M)=4?2p +

1

100

M,

where p is the price of yo-yos and M is income,If M is 100 and p is 1,

(a) What is the income elasticity of demand for yo-yos? 1=3.

(b) What is the price elasticity of demand for yo-yos2=3.

15.8 (0) If the demand function for zarfs is P =10?Q,

(a) At what price will total revenue realized from their sale be at a max-

imum? P =5.

(b) How many zarfs will be sold at that price? Q =5.

15.9 (0) The demand function for football tickets for a typical game at a

large midwestern university is D(p) = 200;000?10;000p,The university

has a clever and avaricious athletic director who sets his ticket prices so

as to maximize revenue,The university’s football stadium holds 100,000

spectators.

(a) Write down the inverse demand function,p(q)=20?

q=10;000.

198 MARKET DEMAND (Ch,15)

(b) Write expressions for total revenue R(q)=20q?q

2

=10;000

and marginal revenue MR =20?q=5;000 as a function of the

number of tickets sold.

(c) On the graph below,use blue ink to draw the inverse demand function

and use red ink to draw the marginal revenue function,On your graph,

also draw a vertical blue line representing the capacity of the stadium.

0 20 40 60 80 100 120 140 160

5

10

15

20

25

30

Price

Quantity x 1000

Red line

Red line

Black line

Blue line

Stadium capacity

(d) What price will generate the maximum revenue? $10,What

quantity will be sold at this price? 100,000.

(e) At this quantity,what is marginal revenue? 0,At this quantity,

what is the price elasticity of demand1,Will the stadium be full?

Yes.

(f) A series of winning seasons caused the demand curve for football

tickets to shift upward,The new demand function is q(p) = 300;000?

10;000p,What is the new inverse demand function? p(q)=30?

q=10;000.

NAME 199

(g) Write an expression for marginal revenue as a function of output.

MR(q)= 30?q=5;000,Use red ink to draw the new demand

function and use black ink to draw the new marginal revenue function.

(h) Ignoring stadium capacity,what price would generate maximum

revenue? $15,What quantity would be sold at this price?

150,000.

(i) As you noticed above,the quantity that would maximize total revenue

given the new higher demand curve is greater than the capacity of the

stadium,Clever though the athletic director is,he cannot sell seats he

hasn’t got,He notices that his marginal revenue is positive for any number

of seats that he sells up to the capacity of the stadium,Therefore,in order

to maximize his revenue,he should sell 100,000 tickets at a price

of $20.

(j) When he does this,his marginal revenue from selling an extra seat

is 10,The elasticity of demand for tickets at this price quantity

combination is =?2.

15.10 (0) The athletic director discussed in the last problem is consid-

ering the extra revenue he would gain from three proposals to expand the

size of the football stadium,Recall that the demand function he is now

facing is given by q(p) = 300;000?10;000p.

(a) How much could the athletic director increase the total revenue per

game from ticket sales if he added 1,000 new seats to the stadium’s capac-

ity and adjusted the ticket price to maximize his revenue? 9,900.

(b) How much could he increase the revenue per game by adding 50,000

new seats? $250,000,60,000 new seats? (Hint,The athletic

director still wants to maximize revenue.) $250,000.

(c) A zealous alumnus o ers to build as large a stadium as the athletic

director would like and donate it to the university,There is only one hitch.

The athletic director must price his tickets so as to keep the stadium full.

If the athletic director wants to maximize his revenue from ticket sales,

how large a stadium should he choose? 150,000 seats.

200 MARKET DEMAND (Ch,15)

Chapter 16 NAME

Equilibrium

Introduction,Supply and demand problems are bread and butter for

economists,In the problems below,you will typically want to solve for

equilibrium prices and quantities by writing an equation that sets supply

equal to demand,Where the price received by suppliers is the same as the

price paid by demanders,one writes supply and demand as functions of

the same price variable,p,and solves for the price that equalizes supply

and demand,But if,as happens with taxes and subsidies,suppliers face

di erent prices from demanders,it is a good idea to denote these two

prices by separate variables,p

s

and p

d

,Then one can solve for equilibrium

by solving a system of two equations in the two unknowns p

s

and p

d

.The

two equations are the equation that sets supply equal to demand and

the equation that relates the price paid by demanders to the net price

received by suppliers.

Example,The demand function for commodity x is q =1;000?10p

d

,

where p

d

is the price paid by consumers,The supply function for x is

q = 100 + 20p

s

,wherep

s

is the price received by suppliers,For each unit

sold,the government collects a tax equal to half of the price paid by con-

sumers,Let us nd the equilibrium prices and quantities,In equilibrium,

supply must equal demand,so that 1;000?10p

d

= 100 + 20p

s

,Since the

government collects a tax equal to half of the price paid by consumers,

it must be that the sellers only get half of the price paid by consumers,

so it must be that p

s

= p

d

=2,Now we have two equations in the two

unknowns,p

s

and p

d

,Substitute the expression p

d

=2forp

s

in the rst

equation,and you have 1;000?10p

d

= 100 + 10p

d

,Solve this equation

to nd p

d

= 45,Then p

s

=22:5andq = 550.

16.1 (0) The demand for yak butter is given by 120?4p

d

and the

supply is 2p

s

30,where p

d

is the price paid by demanders and p

s

is

the price received by suppliers,measured in dollars per hundred pounds.

Quantities demanded and supplied are measured in hundred-pound units.

(a) On the axes below,draw the demand curve (with blue ink) and the

supply curve (with red ink) for yak butter.

202 EQUILIBRIUM (Ch,16)

0 40 60 80 100

Yak butter

20

40

60

80

Price

20 120

Blue line

Red line

p1

q1q2

p2

(b) Write down the equation that you would solve to nd the equilibrium

price,Solve 120?4p =2p?30.

(c) What is the equilibrium price of yak butter? $25,What is the

equilibrium quantity? 20,Locate the equilibrium price and quantity

on the graph,and label them p

1

and q

1

.

(d) A terrible drought strikes the central Ohio steppes,traditional home-

land of the yaks,The supply schedule shifts to 2p

s

60,The demand

schedule remains as before,Draw the new supply schedule,Write down

the equation that you would solve to nd the new equilibrium price of

yak butter,120?4p =2p?60.

(e) The new equilibrium price is 30 and the quantity is 0.

Locate the new equilibrium price and quantity on the graph and label

them p

2

and q

2

.

(f) The government decides to relieve stricken yak butter consumers and

producers by paying a subsidy of $5 per hundred pounds of yak butter

to producers,If p

d

is the price paid by demanders for yak butter,what

is the total amount received by producers for each unit they produce?

p

d

+5,When the price paid by consumers is p

d

,how much yak butter

is produced? 2p

d

50.

NAME 203

(g) Write down an equation that can be solved for the equilibrium price

paid by consumers,given the subsidy program,2p

d

50 =

120? 4p

d

,What are the equilibrium price paid by consumers

and the equilibrium quantity of yak butter now? p

d

= 170=6,

q = 170=3?50 = 20=3.

(h) Suppose the government had paid the subsidy to consumers rather

than producers,What would be the equilibrium net price paid by con-

sumers? 170=6,The equilibrium quantity would be 20=3.

16.2 (0) Here are the supply and demand equations for throstles,where

p is the price in dollars:

D(p)=40?p

S(p)=10+p:

On the axes below,draw the demand and supply curves for throstles,

using blue ink.

010203040

10

20

30

40

Price

Throstles

Demand

Supply

Deadweight

loss

(a) The equilibrium price of throstles is 15 and the equilibrium

quantity is 25.

(b) Suppose that the government decides to restrict the industry to selling

only 20 throstles,At what price would 20 throstles be demanded? 20.

How many throstles would suppliers supply at that price? 30,At what

price would the suppliers supply only 20 units? $10.

204 EQUILIBRIUM (Ch,16)

(c) The government wants to make sure that only 20 throstles are bought,

but it doesn’t want the rms in the industry to receive more than the

minimum price that it would take to have them supply 20 throstles,One

way to do this is for the government to issue 20 ration coupons,Then

in order to buy a throstle,a consumer would need to present a ration

coupon along with the necessary amount of money to pay for the good.

If the ration coupons were freely bought and sold on the open market,

what would be the equilibrium price of these coupons? $10.

(d) On the graph above,shade in the area that represents the deadweight

loss from restricting the supply of throstles to 20,How much is this ex-

pressed in dollars? (Hint,What is the formula for the area of a triangle?)

$25.

16.3 (0) The demand curve for ski lessons is given by D(p

D

) = 100?2p

D

and the supply curve is given by S(p

S

)=3p

S

.

(a) What is the equilibrium price? $20,What is the equilibrium

quantity? 60.

(b) A tax of $10 per ski lesson is imposed on consumers,Write an equation

that relates the price paid by demanders to the price received by suppliers.

p

D

= p

S

+10,Write an equation that states that supply equals

demand,100?2p

D

=3p

S

.

(c) Solve these two equations for the two unknowns p

S

and p

D

.With

the $10 tax,the equilibrium price p

D

paid by consumers would be $26

per lesson,The total number of lessons given would be 48.

(d) A senator from a mountainous state suggests that although ski lesson

consumers are rich and deserve to be taxed,ski instructors are poor and

deserve a subsidy,He proposes a $6 subsidy on production while main-

taining the $10 tax on consumption of ski lessons,Would this policy have

any di erent e ects for suppliers or for demanders than a tax of $4 per

lesson? No.

16.4 (0) The demand curve for salted cod sh is D(P) = 200?5P and

the supply curve S(P)=5P.

NAME 205

(a) On the graph below,use blue ink to draw the demand curve and the

supply curve,The equilibrium market price is $20 and the equilibrium

quantity sold is 100.

0 50 100 150 200

10

20

30

40

Price

Quantity of codfish

Demand

Blue Supply

Deadweight

loss

Red

supply

(b) A quantity tax of $2 per unit sold is placed on salted cod sh,Use red

ink to draw the new supply curve,where the price on the vertical axis

remains the price per unit paid by demanders,The new equilibrium price

paid by the demanders will be $21 and the new price received by the

suppliers will be $19,The equilibrium quantity sold will be 95.

(c) The deadweight loss due to this tax will be 5=2 5=2,On

your graph,shade in the area that represents the deadweight loss.

16.5 (0) The demand function for merino ewes is D(P) = 100=P,and

the supply function is S(P)=P.

(a) What is the equilibrium price? $10.

206 EQUILIBRIUM (Ch,16)

(b) What is the equilibrium quantity? 10.

(c) An ad valorem tax of 300% is imposed on merino ewes so that the

price paid by demanders is four times the price received by suppliers.

What is the equilibrium price paid by the demanders for merino ewes

now? $20,What is the equilibrium price received by the suppliers

for merino ewes? $5,What is the equilibrium quantity? 5.

16.6 (0) Schrecklich and LaMerde are two justi ably obscure nineteenth-

century impressionist painters,The world’s total stock of paintings by

Schrecklich is 100,and the world’s stock of paintings by LaMerde is 150.

The two painters are regarded by connoisseurs as being very similar in

style,Therefore the demand for either painter’s work depends both on its

own price and the price of the other painter’s work,The demand function

for Schrecklichs is D

S

(P) = 200?4P

S

2P

L

,and the demand function for

LaMerdes is D

L

(P) = 200?3P

L

P

S

,whereP

S

and P

L

are respectively

the price in dollars of a Schrecklich painting and a LaMerde painting.

(a) Write down two simultaneous equations that state the equilibrium

condition that the demand for each painter’s work equals supply.

The equations are 200?4P

S

2P

L

= 100 and

200?3P

L

P

S

= 150.

(b) Solving these two equations,one nds that the equilibrium price of

Schrecklichs is 20 and the equilibrium price of LaMerdes is 10.

(c) On the diagram below,draw a line that represents all combinations of

prices for Schrecklichs and LaMerdes such that the supply of Schrecklichs

equals the demand for Schrecklichs,Draw a second line that represents

those price combinations at which the demand for LaMerdes equals the

supply of LaMerdes,Label the unique price combination at which both

markets clear with the letter E.

NAME 207

010203040

10

20

30

40

Pl

Ps

e

Schrecklich

La Mendes

Red line

e'

(d) A re in a bowling alley in Hamtramck,Michigan,destroyed one of

the world’s largest collections of works by Schrecklich,The re destroyed

a total of 10 Schrecklichs,After the re,the equilibrium price of Schreck-

lichs was 23 and the equilibrium price of LaMerdes was 9.

(e) On the diagram you drew above,use red ink to draw a line that shows

the locus of price combinations at which the demand for Schrecklichs

equals the supply of Schrecklichs after the re,On your diagram,label

the new equilibrium combination of prices E

0

.

16.7 (0) The price elasticity of demand for oatmeal is constant and

equal to?1,When the price of oatmeal is $10 per unit,the total amount

demanded is 6,000 units.

(a) Write an equation for the demand function,q =60;000=p.

Graph this demand function below with blue ink,(Hint,If the demand

curve has a constant price elasticity equal to,thenD(p)=ap

for some

constant a,You have to use the data of the problem to solve for the

constants a and that apply in this particular case.)

208 EQUILIBRIUM (Ch,16)

046810

Quantity (thousands)

5

10

15

20

Price

2 12

e

Red lines

Blue lines

(b) If the supply is perfectly inelastic at 5,000 units,what is the equilib-

rium price? $12,Show the supply curve on your graph and label the

equilibrium with an E.

(c) Suppose that the demand curve shifts outward by 10%,Write down

the new equation for the demand function,q =66;000=p,Sup-

pose that the supply curve remains vertical but shifts to the right by 5%.

Solve for the new equilibrium price 12:51 and quantity 5;250.

(d) By what percentage approximately did the equilibrium price rise?

It rose by about 5 percent,Use red ink to draw the

new demand curve and the new supply curve on your graph.

(e) Suppose that in the above problem the demand curve shifts outward

by x% and the supply curve shifts right by y%,By approximately what

percentage will the equilibrium price rise? By about (x?y)

percent.

16.8 (0) An economic historian* reports that econometric studies in-

dicate for the pre{Civil War period,1820{1860,the price elasticity of

demand for cotton from the American South was approximately?1,Due

to the rapid expansion of the British textile industry,the demand curve

for American cotton is estimated to have shifted outward by about 5%

per year during this entire period.

* Gavin Wright,The Political Economy of the Cotton South,W.W.

Norton,1978.

NAME 209

(a) If during this period,cotton production in the United States grew by

3% per year,what (approximately) must be the rate of change of the price

of cotton during this period? It would rise by about 2%

a year.

(b) Assuming a constant price elasticity of?1,and assuming that when

the price is $20,the quantity is also 20,graph the demand curve for

cotton,What is the total revenue when the price is $20? 400,What

is the total revenue when the price is $10? 400.

010203040

10

20

30

40

Price of cotton

Quantity of cotton

(c) If the change in the quantity of cotton supplied by the United States is

to be interpreted as a movement along an upward-sloping long-run supply

curve,what would the elasticity of supply have to be? (Hint,From 1820

to 1860 quantity rose by about 3% per year and price rose by 2 %

per year,[See your earlier answer.] If the quantity change is a movement

along the long-run supply curve,then the long-run price elasticity must

be what?) 1.5 %.

(d) The American Civil War,beginning in 1861,had a devastating e ect

on cotton production in the South,Production fell by about 50% and

remained at that level throughout the war,What would you predict

would be the e ect on the price of cotton? It would double

if demand didn’t change.

210 EQUILIBRIUM (Ch,16)

(e) What would be the e ect on total revenue of cotton farmers in the

South? Since the demand has elasticity of

1,the revenue would stay the same.

(f) The expansion of the British textile industry ended in the 1860s,

and for the remainder of the nineteenth century,the demand curve for

American cotton remained approximately unchanged,By about 1900,

the South approximately regained its prewar output level,What do you

think happened to cotton prices then? They would recover

to their old levels.

16.9 (0) The number of bottles of chardonnay demanded per year is

$1;000;000?60;000P,whereP is the price per bottle (in U.S,dollars).

The number of bottles supplied is 40;000P.

(a) What is the equilibrium price? $10,What is the equilibrium

quantity? 400;000.

(b) Suppose that the government introduces a new tax such that the wine

maker must pay a tax of $5 per bottle for every bottle that he produces.

What is the new equilibrium price paid by consumers? $12,What is

the new price received by suppliers? $7,What is the new equilibrium

quantity? 280,000.

16.10 (0) The inverse demand function for bananas is P

d

=18?3Q

d

and the inverse supply function is P

s

=6+Q

s

,where prices are measured

in cents.

(a) If there are no taxes or subsidies,what is the equilibrium quantity?

3,What is the equilibrium market price? 9 cents.

(b) If a subsidy of 2 cents per pound is paid to banana growers,then

in equilibrium it still must be that the quantity demanded equals the

quantity supplied,but now the price received by sellers is 2 cents higher

than the price paid by consumers,What is the new equilibrium quantity?

3.5,What is the new equilibrium price received by suppliers? 9.5

cents,What is the new equilibrium price paid by demanders? 7.5

cents.

NAME 211

(c) Express the change in price as a percentage of the original price.

-16.66%,If the cross-elasticity of demand between bananas and

apples is +.5,what will happen to the quantity of apples demanded as a

consequence of the banana subsidy,if the price of apples stays constant?

(State your answer in terms of percentage change.) -8.33%.

16.11 (1) King Kanuta rules a small tropical island,Nutting Atoll,

whose primary crop is coconuts,If the price of coconuts is P,thenKing

Kanuta’s subjects will demand D(P)=1;200?100P coconuts per week

for their own use,The number of coconuts that will be supplied per week

by the island’s coconut growers is S(p) = 100P.

(a) The equilibrium price of coconuts will be 6 and the equilib-

rium quantity supplied will be 600.

(b) One day,King Kanuta decided to tax his subjects in order to collect

coconuts for the Royal Larder,The king required that every subject

who consumed a coconut would have to pay a coconut to the king as a

tax,Thus,if a subject wanted 5 coconuts for himself,he would have

to purchase 10 coconuts and give 5 to the king,When the price that

is received by the sellers is p

S

,how much does it cost one of the king’s

subjects to get an extra coconut for himself? 2p

S

.

(c) When the price paid to suppliers is p

S

,how many coconuts will the

king’s subjects demand for their own consumption? (Hint,Express p

D

in terms of p

S

and substitute into the demand function.) Since

p

D

=2p

S

,they consume 1;200?200p

S

.

(d) Since the king consumes a coconut for every coconut consumed by

the subjects,the total amount demanded by the king and his subjects is

twice the amount demanded by the subjects,Therefore,when the price

received by suppliers is p

S

,the total number of coconuts demanded per

week by Kanuta and his subjects is 2;400?400p

S

.

(e) Solve for the equilibrium value of p

S

24/5,the equilibrium total

number of coconuts produced 480,and the equilibrium total number

of coconuts consumed by Kanuta’s subjects,240.

212 EQUILIBRIUM (Ch,16)

(f) King Kanuta’s subjects resented paying the extra coconuts to the

king,and whispers of revolution spread through the palace,Worried by

the hostile atmosphere,the king changed the coconut tax,Now,the

shopkeepers who sold the coconuts would be responsible for paying the

tax,For every coconut sold to a consumer,the shopkeeper would have to

pay one coconut to the king,This plan resulted in 480=2 = 240

coconuts being sold to the consumers,The shopkeepers got 24=5 per

coconut after paying their tax to the king,and the consumers paid a price

of 48=5 per coconut.

Chapter 17 NAME

Auctions

Introduction,An auction is described by a set of rules,The rules

specify bidding procedures for participants and the way in which the

array of bids made determines who gets the object being sold and how

much each bidder pays,Those who are trying to sell an object by auction

typically do not know the willingness to pay of potential buyers but have

some probabilistic expectations,Sellers are interested in nding rules that

maximize their expected revenue from selling the object.

Social planners are often interested not only in the revenue generated

from an auction method,but also in its e ciency,In the absence of

externalities,an auction for a single object will be e cient only if the

object is sold to the buyer who values it most highly.

17.1 (1) At Toivo’s auction house in Ishpemming,Michigan,a beautiful

stu ed moosehead is being sold by auction,There are 5 bidders in atten-

dance,Aino,Erkki,Hannu,Juha,and Matti,The moosehead is worth

$100 to Aino,$20 to Erkki,and $5 to each of the others,The bidders do

not collude and they don’t know each others’ valuations.

(a) If the auctioneer sells it in an English auction,who would get the

moosehead and approximately how much would the buyer pay? Aino

would get it for $20.

(b) If the auctioneer sells it in a sealed-bid,second-price auction and if

no bidder knows the others’ values for the moosehead,how much should

Aino bid in order to maximize his expected gain? $100 How much

should Erkki bid? $20 How much would each of the others bid?

$5 Who would get the moosehead and how much would he pay?

Aino would get it for $20.

17.2 (2) Charlie Plopp sells used construction equipment in a quiet

Oklahoma town,He has run short of cash and needs to raise money

quickly by selling an old bulldozer,If he doesn’t sell his bulldozer to a

customer today,he will have to sell it to a wholesaler for $1,000.

Two kinds of people are interested in buying bulldozers,These are

professional bulldozer operators and people who use bulldozers only for

recreational purposes on weekends,Charlie knows that a professional

bulldozer operator would be willing to pay $6,000 for his bulldozer but no

214 AUCTIONS (Ch,17)

more,while a weekend recreational user would be willing to pay $4;500

but no more,Charlie puts a sign in his window,\Bulldozer Sale Today."

Charlie is disappointed to discover that only two potential buyers

have come to his auction,These two buyers evidently don’t know each

other,Charlie believes that the probability that either is a professional

bulldozer operator is independent of the other’s type and he believes that

each of them has a probability of 1/2 of being a professional bulldozer

operator and a probability of 1/2 of being a recreational user.

Charlie considers the following three ways of selling the bulldozer:

Method 1,Post a price of $6,000 and if nobody takes the bulldozer

at that price,sell it to the wholesaler.

Method 2,Post a price equal to a recreational bulldozer user’s buyer

value and sell it to anyone who o ers that price.

Method 3,Run a sealed-bid auction and sell the bulldozer to the

high bidder at the second highest bid (if there is a tie,choose one of

the high bidders at random and sell the bulldozer to this bidder at

the price bid by both bidders.)

(a) What is the probability that both potential buyers are professional

bulldozer operators? 1/4,What is the probability that both are recre-

ational bulldozer users? 1/4,What is the probability that one of them

is of each type? 1/2.

(b) If Charlie sells by method 1,what is the probability that he will be

able to sell the bulldozer to one of the two buyers? 3/4,What is

the probability that he will have to sell the bulldozer to the wholesaler?

1/4,What is his expected revenue? $(3=4) $6;000 +

(1=4) $1;000 = $4;750:

(c) If Charlie sells by method 2,how much will he receive for his bulldozer?

$4,500.

(d) Suppose that Charlie sells by method 3 and that both potential

buyers bid rationally,If both bidders are professional bulldozer oper-

ators,how much will each bid? $6,000,How much will Char-

lie receive for his bulldozer? $6,000,If one bidder is a profes-

sional bulldozer operator and one is a recreational user,what bids will

Charlie receive? Professional bids $6,000.

Recreational user bids $4,500,Who will get

NAME 215

the bulldozer? The professional,How much money will

Charlie get for his bulldozer? $4,500,If both bidders are recre-

ational bulldozer users,how much will each bid? $4,500,How

much will Charlie receive for his bulldozer? $4,500,What will be

Charlie’s expected revenue from selling the bulldozer by method 3?

$(1=4) $6;000 + (3=4) $4;500 = $4;875.

(e) Which of the three methods will give Charlie the highest expected

revenue? Method 3.

17.3 (2) We revisit our nancially a icted friend,Charlie Plopp,This

time we will look at a slightly generalized version of the same problem,All

else is as before,but the willingness to pay of recreational bulldozers is an

amount C<$6;000 which is known to Charlie,In the previous problem

we dealt with the special case where C =$4;500,Now we want to explore

the way in which the sales method that gives Charlie the highest expected

revenue depends on the size of C.

(a) What will Charlie’s expected revenue be if he posts a price equal to

the reservation price of professional bulldozer operators? $(3=4)

$6;000 + (1=4) $1;000 = $4;750:

(b) If Charlie posts a price equal to the reservation price C of recreational

bulldozer operators,what is his expected revenue? $C.

(c) If Charlie sells his bulldozer by method 3,the second-price sealed-bid

auction,what is his expected revenue? (The answer is a function of C.)

$(1=4) $6;000 + (3=4) $C =$1;500 + (3=4)C.

(d) Show that selling by method 3 will give Charlie a higher expected pay-

o than selling by method 2 if C<$6;000,With method 3,

each bidder will bid his true valuation.

If both bidders have valuations of

$6,000,he will get $6,000,Otherwise,he

will get $C,His expected payoff is then

216 AUCTIONS (Ch,17)

$1;500+(3=4)C,With method 2 he gets $C.

But $1;500 + (3=4)C>Cwhenever C<$6;000.

(e) For what values of C is Charlie better o selling by method 2 than by

method 1? C>$(3=4)6;000 + (1=4)1;000 = 4;759.

(f) For what values of C is Charlie better o selling by method 1 than by

method 3? This happens when 4;750 > 1;500+

3

4C

,

which is the case whenever C<4;333:33.

17.4 (3) Yet again we tread the dusty streets of Charlie Plopp’s home

town,Everything is as in the previous problem,Professional bulldozer

operators are willing to pay $6,000 for a bulldozer and recreational users

are willing to pay C,Charlie is just about to sell his bulldozer when a

third potential buyer appears,Charlie believes that this buyer,like the

other two,is equally likely to be a professional bulldozer operator as a

recreational bulldozer operator and that this probability is independent

of the types of the other two.

(a) With three buyers,Charlie’s expected revenue from using method 1

is 5375,his expected revenue from using method 2 is C,and

his expected revenue from using method 3 is $3;000 + (C=2).

(b) At which values of C would method 1 give Charlie a higher expected

revenue than either of the other two methods of selling proposed above?

C<$4;750.

(c) At which values of C (if any) would method 2 give Charlie a higher

expected revenue than either of the other two methods of selling proposed

above? None.

(d) At which values of C would method 3 give Charlie a higher expected

revenue than either of the other two methods of selling proposed above?

C>$4;750

17.5 (2) General Scooters has decided to replace its old assembly line

with a new one that makes extensive use of robots,There are two con-

tractors who would be able to build the new assembly line,General

Scooters’s industrial spies and engineers have done some exploratory re-

search of their own on the costs of building the new assembly line for each

NAME 217

of the two contractors,They have discovered that for each rm,this cost

will take one of three possible values H,M,andL,whereH>M>L.

Unfortunately,General Scooters has not been able to determine whether

the costs of either of the rms are H,M,orL,The best information that

General Scooters’s investigators have been able to give it is that for each

contractor the probability is 1/3 that the cost is H,1/3 that the cost is

M,and 1/3 that the cost is L and that the probability distribution of

costs is independent between the two contractors,Each contractor knows

its own costs but thinks that the other’s costs are equally likely to be

H,M,orL,General Scooters is con dent that the contractors will not

collude.

(a) Accountants at General Scooters suggested that General Scoooters

accept sealed bids from the two contractors for constructing the assembly

line and that it announce that it will award the contract to the low bidder

but will pay the low bidder the amount bid by the other contractor,(If

there is a tie for low bidder,one of the bidders will be selected at random

to get the contract.) If this is done,what bidding strategy should each

of the contractors use (assuming that they cannot collude) in order to

maximize their expected pro ts? Each would bid his

true valuation.

(b) Suppose that General Scooters uses the bidding mechanism suggested

by the accountants,What is the probability that it will have to pay H

to get the job done? 5/9 What is the probability that it will have

to pay M? 1/3 What is the probability that it will have to pay L?

1/9 Write an expression in terms of the variables H,M,andL for the

expected cost of the project to General Scooters,H

5

9

+M

3

9

+L

1

9

(c) When the distinguished-looking,silver-haired chairman of General

Scooters was told of the accountants’ suggested bidding scheme,he was

outraged,\What a stupid bidding system! Any fool can see that it is

more pro table for us to pay the lower of the two bids,Why on earth

would you ever want to pay the higher bid rather than the lower one?"

he roared.

A timid-looking accountant summoned up his courage and answered

the chairman’s question,What answer would you suggest that he

make? The amount that contractors will bid

depends on the rules of the auction,If

you contract to the low bidder at the low

218 AUCTIONS (Ch,17)

bidder’s bid,then all bidders will bid

a higher amount than they would if you

contract at the second lowest bid.

(d) The chairman ignored the accountants and proposed the following

plan,\Let us award the contract by means of sealed bids,but let us do it

wisely,Since we know that the contractors’ costs are either H,M,orL,

we will accept only bids of H,M,orL,and we will award the contract

to the low bidder at the price he himself bids,(If there is a tie,we will

randomly select one of the bidders and award it to him at his bid.)"

If the chairman’s scheme is adopted,would it ever be worthwhile for

a contractor with costs of L to bid L? No,If he bids L,

he is sure to make zero profits whether

or not he gets the contract,If he bids

higher than L there is a chance that he

might get the contract and make a profit.

(e) Suppose that the chairman’s bidding scheme is adopted and that both

contractors use the strategy of padding their bids in the following way,A

contractor will bid M if her costs are L,and she will bid H if her costs are

H or M,If contractors use this strategy,what is the expected cost of the

project to General Scooters? H

2

3

+ M

1

3

Which of the two schemes

will result in a lower expected cost for General Scooters,the accountants’

scheme or the chairman’s scheme?* The accountants’

scheme.

(f) We have not yet demonstrated that the bid-padding strategies pro-

posed above are equilibrium strategies for bidders,Here we will show that

this is the case for some (but not all) values of H,M,andL,Suppose

that you are one of the two contractors,You believe that the other con-

tractor is equally likely to have costs of H,M,orL andthathewillbid

* The chairman’s scheme might not have worked out so badly for Gen-

eral Scooters if he had not insisted that the only acceptable bids are H,

M,andL,If bidders had been allowed to bid any number between L

and H,then the only equilibrium in bidding strategies would involve the

use of mixed strategies,and if the contractors used these strategies,the

expected cost of the project to General Scooters would be the same as it

is with the second-bidder auction proposed by the accountants.

NAME 219

H when his costs are M or H and he will bid M when his costs are L.

Obviously if your costs are H,you can do no better than to bid H.If

your costs are M,your expected pro ts will be positive if you bid H and

negative or zero if you bid L or M,What if your costs are L? For what

values of H,M,andL will the best strategy available to you be to bid

H? 5M?4H>L

17.6 (3) Late in the day at an antique rug auction there are only two

bidders left,April and Bart,The last rug is brought out and each bidder

takes a look at it,The seller says that she will accept sealed bids from

each bidder and will sell the rug to the highest bidder at the highest

bidder’s bid.

Each bidder believes that the other is equally likely to value the

rug at any amount between 0 and $1,000,Therefore for any number X

between 0 and 1,000,each bidder believes that the probability that the

other bidder values the rug at less than X is X=1;000,The rug is actually

worth $800 to April,If she gets the rug,her pro t will be the di erence

between $800 and what she pays for it,and if she doesn’t get the rug,

her pro t will be zero,She wants to make her bid in such a way as to

maximize her expected pro t.

(a) Suppose that April thinks that Bart will bid exactly what the rug is

worth to him,If she bids $700 for the rug,what is the probability that

she will get the rug? 7/10,If she gets the rug for $700,what is her

pro t? $100,What is her expected pro t if she bids $700? $70.

(b) Suppose that Bart will pay exactly what the rug is worth to him.

If April bids $600 for the rug,what is the probability that she will get

the rug? 6/10,What is her pro t if she gets the rug for $600?

$200,What is her expected pro t if she bids $600? $120.

(c) Again suppose that Bart will bid exactly what the rug is worth to

him,If April bids $x for the rug (where x is a number between 0 and

1,000) what is the probability that she will get the rug? x=1;000

What is her pro t if she gets the rug? $800-x Write a formula for

her expected pro t if she bids $x,$(800?x)(x=1;000),Find

the bid x that maximizes her expected pro t,(Hint,Take a derivative.)

x = 400.

220 AUCTIONS (Ch,17)

(d) Now let us go a little further toward nding a general answer,Suppose

that the value of the rug to April is $V and she believes that Bart will

bid exactly what the rug is worth to him,Write a formula that expresses

her expected pro t in terms of the variables V and x if she bids $x.

$(V?x)(x=1;000) Now calculate the bid $x that will maximize

her expected pro t,(Same hint,Take a derivative.) x = V=2.

17.7 (3) If you did the previous problem correctly,you found that if

April believes that Bart will bid exactly as much as the rug is worth to

him,then she will bid only half as much as the rug is worth to her,If

this is the case,it doesn’t seem reasonable for April to believe that Bart

will bid his full value,Let’s see what would the best thing for April to do

if she believed that Bart would bid only half as much as the rug is worth

to him.

(a) If Bart always bids half of what the rug is worth to him,what is

the highest amount that Bart would ever bid? $500,Why would it

never pay for April to bid more than $500.01? She can get it

for sure by bidding just over $500,since

Bart will never bid more than $500.

(b) Suppose that the the rug is worth $800 to April and she bids $300 for

it,April will only get the rug if the value of the rug to Bart is less than

$600 What is the probability that she will get the rug if she bids $300

for it? 6/10,What is her pro t if she bids $300 and gets the rug?

$500,What is her expected pro t if she bids $300? $300.

(c) Suppose that the rug is worth $800 to April,What is the probability

that she will get it if she bids $x where $x<$500? 2x=1;000,Write

a formula for her expected pro t as a function of her bid $x when the rug

is worth $800 to her,$(800?x)2x=1;000,What bid maximizes

her expected pro t in this case? $400.

NAME 221

(d) Suppose that April values the rug at $V and she believes that Bart

will bid half of his true value,Show that the best thing for April is to

bidhalfofherowntruevalue,Maximize (V?x)2x=1;000:

The derivative with respect to x is 0

when x = V=2.

(e) Suppose that April believes that Bart will bid half of his actual value

and Bart believes that April will bid half of her actual value,Suppose also

that they both act to maximize their expected pro t given these beliefs.

Will these beliefs be self-con rming in the sense that given these beliefs,

each will take the action that the other expects? Yes.

17.8 (2) Rod’s Auction House in Bent Crankshaft,Oregon,holds sealed-

bid used-car auctions every Tuesday,Each used car is sold to the highest

bidder at the second-highest bidder’s bid,On average,half of the cars

that are sold at Rod’s Auction House are lemons and half are good used

cars,A good used car is worth $1,000 to any buyer and a lemon is worth

only $100 to any buyer,Buyers are allowed to look over the used cars for

a few minutes before they are auctioned,Almost all of the buyers who

attend the auctions can do no better than random choice at picking good

cars from among the lemons,The only exception is Al Crankcase,Al can

sometimes,but not always,detect a lemon by licking the oil o of the

dipstick,A good car will never fail Al’s taste test,but 1/3 of the lemons

fail his test,Al attends every auction,licks every dipstick,and taking

into account the results of his taste test,bids his expected value for every

car.

(a) This auction environment is an example of a (common,private)

common value auction.

(b) If a car passes Al’s taste test,what is the probability that it is a good

used car? 3/4

(c) If a car fails Al’s taste test,what is the probability that it is a good

used car? 0

(d) How much will Al bid for a car that passes his taste test? 3=5

1;000 + 2=5 100 = $640 How much will he bid for a car

that fails his taste test? $100

222 AUCTIONS (Ch,17)

(e) Suppose that for each car,a naive bidder at Rod’s Auction House bid

his expected value for a randomly selected car from among those available.

How much would he bid? $550

(f) Given that Al bids his expected value for every used car and the naive

bidders bid the expected value of a randomly selected car,will a naive

bidder ever get a car that passed Al’s taste test? No

(g) What is the expected value of cars that naive bidders get if they always

bid their expected values for a randomly selected car? $100 Will naive

bidders make money,lose money,or break even if they follow this policy?

Lose money.

(h) If the bidders other than Al bid their expected value for a car,given

that it has failed Al’s taste test,how much will they bid? $100

(i) If bidders other than Al bid their expected values for cars that fail Al’s

taste test,and Al bids his expected value for all cars,given the results

of the test,who will get the good cars and at what price? (Recall that

cars are sold to the highest bidder at the second-highest bid.) Al

will get all of the good cars and he will

pay $100 for them.

(j) What will Al’s expected pro t be on a car that passes his test?

$540

17.9 (3) Steve and Leroy buy antique paintings at an art gallery in

Fresno,California,Eighty percent of the paintings that are sold at the

gallery are fakes,and the rest are genuine,After a painting is purchased,

it will be carefully analyzed,and then everybody will know for certain

whether it is genuine or a fake,A genuine antique is worth $1,000,A

fake is worthless,Before they place their bids,buyers are allowed to

inspect the paintings briefly and then must place their bids,Because

they are allowed only a brief inspection,Steve and Leroy each try to

guess whether the paintings are fakes by smelling them,Steve nds that

if a painting fails his sni test,then it is certainly a fake,However,he

cannot detect all fakes,In fact the probability that a fake passes Steve’s

sni test is 1/2,Leroy detects fakes in the same way as Steve,Half of

the fakes fail his sni test and half of them pass his sni test,Genuine

paintings are sure to pass Leroy’s sni test,For any fake,the probability

that Steve recognizes it as a fake is independent of the probability that

Leroy recognizes it as a fake.

NAME 223

The auction house posts a price for each painting,Potential buyers

can submit a written o er to buy at the posted price on the day of the

sale,If more than one person o ers to buy the painting,the auction house

will select one of them at random and sell to that person at the posted

price.

(a) One day,as the auction house is about to close,Steve arrives and

discovers that neither Leroy nor any other bidders have appeared,He

sni s a painting,and it passes his test,Given that it has passed his test,

what is the probability that it is a good painting? (Hint,Since fakes are

much more common than good paintings,the number of fakes that pass

Steve’s test will exceed the number of genuine antiques that pass his test.)

1/3 Steve realizes that he can buy the painting for the posted price

if he wants it,What is the highest posted price at which he would be

willing to buy the painting? $333.33.

(b) On another day,Steve and Leroy see each other at the auction,sni ng

all of the paintings,No other customers have appeared at the auction

house,In deciding how much to bid for a painting that passes his sni

test,Steve considers the following,If a painting is selected at random and

sni ed by both Steve and Leroy,there are ve possible outcomes,Fill in

the blanks for the probability of each.

A,Genuine and passes both dealers’ tests,Probability,.2

B,Fake and passes both dealers’ tests,Probability,.2

C,Fake and passes Steve’s test but fails Leroy’s,Probability,.2

D,Fake and passes Leroy’s test but fails Steve’s,Probability:

.2

E,Fake and fails both dealers’ tests,Probability,.2

(c) On the day when Steve and Leroy are the only customers,the auction

house sets a reserve price of $300,Suppose that Steve believes that Leroy

will o er to buy any painting that passes his sni test,Recall that if Steve

and Leroy both bid on a painting,the probability that Steve gets it is only

1/2,If Steve decides to bid on every painting that passes his own sni

test,what is the probability that a randomly selected painting is genuine

and that Steve is able to buy it?,1 What is the probability that

a randomly selected painting is a fake and that Steve will bid on it and

get it?,3 If Steve o ers to pay $300 for every painting that

passes his sni test,will his expected pro t be positive or negative?

negative Suppose that Steve knows that Leroy is willing to pay the

224 AUCTIONS (Ch,17)

reserve price for any painting that passes Leroy’s sni test,What is the

highest reserve price that Steve should be willing to pay for a painting

that passes his own sni test? $250

17.10 (2) Every day the Repo nance company holds a sealed-bid,

second-price auction in which it sells a repossessed automobile,There are

only three bidders who bid on these cars,Arnie,Barney,and Carny,Each

of these bidders is a used-car dealer whose willingness to pay for another

used car fluctuates randomly from day to day in response to variation in

demand at his car lot,The value of one of these used cars to any dealer,

on any given day is a random variable which takes a high value $H with

probability 1=2andalowvalue$L with probability 1=2,The value that

each dealer places on a car on a given day is independent of the values

placed by the other dealers.

Each day the used-car dealers submit written bids for the used car

being auctioned,The Repo nance company will sell the car to the dealer

with the highest bid at the price bid by the second-highest bidder,If there

is a tie for the highest bid,then the second-highest bid is equal to the

highest bid and so that day’s car will be sold to a randomly selected top

bidder at the price bid by all top bidders.

(a) How much should a dealer bid for a used car on a day when he places

a value of $H on a used car? $H How much should a dealer bid for a

used car on a day when he places a value of $L on a used car? $L

(b) If the dealers do not collude,how much will Repo get for a used car

on days when two or three dealers value the car at $H? $H How much

will Repo get for a used car on days when fewer than two dealers value

the car at $H? $L

(c) On any given day,what is the probability that Repo receives $H for

that day’s used car? 1/2 What is the probability that Repo receives

$L for that day’s used car? 1/2 What is Repo’s expected revenue from

the sale? $(H +L)=2

(d) If there is no collusion and every dealer bids his actual valuation for

every used car,what is the probability on any given day that Arnie gets

a car for a lower price than the value he places on it? (Hint,This will

happen only if the car is worth $H to Arnie and $L to the other dealers.)

1/8 Suppose that we measure a car dealer’s pro t by the di erence

NAME 225

between what a car is worth to him and what he pays for it,On a

randomly selected day,what is Arnie’s expected pro t? (H?L)=8

(e) The expected total pro t of all participants in the market is the sum

of the expected pro ts of the three car dealers and the expected revenue

realized by Repo,Used cars are sold by a sealed-bid,second-price auction

and the dealers do not collude,What is the sum of the expected pro ts of

all participants in the market? (3(H?L)=8)+((H+L)=2) =

(7=8)H +(1=8)L

17.11 (3) This problem (and the two that follow) concerns collusion

among bidders in sealed-bid auctions,Many writers have found evidence

that collusive bidding occurs,The common name for a group that prac-

tices collusive bidding is a \bidding ring."*

Arnie,Barney,and Carny of the previous problem happened to meet

at a church social and got to talking about the high prices they were

paying for used cars and the low pro ts they were making,Carny com-

plained,\About half the time the used cars go for $H,and when that

happens,none of us makes any money." Arnie got a thoughtful look and

then whispered,\Why don’t we agree to always bid $L in Repo’s used-car

auctions?" Barney said,\I’m not so sure that’s a good idea,If we all bid

$L,then we will save some money,but the trouble is,when we all bid the

same,we are just as likely to get the car if we have a low value as we are

to get it if we have a high value,When we bid what we think its worth,

then it always goes to one of the people who value it most."

(a) If Arnie,Barney,and Carny agree to always bid $L,thenonanygiven

day,what is the probability that Barney gets the car for $L when it is

actually worth $H to him? 1/6 What is Barney’s expected pro t per

day? $(H?L)=6

(b) Do the three dealers make higher expected pro ts with this collusive

agreement than they would if they did not collude? Explain,Yes.

(H?L)=6 > (H?L)=8

* Our discussion draws extensively on a paper,\Collusive Bidder Be-

havior at Single-Object,Second-price,and English Auctions" by Daniel

Graham and Robert Marshall in the Journal of Political Economy,1987.

226 AUCTIONS (Ch,17)

(c) Calculate the expected total pro ts of all participants in the market

(including Repo as well as the three dealers) in the case where the dealers

collude,3(H?L)=6+L =(H+L)=2 Are these expected total

pro ts larger or smaller than they are when the dealers do not collude?

Smaller

(d) The cars are said to be allocated e ciently if a car never winds up

in the hands of a dealer who values it less than some other dealer values

it,With a sealed-bid,second-price auction,if there is no collusion,are

the cars allocated e ciently? Yes,If the dealers collude as in this

problem,are the cars allocated e ciently? No.

17.12 (2) Arnie,Barney,and Carny happily practiced the strategy of

\always bid low" for several weeks,until one day Arnie had another idea.

Arnie proposed to the others,\When we all bid $L,it sometimes happens

that the one who gets the week’s car values it at only $L although it is

worth $H to somebody else,I’ve thought of a scheme that will increase

pro ts for all of us." Here is Arnie’s scheme,Every day,before Repo holds

its auction,Arnie,Barney and Carny will hold a sealed-bid,second-price

preauction auction among themselves in which they bid for the right to

be the only high bidder in that day’s auction,The dealer who wins this

preauction bidding can bid anything he likes,while the other two bidders

must bid $L,A preauction auction like this is known is a \knockout."

The revenue that is collected from the \knockout" auction is divided

equally among Arnie,Barney,and Carny,For this problem,assume that

in the knockout auction,each bidder bids his actual value of winning the

knockout auction.*

(a) If the winner of the knockout auction values the day’s used car at

$H,then he knows that he can bid $H for this car in Repo’s second-price

sealed-bid auction and he will get it for a price of $L,Therefore the value

of winning the knockout auction to someone who values a used car at $H

must be $H?L,The value of winning the knockout auction to

someone who values a used car at $L is 0

* It is not necessarily the case that this is the best strategy in the

knockout auction,since one’s bids a ect the revenue redistributed from

the auction as well as who gets the right to bid,Graham and Marshall

present a variation on this mechanism that ensures \honest" bidding in

the knockout auction.

NAME 227

(b) On a day when one dealer values the used car at $H and the other

two value it at $L,the dealer with value $H will bid $H?L in

the knockout auction and the other two dealers will bid 0 In this

case,in the knockout auction,the dealer pays 0 for the right to

be the only high bidder in Repo’s auction,In this case,the day’s used

car will go to the only dealer with value $H and he pays Repo $L for

it,On this day,the dealer with the high buyer value makes a total pro t

of $H?L

(c) We continue to assume that in the knockout auction,dealers bid

their actual values of winning the knockout,On days when two or more

buyers value the used car at $H,the winner of the knockout auction pays

H?L for the right to be the only high bidder in Repo’s auction.

(d) If Arnie’s scheme is adopted,what is the expected total pro t of each

of the three car dealers? (Remember to include each dealer’s share of the

revenue from the knockout auction.) 7( H-L)/8

17.13 (2) After the passage of several weeks during which Repo never got

more than one high bid for a car,the Repo folks guessed that something

was amiss,Some members of the board of directors proposed hiring a hit

man to punish Arnie,Barney,and Carny,but cooler heads prevailed and

they decided instead to hire an economist who had studied Intermediate

Microeconomics,The economist suggested,\Why don’t you set a reserve

price $R which is just a little bit lower than $H (but of course much

larger than $L)? If you get at least one bid of $R,sellitfor$R to one

of these bidders,and if you don’t get a bid as large as your $R,then just

dump that day’s car into the river,(Sadly,the environmental protection

authorities in Repo’s hometown are less than vigilant.) \But what a

waste," said a Repo o cial,\Just do the math," replied the economist.

(a) The economist continued,\If Repo sticks to its guns and refuses to

sell at any price below $R,then even if Arnie,Barney,and Carny collude,

the best they can do is for each to bid $R when they value a car at $H

and to bid nothing when they value it at $L." If they follow this strategy,

the probability that Repo can sell a given car for $R is 7/8,soRepo’s

expected pro t will be $(7=8)R.

(b) Setting a reserve price that is just slightly below $H and destroying

cars for which it gets no bid will be more pro table for Repo than setting

no reservation price if the ratio H=L is greater than 7/8 and less

pro table if H=L is less than 7/8

228 AUCTIONS (Ch,17)

Chapter 18 NAME

Technology

Introduction,In this chapter you work with production functions,re-

lating output of a rm to the inputs it uses,This theory will look familiar

to you,because it closely parallels the theory of utility functions,In utility

theory,an indi erence curve is a locus of commodity bundles,all of which

give a consumer the same utility,In production theory,an isoquant is a lo-

cus of input combinations,all of which give the same output,In consumer

theory,you found that the slope of an indi erence curve at the bundle

(x

1;x

2

) is the ratio of marginal utilities,MU

1

(x

1;x

2

)=MU

2

(x

1;x

2

),In

production theory,the slope of an isoquant at the input combination

(x

1;x

2

) is the ratio of the marginal products,MP

1

(x

1;x

2

)=MP

2

(x

1;x

2

).

Most of the functions that we gave as examples of utility functions can

also be used as examples of production functions.

There is one important di erence between production functions and

utility functions,Remember that utility functions were only \unique up to

monotonic transformations." In contrast,two di erent production func-

tions that are monotonic transformations of each other describe di erent

technologies.

Example,If the utility function U(x

1;x

2

)=x

1

+x

2

represents a person’s

preferences,then so would the utility function U

(x

1;x

2

)=(x

1

+ x

2

)

2

.

A person who had the utility function U

(x

1;x

2

)wouldhavethesame

indi erence curves as a person with the utility function U(x

1;x

2

)and

would make the same choices from every budget,But suppose that one

rm has the production function f(x

1;x

2

)=x

1

+x

2

,and another has the

production function f

(x

1;x

2

)=(x

1

+x

2

)

2

.Itistruethatthetwo rms

will have the same isoquants,but they certainly do not have the same

technology,If both rms have the input combination (x

1;x

2

)=(1;1),

then the rst rm will have an output of 2 and the second rm will have

an output of 4.

Now we investigate \returns to scale." Here we are concerned with

the change in output if the amount of every input is multiplied by a

number t>1,If multiplying inputs by t multiplies output by more than

t,then there are increasing returns to scale,If output is multiplied by

exactly t,there are constant returns to scale,If output is multiplied by

less than t,then there are decreasing returns to scale.

Example,Consider the production function f(x

1;x

2

)=x

1=2

1

x

3=4

2

.Ifwe

multiply the amount of each input by t,then output will be f(tx

1;tx

2

)=

(tx

1

)

1=2

(tx

2

)

3=4

.Tocomparef(tx

1;tx

2

)tof(x

1;x

2

),factor out the

expressions involving t from the last equation,You get f(tx

1;tx

2

)=

t

5=4

x

1=2

1

x

3=4

2

= t

5=4

f(x

1;x

2

),Therefore when you multiply the amounts

of all inputs by t,you multiply the amount of output by t

5=4

,This means

there are increasing returns to scale.

230 TECHNOLOGY (Ch,18)

Example,Let the production function be f(x

1;x

2

)=minfx

1;x

2

g.Then

f(tx

1;tx

2

)=minftx

1;tx

2

g=mintfx

1;x

2

g= tminfx

1;x

2

g= tf(x

1;x

2

):

Therefore when all inputs are multiplied by t,output is also multiplied by

t,It follows that this production function has constant returns to scale.

You will also be asked to determine whether the marginal product

of each single factor of production increases or decreases as you increase

the amount of that factor without changing the amount of other factors.

Those of you who know calculus will recognize that the marginal product

of a factor is the rst derivative of output with respect to the amount

of that factor,Therefore the marginal product of a factor will decrease,

increase,or stay constant as the amount of the factor increases depending

on whether the second derivative of the production function with respect

to the amount of that factor is negative,positive,or zero.

Example,Consider the production function f(x

1;x

2

)=x

1=2

1

x

3=4

2

.The

marginal product of factor 1 is

1

2

x

1=2

1

x

3=4

2

,This is a decreasing function

of x

1

,as you can verify by taking the derivative of the marginal product

with respect to x

1

,Similarly,you can show that the marginal product of

x

2

decreases as x

2

increases.

18.0 Warm Up Exercise,The rst part of this exercise is to cal-

culate marginal products and technical rates of substitution for several

frequently encountered production functions,As an example,consider

the production function f(x

1;x

2

)=2x

1

+

p

x

2

,The marginal product of

x

1

is the derivative of f(x

1;x

2

) with respect to x

1

,holding x

2

xed,This

is just 2,The marginal product of x

2

is the derivative of f(x

1;x

2

)with

respect to x

2

,holding x

1

xed,which in this case is

1

2

p

x

2

.TheTRS is

MP

1

=MP

2

=?4

p

x

2

,Those of you who do not know calculus should

ll in this table from the answers in the back,The table will be a useful

reference for later problems.

NAME 231

Marginal Products and Technical Rates of Substitution

f(x

1;x

2

) MP

1

(x

1;x

2

) MP

2

(x

1;x

2

) TRS(x

1;x

2

)

x

1

+2x

2

1 2?1=2

ax

1

+bx

2

a b?a=b

50x

1

x

2

50x

2

50x

1

x

2

x

1

x

1=4

1

x

3=4

2

1

4

x

3=4

1

x

3=4

2

3

4

x

1=4

1

x

1=4

2

x

2

3x

1

Cx

a

1

x

b

2

Cax

a?1

1

x

b

2

Cbx

a

1

x

b?1

2

ax

2

bx

1

(x

1

+2)(x

2

+1) x

2

+1 x

1

+2?

x

2

+1

x

1

+2

(x

1

+a)(x

2

+b) x

2

+b x

1

+a?

x

2

+b

x

1

+a

ax

1

+b

p

x

2

a

b

2

p

x

2

2a

p

x

2

b

x

a

1

+x

a

2

ax

a?1

1

ax

a?1

2

x

1

x

2

a?1

(x

a

1

+x

a

2

)

b

bax

a?1

1

(x

a

1

+x

a

2

)

b?1

bax

a?1

2

(x

a

1

+x

a

2

)

b?1

x

1

x

2

a?1

232 TECHNOLOGY (Ch,18)

Returns to Scale and Changes in Marginal Products

For each production function in the table below,put an I,C,orD in

the rst column if the production function has increasing,constant,or

decreasing returns to scale,Put an I,C,orD in the second (third)

column,depending on whether the marginal product of factor 1 (factor

2) is increasing,constant,or decreasing,as the amount of that factor

alone is varied.

f(x

1;x

2

) Scale MP

1

MP

2

x

1

+2x

2

C C C

p

x

1

+2x

2

D D D

:2x

1

x

2

2

I C I

x

1=4

1

x

3=4

2

C D D

x

1

+

p

x

2

D C D

(x

1

+1)

:5

(x

2

)

:5

D D D

x

1=3

1

+x

1=3

2

3

C D D

18.1 (0) Prunella raises peaches,Where L is the number of units of

labor she uses and T is the number of units of land she uses,her output

is f(L;T)=L

1

2

T

1

2

bushels of peaches.

(a) On the graph below,plot some input combinations that give her an

output of 4 bushels,Sketch a production isoquant that runs through these

points,The points on the isoquant that gives her an output of 4 bushels

all satisfy the equation T = 16=L.

NAME 233

0481216

2

4

6

L

T

8

(b) This production function exhibits (constant,increasing,decreasing)

returns to scale,Constant returns to scale.

(c) In the short run,Prunella cannot vary the amount of land she uses.

On the graph below,use blue ink to draw a curve showing Prunella’s

output as a function of labor input if she has 1 unit of land,Locate the

points on your graph at which the amount of labor is 0,1,4,9,and

16 and label them,The slope of this curve is known as the marginal

product of labor,Is this curve getting steeper or flatter

as the amount of labor increase? Flatter.

0481216

2

4

6

Labour

Output

8

Blue line

Red line

Red MPL line

234 TECHNOLOGY (Ch,18)

(d) Assuming she has 1 unit of land,how much extra output does she

get from adding an extra unit of labor when she previously used 1 unit of

labor?

p

2?1,41,4 units of labor?

p

5?2,24,If

you know calculus,compute the marginal product of labor at the input

combination (1;1) and compare it with the result from the unit increase

in labor output found above,Derivative is 1=2

p

L,so

the MP is,5 when L =1 and,25 when L =4.

(e) In the long run,Prunella can change her input of land as well as

of labor,Suppose that she increases the size of her orchard to 4 units

of land,Use red ink to draw a new curve on the graph above showing

output as a function of labor input,Also use red ink to draw a curve

showing marginal product of labor as a function of labor input when the

amount of land is xed at 4.

18.2 (0) Supposex

1

and x

2

areusedin xedproportionsandf(x

1;x

2

)=

minfx

1;x

2

g.

(a) Suppose that x

1

<x

2

,The marginal product for x

1

is 1

and (increases,remains constant,decreases) remains constant

for small increases in x

1

.Forx

2

the marginal product is 0,

and (increases,remains constant,decreases) remains constant

for small increases in x

2

,The technical rate of substitution between x

2

and x

1

is infinity,This technology demonstrates (increasing,

constant,decreasing) constant returns to scale.

(b) Suppose that f(x

1;x

2

)=minfx

1;x

2

g and x

1

= x

2

= 20,What is

the marginal product of a small increase in x

1

0,What is the

marginal product of a small increase in x

2

0,The marginal

product of x

1

will (increase,decrease,stay constant) increase if

the amount of x

2

is increased by a little bit.

Calculus 18.3 (0) Suppose the production function is Cobb-Douglas and

f(x

1;x

2

)=x

1=2

1

x

3=2

2

.

(a) Write an expression for the marginal product of x

1

at the point

(x

1;x

2

).

1

2

x

1=2

1

x

3=2

2

.

NAME 235

(b) The marginal product of x

1

(increases,decreases,remains constant)

decreases for small increases in x

1

,holding x

2

xed.

(c) The marginal product of factor 2 is 3=2x

1=2

1

x

1=2

2

,and it (in-

creases,remains constant,decreases) increases for small increases

in x

2

.

(d) An increase in the amount of x

2

(increases,leaves unchanged,de-

creases) increases the marginal product of x

1

.

(e) The technical rate of substitution between x

2

and x

1

is?x

2

=3x

1

.

(f) Does this technology have diminishing technical rate of substitution?

Yes.

(g) This technology demonstrates (increasing,constant,decreasing)

increasing returns to scale.

18.4 (0) The production function for fragles is f(K;L)=L=2+

p

K,

where L is the amount of labor used and K the amount of capital used.

(a) There are (constant,increasing,decreasing) decreasing re-

turns to scale,The marginal product of labor is constant (con-

stant,increasing,decreasing).

(b) In the short run,capital is xed at 4 units,Labor is variable,On the

graph below,use blue ink to draw output as a function of labor input in

the short run,Use red ink to draw the marginal product of labor as a

function of labor input in the short run,The average product of labor is

de ned as total output divided by the amount of labor input,Use black

ink to draw the average product of labor as a function of labor input in

the short run.

236 TECHNOLOGY (Ch,18)

0481216

2

4

6

Labour

Fragles

8

Black line

Blue

line

Red line

18.5 (0) General Monsters Corporation has two plants for producing

juggernauts,one in Flint and one in Inkster,The Flint plant produces

according to f

F

(x

1;x

2

)=minfx

1;2x

2

g and the Inkster plant produces

according to f

I

(x

1;x

2

)=minf2x

1;x

2

g,wherex

1

and x

2

are the inputs.

(a) On the graph below,use blue ink to draw the isoquant for 40 jugger-

nauts at the Flint plant,Use red ink to draw the isoquant for producing

40 juggernauts at the Inkster plant.

NAME 237

020406080

20

40

60

X2

80

,

a

b

c

Blue isoquant

Red

isoquant

Black isoquant

X1

(b) Suppose that the rm wishes to produce 20 juggernauts at each plant.

How much of each input will the rm need to produce 20 juggernauts

at the Flint plant? x

1

=20;x

2

=10,How much of each

input will the rm need to produce 20 juggernauts at the Inkster plant?

x

1

=10;x

2

=20,Label with an a on the graph,the point

representing the total amount of each of the two inputs that the rm

needs to produce a total of 40 juggernauts,20 at the Flint plant and 20

at the Inkster plant.

(c) Label with a b on your graph the point that shows how much of each

of the two inputs is needed in toto if the rm is to produce 10 juggernauts

in the Flint plant and 30 juggernauts in the Inkster plant,Label with a

c the point that shows how much of each of the two inputs that the rm

needs in toto if it is to produce 30 juggernauts in the Flint plant and

10 juggernauts in the Inkster plant,Use a black pen to draw the rm’s

isoquant for producing 40 units of output if it can split production in any

manner between the two plants,Is the technology available to this rm

convex? Yes.

18.6 (0) You manage a crew of 160 workers who could be assigned to

make either of two products,Product A requires 2 workers per unit of

output,Product B requires 4 workers per unit of output.

(a) Write an equation to express the combinations of products A and

B that could be produced using exactly 160 workers,2A +4B =

160,On the diagram below,use blue ink to shade in the area depicting

238 TECHNOLOGY (Ch,18)

the combinations of A and B that could be produced with 160 workers.

(Assume that it is also possible for some workers to do nothing at all.)

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

,

,

,

,

,

,

,

,

,

,

,

,

,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

,,,,,,,,,,,,,,

020406080

20

40

60

B

80

,

A

Red

shading

Blue

shading

Black

shading

a

(b) Suppose now that every unit of product A that is produced requires

the use of 4 shovels as well as 2 workers and that every unit of product B

produced requires 2 shovels and 4 workers,On the graph you have just

drawn,use red ink to shade in the area depicting combinations of A and

B that could be produced with 180 shovels if there were no worries about

the labor supply,Write down an equation for the set of combinations of

A and B that require exactly 180 shovels,4A+2B = 180.

(c) On the same diagram,use black ink to shade the area that repre-

sents possible output combinations when one takes into account both the

limited supply of labor and the limited supply of shovels.

(d) On your diagram locate the feasible combination of inputs that use up

all of the labor and all of the shovels,If you didn’t have the graph,what

equations would you solve to determine this point? 2A +4B =

160 and 4A+2B = 180.

(e) If you have 160 workers and 180 shovels,what is the largest amount of

product A that you could produce? 45 units,If you produce this

amount,you will not use your entire supply of one of the inputs,Which

one? Workers,How many will be left unused? 70.

18.7 (0) A rm has the production function f(x;y)=minf2x;x + yg.

On the graph below,use red ink to sketch a couple of production isoquants

for this rm,A second rm has the production function f(x;y)=x +

minfx;yg,Do either or both of these rms have constant returns to scale?

NAME 239

Both do,On the same graph,use black ink to draw a couple of

isoquants for the second rm.

010203040

10

20

30

40

y

x

Black

isoquants

Red

isoquants

18.8 (0) Suppose the production function has the form

f(x

1;x

2;x

3

)=Ax

a

1

x

b

2

x

c

3;

where a+b+c>1.Provethatthereareincreasingreturnstoscale.

For any t>1,f(tx

1;tx

2;tx

3

)=A(tx

1

)

a

(tx

2

)

b

(tx

3

)

c

=

t

a+b+c

f(x

1;x

2;x

3

) >tf(x

1;x

2;x

3

).

18.9 (0) Suppose that the production function is f(x

1;x

2

)=Cx

a

1

x

b

2

,

where a,b,andC are positive constants.

(a) For what positive values of a,b,andC are there decreasing returns

to scale? All C>0 and a + b<1,constant returns to

scale? All C>0 and a + b =1,increasing returns to

scale? All C>0 and a+b>1.

(b) For what positive values of a,b,andC is there decreasing marginal

product for factor 1? All C>0 and b>0 and a<1.

240 TECHNOLOGY (Ch,18)

(c) For what positive values of a,b,andC is there diminishing technical

rate of substitution? For all positive values.

18.10 (0) Suppose that the production function is f(x

1;x

2

)=

(x

a

1

+x

a

2

)

b

,wherea and b are positive constants.

(a) For what positive values of a and b are there decreasing returns to

scale? ab< 1,Constant returns to scale? ab =1,Increasing

returns to scale? ab> 1.

18.11 (0) Suppose that a rm has the production function f(x

1;x

2

)=

p

x

1

+x

2

2

.

(a) The marginal product of factor 1 (increases,decreases,stays constant)

decreases as the amount of factor 1 increases,The marginal

product of factor 2 (increases,decreases,stays constant) increases

as the amount of factor 2 increases.

(b) This production function does not satisfy the de nition of increasing

returns to scale,constant returns to scale,or decreasing returns to scale.

How can this be? Returns to scale are different

depending on the ratio in which the factors

are used,Find a combination of inputs such that doubling the

amount of both inputs will more than double the amount of output.

x

1

=1,x

2

=4,for example,Find a combination of

inputs such that doubling the amount of both inputs will less than double

output,x

1

=4,x

2

=0,for example.

Chapter 19 NAME

Profit Maximization

Introduction,A rm in a competitive industry cannot charge more than

the market price for its output,If it also must compete for its inputs,then

it has to pay the market price for inputs as well,Suppose that a pro t-

maximizing competitive rm can vary the amount of only one factor and

that the marginal product of this factor decreases as its quantity increases.

Then the rm will maximize its pro ts by hiring enough of the variable

factor so that the value of its marginal product is equal to the wage,Even

if a rm uses several factors,only some of them may be variable in the

short run.

Example,A rm has the production function f(x

1;x

2

)=x

1=2

1

x

1=2

2

,Sup-

pose that this rm is using 16 units of factor 2 and is unable to vary this

quantity in the short run,In the short run,the only thing that is left for

the rm to choose is the amount of factor 1,Let the price of the rm’s

output be p,and let the price it pays per unit of factor 1 be w

1

.We

want to nd the amount of x

1

that the rm will use and the amount of

output it will produce,Since the amount of factor 2 used in the short run

must be 16,we have output equal to f(x

1;16) = 4x

1=2

1

.Themarginal

product of x

1

is calculated by taking the derivative of output with respect

to x

1

,This marginal product is equal to 2x

1=2

1

,Setting the value of the

marginal product of factor 1 equal to its wage,we have p2x

1=2

1

= w

1

.

Now we can solve this for x

1

,We nd x

1

=(2p=w

1

)

2

,Plugging this

into the production function,we see that the rm will choose to produce

4x

1=2

1

=8p=w

1

units of output.

In the long run,a rm is able to vary all of its inputs,Consider

the case of a competitive rm that uses two inputs,Then if the rm is

maximizing its pro ts,it must be that the value of the marginal product

of each of the two factors is equal to its wage,This gives two equations in

the two unknown factor quantities,If there are decreasing returns to scale,

these two equations are enough to determine the two factor quantities,If

there are constant returns to scale,it turns out that these two equations

are only su cient to determine the ratio in which the factors are used.

In the problems on the weak axiom of pro t maximization,you are

asked to determine whether the observed behavior of rms is consistent

with pro t-maximizing behavior,To do this you will need to plot some of

the rm’s isopro t lines,An isopro t line relates all of the input-output

combinations that yield the same amount of pro t for some given input

and output prices,To get the equation for an isopro t line,just write

down an equation for the rm’s pro ts at the given input and output

prices,Then solve it for the amount of output produced as a function

of the amount of the input chosen,Graphically,you know that a rm’s

behavior is consistent with pro t maximization if its input-output choice

242 PROFIT MAXIMIZATION (Ch,19)

in each period lies below the isopro t lines of the other periods.

19.1 (0) The short-run production function of a competitive rm is

given by f(L)=6L

2=3

,whereL istheamountoflaborituses,(For

those who do not know calculus|if total output is aL

b

,wherea and b

are constants,and where L is the amount of some factor of production,

then the marginal product of L is given by the formula abL

b?1

.) The cost

per unit of labor is w = 6 and the price per unit of output is p =3.

(a) Plot a few points on the graph of this rm’s production function and

sketch the graph of the production function,using blue ink,Use black

ink to draw the isopro t line that passes through the point (0;12),the

isopro t line that passes through (0;8),and the isopro t line that passes

through the point (0;4),What is the slope of each of the isopro t lines?

They all have slope 2,How many points on the isopro t

line through (0;12) consist of input-output points that are actually pos-

sible? None,Make a squiggly line over the part of the isopro t line

through (0;4) that consists of outputs that are actually possible.

(b) How many units of labor will the rm hire? 8,How much

output will it produce? 24,If the rm has no other costs,how much

will its total pro ts be? 24.

0 8 12 16 20

Labour input

12

24

36

48

Output

424

8

4

Black lines

Blue curve

Squiggly line

13.3

Red line

NAME 243

(c) Suppose that the wage of labor falls to 4,and the price of output

remains at p,On the graph,use red ink to draw the new isopro t line

for the rm that passes through its old choice of input and output,Will

the rm increase its output at the new price? Yes,Explain why,

referring to your diagram,As the diagram shows,the

firm can reach a higher isoprofit line by

increasing output.

Calculus 19.2 (0) A Los Angeles rm uses a single input to produce a recreational

commodity according to a production function f(x)=4

p

x,wherex is

the number of units of input,The commodity sells for $100 per unit,The

input costs $50 per unit.

(a) Write down a function that states the rm’s pro t as a function of

the amount of input,= 400

p

x?50x.

(b) What is the pro t-maximizing amount of input? 16,of output?

16,How much pro ts does it make when it maximizes pro ts?

$800.

(c) Suppose that the rm is taxed $20 per unit of its output and the price

of its input is subsidized by $10,What is its new input level? 16.

What is its new output level? 16,How much pro t does it make now?

$640,(Hint,A good way to solve this is to write an expression for the

rm’s pro t as a function of its input and solve for the pro t-maximizing

amount of input.)

(d) Suppose that instead of these taxes and subsidies,the rm is taxed

at 50% of its pro ts,Write down its after-tax pro ts as a function of the

amount of input,=,50 (400

p

x?50x),What is the

pro t-maximizing amount of output? 16,How much pro t does it

make after taxes? $400.

19.3 (0) Brother Jed takes heathens and reforms them into righteous

individuals,There are two inputs needed in this process,heathens (who

are widely available) and preaching,The production function has the

following form,r

p

=minfh;pg,wherer

p

is the number of righteous

244 PROFIT MAXIMIZATION (Ch,19)

persons produced,h is the number of heathens who attend Jed’s sermons,

and p is the number of hours of preaching,For every person converted,

Jed receives a payment of s from the grateful convert,Sad to say,heathens

do not flock to Jed’s sermons of their own accord,Jed must o er heathens

apaymentofw to attract them to his sermons,Suppose the amount of

preaching is xed at p and that Jed is a pro t-maximizing prophet.

(a) If h< p,what is the marginal product of heathens? 1,What

is the value of the marginal product of an additional heathen? s.

(b) If h> p,what is the marginal product of heathens? 0,What

is the value of the marginal product of an additional heathen in this case?

0.

(c) Sketch the shape of this production function in the graph below,Label

the axes,and indicate the amount of the input where h = p.

r

hp

p

_

(d) If w<s,how many heathens will be converted? p,If w>s,

how many heathens will be converted? 0.

19.4 (0) Allie’s Apples,Inc,purchases apples in bulk and sells two prod-

ucts,boxes of apples and jugs of cider,Allie’s has capacity limitations of

three kinds,warehouse space,crating facilities,and pressing facilities,A

box of apples requires 6 units of warehouse space,2 units of crating facili-

ties,and no pressing facilities,A jug of cider requires 3 units of warehouse

space,2 units of crating facilities,and 1 unit of pressing facilities,The

total amounts available each day are,1,200 units of warehouse space,600

units of crating facilities,and 250 units of pressing facilities.

(a) If the only capacity limitations were on warehouse facilities,and if all

warehouse space were used for the production of apples,how many boxes

of apples could be produced in one day? 200,How many jugs of cider

could be produced each day if,instead,all warehouse space were used in

NAME 245

the production of cider and there were no other capacity constraints?

400,Draw a blue line in the following graph to represent the warehouse

space constraint on production combinations.

(b) Following the same reasoning,draw a red line to represent the con-

straints on output to limitations on crating capacity,How many boxes of

apples could Allie produce if he only had to worry about crating capacity?

300,Howmanyjugsofcider? 300.

(c) Finally draw a black line to represent constraints on output combina-

tions due to limitations on pressing facilities,How many boxes of apples

could Allie produce if he only had to worry about the pressing capacity

and no other constraints? An infinite number,How many

jugs of cider? 250.

(d) Now shade the area that represents feasible combinations of daily

production of apples and cider for Allie’s Apples.

0 300 400 500

100

200

300

400

500

600

100

Cider

200

Apples

600

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Blue line

Black revenue line

Red line

Black line

(e) Allie’s can sell apples for $5 per box of apples and cider for $2 per

jug,Draw a black line to show the combinations of sales of apples and

cider that would generate a revenue of $1,000 per day,At the pro t-

maximizing production plan,Allie’s is producing 200 boxes of apples

and 0 jugs of cider,Total revenues are $1,000.

246 PROFIT MAXIMIZATION (Ch,19)

19.5 (0) A pro t-maximizing rm produces one output,y,and uses one

input,x,to produce it,The price per unit of the factor is denoted by

w and the price of the output is denoted by p,You observe the rm’s

behavior over three periods and nd the following:

Period y x w p

1 1 1 1 1

2 2.5 3,5 1

3 4 8,25 1

(a) Write an equation that gives the rm’s pro ts,,as a function of the

amount of inputxit uses,the amount of outputy it produces,the per-unit

cost of the input w,and the price of output p,= py?wx.

(b) In the diagram below,draw an isopro t line for each of the three

periods,showing combinations of input and output that would yield the

same pro ts that period as the combination actually chosen,What are

the equations for these three lines? y = x,y =1+:5x,

y =2+:25x,Using the theory of revealed pro tability,shade in

the region on the graph that represents input-output combinations that

could be feasible as far as one can tell from the evidence that is available.

How would you describe this region in words? The region that

is below all 3 isoprofit lines.

06810

2

4

6

8

10

12

2

Output

4

Input

12

Period 3

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

Period 2

Period 1

NAME 247

19.6 (0) T-bone Pickens is a corporate raider,This means that he looks

for companies that are not maximizing pro ts,buys them,and then tries

to operate them at higher pro ts,T-bone is examining the nancial

records of two re neries that he might buy,the Shill Oil Company and

the Golf Oil Company,Each of these companies buys oil and produces

gasoline,During the time period covered by these records,the price of

gasoline fluctuated signi cantly,while the cost of oil remained constant

at $10 a barrel,For simplicity,we assume that oil is the only input to

gasoline production.

Shill Oil produced 1 million barrels of gasoline using 1 million barrels

of oil when the price of gasoline was $10 a barrel,When the price of

gasoline was $20 a barrel,Shill produced 3 million barrels of gasoline

using 4 million barrels of oil,Finally,when the price of gasoline was $40

a barrel,Shill used 10 million barrels of oil to produce 5 million barrels

of gasoline.

Golf Oil (which is managed by Martin E,Lunch III) did exactly the

same when the price of gasoline was $10 and $20,but when the price

of gasoline hit $40,Golf produced 3.5 million barrels of gasoline using 8

million barrels of oil.

(a) Using black ink,plot Shill Oil’s isopro t lines and choices for the three

di erent periods,Label them 10,20,and 40,Using red ink draw Golf

Oil’s isopro t line and production choice,Label it with a 40 in red ink.

06810

2

4

6

8

10

12

2

Million barrels of gasoline

4

Million barrels of oil

12

10

20

40

Red 40

248 PROFIT MAXIMIZATION (Ch,19)

(b) How much pro ts could Golf Oil have made when the price of gasoline

was $40 a barrel if it had chosen to produce the same amount that it did

when the price was $20 a barrel? $80 million,What pro ts

did Golf actually make when the price of gasoline was $40? $60

million.

(c) Is there any evidence that Shill Oil is not maximizing pro ts? Explain.

No,The data satisfy WAPM.

(d) Is there any evidence that Golf Oil is not maximizing pro ts? Explain.

Yes,When price of gas was $40,Golf could

have made more money by acting as it did

when price of gas was $20.

19.7 (0) After carefully studying Shill Oil,T-bone Pickens decides that

it has probably been maximizing its pro ts,But he still is very interested

in buying Shill Oil,He wants to use the gasoline they produce to fuel his

delivery fleet for his chicken farms,Capon Truckin’,In order to do this

Shill Oil would have to be able to produce 5 million barrels of gasoline

from 8 million barrels of oil,Mark this point on your graph,Assuming

that Shill always maximizes pro ts,would it be technologically feasible

for it to produce this input-output combination? Why or why not?

No,If it could,then it would have made

more profits by choosing this combination

than what it chose when price of oil was

$40.

19.8 (0) Suppose that rms operate in a competitive market,attempt to

maximize pro ts,and only use one factor of production,Then we know

that for any changes in the input and output price,the input choice and

the output choice must obey the Weak Axiom of Pro t Maximization,

p y? w x 0.

Which of the following propositions can be proven by the Weak Ax-

iom of Pro t Maximizing Behavior (WAPM)? Respond yes or no,and

give a short argument.

NAME 249

(a) If the price of the input does not change,then a decrease in the price

of the output will imply that the rm will produce the same amount or

less output,Yes,If price of input doesn’t

change,w =0,so WAPM says p y 0.

(b) If the price of the output remains constant,then a decrease in the

input price will imply that the rm will use the same amount or more

of the input,Yes,If price of output doesn’t

change,p =0,so WAPM says? w x 0.

(c) If both the price of the output and the input increase and the rm

produces less output,then the rm will use more of the input,No.

Sign pattern is (+)(?)?(+)(+) 0,which

cannot happen.

19.9 (1) Farmer Hoglund has discovered that on his farm,he can get

30 bushels of corn per acre if he applies no fertilizer,When he applies N

pounds of fertilizer to an acre of land,the marginal product of fertilizer is

1?N=200 bushels of corn per pound of fertilizer.

(a) If the price of corn is $3 a bushel and the price of fertilizer is $p per

pound (where p<3),how many pounds of fertilizer should he use per

acre in order to maximize pro ts? 200?66:66p.

(b) (Only for those who remember a bit of easy integral calculus.) Write

down a function that states Farmer Hoglund’s yield per acre as a function

of the amount of fertilizer he uses,30 +N?N

2

=400.

(c) Hoglund’s neighbor,Skoglund,has better land than Hoglund,In fact,

for any amount of fertilizer that he applies,he gets exactly twice as much

corn per acre as Hoglund would get with the same amount of fertilizer.

How much fertilizer will Skoglund use per acre when the price of corn is

$3 a bushel and the price of fertilizer is $p a pound? 200?33:33p.

(Hint,Start by writing down Skoglund’s marginal product of fertilizer as

a function of N.)

250 PROFIT MAXIMIZATION (Ch,19)

(d) When Hoglund and Skoglund are both maximizing pro ts,will

Skoglund’s output be more than twice as much,less than twice as much

or exactly twice as much as Hoglund’s? Explain,More than

twice as much,S,would produce twice as

much as H,if they used equal amounts of

fertilizer,but S,uses more fertilizer

than H,does.

(e) Explain how someone who looked at Hoglund’s and Skoglund’s corn

yields and their fertilizer inputs but couldn’t observe the quality of

their land,would get a misleading idea of the productivity of fertil-

izer,Fertilizer did not cause the entire

difference in yield,The best land got the

most fertilizer.

19.10 (0) A rm has two variable factors and a production function,

f(x

1;x

2

)=x

1=2

1

x

1=4

2

,The price of its output is 4,Factor 1 receives a

wage of w

1

and factor 2 receives a wage of w

2

.

(a) Write an equation that says that the value of the marginal product

of factor 1 is equal to the wage of factor 1 2x

1=2

1

x

1=4

2

= w

1

and

an equation that says that the value of the marginal product of factor

2 is equal to the wage of factor 2,x

1=2

1

x

3=4

2

= w

2

,Solve two

equations in the two unknowns,x

1

and x

2

,to give the amounts of factors

1 and 2 that maximize the rm’s pro ts as a function of w

1

and w

2

.This

gives x

1

= 8=(w

3

1

w

2

) and x

2

= 4=(w

2

1

w

2

2

),(Hint,You could

use the rst equation to solve for x

1

as a function of x

2

and of the factor

wages,Then substitute the answer into the second equation and solve for

x

2

as a function of the two wage rates,Finally use your solution for x

2

to nd the solution for x

1

.)

(b) If the wage of factor 1 is 2,and the wage of factor 2 is 1,how many

units of factor 1 will the rm demand? 1,How many units of

factor 2 will it demand? 1,How much output will it produce?

1,How much pro t will it make? 1.

19.11 (0) A rm has two variable factors and a production function

f(x

1;x

2

)=x

1=2

1

x

1=2

2

,The price of its output is 4,the price of factor 1 is

w

1

,and the price of factor 2 is w

2

.

NAME 251

(a) Write the two equations that say that the value of the marginal prod-

uct of each factor is equal to its wage,2x

1=2

1

x

1=2

2

= w

1

and

2x

1=2

1

x

1=2

2

= w

2

,If w

1

=2w

2

,these two equations imply that

x

1

=x

2

= 1/2.

(b) For this production function,is it possible to solve the two marginal

productivity equations uniquely for x

1

and x

2

No.

19.12 (1) A rm has two variable factors and a production function

f(x

1;x

2

)=

p

2x

1

+4x

2

,On the graph below,draw production isoquants

corresponding to an ouput of 3 and to an output of 4.

(a) If the price of the output good is 4,the price of factor 1 is 2,and

the price of factor 2 is 3,nd the pro t-maximizing amount of factor 1

0,the pro t-maximizing amount of factor 2 16/9,andthe

pro t-maximizing output 8/3.

0481216

4

8

12

Factor 1

Factor 2

16

9

_

4

252 PROFIT MAXIMIZATION (Ch,19)

Chapter 20 NAME

Cost Minimization

Introduction,In the chapter on consumer choice,you studied a con-

sumer who tries to maximize his utility subject to the constraint that he

has a xed amount of money to spend,In this chapter you study the

behavior of a rm that is trying to produce a xed amount of output

in the cheapest possible way,In both theories,you look for a point of

tangency between a curved line and a straight line,In consumer theory,

there is an \indi erence curve" and a \budget line." In producer theory,

there is a \production isoquant" and an \isocost line." As you recall,

in consumer theory,nding a tangency gives you only one of the two

equations you need to locate the consumer’s chosen point,The second

equation you used was the budget equation,In cost-minimization theory,

again the tangency condition gives you one equation,This time you don’t

know in advance how much the producer is spending; instead you are told

how much output he wants to produce and must nd the cheapest way

to produce it,So your second equation is the equation that tells you that

the desired amount is being produced.

Example,A rm has the production function f(x

1;x

2

)=(

p

x

1

+

3

p

x

2

)

2

,The price of factor 1 is w

1

= 1 and the price of factor 2

is w

2

= 1,Let us nd the cheapest way to produce 16 units of out-

put,We will be looking for a point where the technical rate of sub-

stitution equals?w

1

=w

2

,If you calculate the technical rate of sub-

stitution (or look it up from the warm up exercise in Chapter 18),

you nd TRS(x

1;x

2

)=?(1=3)(x

2

=x

1

)

1=2

,Therefore we must have

(1=3)(x

2

=x

1

)

1=2

=?w

1

=w

2

=?1,This equation can be simpli ed

to x

2

=9x

1

,So we know that the combination of inputs chosen has to

lie somewhere on the line x

2

=9x

1

,We are looking for the cheapest way

to produce 16 units of output,So the point we are looking for must sat-

isfy the equation (

p

x

1

+3

p

x

2

)

2

= 16,or equivalently

p

x

1

+3

p

x

2

=4.

Since x

2

=9x

1

,we can substitute for x

2

in the previous equation to get

p

x

1

+3

p

9x

1

= 4,This equation simpli es further to 10

p

x

1

=4,Solving

this for x

1

,wehavex

1

=16=100,Then x

2

=9x

1

= 144=100.

The amounts x

1

and x

2

that we solved for in the previous para-

graph are known as the conditional factor demands for factors 1 and 2,

conditional on the wages w

1

=1,w

2

= 1,and output y = 16,We ex-

press this by saying x

1

(1;1;16) = 16=100 and x

2

(1;1;16) = 144=100.

Since we know the amount of each factor that will be used to pro-

duce 16 units of output and since we know the price of each factor,

we can now calculate the cost of producing 16 units,This cost is

c(w

1;w

2;16) = w

1

x

1

(w

1;w

2;16)+w

2

x

2

(w

1;w

2;16),In this instance since

w

1

= w

2

=1,wehavec(1;1;16) = x

1

(1;1;16) +x

2

(1;1;16) = 160=100.

In consumer theory,you also dealt with cases where the consumer’s

indi erence \curves" were straight lines and with cases where there were

254 COST MINIMIZATION (Ch,20)

kinks in the indi erence curves,Then you found that the consumer’s

choice might occur at a boundary or at a kink,Usually a careful look

at the diagram would tell you what is going on,The story with kinks

and boundary solutions is almost exactly the same in the case of cost-

minimizing rms,You will nd some exercises that show how this works.

20.1 (0) Nadine sells user-friendly software,Her rm’s production func-

tion is f(x

1;x

2

)=x

1

+2x

2

,wherex

1

is the amount of unskilled labor

and x

2

is the amount of skilled labor that she employs.

(a) In the graph below,draw a production isoquant representing input

combinations that will produce 20 units of output,Draw another isoquant

representing input combinations that will produce 40 units of output.

010203040

10

20

30

40

x2

x1

20 units

40 units

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Constant.

(c) If Nadine uses only unskilled labor,how much unskilled labor would

she need in order to produce y units of output? y.

(d) If Nadine uses only skilled labor to produce output,how much skilled

labor would she need in order to produce y units of output?

y

2

.

(e) If Nadine faces factor prices (1;1),what is the cheapest way for her

to produce 20 units of output? x

1

= 0,x

2

= 10.

NAME 255

(f) If Nadine faces factor prices (1;3),what is the cheapest way for her

to produce 20 units of output? x

1

= 20,x

2

= 0.

(g) If Nadine faces factor prices (w

1;w

2

),what will be the minimal cost

of producing 20 units of output? c =minf20w

1;10w

2

g =

10 minf2w

1;w

2

g.

(h) If Nadine faces factor prices (w

1;w

2

),what will be the mini-

mal cost of producing y units of output? c(w

1;w

2;y)=

minfw

1;w

2

=2gy.

20.2 (0) The Ontario Brassworks produces brazen e ronteries,As you

know brass is an alloy of copper and zinc,used in xed proportions,The

production function is given by,f(x

1;x

2

)=minfx

1;2x

2

g,wherex

1

is

theamountofcopperitusesandx

2

istheamountofzincthatitusesin

production.

(a) Illustrate a typical isoquant for this production function in the graph

below.

010203040

10

20

30

40

x2

x1

x

2

=

1_

2

x

1

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Constant.

(c) If the rm wanted to produce 10 e ronteries,how much copper would

it need? 10 units,How much zinc would it need? 5 units.

256 COST MINIMIZATION (Ch,20)

(d) If the rm faces factor prices (1;1),what is the cheapest way for it

to produce 10 e ronteries? How much will this cost? It can

only produce 10 units of output by using the

bundle (10;5),so this is the cheapest way.

It will cost $15.

(e) If the rm faces factor prices (w

1;w

2

),what is the cheapest cost to

produce 10 e ronteries? c(w

1;w

2;10) = 10w

1

+5w

2

.

(f) If the rm faces factor prices (w

1;w

2

),what will be the minimal cost

of producing y e ronteries? (w

1

+w

2

=2)y.

Calculus 20.3 (0) A rm uses labor and machines to produce output according to

the production function f(L;M)=4L

1=2

M

1=2

,whereL is the number of

units of labor used and M is the number of machines,The cost of labor

is $40 per unit and the cost of using a machine is $10.

(a) On the graph below,draw an isocost line for this rm,showing com-

binations of machines and labor that cost $400 and another isocost line

showing combinations that cost $200,What is the slope of these isocost

lines? -4.

(b) Suppose that the rm wants to produce its output in the cheapest

possible way,Find the number of machines it would use per worker.

(Hint,The rm will produce at a point where the slope of the production

isoquant equals the slope of the isocost line.) 4.

(c) On the graph,sketch the production isoquant corresponding to an

output of 40,Calculate the amount of labor 5 units and the

number of machines 20 that are used to produce 40 units of output

in the cheapest possible way,given the above factor prices,Calculate the

cost of producing 40 units at these factor prices,c(40;10;40) = 400.

(d) How many units of labor y/8 and how many machines y/2

would the rm use to produce y units in the cheapest possible way? How

much would this cost? 10y,(Hint,Notice that there are constant

returns to scale.)

NAME 257

010203040

10

20

30

40

Machines

Labour

$400 isocost line

$200 isocost line

20.4 (0) Earl sells lemonade in a competitive market on a busy street

corner in Philadelphia,His production function is f(x

1;x

2

)=x

1=3

1

x

1=3

2

,

where output is measured in gallons,x

1

is the number of pounds of lemons

he uses,and x

2

is the number of labor-hours spent squeezing them.

(a) Does Earl have constant returns to scale,decreasing returns to scale,

or increasing returns to scale? Decreasing.

(b) Where w

1

is the cost of a pound of lemons and w

2

is the wage rate

for lemon-squeezers,the cheapest way for Earl to produce lemonade is to

use w

1

=w

2

hours of labor per pound of lemons,(Hint,Set the slope

of his isoquant equal to the slope of his isocost line.)

(c) If he is going to produce y units in the cheapest way possible,

then the number of pounds of lemons he will use is x

1

(w

1;w

2;y)=

w

1=2

2

y

3=2

=w

1=2

1

and the number of hours of labor that he will use

is x

2

(w

1;w

2;y)= w

1=2

1

y

3=2

=w

1=2

2

,(Hint,Use the production func-

tion and the equation you found in the last part of the answer to solve

for the input quantities.)

(d) The cost to Earl of producing y units at factor prices w

1

and w

2

is

c(w

1;w

2;y)=w

1

x

1

(w

1;w

2;y)+w

2

x

2

(w

1;w

2;y)= 2w

1=2

1

w

1=2

2

y

3=2

.

20.5 (0) The prices of inputs (x

1;x

2;x

3;x

4

)are(4;1;3;2).

258 COST MINIMIZATION (Ch,20)

(a) If the production function is given by f(x

1;x

2

)=minfx

1;x

2

g,what

is the minimum cost of producing one unit of output? $5.

(b) If the production function is given by f(x

3;x

4

)=x

3

+x

4

,what is the

minimum cost of producing one unit of output? $2.

(c) If the production function is given by f(x

1;x

2;x

3;x

4

)=minfx

1

+

x

2;x

3

+x

4

g,what is the minimum cost of producing one unit of output?

$3.

(d) If the production function is given by f(x

1;x

2

)=minfx

1;x

2

g +

minfx

3;x

4

g,what is the minimum cost of producing one unit of output?

$5.

20.6 (0) Joe Grow,an avid indoor gardener,has found that the number

of happy plants,h,depends on the amount of light,l,and water,w.In

fact,Joe noticed that plants require twice as much light as water,and any

more or less is wasted,Thus,Joe’s production function is h =minfl;2wg.

(a) Suppose Joe is using 1 unit of light,what is the least amount of

water he can use and still produce a happy plant? 1=2 unit of

water.

(b) If Suppose Joe wants to produce 4 happy plants,what are the mini-

mum amounts of light and water required? (4;2).

(c) Joe’s conditional factor demand function for light is l(w

1;w

2;h)=

h and his conditional factor demand function for water is

w(w

1;w

2;h)= h=2.

(d) If each unit of light costs w

1

and each unit of water costs w

2

,Joe’s

cost function is c(w

1;w

2;h)= w

1

h+

w

2

2

h.

20.7 (1) Joe’s sister,Flo Grow,is a university administrator,She uses

an alternative method of gardening,Flo has found that happy plants

only need fertilizer and talk,(Warning,Frivolous observations about

university administrators’ talk being a perfect substitute for fertilizer is

in extremely poor taste.) Where f is the number of bags of fertilizer used

and t is the number of hours she talks to her plants,the number of happy

plants produced is exactly h = t+2f,Suppose fertilizer costs w

f

per bag

and talk costs w

t

per hour.

NAME 259

(a) If Flo uses no fertilizer,how many hours of talk must she devote if she

wants one happy plant? 1 hour,If she doesn’t talk to her plants

at all,how many bags of fertilizer will she need for one happy plant?

1=2 bag.

(b) If w

t

<w

f

=2,would it be cheaper for Flo to use fertilizer or talk to

raise one happy plant? It would be cheaper to talk.

(c) Flo’s cost function is c(w

f;w

t;h)= minf

w

f

2;w

t

gh.

(d) Her conditional factor demand for talk is t(w

f;w

t;h)= h if

w

t

<w

f

=2and 0 if w

t

>w

f

=2.

20.8 (0) Remember T-bone Pickens,the corporate raider? Now he’s con-

cerned about his chicken farms,Pickens’s Chickens,He feeds his chickens

on a mixture of soybeans and corn,depending on the prices of each,Ac-

cording to the data submitted by his managers,when the price of soybeans

was $10 a bushel and the price of corn was $10 a bushel,they used 50

bushels of corn and 150 bushels of soybeans for each coop of chickens.

When the price of soybeans was $20 a bushel and the price of corn was

$10 a bushel,they used 300 bushels of corn and no soybeans per coop

of chickens,When the price of corn was $20 a bushel and the price of

soybeans was $10 a bushel,they used 250 bushels of soybeans and no corn

for each coop of chickens.

(a) Graph these three input combinations and isocost lines in the following

diagram.

0 100 200 300 400

100

200

300

400

Corn

Soybeans

125

260 COST MINIMIZATION (Ch,20)

(b) How much money did Pickens’ managers spend per coop of chickens

when the prices were (10;10)? $2,000,When the prices were

(10;20)? $2,500,When the prices were (20;10)? $3,000.

(c) Is there any evidence that Pickens’s managers were not minimizing

costs? Why or why not?

There is no such evidence,since the data

satisfy WACM.

(d) Pickens wonders whether there are any prices of corn and soybeans at

which his managers will use 150 bushels of corn and 50 bushels of soybeans

to produce a coop of chickens,How much would this production method

cost per coop of chickens if the prices were p

s

=10andp

c

= 10?

$2,000,if the prices were p

s

= 10,p

c

= 20? $3,500,if the

prices were p

s

= 20,p

c

= 10? $2,500.

(e) If Pickens’s managers were always minimizing costs,can it be pos-

sible to produce a coop of chickens using 150 bushels and 50 bushels of

soybeans? No,At prices (20;10),this bundle

costs less than the bundle actually used

at prices (20;10),If it produced as much

as that bundle,the chosen bundle wouldn’t

have been chosen.

20.9 (0) A genealogical rm called Roots produces its output using only

one input,Its production function is f(x)=

p

x.

(a) Does the rm have increasing,constant,or decreasing returns to scale?

Decreasing.

(b) How many units of input does it take to produce 10 units of output?

100 units,If the input costs w per unit,what does it cost to

produce 10 units of output? 100w.

NAME 261

(c) How many units of input does it take to produce y units of output?

y

2

,If the input costs w per unit,what does it cost to produce y units

of output? y

2

w.

(d) If the input costs w per unit,what is the average cost of producing y

units? AC(w;y)= yw.

20.10 (0) A university cafeteria produces square meals,using only one

input and a rather remarkable production process,We are not allowed to

say what that ingredient is,but an authoritative kitchen source says that

\fungus is involved." The cafeteria’s production function is f(x)=x

2

,

where x is the amount of input and f(x) is the number of square meals

produced.

(a) Does the cafeteria have increasing,constant,or decreasing returns to

scale? Increasing.

(b) How many units of input does it take to produce 144 square meals?

12,If the input costs w per unit,what does it cost to produce 144

square meals? 12w.

(c) How many units of input does it take to produce y square meals?

p

y,If the input costs w per unit,what does it cost to produce y

square meals? w

p

y.

(d) If the input costs w per unit,what is the average cost of producing y

square meals? AC(w;y)= w=

p

y.

20.11 (0) Irma’s Handicrafts produces plastic deer for lawn ornaments.

\It’s hard work," says Irma,\but anything to make a buck." Her produc-

tion function is given by f(x

1;x

2

)=(minfx

1;2x

2

g)

1=2

,wherex

1

is the

amount of plastic used,x

2

is the amount of labor used,and f(x

1;x

2

)is

the number of deer produced.

(a) In the graph below,draw a production isoquant representing input

combinations that will produce 4 deer,Draw another production isoquant

representing input combinations that will produce 5 deer.

262 COST MINIMIZATION (Ch,20)

010203040

10

20

30

40

x2

x1

x

2

=

1_

2

x

1

Output

of 5

deer

Output

of 4

deer

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Decreasing returns to scale.

(c) If Irma faces factor prices (1;1),what is the cheapest way for her to

produce 4 deer? Use (16,8),How much does this cost? $24.

(d) At the factor prices (1;1),what is the cheapest way to produce 5 deer?

Use (25,12.5),How much does this cost? $37.50.

(e) At the factor prices (1;1),the cost of producing y deer with this

technology is c(1;1;y)= 3y

2

=2.

(f) At the factor prices (w

1;w

2

),the cost of producing y deer with this

technology is c(w

1;w

2;y)= (w

1

+w

2

=2)y

2

.

20.12 (0) Al Deardwarf also makes plastic deer for lawn ornaments.

Al has found a way to automate the production process completely,He

doesn’t use any labor{only wood and plastic,Al says he likes the business

\because I need the doe." Al’s production function is given by f(x

1;x

2

)=

(2x

1

+ x

2

)

1=2

,wherex

1

is the amount of plastic used,x

2

is the amount

of wood used,and f(x

1;x

2

) is the number of deer produced.

NAME 263

(a) In the graph below,draw a production isoquant representing input

combinations that will produce 4 deer,Draw another production isoquant

representing input combinations that will produce 6 deer.

010203040

10

20

30

40

x2

x1

Output

of 4

deer

Output

of 6

deer

36

16

8 18

(b) Does this production function exhibit increasing,decreasing,or con-

stant returns to scale? Decreasing returns to scale.

(c) If Al faces factor prices (1;1),what is the cheapest way for him to

produce 4 deer? (8;0),How much does this cost? $8.

(d) At the factor prices (1;1),what is the cheapest way to produce 6

deer? (18;0),How much does this cost? $18.

(e) At the factor prices (1;1),the cost of producing y deer with this

technology is c(1;1;y)= y

2

=2.

(f) At the factor prices (3;1),the cost of producing y deer with this

technology is c(3;1;y)= y

2

.

20.13 (0) Suppose that Al Deardwarf from the last problem cannot vary

the amount of wood that he uses in the short run and is stuck with using

20 units of wood,Suppose that he can change the amount of plastic that

he uses,even in the short run.

(a) How much plastic would Al need in order to make 100 deer? 4,990

units.

264 COST MINIMIZATION (Ch,20)

(b) If the cost of plastic is $1 per unit and the cost of wood is $1 per unit,

how much would it cost Al to make 100 deer? $5,010.

(c) Write down Al’s short-run cost function at these factor prices.

c(1;1;y)=20+(y

2

20)=2.

Chapter 21 NAME

Cost Curves

Introduction,Here you continue to work on cost functions,Total cost

can be divided into xed cost,the part that doesn’t change as output

changes,and variable cost,To get the average (total) cost,average xed

cost,and average variable cost,just divide the appropriate cost function

by y,the level of output,The marginal cost function is the derivative of

the total cost function with respect to output|or the rate of increase in

cost as output increases,if you don’t know calculus.

Remember that the marginal cost curve intersects both the average

cost curve and the average variable cost curve at their minimum points.

So to nd the minimum point on the average cost curve,you simply set

marginal cost equal to average cost and similarly for the minimum of

average variable cost.

Example,A rm has the total cost function C(y) = 100 + 10y.Letus

nd the equations for its various cost curves,Total xed costs are 100,so

the equation of the average xed cost curve is 100=y,Total variable costs

are 10y,so average variable costs are 10y=y = 10 for all y,Marginal cost

is 10 for all y,Average total costs are (100 + 10y)=y =10+10=y.Notice

that for this rm,average total cost decreases as y increases,Notice also

that marginal cost is less than average total cost for all y.

21.1 (0) Mr,Otto Carr,owner of Otto’s Autos,sells cars,Otto buys

autos for $c each and has no other costs.

(a) What is his total cost if he sells 10 cars? 10c,What if he sells 20

cars? 20c,Write down the equation for Otto’s total costs assuming

he sells y cars,TC(y)= cy.

(b) What is Otto’s average cost function? AC(y)= c,For every

additional auto Otto sells,by how much do his costs increase? c.

Write down Otto’s marginal cost function,MC(y)= c.

(c) In the graph below draw Otto’s average and marginal cost curves if

c = 20.

266 COST CURVES (Ch,21)

010203040

10

20

30

40

AC,MC

Red line

AC=MC=20

Output

(d) Suppose Otto has to pay $b a year to produce obnoxious television

commercials,Otto’s total cost curve is now TC(y)= cy + b,his

average cost curve is now AC(y)= c + b=y,and his marginal cost

curve is MC(y)= c.

(e) If b = $100,use red ink to draw Otto’s average cost curve on the

graph above.

21.2 (0) Otto’s brother,Dent Carr,is in the auto repair business,Dent

recently had little else to do and decided to calculate his cost conditions.

He found that the total cost of repairing s cars is TC(s)=2s

2

+ 10,But

Dent’s attention was diverted to other things,:,and that’s where you

come in,Please complete the following:

Dent’s Total Variable Costs,2s

2

.

Total Fixed Costs,10.

Average Variable Costs,2s.

Average Fixed Costs,10=s.

Average Total Costs,2s+10=s.

Marginal Costs,4s.

NAME 267

21.3 (0) A third brother,Rex Carr,owns a junk yard,Rex can use one

of two methods to destroy cars,The rst involves purchasing a hydraulic

car smasher that costs $200 a year to own and then spending $1 for every

car smashed into oblivion; the second method involves purchasing a shovel

that will last one year and costs $10 and paying the last Carr brother,

Scoop,to bury the cars at a cost of $5 each.

(a) Write down the total cost functions for the two methods,where y is

output per year,TC

1

(y)= y + 200,TC

2

(y)= 5y +10.

(b) The rst method has an average cost function 1 + 200=y and a

marginal cost function 1,For the second method these costs are

5+10=y and 5.

(c) If Rex wrecks 40 cars per year,which method should he use?

Method 2,If Rex wrecks 50 cars per year,which method should

he use? Method 1,What is the smallest number of cars per year

for which it would pay him to buy the hydraulic smasher? 48 cars

per year.

21.4 (0) Mary Magnolia wants to open a flower shop,the Petal Pusher,

in a new mall,She has her choice of three di erent floor sizes,200 square

feet,500 square feet,or 1,000 square feet,The monthly rent will be $1 a

square foot,Mary estimates that if she has F square feet of floor space

and sells y bouquets a month,her variable costs will be c

v

(y)=y

2

=F per

month.

(a) If she has 200 square feet of floor space,write down her marginal cost

function,MC =

y

100

and her average cost function,AC =

200

y

+

y

200

,At what amount of output is average cost minimized?

200,At this level of output,how much is average cost? $2.

(b) If she has 500 square feet,write down her marginal cost function:

MC = y=250 and her average cost function,AC =

(500=y)+y=500,At what amount of output is average cost min-

268 COST CURVES (Ch,21)

imized? 500,At this level of output,how much is average cost?

$2.

(c) If she has 1,000 square feet of floor space,write down her marginal

cost function,MC = y=500 and her average cost function:

AC =(1;000=y)+y=1;000,At what amount of output is

average cost minimized? 1,000,At this level of output,how much

is average cost? $2.

(d) Use red ink to show Mary’s average cost curve and her marginal cost

curves if she has 200 square feet,Use blue ink to show her average cost

curve and her marginal cost curve if she has 500 square feet,Use black

ink to show her average cost curve and her marginal cost curve if she has

1,000 square feet,Label the average cost curves AC and the marginal

cost curves MC.

0 400 600 800 1000

Bouquents

1

2

3

4

Dollars

200 1200

mc

ac

mc

ac

mc

ac

Red

lines

Blue

lines

Black

lines

LRMC=LRAC (yellow line)

(e) Use yellow marker to show Mary’s long-run average cost curve and

her long-run marginal cost curve in your graph,Label them LRAC and

LRMC.

21.5 (0) Touchie MacFeelie publishes comic books,The only inputs he

needs are old jokes and cartoonists,His production function is

Q =,1J

1

2

L

3=4;

NAME 269

whereJ is the number of old jokes used,L the number of hours of cartoon-

ists’ labor used as inputs,and Q is the number of comic books produced.

(a) Does this production process exhibit increasing,decreasing,or con-

stant returns to scale? Explain your answer,It exhibits

increasing returns to scale since f(tJ;tL)=

t

5=4

f(J;L) >tf(J;L).

(b) If the number of old jokes used is 100,write an expression for the

marginal product of cartoonists’ labor as a function of L,MP =

3

4L

1=4

Is the marginal product of labor decreasing or increasing as the

amount of labor increases? Decreasing.

21.6 (0) Touchie MacFeelie’s irascible business manager,Gander Mac-

Grope,announces that old jokes can be purchased for $1 each and that

the wage rate of cartoonists’ labor is $2.

(a) Suppose that in the short run,Touchie is stuck with exactly 100 old

jokes (for which he paid $1 each) but is able to hire as much labor as he

wishes,How much labor would he have to hire in order produce Q comic

books? Q

4=3

.

(b) Write down Touchie’s short-run total cost as a function of his output

2Q

4=3

+ 100.

(c) His short-run marginal cost function is 8Q

1=3

=3.

(d) His short-run average cost function is 2Q

1=3

+ 100=Q.

Calculus 21.7 (1) Touchie asks his brother,Sir Francis MacFeelie,to study the

long-run picture,Sir Francis,who has carefully studied the appendix to

Chapter 19 in your text,prepared the following report.

(a) If all inputs are variable,and if old jokes cost $1 each and car-

toonist labor costs $2 per hour,the cheapest way to produce exactly

one comic book is to use 10

4=5

(4=3)

3=5

7:4 jokes and

10

4=5

(3=4)

2=5

5:6 hours of labor,(Fractional jokes are cer-

tainly allowable.)

270 COST CURVES (Ch,21)

(b) This would cost 18.7 dollars.

(c) Given our production function,the cheapest proportions in which to

use jokes and labor are the same no matter how many comic books we

print,But when we double the amount of both inputs,the number of

comic books produced is multiplied by 2

5=4

.

21.8 (0) Consider the cost function c(y)=4y

2

+ 16.

(a) The average cost function is AC =4y +

16

y

.

(b) The marginal cost function is MC =8y.

(c) The level of output that yields the minimum average cost of production

is y =2.

(d) The average variable cost function is AVC =4y.

(e) At what level of output does average variable cost equal marginal

cost? At y =0.

21.9 (0) A competitive rm has a production function of the form

Y =2L +5K.Ifw =$2andr = $3,what will be the minimum cost of

producing 10 units of output? $6.

Chapter 22 NAME

Firm Supply

Introduction,The short-run supply curve of a competitive rm is the

portion of its short-run marginal cost curve that is upward sloping and

lies above its average variable cost curve,The long-run supply curve of a

competitive rm is the portion of its short-run marginal cost curve that

is upward-sloping and lies above its long-run average cost curve.

Example,A rm has the long-run cost function c(y)=2y

2

+ 200 for

y>0andc(0) = 0,Let us nd its long-run supply curve,The rm’s

marginal cost when its output is y is MC(y)=4y,If we graph output on

the horizontal axis and dollars on the vertical axis,then we nd that the

long-run marginal cost curve is an upward-sloping straight line through

the origin with slope 4,The long-run supply curve is the portion of this

curve that lies above the long-run average cost curve,When output is y,

long-run average costs of this rm are AC(y)=2y + 200=y.ThisisaU-

shaped curve,As y gets close to zero,AC(y) becomes very large because

200=y becomes very large,When y is very large,AC(y) becomes very

large because 2y is very large,When is it true that AC(y) <MC(y)?

This happens when 2y+ 200=y < 4y,Simplify this inequality to nd that

AC(y) <MC(y)wheny>10,Therefore the long-run supply curve is

the piece of the long-run marginal cost curve for which y>10,So the

long-run supply curve has the equation p =4y for y>10,If we want to

nd quantity supplied as a function of price,we just solve this expression

for y as a function of p.Thenwehavey = p=4 whenever p>40.

Suppose that p<40,For example,what if p = 20,how much will

the rm supply? At a price of 20,if the rm produces where price equals

long-run marginal cost,it will produce 5 = 20=4 units of output,When

the rm produces only 5 units,its average costs are 2 5 + 200=5 = 50.

Therefore when the price is 20,the best the rm can do if it produces a

positive amount is to produce 5 units,But then it will have total costs of

5 50 = 250 and total revenue of 5 20 = 100,It will be losing money,It

would be better o producing nothing at all,In fact,for any price p<40,

the rm will choose to produce zero output.

22.1 (0) Remember Otto’s brother Dent Carr,who is in the auto repair

business? Dent found that the total cost of repairing s cars is c(s)=

2s

2

+ 100.

(a) This implies that Dent’s average cost is equal to 2s + 100=s,

his average variable cost is equal to 2s,and his marginal cost is

equal to 4s,On the graph below,plot the above curves,and also plot

Dent’s supply curve.

272 FIRM SUPPLY (Ch,22)

0 5 10 15 20

20

40

60

Output

Dollars

80

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Supply

mc

ac

avc

Revenue

Costs

Profit

(b) If the market price is $20,how many cars will Dent be willing to

repair? 5,If the market price is $40,how many cars will Dent

repair? 10.

(c) Suppose the market price is $40 and Dent maximizes his pro ts,On

the above graph,shade in and label the following areas,total costs,total

revenue,and total pro ts.

Calculus 22.2 (0) A competitive rm has the following short-run cost function:

c(y)=y

3

8y

2

+30y +5.

(a) The rm’s marginal cost function is MC(y)= 3y

2

16y+30.

(b) The rm’s average variable cost function is AVC(y)= y

2

8y+

30,(Hint,Notice that total variable costs equal c(y)?c(0).)

(c) On the axes below,sketch and label a graph of the marginal cost

function and of the average variable cost function.

(d) Average variable cost is falling as output rises if output is less than

4 and rising as output rises if output is greater than 4.

(e) Marginal cost equals average variable cost when output is 4.

NAME 273

(f) The rm will supply zero output if the price is less than 14.

(g) The smallest positive amount that the rm will ever supply at any

price is 4,At what price would the rm supply exactly 6 units

of output? 42.

02468

10

20

30

y

Costs

40

mc

avc

Calculus 22.3 (0) Mr,McGregor owns a 5-acre cabbage patch,He forces his

wife,Flopsy,and his son,Peter,to work in the cabbage patch without

wages,Assume for the time being that the land can be used for nothing

other than cabbages and that Flopsy and Peter can nd no alternative

employment,The only input that Mr,McGregor pays for is fertilizer,If

he uses x sacks of fertilizer,the amount of cabbages that he gets is 10

p

x.

Fertilizer costs $1 per sack.

(a) What is the total cost of the fertilizer needed to produce 100 cabbages?

$100,What is the total cost of the amount of fertilizer needed to

produce y cabbages? y

2

=100.

(b) If the only way that Mr,McGregor can vary his output is by varying

the amount of fertilizer applied to his cabbage patch,write an expression

for his marginal cost,as a function of y,MC(y)= y=50.

(c) If the price of cabbages is $2 each,how many cabbages will Mr,Mc-

Gregor produce? 100,How many sacks of fertilizer will he buy?

100,How much pro t will he make? $100.

274 FIRM SUPPLY (Ch,22)

(d) The price of fertilizer and of cabbages remain as before,but Mr,Mc-

Gregor learns that he could nd summer jobs for Flopsy and Peter in

a local sweatshop,Flopsy and Peter would together earn $300 for the

summer,which Mr,McGregor could pocket,but they would have no time

to work in the cabbage patch,Without their labor,he would get no cab-

bages,Now what is Mr,McGregor’s total cost of producing y cabbages?

c(y) = 300 + (y=10)

2

.

(e) Should he continue to grow cabbages or should he put Flopsy and

Peter to work in the sweatshop? Sweatshop.

22.4 (0) Severin,the herbalist,is famous for his hepatica,His total cost

function is c(y)=y

2

+10 fory>0andc(0) = 0,(That is,his cost of

producing zero units of output is zero.)

(a) What is his marginal cost function? 2y,What is his average cost

function? y +10=y.

(b) At what quantity is his marginal cost equal to his average cost?

p

10,At what quantity is his average cost minimized?

p

10.

(c) In a competitive market,what is the lowest price at which he will

supply a positive quantity in long-run equilibrium? 2

p

10,How

much would he supply at that price?

p

10.

22.5 (1) Stanley Ford makes mountains out of molehills,He can do this

with almost no e ort,so for the purposes of this problem,let us assume

that molehills are the only input used in the production of mountains.

Suppose mountains are produced at constant returns to scale and that

it takes 100 molehills to make 1 mountain,The current market price of

molehills is $20 each,A few years ago,Stan bought an \option" that

permits him to buy up to 2,000 molehills at $10 each,His option contract

explicitly says that he can buy fewer than 2,000 molehills if he wishes,but

he can not resell the molehills that he buys under this contract,In or-

der to get governmental permission to produce mountains from molehills,

Stanley would have to pay $10,000 for a molehill-masher’s license.

(a) The marginal cost of producing a mountain for Stanley is $1,000

if he produces fewer than 20 mountains,The marginal cost of producing

a mountain is $2,000 if he produces more than 20 mountains.

NAME 275

(b) On the graph below,show Stanley Ford’s marginal cost curve (in blue

ink) and his average cost curve (in red ink).

010203040

1000

2000

3000

Output

Dollars

4000

Blue mc curve

Red ac

curve

Pencil

mc

curve

(c) If the price of mountains is $1,600,how many mountains will Stanley

produce? 20 mountains.

(d) The government is considering raising the price of a molehill-masher’s

license to $11,000,Stanley claims that if it does so he will have to go out

of business,Is Stanley telling the truth? No,What is the highest

fee for a license that the government could charge without driving him

out of business? The maximum they could charge

is the amount of his profits excluding the

license fee,$12,000.

(e) Stanley’s lawyer,Eliot Sleaze,has discovered a clause in Stanley’s

option contract that allows him to resell the molehills that he purchased

under the option contract at the market price,On the graph above,

use a pencil to draw Stanley’s new marginal cost curve,If the price of

mountains remains $1,600,how many mountains will Stanley produce

now? He will sell all of his molehills and

produce zero mountains.

22.6 (1) Lady Wellesleigh makes silk purses out of sows’ ears,She is

the only person in the world who knows how to do so,It takes one sow’s

ear and 1 hour of her labor to make a silk purse,She can buy as many

276 FIRM SUPPLY (Ch,22)

sows’ ears as she likes for $1 each,Lady Wellesleigh has no other source

of income than her labor,Her utility function is a Cobb-Douglas function

U(c;r)=c

1=3

r

2=3

,wherec is the amount of money per day that she has

to spend on consumption goods and r is the amount of leisure that she

has,Lady Wellesleigh has 24 hours a day that she can devote either to

leisure or to working.

(a) Lady Wellesleigh can either make silk purses or she can earn $5 an

hour as a seamstress in a sweatshop,If she worked in the sweat shop,how

many hours would she work? 8,(Hint,To solve for this amount,

write down Lady Wellesleigh’s budget constraint and recall how to nd

the demand function for someone with a Cobb-Douglas utility function.)

(b) If she could earn a wage of $w an hour as a seamstress,how much

would she work? 8 hours.

(c) If the price of silk purses is $p,how much money will Lady Wellesleigh

earn per purse after she pays for the sows’ ears that she uses? p?1.

(d) If she can earn $5 an hour as a seamstress,what is the lowest price

at which she will make any silk purses? $6.

(e) What is the supply function for silk purses? (Hint,The price of silk

purses determines the \wage rate" that Lady W,can earn by making silk

purses,This determines the number of hours she will choose to work and

hence the supply of silk purses.) S(p)=8for p>6,0

otherwise.

Calculus 22.7 (0) Remember Earl,who sells lemonade in Philadelphia? You

met him in the chapter on cost functions,Earl’s production function is

f(x

1;x

2

)=x

1=3

1

x

1=3

2

,wherex

1

is the number of pounds of lemons he

uses and x

2

is the number of hours he spends squeezing them,As you

found out,his cost function is c(w

1;w

2;y)=2w

1=2

1

w

1=2

2

y

3=2

,wherey is

the number of units of lemonade produced.

(a) If lemons cost $1 per pound,the wage rate is $1 per hour,and the

price of lemonade is p,Earl’s marginal cost function is MC(y)= 3y

1=2

and his supply function is S(p)= p

2

=9,If lemons cost $4 per pound

and the wage rate is $9 per hour,his supply function will be S(p)=

p

2

=324.

NAME 277

(b) In general,Earl’s marginal cost depends on the price of lemons and

the wage rate,At prices w

1

for lemons and w

2

for labor,his mar-

ginal cost when he is producing y units of lemonade is MC(w

1;w

2;y)=

3w

1=2

1

w

1=2

2

y

1=2

,The amount that Earl will supply depends on the

three variables,p,w

1

,w

2

,As a function of these three variables,Earl’s

supply is S(p;w

1;w

2

)= p

2

=9w

1

w

2

.

Calculus 22.8 (0) As you may recall from the chapter on cost functions,Irma’s

handicrafts has the production function f(x

1;x

2

)=(minfx

1;2x

2

g)

1=2

,

where x

1

is the amount of plastic used,x

2

is the amount of labor used,

and f(x

1;x

2

) is the number of lawn ornaments produced,Let w

1

be the

price per unit of plastic and w

2

be the wage per unit of labor.

(a) Irma’s cost function is c(w

1;w

2;y)= (w

1

+w

2

=2)y

2

.

(b) If w

1

= w

2

= 1,then Irma’s marginal cost of producing y units of

output is MC(y)= 3y,The number of units of output that she would

supply at price p is S(p)= p=3,At these factor prices,her average

cost per unit of output would be AC(y)= 3y=2.

(c) If the competitive price of the lawn ornaments she sells is p = 48,and

w

1

= w

2

= 1,how many will she produce? 16,How much pro t will

she make? 384.

(d) More generally,at factor prices w

1

and w

2

,her marginal cost is a

function MC(w

1;w

2;y)= (2w

1

+w

2

)y,At these factor prices and

an output price of p,the number of units she will choose to supply is

S(p;w

1;w

2

)= p=(2w

1

+w

2

).

22.9 (0) Jack Benny can get blood from a stone,If he has x stones,the

number of pints of blood he can extract from them is f(x)=2x

1

3

.Stones

cost Jack $w each,Jack can sell each pint of blood for $p.

(a) How many stones does Jack need to extract y pintsofblood?

y

3

=8.

(b) What is the cost of extracting y pints of blood? wy

3

=8.

278 FIRM SUPPLY (Ch,22)

(c) What is Jack’s supply function when stones cost $8 each? y =

(p=3)

1=2

,When stones cost $w each? y =(8p=3w)

1=2

.

(d) If Jack has 19 relatives who can also get blood from a stone in the

same way,what is the aggregate supply function for blood when stones

cost $w each? Y = 20(8p=3w)

1=2

.

22.10 (1) The Miss Manners Re nery in Dry Rock,Oklahoma,converts

crude oil into gasoline,It takes 1 barrel of crude oil to produce 1 barrel of

gasoline,In addition to the cost of oil there are some other costs involved

in re ning gasoline,Total costs of producing y barrels of gasoline are

described by the cost function c(y)=y

2

=2+p

o

y,wherep

o

is the price of

a barrel of crude oil.

(a) Express the marginal cost of producing gasoline as a function of p

o

and y,y +p

o

.

(b) Suppose that the re nery can buy 50 barrels of crude oil for $5 a

barrel but must pay $15 a barrel for any more that it buys beyond 50

barrels,The marginal cost curve for gasoline will be y +5 up to 50

barrels of gasoline and y +15 thereafter.

(c) Plot Miss Manners’ supply curve in the diagram below using blue ink.

0255075100

20

40

60

Barrels of gasoline

Price of gasoline

80

30

Red line

Black line Blue lines

NAME 279

(d) Suppose that Miss Manners faces a horizontal demand curve for gaso-

line at a price of $30 per barrel,Plot this demand curve on the graph

above using red ink,How much gasoline will she supply? 25

barrels.

(e) If Miss Manners could no longer get the rst 50 barrels of crude for

$5,but had to pay $15 a barrel for all crude oil,how would her output

change? It would decrease to 15 barrels.

(f) Now suppose that an entitlement program is introduced that permits

re neries to buy one barrel of oil at $5 for each barrel of oil that they

buy for $15,What will Miss Manners’ supply curve be now? S(p)=

p?10,Assume that it can buy fractions of a barrel in the same

manner,Plot this supply curve on the graph above using black ink,If

the demand curve is horizontal at $30 a barrel,how much gasoline will

Miss Manners supply now? 20 barrels.

22.11 (2) Suppose that a farmer’s cost of growing y bushels of corn is

given by the cost function c(y)=(y

2

=20) +y.

(a) If the price of corn is $5 a bushel,how much corn will this farmer

grow? 40 bushels.

(b) What is the farmer’s supply curve of corn as a function of the price

of corn? S(p)= 10p?10.

(c) The government now introduces a Payment in Kind (PIK) program,If

the farmer decides to grow y bushels of corn,he will get (40?y)=2 bushels

from the government stockpiles,Write an expression for the farmer’s

pro ts as a function of his output and the market price of corn,taking

into account the value of payments in kind received,py?c(y)+

p(40?y)=2=py?y

2

=20?y +p(40?y)=2.

(d) At the market price p,what will be the farmer’s pro t-maximizing

output of corn? S(p)=5p?10,Plot a supply curve for corn in

the graph below.

280 FIRM SUPPLY (Ch,22)

030450

1

2

3

4

5

6

10

Price

20

Bushels of corn

60

Red line

(e) If p = $2,how many bushels of corn will he produce? 0,How

many bushels will he get from the government stockpiles? 20.

(f) If p = $5,how much corn will he supply? 15 bushels,How

many bushels of corn will he get from the government stockpiles,assuming

he chooses to be in the PIK program? $12.50.

(g) At any price between p =$2andp = $5,write a formula for the

size of the PIK payment,His supply curve is S(p)=

5p?10,and his payment is (40?y)=2.So

he gets 25?2:5p.

(h) How much corn will he supply to the market,counting both pro-

duction and PIK payment,as a function of the market price p?

Sum supply curve and PIK payment to get

TS(p)=2:5p+15.

(i) Use red ink to illustrate the total supply curve of corn (including the

corn from the PIK payment) in your graph above.

Chapter 23 NAME

Industry Supply

Introduction,To nd the industry supply of output,just add up the

supply of output coming from each individual rm,Remember to add

quantities,not prices,The industry supply curve will have a kink in it

where the market price becomes low enough that some rm reduces its

quantity supplied to zero.

The last three questions of this chapter apply supply and demand

analysis to some problems in the economics of illegal activities,In these

examples,you will make use of your knowledge of where supply functions

come from.

23.0 Warm Up Exercise,Here are some drills for you on nding

market supply functions from linear rm supply functions,The trick here

is to remember that the market supply function may have kinks in it,For

example,if the rm supply functions are s

1

(p)=p and s

2

(p)=p?2,

then the market supply function is S(p)=p for p 2andS(p)=2p?2

for p>2; that is,only the rst rm supplies a positive output at prices

below $2,and both rms supply output at prices above $2,Now try to

construct the market supply function in each of the following cases.

(a) s

1

(p)=p;s

2

(p)=2p;s

3

(p)=3p,S(p)=6p.

(b) s

1

(p)=2p;s

2

(p)=p?1,S(p)=2p for p 1;S(p)=

3p?1 for p>1.

(c) 200 rms each have a supply function s

1

(p)=2p?8 and 100 rms

each have a supply function s

2

(p)=p?3,S(p)=0for

p<3,S(p) = 100p?300 for 3 p 4,S(p)=

500p?1;900 for p>4.

(d) s

1

(p)=3p?12;s

2

(p)=2p?8;s

3

(p)=p?4,S(p)=6p?24

for p>4.

23.1 (1) Al Deardwarf’s cousin,Zwerg,makes plaster garden gnomes.

The technology in the garden gnome business is as follows,You need a

gnome mold,plaster,and labor,A gnome mold is a piece of equipment

that costs $1,000 and will last exactly one year,After a year,a gnome

282 INDUSTRY SUPPLY (Ch,23)

mold is completely worn out and has no scrap value,With a gnome

mold,you can make 500 gnomes per year,For every gnome that you

make,you also have to use a total of $7 worth of plaster and labor,The

total amounts of plaster and labor used are variable in the short run,If

you want to produce only 100 gnomes a year with a gnome mold,you

spend only $700 a year on plaster and labor,and so on,The number

of gnome molds in the industry cannot be changed in the short run,To

get a newly built one,you have to special-order it from the gnome-mold

factory,The gnome-mold factory only takes orders on January 1 of any

given year,and it takes one whole year from the time a gnome mold is

ordered until it is delivered on the next January 1,When a gnome mold

is installed in your plant,it is stuck there,To move it would destroy it.

Gnome molds are useless for anything other than making garden gnomes.

For many years,the demand function facing the garden-gnome in-

dustry has been D(p)=60;000?5;000p,whereD(p) is the total number

of garden gnomes sold per year and p is the price,Prices of inputs have

been constant for many years and the technology has not changed,No-

body expects any changes in the future,and the industry is in long-run

equilibrium,The interest rate is 10%,When you buy a new gnome mold,

you have to pay for it when it is delivered,For simplicity of calculations,

we will assume that all of the gnomes that you build during the one-year

life of the gnome mold are sold at Christmas and that the employees and

plaster suppliers are paid only at Christmas for the work they have done

during the past year,Also for simplicity of calculations,let us approxi-

mate the date of Christmas by December 31.

(a) If you invested $1,000 in the bank on January 1,how much money

could you expect to get out of the bank one year later? $1,100,If

you received delivery of a gnome mold on January 1 and paid for it at that

time,by how much would your revenue have to exceed the costs of plaster

and labor if it is to be worthwhile to buy the machine? (Remember that

the machine will be worn out and worthless at the end of the year.)

$1,100.

(b) Suppose that you have exactly one newly installed gnome mold in

your plant; what is your short-run marginal cost of production if you

produce up to 500 gnomes? $7,What is your average variable cost

for producing up to 500 gnomes? $7,With this equipment,is it

possible in the short run to produce more than 500 gnomes? No.

(c) If you have exactly one newly installed gnome mold,you would pro-

duce 500 gnomes if the price of gnomes is above 7 dollars,You

would produce no gnomes if the price of gnomes is below 7 dol-

NAME 283

lars,You would be indi erent between producing any number of gnomes

between 0 and 500 if the price of gnomes is 7 dollars.

(d) If you could sell as many gnomes as you liked for $10 each and none

at a higher price,what rate of return would you make on your $1,000 by

investing in a gnome mold? 50%,Is this higher than the return from

putting your money in the bank? Yes,What is the lowest price for

gnomes at which investing in a gnome mold gives the same rate of return

as you get from the bank? $9.20,Could the long-run equilibrium

price be lower than this? No.

(e) At the price you found in the last section,how many gnomes would

be demanded each year? 14,000,How many molds would be

purchased each year? 28,Is this a long-run equilibrium price?

Yes.

23.2 (1) We continue our study of the garden-gnome industry,Suppose

that initially everything was as described in the previous problem,To

the complete surprise of everyone in the industry,on January 1,1993,the

invention of a new kind of plaster was announced,This new plaster made

it possible to produce garden gnomes using the same molds,but it reduced

the cost of the plaster and labor needed to produce a gnome from $7 to $5

per gnome,Assume that consumers’ demand function for gnomes in 1993

was not changed by this news,The announcement came early enough in

the day for everybody to change his order for gnome molds to be delivered

on January 1,1994,but of course,the number of molds available to be

used in 1993 is already determined from orders made one year ago,The

manufacturer of garden gnome molds contracted to sell them for $1,000

a year ago,so it can’t change the price it charges on delivery.

(a) In 1993,what will be the equilibrium total output of garden gnomes?

14,000,What will be the equilibrium price of garden gnomes?

$9.20,Cousin Zwerg bought a gnome mold that was delivered on

January 1,1993,and,as had been agreed,he paid $1,000 for it on that

day,On January 1,1994,when he sold the gnomes he had made during

the year and when he paid the workers and the suppliers of plaster,he

received a net cash flow of $ 2,100,Did he make more than a 10%

rate of return on his investment in the gnome mold? Yes,What rate

of return did he make? 110%.

284 INDUSTRY SUPPLY (Ch,23)

(b) Zwerg’s neighbor,Munchkin,also makes garden gnomes,and he has

a gnome mold that is to be delivered on January 1,1993,On this day,

Zwerg,who is looking for a way to invest some more money,is considering

buying Munchkin’s new mold from Munchkin and installing it in his own

plant,If Munchkin keeps his mold,he will get a net cash flow of $

2,100 in one year,If the interest rate that Munchkin faces,both

for borrowing and lending is 10%,then should he be willing to sell his

mold for $1,000? No,What is the lowest price that he would be

willing to sell it for? $1,909,If the best rate of return that Zwerg

can make on alternative investments of additional funds is 10%,what is

the most that Zwerg would be willing to pay for Munchkin’s new mold?

$1,909.

(c) What do you think will happen to the number of garden gnomes or-

dered for delivery on January 1,1994? Will it be larger,smaller,or the

same as the number ordered the previous year? Larger,After the

passage of su cient time,the industry will reach a new long-run equilib-

rium,What will be the new equilibrium price of gnomes? $7.20.

23.3 (1) On January 1,1993,there were no changes in technology or

demand functions from that in our original description of the industry,

but the government astonished the garden gnome industry by introducing

a tax on the production of garden gnomes,For every garden gnome

produced,the manufacturer must pay a $1 tax,The announcement came

early enough in the day so that there was time for gnome producers to

change their orders of gnome molds for 1994,Of course the gnome molds

to be used in 1993 had been already ordered a year ago,Gnome makers

had signed contracts promising to pay $1,000 for each gnome mold that

they ordered,and they couldn’t back out of these promises.

(a) Recalling from previous problems the number of gnome molds ordered

for delivery on January 1,1993,we see that if gnome makers produce up

to capacity in 1993,they will produce 14,000 gnomes,Given the

demand function,we see that the market price would then have to be

$9.20.

(b) If you have a garden gnome mold,the marginal cost of producing a

garden gnome,including the tax,is $8,Therefore all gnome molds

(will,will not) will be used up to capacity in 1993.

NAME 285

(c) In 1993,what will be the total output of garden gnomes?

14,000,What will be the price of garden gnomes? $9.20.

What rate of return will Deardwarf’s cousin Zwerg make on his invest-

ment in a garden gnome mold that he ordered a year ago and paid $1,000

foratthattime40%.

(d) Remember that Zwerg’s neighbor,Munchkin,also has a gnome mold

that is to be delivered on January 1,1993,Knowing about the tax makes

Munchkin’s mold a less attractive investment than it was without the

tax,but still Zwerg would buy it if he can get it cheap enough so that he

makes a 10% rate of return on his investment,How much should he be

willing to pay for Munchkin’s new mold? $545.45.

(e) What do you think will happen to the number of gnome molds ordered

for delivery on January 1,1994? Will it be larger,smaller,or the same

as the number ordered the previous year? Smaller.

(f) The tax on garden gnomes was left in place for many years,and no-

body expected any further changes in the tax or in demand or supply con-

ditions,After the passage of su cient time,the industry reached a new

long-run equilibrium,What was the new equilibrium price of gnomes?

$10.20.

(g) In the short run,who would end up paying the tax on garden gnomes,

the producers or the consumers? Producers,In the long run,did

the price of gnomes go up by more,less,or the same amount as the tax

per gnome? Same amount.

(h) Suppose that early in the morning of January 1,1993,the government

had announced that there would be a $1 tax on garden gnomes,but

that the tax would not go into e ect until January 1,1994,Would the

producers of garden gnomes necessarily be worse o than if there were

no tax? Why or why not? No,The producers would

anticipate the tax increase and restrict

supply,thereby raising prices.

286 INDUSTRY SUPPLY (Ch,23)

(i) Is it reasonable to suppose that the government could introduce \sur-

prise" taxes without making rms suspicious that there would be similar

\surprises" in the future? Suppose that the introduction of the tax in Jan-

uary 1993 makes gnome makers suspicious that there will be more taxes

introduced in later years,Will this a ect equilibrium prices and supplies?

How? If a surprise tax makes gnome makers

expect similar ‘‘surprises’’ in future,it

will take a higher current price to get

them to enter the industry,This will raise

the price paid by consumers.

23.4 (0) Consider a competitive industry with a large number of rms,

all of which have identical cost functions c(y)=y

2

+1 fory>0and

c(0) = 0,Suppose that initially the demand curve for this industry is

given by D(p)=52?p,(The output of a rm does not have to be an

integer number,but the number of rms does have to be an integer.)

(a) What is the supply curve of an individual rm? S(p)= p=2,If

there are n rms in the industry,what will be the industry supply curve?

Y = np=2.

(b) What is the smallest price at which the product can be sold? p

=

2.

(c) What will be the equilibrium number of rms in the industry? (Hint:

Take a guess at what the industry price will be and see if it works.)

Guess at p

=2,This gives D(p)=52?2=

n2=2,which says n

=50.

(d) What will be the equilibrium price? p

=2,What will be the

equilibrium output of each rm? y

=1.

(e) What will be the equilibrium output of the industry? Y

=50.

NAME 287

(f) Now suppose that the demand curve shifts to D(p)=52:5?p.

What will be the equilibrium number of rms? (Hint,Can a new rm

enter the market and make nonnegative pro ts?) If a new

firm entered,there would be 51 firms,The

supply-demand equation would be 52:5?p =

51p=2,Solve for p

= 105=53 < 2,A new firm

would lose money,Therefore in equilibrium

there would be 50 firms.

(g) What will be the equilibrium price? Solve 52:5?p =

50p=2 to get p

=2:02,What will be the equilibrium

output of each rm? y

=1:01,What will be the equilibrium

pro ts of each rm? Around,02.

(h) Now suppose that the demand curve shifts to D(p)=53?p,What will

be the equilibrium number of rms? 51,What will be the equilibrium

price? 2.

(i) What will be the equilibrium output of each rm? y =1,What

will be the equilibrium pro ts of each rm? Zero.

23.5 (3) In 1990,the town of Ham Harbor had a more-or-less free market

in taxi services,Any respectable rm could provide taxi service as long

as the drivers and cabs satis ed certain safety standards.

Let us suppose that the constant marginal cost per trip of a taxi ride

is $5,and that the average taxi has a capacity of 20 trips per day,Let

the demand function for taxi rides be given by D(p)=1;200?20p,where

demand is measured in rides per day,and price is measured in dollars.

Assume that the industry is perfectly competitive.

(a) What is the competitive equilibrium price per ride? (Hint,In com-

petitive equilibrium,price must equal marginal cost.) 5,What

is the equilibrium number of rides per day? 1,100,How many

taxicabs will there be in equilibrium? 55.

288 INDUSTRY SUPPLY (Ch,23)

(b) In 1990 the city council of Ham Harbor created a taxicab licensing

board and issued a license to each of the existing cabs,The board stated

that it would continue to adjust the taxicab fares so that the demand for

rides equals the supply of rides,but no new licenses will be issued in the

future,In 1995 costs had not changed,but the demand curve for taxicab

rides had become D(p)=1;220?20p,What was the equilibrium price

of a ride in 1995? $6.

(c) What was the pro t per ride in 1995,neglecting any costs associated

with acquiring a taxicab license? $1,What was the pro t per taxicab

license per day? 20,If the taxi operated every day,what was the

pro t per taxicab license per year? $7,300.

(d) If the interest rate was 10% and costs,demand,and the number of

licenses were expected to remain constant forever,what would be the

market price of a taxicab license? $73,000.

(e) Suppose that the commission decided in 1995 to issue enough new

licenses to reduce the taxicab price per ride to $5,How many more

licenses would this take? 1.

(f) Assuming that demand in Ham Harbor is not going to grow any

more,how much would a taxicab license be worth at this new fare?

Nothing.

(g) How much money would each current taxicab owner be willing to

pay to prevent any new licenses from being issued? $73,000

each,What is the total amount that all taxicab owners together would

be willing to pay to prevent any new licences from ever being issued?

$4,015,000,The total amount that consumers would be willing

to pay to have another taxicab license issued would be (more than,less

than,the same as) more than this amount.

23.6 (2) In this problem,we will determine the equilibrium pattern

of agricultural land use surrounding a city,Think of the city as being

located in the middle of a large featureless plain,The price of wheat at

the market at the center of town is $10 a bushel,and it only costs $5 a

bushel to grow wheat,However,it costs 10 cents a mile to transport a

bushel of wheat to the center of town.

NAME 289

(a) If a farm is located t miles from the center of town,write down

a formula for its pro t per bushel of wheat transported to market.

Profit per bushel =5?:10t.

(b) Suppose you can grow 1,000 bushels on an acre of land,How much

will an acre of land located t miles from the market rent for? Rent =

5;000?100t.

(c) How far away from the market do you have to be for land to be worth

zero? 50 miles.

23.7 (1) Consider an industry with three rms,Suppose the rms have

the following supply functions,S

1

(p)=p,S

2

(p)=p?5,and S

3

(p)=2p

respectively,On the graph below plot each of the three supply curves and

the resulting industry supply curve.

010203040

5

10

15

Quantity

Price

20

S

2

S

1

S

3

Industry

supply

(a) If the market demand curve has the form D(p) = 15,what is the

resulting market price? 5,Output? 15,What is the output

level for rm 1 at this price? 5,Firm 2? 0,Firm 3?

10.

23.8 (0) Suppose all rms in a given industry have the same supply

curve given by S

i

(p)=p=2,Plot and label the four industry supply

curves generated by these rms if there are 1,2,3,or 4 rms operating

in the industry.

290 INDUSTRY SUPPLY (Ch,23)

010203040

5

10

15

Quantity

Price

20

S

2

S

1

S

3

S

4

(a) If all of the rms had a cost structure such that if the price was below

$3,they would be losing money,what would be the equilibrium price and

output in the industry if the market demand was equal to D(p)=3:5?

Answer,price = $3.50,quantity= 3.5,How many rms would

exist in such a market? 2.

(b) What if the identical conditions as above held except that the market

demand was equal to D(p)=8?p? Now,what would be the equilibrium

price and output? $3.20 and 4.8,How many rms would

operate in such a market? 3.

23.9 (0) There is free entry into the pollicle industry,Anybody can

enter this industry and have the same U-shaped average cost curve as all

of the other rms in the industry.

(a) On the diagram below,draw a representative rm’s average and mar-

ginal cost curves using blue ink,Also,indicate the long-run equilibrium

level of the market price.

NAME 291

010203040

5

10

15

Quantity

Price

20

P

P+t

P+l

Blue

mc

Blue ac

Red ac

Red mc

Black ac

(b) Suppose the government imposes a tax,t,on every unit of output sold

by the industry,Use red ink to draw the new conditions on the above

graph,After the industry has adjusted to the imposition of the tax,the

competitive model would predict the following,the market price would

(increase,decrease) increase by amount t,there would

be (more,the same,fewer) fewer rms operating in the industry,and

the output level for each rm operating in the industry would Stay

the same,(increase,stay the same,decrease).

(c) What if the government imposes a tax,l,onevery rm in the in-

dustry,Draw the new cost conditions on the above graph using black

ink,After the industry has adjusted to the imposition of the tax the

competitive model would predict the following,the market price would

(increase,decrease) increase,there would be (more,the same,

fewer) fewer rms operating in the industry,and the output level

for each rm operating in the industry would increase (increase,

stay the same,decrease).

23.10 (0) In many communities,a restaurant that sells alcoholic bev-

erages is required to have a license,Suppose that the number of licenses

is limited and that they may be easily transferred to other restaurant

owners,Suppose that the conditions of this industry closely approximate

perfect competition,If the average restaurant’s revenue is $100,000 a

year,and if a liquor license can be leased for a year for $85,000 from an

existing restaurant,what is the average variable cost in the industry?

$15,000.

292 INDUSTRY SUPPLY (Ch,23)

23.11 (2) In order to protect the wild populations of cockatoos,the

Australian authorities have outlawed the export of these large parrots.

An illegal market in cockatoos has developed,The cost of capturing an

Australian cockatoo and shipping him to the United States is about $40

per bird,Smuggled parrots are drugged and shipped in suitcases,This is

extremely traumatic for the birds and about 50% of the cockatoos shipped

die in transit,Each smuggled cockatoo has a 10% chance of being discov-

ered,in which case the bird is con scated and a ne of $500 is charged.

Con scated cockatoos that are alive are returned to the wild,Con scated

cockatoos that are found dead are donated to university cafeterias.

(a) The probability that a smuggled parrot will reach the buyer alive and

uncon scated is,45,Therefore when the price of smuggled parrots is

p,what is the expected gross revenue to a parrot-smuggler from shipping

a parrot?,45p.

(b) What is the expected cost,including expected nes and the cost of

capturing and shipping,per parrot? $:10 500 + 40 = $90.

(c) The supply schedule for smuggled parrots will be a horizontal line at

the market price $200,(Hint,At what price does a parrot-smuggler

just break even?)

(d) The demand function for smuggled cockatoos in the United States is

D(p)=7;200?20p per year,How many smuggled cockatoos will be sold

in the United States per year at the equilibrium price? 3,200,How

many cockatoos must be caught in Australia in order that this number of

live birds reaches U.S,buyers? 3;200=:45 = 7;111.

(e) Suppose that instead of returning live con scated cockatoos to the

wild,the customs authorities sold them in the American market,The

pro ts from smuggling a cockatoo do not change from this policy change.

Since the supply curve is horizontal,it must be that the equilibrium price

of smuggled cockatoos will have to be the same as the equilibrium price

when the con scated cockatoos were returned to nature,How many live

cockatoos will be sold in the United States in equilibrium? 3,200.

How many cockatoos will be permanently removed from the Australian

wild? 6,400.

The story behind this problem is based on actual fact,but the num-

bers we use are just made up for illustration,It would be very interesting

to have some good estimates of the actual demand functions and cost

functions.

NAME 293

(f) Suppose that the trade in cockatoos is legalized,Suppose that it

costs about $40 to capture and ship a cockatoo to the United States

in a comfortable cage and that the number of deaths in transit by this

method is negligible,What would be the equilibrium price of cockatoos

in the United States? $40,How many cockatoos would be sold in

the United States? 6,400,How many cockatoos would have to be

caught in Australia for the U.S,market? 6,400.

23.12 (0) The horn of the rhinoceros is prized in Japan and China for its

alleged aphrodisiac properties,This has proved to be most unfortunate for

the rhinoceroses of East Africa,Although it is illegal to kill rhinoceroses

in the game parks of Kenya,the rhinoceros population of these parks has

been almost totally depleted by poachers,The price of rhinoceros horns

in recent years has risen so high that a poacher can earn half a year’s

wages by simply killing one rhinoceros,Such high rewards for poaching

have made laws against poaching almost impossible to enforce in East

Africa,There are also large game parks with rhinoceros populations in

South Africa,Game wardens there were able to prevent poaching almost

completely and the rhinoceros population of South Africa has prospered.

In a recent program from the television series Nova,a South African game

warden explained that some rhinoceroses even have to be \harvested" in

order to prevent overpopulation of rhinoceroses,\What then," asked the

interviewer,\do you do with the horns from the animals that are harvested

or that die of natural causes?" The South African game warden proudly

explained that since international trade in rhinoceros horns was illegal,

South Africa did not contribute to international crime by selling these

horns,Instead the horns were either destroyed or stored in a warehouse.

(a) Suppose that all of the rhinoceros horns produced in South Africa

are destroyed,Label the axes below and draw world supply and demand

curves for rhinoceros horns with blue ink,Label the equilibrium price

and quantity.

294 INDUSTRY SUPPLY (Ch,23)

Price

Quantity

P

P

a

b

Q

a

Q

b

D (Blue)

S (Blue) S (Red)

(b) If South Africa were to sell its rhinoceros horns on the world mar-

ket,which of the curves in your diagram would shift and in what di-

rection? Supply curve to the right,Use red ink to

illustrate the shifted curve or curves,If South Africa were to do this,

would world consumption of rhinoceros horns be increased or decreased?

Increased,Would the world price of rhinoceros horns be increased

or decreased? Decreased,Would the amount of rhinoceros poach-

ing be increased or decreased? Decreased.

23.13 (1) The sale of rhinoceros horns is not prohibited because of con-

cern about the wicked pleasures of aphrodisiac imbibers,but because the

supply activity is bad for rhinoceroses,Similarly,the Australian reason

for restricting the exportation of cockatoos to the United States is not be-

cause having a cockatoo is bad for you,Indeed it is legal for Australians

to have cockatoos as pets,The motive for the restriction is simply to

protect the wild populations from being overexploited,In the case of

other commodities,it appears that society has no particular interest in

restricting the supply activities but wishes to restrict consumption,A

good example is illicit drugs,The growing of marijuana,for example,is a

simple pastoral activity,which in itself is no more harmful than growing

sweet corn or brussels sprouts,It is the consumption of marijuana to

which society objects.

Suppose that there is a constant marginal cost of $5 per ounce for

growing marijuana and delivering it to buyers,But whenever the mari-

juana authorities nd marijuana growing or in the hands of dealers,they

seize the marijuana and ne the supplier,Suppose that the probability

NAME 295

that marijuana is seized is,3 and that the ne if you are caught is $10

per ounce.

(a) If the \street price" is $p per ounce,what is the expected revenue net

of nes to a dealer from selling an ounce of marijuana?,7p?3.

What then would be the equilibrium price of marijuana? $11.4.

(b) Suppose that the demand function for marijuana has the equation

Q = A?Bp,If all con scated marijuana is destroyed,what will be the

equilibrium consumption of marijuana? A?11:4B,Suppose that

con scated marijuana is not destroyed but sold on the open market,What

will be the equilibrium consumption of marijuana? A?11:4B.

(c) The price of marijuana will (increase,decrease,stay the same)

Stay the same.

(d) If there were increasing rather than constant marginal cost in mar-

ijuana production,do you think that consumption would be greater

if con scated marijuana were sold than if it were destroyed? Ex-

plain,Consumption will increase because

the supply curve will shift to the right,

lowering the price.

296 INDUSTRY SUPPLY (Ch,23)

Chapter 24 NAME

Monopoly

Introduction,The pro t-maximizing output of a monopolist is found by

solving for the output at which marginal revenue is equal to marginal cost.

Having solved for this output,you nd the monopolist’s price by plugging

the pro t-maximizing output into the demand function,In general,the

marginal revenue function can be found by taking the derivative of the

total revenue function with respect to the quantity,But in the special case

of linear demand,it is easy to nd the marginal revenue curve graphically.

With a linear inverse demand curve,p(y)=a?by,the marginal revenue

curve always takes the form MR(y)=a?2by.

24.1 (0) Professor Bong has just written the rst textbook in Punk

Economics,It is called Up Your Isoquant,Market research suggests that

the demand curve for this book will be Q =2;000?100P,whereP is

its price,It will cost $1,000 to set the book in type,This setup cost is

necessary before any copies can be printed,In addition to the setup cost,

there is a marginal cost of $4 per book for every book printed.

(a) The total revenue function for Professor Bong’s book is R(Q)=

20Q?Q

2

=100.

(b) The total cost function for producing Professor Bong’s book is C(Q)=

1;000 + 4Q.

(c) The marginal revenue function is MR(Q)= 20?Q=50 and

the marginal cost function is MC(Q)= 4,The pro t-maximizing

quantity of books for professor Bong to sell is Q

= 800.

24.2 (0) Peter Morgan sells pigeon pies from a pushcart in Central Park.

Morgan is the only supplier of this delicacy in Central Park,His costs are

zero due to the abundant supplies of raw materials available in the park.

(a) When he rst started his business,the inverse demand curve for pigeon

pies was p(y) = 100?y,where the price is measured in cents and y

measures the number of pies sold,Use black ink to plot this curve in

the graph below,On the same graph,use red ink to plot the marginal

revenue curve.

298 MONOPOLY (Ch,24)

0 50 75 100 125

Pigeon pies

25

50

75

100

Cents

25 150

Black

lines

Blue line

Red line

(b) What level of output will maximize Peter’s pro ts? 50,What

price will Peter charge per pie? 50 cents.

(c) After Peter had been in business for several months,he noticed that

the demand curve had shifted to p(y)=75?y=2,Useblueinktoplot

this curve in the graph above,Plot the new marginal revenue curve on

the same graph with black ink.

(d) What is his pro t-maximizing output at this new price? 75,What

is the new pro t-maximizing price? 37.5 cents per pie.

24.3 (0) Suppose that the demand function for Japanese cars in the

United States is such that annual sales of cars (in thousands of cars) will

be 250?2P,whereP is the price of Japanese cars in thousands of dollars.

(a) If the supply schedule is horizontal at a price of $5,000 what will

be the equilibrium number of Japanese cars sold in the United States?

240 thousand,How much money will Americans spend in total on

Japanese cars? 1.2 billion dollars.

(b) Suppose that in response to pressure from American car manufactur-

ers,the United States imposes an import duty on Japanese cars in such a

way that for every car exported to the United States the Japanese man-

ufacturers must pay a tax to the U.S,government of $2,000,How many

Japanese automobiles will now be sold in the United States? 236

thousand,At what price will they be sold? 7 thousand dollars.

NAME 299

(c) How much revenue will the U.S,government collect with this tari?

472 million dollars.

(d) On the graph below,the price paid by American consumers is mea-

sured on the vertical axis,Use blue ink to show the demand and supply

schedules before the import duty is imposed,After the import duty is

imposed,the supply schedule shifts and the demand schedule stays as

before,Use red ink to draw the new supply schedule.

0 100 150 200 250

Japanese autos (thousands)

2

4

6

8

Price (thousands)

50 300

7

5

Blue

lines

Red line

Demand

Supply

Supply with duty

(e) Suppose that instead of imposing an import duty,the U.S,government

persuades the Japanese government to impose \voluntary export restric-

tions" on their exports of cars to the United States,Suppose that the

Japanese agree to restrain their exports by requiring that every car ex-

ported to the United States must have an export license,Suppose further

that the Japanese government agrees to issue only 236,000 export licenses

and sells these licenses to the Japanese rms,If the Japanese rms know

the American demand curve and if they know that only 236,000 Japanese

cars will be sold in America,what price will they be able to charge in

America for their cars? 7 thousand dollars.

(f) How much will a Japanese rm be willing to pay the Japanese govern-

ment for an export license? 2 thousand dollars,(Hint,Think

about what it costs to produce a car and how much it can be sold for if

youhaveanexportlicense.)

(g) How much will be the Japanese government’s total revenue from the

sale of export licenses? 472 million dollars.

300 MONOPOLY (Ch,24)

(h) How much money will Americans spend on Japanese cars? 1.652

billion dollars.

(i) Why might the Japanese \voluntarily" submit to export controls?

Total revenue of Japanese companies and

government is greater with export controls

than without them,Since there is less

output,costs are lower,Higher revenue,

lower costs imply more profit.

24.4 (0) A monopolist has an inverse demand curve given by p(y)=

12?y and a cost curve given by c(y)=y

2

.

(a) What will be its pro t-maximizing level of output? 3.

(b) Suppose the government decides to put a tax on this monopolist so

that for each unit it sells it has to pay the government $2,What will be

its output under this form of taxation? 2.5.

(c) Suppose now that the government puts a lump sum tax of $10 on the

pro ts of the monopolist,What will be its output? 3.

24.5 (1) In Gomorrah,New Jersey,there is only one newspaper,the

Daily Calumny,The demand for the paper depends on the price and the

amount of scandal reported,The demand function is Q =15S

1=2

P

3

,

where Q is the number of issues sold per day,S is the number of column

inches of scandal reported in the paper,and P is the price,Scandals

are not a scarce commodity in Gomorrah,However,it takes resources to

write,edit,and print stories of scandal,The cost of reporting S units

of scandal is $10S,These costs are independent of the number of papers

sold,In addition it costs money to print and deliver the paper,These

cost $:10 per copy and the cost per unit is independent of the amount

of scandal reported in the paper,Therefore the total cost of printing Q

copies of the paper with S column inches of scandal is $10S +:10Q.

(a) Calculate the price elasticity of demand for the Daily Calumny.

3,Does the price elasticity depend on the amount of scandal re-

ported? No,Is the price elasticity constant over all prices? Yes.

NAME 301

(b) Remember that MR = P(1 +

1

),To maximize pro ts,the Daily

Calumny will set marginal revenue equal to marginal cost,Solve for

the pro t-maximizing price for the Calumny to charge per newspaper.

$.15,When the newspaper charges this price,the di erence between

the price and the marginal cost of printing and delivering each newspaper

is $.05.

(c) If the Daily Calumny charges the pro t-maximizing price and prints

100 column inches of scandal,how many copies would it sell? (Round

to the nearest integer.) 44,444,Write a general expression

for the number of copies sold as a function of S,Q(S)= Q =

15S

1=2

(:15)

3

=4;444:44S

1=2

.

(d) Assuming that the paper charges the pro t-maximizing price,write

an expression for pro ts as a function of Q and S,Profits=

:15Q?:10Q?10S,Using the solution for Q(S) that you found

in the last section,substitute Q(S)forQ to write an expression for pro ts

as a function of S alone,Profits =:05(4;444:44S

1=2

)?

10S = 222:22S

1=2

10S.

(e) If the Daily Calumny charges its pro t-maximizing price,and prints

the pro t-maximizing amount of scandal,how many column inches of

scandal should it print? 123.456 inches,How many copies

are sold 49,383 and what is the amount of pro t for the Daily

Calumny if it maximizes its pro ts? 1,234.5.

24.6 (0) In the graph below,use black ink to draw the inverse demand

curve,p

1

(y) = 200?y.

(a) If the monopolist has zero costs,where on this curve will it choose to

operate? At y = 100,p = 100.

(b) Now draw another demand curve that passes through the pro t-

maximizing point and is flatter than the original demand curve,Use

a red pen to mark the part of this new demand curve on which the mo-

nopolist would choose to operate,(Hint,Remember the idea of revealed

preference?)

302 MONOPOLY (Ch,24)

(c) The monopolist would have (larger,smaller) pro ts at the new demand

curve than it had at the original demand curve,Larger.

0 50 100 150 200

50

100

150

Quantity

Price

200

Red

Line

Black Line

Chapter 25 NAME

Monopoly Behavior

Introduction,Problems in this chapter explore the possibilities of price

discrimination by monopolists,There are also problems related to spatial

markets,where transportation costs are accounted for and we show that

lessons learned about spatial models give us a useful way of thinking about

competition under product di erentiation in economics and in politics.

Remember that a price discriminator wants the marginal revenue in

each market to be equal to the marginal cost of production,Since he

produces all of his output in one place,his marginal cost of production

is the same for both markets and depends on his total output,The trick

for solving these problems is to write marginal revenue in each market as

a function of quantity sold in that market and to write marginal cost as

a function of the sum of quantities sold in the two markets,The pro t-

maximizing conditions then become two equations that you can solve

for the two unknown quantities sold in the two markets,Of course,if

marginal cost is constant,your job is even easier,since all you have to do

is nd the quantities in each market for which marginal revenue equals

the constant marginal cost.

Example,A monopolist sells in two markets,The inverse demand curve

in market 1 is p

1

= 200?q

1

,The inverse demand curve in market 2 is

p

2

= 300?q

2

,The rm’s total cost function is C(q

1

+q

2

)=(q

1

+q

2

)

2

.The

rm is able to price discriminate between the two markets,Let us nd the

prices that it will charge in each market,In market 1,the rm’s marginal

revenue is 200?2q

1

,In market 2,marginal revenue is 300?2q

2

.The

rm’s marginal costs are 2(q

1

+q

2

),To maximize its pro ts,the rm sets

marginal revenue in each market equal to marginal cost,This gives us the

two equations 200?2q

1

=2(q

1

+q

2

) and 300?2q

2

=2(q

1

+q

2

),Solving

these two equations in two unknowns for q

1

and q

2

,we nd q

1

=16:67

and q

2

=66:67,We can nd the price charged in each market by plugging

these quantities into the demand functions,The price charged in market

1 will be 183.33,The price charged in market 2 will be 233.33.

25.1 (0) Ferdinand Sludge has just written a disgusting new book,Orgy

in the Piggery,His publisher,Graw McSwill,estimates that the demand

for this book in the United States is Q

1

=50;000? 2;000P

1

,where

P

1

is the price in America measured in U.S,dollars,The demand for

Sludge’s opus in England is Q

2

=10;000?500P

2

,whereP

2

is its price

in England measured in U,S,dollars,His publisher has a cost function

C(Q) = $50;000 + $2Q,whereQ is the total number of copies of Orgy

that it produces.

(a) If McSwill must charge the same price in both countries,how many

copies should it sell 27,500,and what price should it charge

304 MONOPOLY BEHAVIOR (Ch,25)

$13 to maximize its pro ts,and how much will those pro ts be?

$252,500.

(b) If McSwill can charge a di erent price in each country,and wants to

maximize pro ts,how many copies should it sell in the United States?

23,000,What price should it charge in the United States?

$13.50,How many copies should it sell in England? 4,500.

What price should it charge in England? $11,How much will its

total pro ts be? $255,000.

25.2 (0) A monopoly faces an inverse demand curve,p(y) = 100?2y,

and has constant marginal costs of 20.

(a) What is its pro t-maximizing level of output? 20.

(b) What is its pro t-maximizing price? $60.

(c) What is the socially optimal price for this rm? $20.

(d) What is the socially optimal level of output for this rm? 40.

(e) What is the deadweight loss due to the monopolistic behavior of this

rm? 400.

(f) Suppose this monopolist could operate as a perfectly discriminating

monopolist and sell each unit of output at the highest price it would fetch.

The deadweight loss in this case would be 0.

Calculus 25.3 (1) Banana Computer Company sells Banana computers both in

the domestic and foreign markets,Because of di erences in the power

supplies,a Banana purchased in one market cannot be used in the other

market,The demand and marginal revenue curves associated with the

two markets are as follows:

P

d

=20;000?20QP

f

=25;000?50Q

MR

d

=20;000?40QMR

f

=25;000?100Q:

Banana’s production process exhibits constant returns to scale and it

takes $1,000,000 to produce 100 computers.

NAME 305

(a) Banana’s long-run average cost function is AC(Q)= $10,000

and its long-run marginal cost function is MC(Q)= $10,000.

(Hint,If there are constant returns to scale,does long-run average cost

change as output changes?) Draw the average and marginal cost curves

on the graph.

(b) Draw the demand curve for the domestic market in black ink and

the marginal revenue curve for the domestic market in pencil,Draw the

demand curve for the foreign market in red ink and the marginal revenue

curve for the foreign market in blue ink.

0 100 200 300 400 500 600 700 800

10

20

30

40

50

60

Dollars (1,000s)

Red line

Blue line

Black line

Pencil line

LRAC

LRMC

Banana Computers

(c) If Banana is maximizing its pro ts,it will sell 250 computers in

the domestic market at 15,000 dollars each and 150 computers

in the foreign market at 17,500 dollars each,What are Banana’s

total pro ts? $2,375,000.

306 MONOPOLY BEHAVIOR (Ch,25)

(d) At the pro t-maximizing price and quantity,what is the price elas-

ticity of demand in the domestic market3,What is the price

elasticity of demand in the foreign market2:33,Is demand more

or less elastic in the market where the higher price is charged? Less

elastic.

(e) Suppose that somebody gures out a wiring trick that allows a Banana

computer built for either market to be costlessly converted to work in the

other,(Ignore transportation costs.) On the graph below,draw the new

inverse demand curve (with blue ink) and marginal revenue curve (with

black ink) facing Banana.

0 100 200 300 400 500 600 700 800

10

20

30

40

Dollars (1,000s)

Blue line

LRAC

LRMC

Banana Computers

Black line

(f) Given that costs haven’t changed,how many Banana computers

should Banana sell? 400,What price will it charge? $15,714.

How will Banana’s pro ts change now that it can no longer practice price

discrimination? Decrease by $89,284.

25.4 (0) A monopolist has a cost function given by c(y)=y

2

and faces

a demand curve given by P(y) = 120?y.

(a) What is his pro t-maximizing level of output? 30,What price

will the monopolist charge? $90.

NAME 307

(b) If you put a lump sum tax of $100 on this monopolist,what would its

output be? 30.

(c) If you wanted to choose a price ceiling for this monopolist so as to

maximize consumer plus producer surplus,what price ceiling should you

choose? $80.

(d) How much output will the monopolist produce at this price ceiling?

40.

(e) Suppose that you put a speci c tax on the monopolist of $20 per unit

output,What would its pro t-maximizing level of output be? 25.

25.5 (1) The Grand Theater is a movie house in a medium-sized college

town,This theater shows unusual lms and treats early-arriving movie

goers to live organ music and Bugs Bunny cartoons,If the theater is

open,the owners have to pay a xed nightly amount of $500 for lms,

ushers,and so on,regardless of how many people come to the movie.

For simplicity,assume that if the theater is closed,its costs are zero,The

nightly demand for Grand Theater movies by students is Q

S

= 220?40P

S

,

where Q

S

is the number of movie tickets demanded by students at price

P

S

,The nightly demand for nonstudent moviegoers is Q

N

= 140?20P

N

.

(a) If the Grand Theater charges a single price,P

T

,toeverybody,then

at prices between 0 and $5.50,the aggregate demand function for movie

tickets is Q

T

(P

T

)= 360?60P

T

,Over this range of prices,the

inverse demand function is then P

T

(Q

T

)= 6?Q

T

=60.

(b) What is the pro t-maximizing number of tickets for the Grand The-

ater to sell if it charges one price to everybody? 180,At what price

would this number of tickets be sold? $3,How much pro ts would

the Grand make? $40,How many tickets would be sold to students?

100,To nonstudents? 80.

(c) Suppose that the cashier can accurately separate the students from

the nonstudents at the door by making students show their school ID

cards,Students cannot resell their tickets and nonstudents do not have

access to student ID cards,Then the Grand can increase its pro ts by

charging students and nonstudents di erent prices,What price will be

charged to students? $2.75,How many student tickets will be sold?

308 MONOPOLY BEHAVIOR (Ch,25)

110,What price will be charged to nonstudents? $3.50,How

many nonstudent tickets will be sold? 70,How much pro t will the

Grand Theater make? $47.50.

(d) If you know calculus,see if you can do this part,Suppose that

the Grand Theater can hold only 150 people and that the manager

wants to maximize pro ts by charging separate prices to students and

to nonstudents,If the capacity of the theater is 150 seats and Q

S

tickets are sold to students,what is the maximum number of tickets

that can be sold to nonstudents? Q

N

= 150?Q

S

,Write

an expression for the price of nonstudent tickets as a function of the

number of student tickets sold,(Hint,First nd the inverse nonstu-

dent demand function.) P

N

=?1=2+Q

S

=20,Write

an expression for Grand Theater pro ts as a function of the number

Q

S

only,(Hint,Make substitutions using your previous answers.)

Q

S

(11=2?Q

S

=40) + (?1=2+20=Q

S

)(150?Q

S

)?

500 =?3Q

2

S

=40 + 27Q

S

=2?575,How many student

tickets should the Grand sell to maximize pro ts? 90,What price

is charged to students? $3.25,How many nonstudent tickets are

sold? 60,What price is charged to nonstudents? $4,How much

pro t does the Grand make under this arrangement? $32.50.

25.6 (2) The Mall Street Journal is considering o ering a new service

which will send news articles to readers by email,Their market research

indicates that there are two types of potential users,impecunious under-

graduates studying microeconomics and high-level executives,Let x be

the number of articles that a user requests per year,The executives have

an inverse demand function P

E

(x) = 100?x and the undergraduates

have an inverse demand function P

U

(x)=80?x,(Prices are measured

in cents.) The Journal has a zero marginal cost of sending articles via

email,Please draw these demand functions in the graph below and label

them.

NAME 309

20 40 60 80 100 120

20

40

60

80

100

120

Quantity

Price

0

P (X) = 100 - X

E

P (X) = 80 - X

U

(a) Suppose that the Journal can identify which of the users are under-

graduates and which are executives,It decides to o er a plan where users

can buy a xed number of articles per year for a xed price per year.

If it wants to maximize total pro ts it will o er 100 articles to the

executives and 80 articles per year to the students.

(b) It will charge $50 per year to the executives and $32 per year

to the students.

(c) Suppose that the Journal cannot identify which users are executives

and which are undergraduates,In this case it simply o ers two packages,

and lets the users self-select the one that is optimal for them,Suppose

that it o ers two packages,one that allows up to 80 articles per year the

other that allows up to 100 articles per year,What’s the highest price

that the undergraduates will pay for the 80-article subscription? $32.

(d) What (gross) consumer surplus would the executives get if they con-

sumed 80 articles per year? $48.

(e) What is the the maximum price that the Journal can charge for 100

articles per year if it o ers 80 a year at the highest price the undergradu-

ates are willing to pay? Solve 50?p =48?32 to find

p = $34.

310 MONOPOLY BEHAVIOR (Ch,25)

(f) Suppose that the Mall Street Journal decides to include only 60 articles

in the student package,What is the most it could charge and still get

student to buy this package? $30.

(g) If the Mall Street Journal o ers a \student package" of 60 articles

at this price,how much net consumer surplus would executives get from

buying the student package? $12.

(h) What is the most that the Mall Street Journal could charge for 100

article package and expect executives to buy this package rather than the

student package? $38.

(i) If the number of executives in the population equals the number of

students,would the Mall Street Journal make higher pro ts by o ering a

student package of 80 articles or a student package of 60 articles? 60.

25.7 (2) Bill Barriers,CEO of MightySoft software,is contemplating

a new marketing strategy,bundling their best-selling wordprocessor and

their spreadsheet together and selling the pair of software products for

one price.

From the viewpoint of the company,bundling software and selling it

at a discounted price has two e ects on sales,1) revenues go up due to

to additional sales of the bundle; and 2) revenues go down since there is

less of a demand for the individual components of the bundle.

The pro tability of bundling depends on which of these two e ects

dominates,Suppose that MightySoft sells the wordprocessor for $200 and

the spreadsheet for $250,A marketing survey of 100 people who purchased

either of these packages in the last year turned up the following facts:

1) 20 people bought both.

2) 40 people bought only the wordprocessor,They would be willing to

spend up to $120 more for the spreadsheet.

3) 40 people bought only the spreadsheet,They would be willing to

spend up to $100 more for the wordprocessor.

In answering the following questions you may assume the following:

1) New purchasers of MightySoft products will have the same charac-

teristics as this group.

2) There is a zero marginal cost to producing extra copies of either

software package.

3) There is a zero marginal cost to creating a bundle.

(a) Let us assume that MightySoft also o ers the products separately

as well as bundled,In order to determine how to price the bundle,Bill

Barriers asks himself the following questions,In order to sell the bundle

to the wordprocessor purchasers,the price would have to be less than

200 + 120 = 320.

NAME 311

(b) In order to sell to the spreadsheet users,the price would have to be

less than 250 + 100 = 350.

(c) What would MightySoft’s pro ts be on a group of 100 users if it priced

the bundle at $320? Everyone buys the bundle so

profits are 100 320 = $32;000.

(d) What would MightySoft’s pro ts be on a group of 100 users if it

priced the bundle at $350? 20 people would buy both

anyway,40 people bought spreadsheet only

and would be willing to buy the bundle,

40 people buy the wordprocessor,but not

the spreadsheet,Total profits are 20

350 + 40 350 + 40 200 = 29;000.

(e) If MightySoft o ers the bundle,what price should it set? $320

is the more profitable price.

(f) What would pro ts be without o ering the bundle? Without

the bundle,profits would be 20 (200+250)+

40 200 + 40 250 = 27;000.

(g) What would be the pro ts with the bundle? 100 320 =

32;000

(h) Is it more pro table to bundle or not bundle? bundle.

(i) Suppose that MightySoft worries about the reliability of their market

survey and decides that they believe that without bundling t of the 100

people will buy both products,and (100?t)=2 will buy the wordprocessor

only and (100?t)=2 will buy the spreadsheet only,Calculate pro ts as a

function of t if there is no bundling,225 (100?t)+450 t.

312 MONOPOLY BEHAVIOR (Ch,25)

(j) What are pro ts with the bundle? $32000.

(k) At what values of t would it be unpro table to o er the bundle?

Solve for the t that equates the two

profits to find t =42:22,So if more than

42 of the 100 new purchasers would buy both

products anyway,it is not profitable to

bundle them.

(l) This analysis so far has been concerned only with customers who

would purchase at least one of the programs at the original set of prices.

Is there any additional source of demand for the bundle? What does

this say about the calculations we have made about the pro tability of

bundling? Yes,it may be that there are

some consumers who were not willing to pay

$200 for the wordprocessor or $250 for

the spreadsheet,but would be willing to

pay $320 for the bundle,This means that

bundling would be even more profitable

than the calculations above indicate.

25.8 (0) Col,Tom Barker is about to open his newest amusement park,

Elvis World,Elvis World features a number of exciting attractions,you

can ride the rapids in the Blue Suede Chutes,climb the Jailhouse Rock

and eat dinner in the Heartburn Hotel,Col,Tom gures that Elvis World

will attract 1,000 people per day,and each person will take x =50?50p

rides,where p is the price of a ride,Everyone who visits Elvis World is

pretty much the same and negative rides are not allowed,The marginal

cost of a ride is essentially zero.

(a) What is each person’s inverse demand function for rides? p(x)=

1?x=50.

(b) If Col,Tom sets the price to maximize pro t,how many rides will be

taken per day by a typical visitor? 25.

NAME 313

(c) What will the price of a ride be? 50 cents.

(d) What will Col,Tom’s pro ts be per person? $12.50

(e) What is the Pareto e cient price of a ride? Zero.

(f) If Col,Tom charged the Pareto e cient price for a ride,how many

rides would be purchased? 50.

(g) How much consumers’ surplus would be generated at this price and

quantity? 25.

(h) If Col,Tom decided to use a two-part tari,he would set an admission

fee of $25 and charge a price per ride of 0.

25.9 (1) The city of String Valley is squeezed between two mountains

and is 36 miles long,running from north to south,and only about 1

block wide,Within the town,the population has a uniform density of

100 people per mile,Because of the rocky terrain,nobody lives outside

the city limits on either the north or the south edge of town,Because of

strict zoning regulations,the city has only three bowling alleys,One of

these is located at the city limits on the north edge of town,one of them is

located at the city limits on the south edge of town,and one is located at

the exact center of town,Travel costs including time and gasoline are $1

per mile,All of the citizens of the town have the same preferences,They

are willing to bowl once a week if the cost of bowling including travel

costs and the price charged by the bowling alley does not exceed $15.

(a) Consider one of the bowling alleys at either edge of town,If it charges

$10 for a night of bowling,how far will a citizen of String Valley be willing

to travel to bowl there? Up to 5 miles,How many customers

would this bowling alley have per week if it charged $10 per night of

bowling? 500.

(b) Write a formula for the number of customers that a bowling alley

at the edge of town will have if it charges $p per night of bowling.

100 (15?p).

(c) Write a formula for this bowling alley’s inverse demand function.

p =15?q=100.

314 MONOPOLY BEHAVIOR (Ch,25)

(d) Suppose that the bowling alleys at the end of town have a marginal

cost of $3 per customer and set their prices to maximize pro ts,(For

the time being assume that these bowling alleys face no competition from

the other bowling alleys in town.) How many customers will they have?

600,What price will they charge? $9,How far away from the edge

of town does their most distant customer live? 6 miles.

(e) Now consider the bowling alley in the center of town,If it charges a

price of $p,how many customers will it have per week? 2*(15-p).

(f) If the bowling alley in the center of town also has marginal costs

of $3 per customer and maximizes its pro ts,what price will it charge?

$9,How many customers will it have per week? 1,200,How far

away from the center of town will its most distant customers live? 6

miles.

(g) Suppose that the city relaxes its zoning restrictions on where the

bowling alleys can locate,but continues to issue operating licenses to

only 3 bowling alleys,Both of the bowling alleys at the end of town

are about to lose their leases and can locate anywhere in town that they

like at about the same cost,The bowling alley in the center of town is

committed to stay where it is,Would either of the alleys at the edge of

town improve its pro ts by locating next to the existing bowling alley in

the center of town? No,What would be a pro t-maximizing location

for each of these two bowling alleys? One would be located

12 miles north of the town center and one

12 miles south.

25.10 (1) In a congressional district somewhere in the U.S,West a

new representative is being elected,The voters all have one-dimensional

political views that can be neatly arrayed on a left-right spectrum,We

can de ne the \location" of a citizen’s political views in the following way.

The citizen with the most extreme left-wing views is said to be at point

0 and the citizen with the most estreme right-wing views is said to be at

point 1,If a citizen has views that are to the right of the views of the

fraction x of the state’s population,that citizen’s views are said to be

located at the point x,Candidates for o ce are forced to publically state

their own political position on the zero-one left-right scale,Voters always

vote for the candidate whose stated position is nearest to their own views.

NAME 315

(If there is a tie for nearest candidate,voters flip a coin to decide which

to vote for.)

(a) There are two candidates for the congressional seat,Suppose that

each candidate cares only about getting as many votes as possible,Is

there an equilibrium in which each candidate chooses the best position

given the position of the other candidate? If so,describe this equilibrium.

The only equilibrium is one in which both

candidates choose the same position,and

that position is at the point 1/2.

25.11 (2) In the congressional district described by the previous problem,

let us investigate what will happen if the two candidates do not care

about the number of votes that they get but only about the amount

of campaign contributions that they receive,Therefore each candidate

chooses his ideological location in such a way as to maximize the amount

of campaign contributions he receives,given the position of the other.*

Let us de ne a left-wing extremist as a voter whose political views

lie to the left of the leftmost candidate,a right-wing extremist as a voter

whose political views lie to the right of the rightmost candidate,and a

moderate voter as one whose political views lie between the positions

of the two candidates,Assume that each extremist voter contributes to

the candidate whose position is closest to his or her own views and that

moderate voters make no campaign contributions,The number of dol-

lars that an extremist voter contributes to his or her favorite candidate

is proportional to the distance between the two candidates,Speci cally,

we assume that there is some constant C such that if the left-wing can-

didate is located at x and the right-wing candidate is located at y,then

total campaign contributions received by the left-wing candidate will be

$Cx(y?x) and total campaign contributions received by the right-wing

candidate will be $C(1?y)(y?x).

(a) If the right-wing candidate is located at y,the contribution-

maximizing position for the left-wing candidate is x = y=2 If the

left-wing candidate is located at x,the contribution-maximizing position

for the right-wing candidate is y = ((1 + x)=2 (Hint,Take a

derivative and set it equal to zero.)

(b) Solve for the unique pair of ideological positions for the two can-

didates such that each takes the position that maximizes his campaign

contributions given the position of the other,x =1=3,y =2=3

* This assumption is a bit extreme,Candidates typically spend at least

some of their campaign contributions on advertising for votes,and this

advertising a ects the voting outcomes.

316 MONOPOLY BEHAVIOR (Ch,25)

(c) Suppose that in addition to collecting contributions from extremists

on his side,candidates can also collect campaign contributions from mod-

erates whose views are closer to their position than to that of their rival’s

position,Suppose that moderates,like extremists,contribute to their

preferred candidate and that they contribute in proportion to the dif-

ference between their own ideological distance from their less-preferred

candidate and their ideological distance from their more-preferred can-

didate,Show that in this case the unique positions in which the left-

and right-wing candidates are each maximizing their campaign contribu-

tions,given the position of the other candidate,occurs where x =1=4

and y =3=4,Total contributions received

by the left wing candidate will be

C

parenleftbig

x(y?x)+(y?x)

2

=4

Total contributions

received by the right-wing candidate

will be C

parenleftbig

(1?y)(y?x)+(y?x)

2

=4

.

Differentiating the former expression with

respect to x and the latter with respect

to y and solving the resulting simultaneous

equations yields x =1=4 and y =3=4.

Chapter 26 NAME

Factor Markets

Introduction,In this chapter you will examine the factor demand de-

cision of a monopolist,If a rm is a monopolist in some industry,it

will produce less output than if the industry were competitively orga-

nized,Therefore it will in general want to use less inputs than does a

competitive rm,The value marginal product is just the value of the ex-

tra output produced by hiring an extra unit of the factor,The ordinary

logic of competitive pro t maximization implies that a competitive rm

will hire a factor up until the point where the value marginal product

equals the price of the factor.

The marginal revenue product is the extra revenue produced by

hiring an extra unit of a factor,For a competitive rm,the marginal

revenue product is the same as the value of the marginal product,but

they di er for monopolist,A monopolist has to take account of the fact

that increasing its production will force the price down,so the marginal

revenue product of an extra unit of a factor will be less than the value

marginal product.

Another thing we study in this chapters is monopsony,whichisthe

case of a market dominated by a single buyer of some good,The case of

monopsony is very similar to the case of a monopoly,The monopsonist

hires less of a factor than a similar competitive rm because the monop-

sony recognizes that the price it has to pay for the factor depends on how

much it buys.

Finally,we consider an interesting example of factor supply,in which

a monopolist produces a good that is used by another monopolist.

Example,Suppose a monopolist faces a demand curve for output of the

form p(y) = 100?2y,The production function takes the simple form

y =2x,and the factor costs $4 per unit,How much of the factor of

production will the monopolist want to employ? How much of the factor

would a competitive industry employ if all the rms in the industry had

the same production function?

Answer,The monopolist will employ the factor up to the point where

the marginal revenue product equals the price of the factor,Revenue as

a function of output is R(y)=p(y)y = (100?2y)y,To nd revenue as a

function of the input,we substitute y =2x:

R(x) = (100?4x)2x = (200?8x)x:

The marginal revenue product function will have the form MRP

x

= 200?

16x,Setting marginal revenue product equal to factor price gives us the

equation

200?16x =4:

Solving this equation gives us x

=12:25:

318 FACTOR MARKETS (Ch,26)

If the industry were competitive,then the industry would employ the

factor up to the point where the value of the marginal product was equal

to 4,This gives us the equation

p2=4;

so p

= 2,How much output would be demanded at this price? We plug

this into the demand function to get the equation 2 = 100?2y,which

implies y

= 49,Since the production function is y =2x,wecansolve

for x

= y

=2=24:5:

26.1 (0) Gargantuan Enterprises has a monopoly in the production of

antimacassars,Its factory is located in the town of Pantagruel,There is

no other industry in Pantagruel,and the labor supply equation there is

W =10+:1L,whereW is the daily wage and L is the number of person-

days of work performed,Antimacassars are produced with a production

function,Q =10L,whereL is daily labor supply and Q is daily output.

The demand curve for antimacassars is P =41?

Q

1;000

,whereP is the

price and Q is the number of sales per day.

(a) Find the pro t-maximizing output for Gargantuan,(Hint,Use the

production function to nd the labor input requirements for any level of

output,Make substitutions so you can write the rm’s total costs as a

function of its output and then its pro t as a function of output,Solve

for the pro t{maximizing output.) 10,000.

(b) How much labor does it use? 1,000,What is the wage rate that

it pays? $110.

(c) What is the price of antimacassars? $31,How much pro t is

made? $200,000.

26.2 (0) The residents of Seltzer Springs,Michigan,consume bottles of

mineral water according to the demand function D(p)=1;000?p.Here

D(p) is the demand per year for bottles of mineral water if the price per

bottle is p.

The sole distributor of mineral water in Seltzer Springs,Bubble Up,

purchases mineral water at c per bottle from their supplier Perry Air.

Perry Air is the only supplier of mineral water in the area and behaves

as a pro t-maximizing monopolist,For simplicity we suppose that it has

zero costs of production.

(a) What is the equilibrium price charged by the distributor Bubble Up?

p

=

1;000+c

2

.

NAME 319

(b) What is the equilibrium quantity sold by Bubble Up? D(p

)=

1;000?c

2

.

(c) What is the equilibrium price charged by the producer Perry Air?

c

= 500.

(d) What is the equilibrium quantity sold by Perry Air? D(c

)=

250.

(e) What are the pro ts of Bubble Up?

b

= (500?250)(750?

500) = 250

2

.

(f) What are the pro ts of Perry Air?

p

= 500 250.

(g) How much consumer’s surplus is generated in this market? CS

e

=

250

2

=2.

(h) Suppose that this situation is expected to persist forever and that

the interest rate is expected to be constant at 10% per year,What is the

minimum lump sum payment that Perry Air would need to pay to Bubble

Up to buy it out? 10 250

2

.

(i) Suppose that Perry Air does this,What will be the new price and

quantity for mineral water? p

= 500 and D(p

) = 500.

(j) What are the pro ts of the new merged rm?

p

= 500

2

.

(k) What is the total amount of consumers’ surplus generated? How does

this compare with the previous level of consumers’ surplus? CS

i

=

500

2

=2 >CS

e

.

Calculus 26.3 (0) Upper Peninsula Underground Recordings (UPUR) has a mon-

opoly on the recordings of the famous rock group Moosecake,Moosecake’s

music is only provided on digital tape,and blank digital tapes cost them

c per tape,There are no other manufacturing or distribution costs,Let

p(x) be the inverse demand function for Moosecake’s music as a function

of x,the number of tapes sold.

320 FACTOR MARKETS (Ch,26)

(a) What is the rst-order condition for pro t maximization? For future

reference,let x

be the pro t-maximizing amount produced and p

be the

price at which it sells,(In this part,assume that tapes cannot be copied.)

p(x

)+p

0

(x

)x

= c.

Now a new kind of consumer digital tape recorder becomes widely

available that allows the user to make 1 and only 1 copy of a prerecorded

digital tape,The copies are a perfect substitute in consumption value for

the original prerecorded tape,and there are no barriers to their use or

sale,However,everyone can see the di erence between the copies and the

orginals and recognizes that the copies cannot be used to make further

copies,Blank tapes cost the consumers c per tape,the same price the

monopolist pays.

(b) All Moosecake fans take advantage of the opportunity to make a single

copy of the tape and sell it on the secondary market,How is the price of an

original tape related to the price of a copy? Derive the inverse demand

curve for original tapes facing UPUR,(Hint,There are two sources of

demand for a new tape,the pleasure of listening to it,and the pro ts

from selling a copy.) If UPUR produces x tapes,2x

tapes reach the market,so UPUR can sell

a single tape for p(2x)+[p(2x)?c],The

first term is the willingness-to-pay for

listening; the second term is profit from

selling a copy.

(c) Write an expression for UPUR’s pro ts if it produces x tapes.

[p(2x)+p(2x)?c]x?cx =2p(2x)x?2cx.

(d) Let x

be the pro t-maximizing level of production by UPUR,How

does it compare to the former pro t-maximizing level of production?

From the two profit functions,one sees

that 2x

= x

,so x

= x

=2.

(e) How does the price of a copy of a Moosecake tape compare to the

price determined in Part (a)? The prices are the same.

(f) If p

is the price of a copy of a Moosecake tape,how much will a new

Moosecake tape sell for? 2p

c.

Chapter 27 NAME

Oligopoly

Introduction,In this chapter you will solve problems for rm and indus-

try outcomes when the rms engage in Cournot competition,Stackelberg

competition,and other sorts of oligopoly behavior,In Cournot competi-

tion,each rm chooses its own output to maximize its pro ts given the

output that it expects the other rm to produce,The industry price de-

pends on the industry output,say,q

A

+q

B

,where A and B are the rms.

To maximize pro ts,rm A sets its marginal revenue (which depends on

the output of rm A and the expected output of rm B since the expected

industry price depends on the sum of these outputs) equal to its marginal

cost,Solving this equation for rm A’s output as a function of rm B’s

expected output gives you one reaction function; analogous steps give you

rm B’s reaction function,Solve these two equations simultaneously to

get the Cournot equilibrium outputs of the two rms.

Example,In Heifer’s Breath,Wisconsin,there are two bakers,Anderson

and Carlson,Anderson’s bread tastes just like Carlson’s|nobody can

tell the di erence,Anderson has constant marginal costs of $1 per loaf of

bread,Carlson has constant marginal costs of $2 per loaf,Fixed costs are

zero for both of them,The inverse demand function for bread in Heifer’s

Breath is p(q)=6?:01q,whereq is the total number of loaves sold per

day.

Let us nd Anderson’s Cournot reaction function,If Carlson bakes

q

C

loaves,then if Anderson bakes q

A

loaves,total output will be q

A

+

q

C

and price will be 6?:01(q

A

+ q

C

),For Anderson,the total cost of

producing q

A

units of bread is just q

A

,so his pro ts are

pq

A

q

A

=(6?:01q

A

:01q

C

)q

A

q

A

=6q

A

:01q

2

A

:01q

C

q

A

q

A

:

Therefore if Carlson is going to bake q

C

units,then Anderson will choose

q

A

to maximize 6q

A

:01q

2

A

:01q

C

q

A

q

A

,This expression is maximized

when 6?:02q

A

:01q

C

= 1,(You can nd this out either by setting

A’s marginal revenue equal to his marginal cost or directly by setting

the derivative of pro ts with respect to q

A

equal to zero.) Anderson’s

reaction function,R

A

(q

C

) tells us Anderson’s best output if he knows

that Carlson is going to bake q

C

,We solve from the previous equation to

nd R

A

(q

C

)=(5?:01q

C

)=:02 = 250?:5q

C

.

We can nd Carlson’s reaction function in the same way,If Carlson

knows that Anderson is going to produce q

A

units,then Carlson’s pro ts

will be p(q

A

+q

C

)?2q

C

=(6?:01q

A

:01q

C

)q

C

2q

C

=6q

C

:01q

A

q

C

:01q

2

C

2q

C

,Carlson’s pro ts will be maximized if he chooses q

C

to satisfy

the equation 6?:01q

A

:02q

C

= 2,Therefore Carlson’s reaction function

is R

C

(q

A

)=(4?:01q

A

)=:02 = 200?:5q

A

.

322 OLIGOPOLY (Ch,27)

Let us denote the Cournot equilibrium quantities by q

A

and q

C

.The

Cournot equilibrium conditions are that q

A

= R

A

( q

C

)and q

C

= R

C

( q

A

).

Solving these two equations in two unknowns we nd that q

A

= 200 and

q

C

= 100,Now we can also solve for the Cournot equilibrium price and for

the pro ts of each baker,The Cournot equilibrium price is 6?:01(200 +

100) = $3,Then in Cournot equilibrium,Anderson makes a pro t of $2

on each of 200 loaves and Carlson makes $1 on each of 100 loaves.

In Stackelberg competition,the follower’s pro t-maximizing output

choice depends on the amount of output that he expects the leader to

produce,His reaction function,R

F

(q

L

),is constructed in the same way

as for a Cournot competitor,The leader knows the reaction function of

the follower and gets to choose her own output,q

L

,rst,So the leader

knows that the industry price depends on the sum of her own output and

the follower’s output,that is,on q

L

+ R

F

(q

L

),Since the industry price

can be expressed as a function of q

L

only,so can the leader’s marginal

revenue,So once you get the follower’s reaction function and substitute it

into the inverse demand function,you can write down an expression that

depends on just q

L

and that says marginal revenue equals marginal cost

for the leader,You can solve this expression for the leader’s Stackelberg

output and plug in to the follower’s reaction function to get the follower’s

Stackelberg output.

Example,Suppose that one of the bakers of Heifer’s Breath plays the role

of Stackelberg leader,Perhaps this is because Carlson always gets up an

hour earlier than Anderson and has his bread in the oven before Anderson

gets started,If Anderson always nds out how much bread Carlson has

in his oven and if Carlson knows that Anderson knows this,then Carlson

can act like a Stackelberg leader,Carlson knows that Anderson’s reaction

function is R

A

(q

C

) = 250?:5q

c

,Therefore Carlson knows that if he bakes

q

C

loaves of bread,then the total amount of bread that will be baked in

Heifer’s Breath will be q

C

+R

A

(q

C

)=q

C

+250?:5q

C

= 250+:5q

C

.Since

Carlson’s production decision determines total production and hence the

price of bread,we can write Carlson’s pro t simply as a function of his

own output,Carlson will choose the quantity that maximizes this pro t.

If Carlson bakes q

C

loaves,the price will be p =6?:01(250 +,5q

C

)=

3:5?:005q

C

,Then Carlson’s pro ts will be pq

C

2q

C

=(3:5?:005q

C

)q

C

2q

C

=1:5q

C

:005q

2

C

,His pro ts are maximized when q

C

= 150,(Find

this either by setting marginal revenue equal to marginal cost or directly

by setting the derivative of pro ts to zero and solving for q

C

.) If Carlson

produces 150 loaves,then Anderson will produce 250?:5 150 = 175

loaves,The price of bread will be 6?:01(175 + 150) = 2:75,Carlson will

now make $.75 per loaf on each of 150 loaves and Anderson will make

$1.75 on each of 175 loaves.

27.1 (0) Carl and Simon are two rival pumpkin growers who sell their

pumpkins at the Farmers’ Market in Lake Witchisit,Minnesota,They are

the only sellers of pumpkins at the market,where the demand function

for pumpkins is q =3;200?1;600p,The total number of pumpkins sold

at the market is q = q

C

+ q

S

,whereq

C

is the number that Carl sells

NAME 323

and q

S

is the number that Simon sells,The cost of producing pumpkins

for either farmer is $.50 per pumpkin no matter how many pumpkins he

produces.

(a) The inverse demand function for pumpkins at the Farmers’ Market is

p = a?b(q

C

+ q

S

),where a = 2 and b = 1=1;600,The

marginal cost of producing a pumpkin for either farmer is $.50.

(b) Every spring,each of the farmers decides how many pumpkins to

grow,They both know the local demand function and they each know

how many pumpkins were sold by the other farmer last year,In fact,

each farmer assumes that the other farmer will sell the same number this

year as he sold last year,So,for example,if Simon sold 400 pumpkins

last year,Carl believes that Simon will sell 400 pumpkins again this year.

If Simon sold 400 pumpkins last year,what does Carl think the price of

pumpkins will be if Carl sells 1,200 pumpkins this year? 1,If

Simon sold q

t?1

S

pumpkins in year t?1,then in the spring of year t,Carl

thinks that if he,Carl,sells q

t

C

pumpkins this year,the price of pumpkins

this year will be 2?(q

t?1

S

+q

t

C

)=1;600.

(c) If Simon sold 400 pumpkins last year,Carl believes that if he sells

q

t

C

pumpkins this year then the inverse demand function that he faces is

p =2?400=1;600?q

t

C

=1;600 = 1:75?q

t

C

=1;600,Therefore if Simon

sold 400 pumpkins last year,Carl’s marginal revenue this year will be

1:75?q

t

C

=800,More generally,if Simon sold q

t?1

S

pumpkins last year,

then Carl believes that if he,himself,sells q

t

C

pumpkins this year,his

marginal revenue this year will be 2?q

t?1

S

=1;600?q

t

C

=800.

(d) Carl believes that Simon will never change the amount of pumpkins

that he produces from the amount q

t?1

S

that he sold last year,Therefore

Carl plants enough pumpkins this year so that he can sell the amount

that maximizes his pro ts this year,To maximize this pro t,he chooses

the output this year that sets his marginal revenue this year equal to

his marginal cost,This means that to nd Carl’s output this year when

Simon’s output last year was q

t?1

S

,Carl solves the following equation.

2?q

t?1

S

=1;600?q

t

C

=800 =,5.

(e) Carl’s Cournot reaction function,R

t

C

(q

t?1

S

),is a function that tells us

what Carl’s pro t-maximizing output this year would be as a function of

Simon’s output last year,Use the equation you wrote in the last answer to

nd Carl’s reaction function,R

t

C

(q

t?1

S

)= 1;200?q

t?1

S

=2,(Hint:

This is a linear expression of the form a?bq

t?1

S

,You have to nd the

constants a and b.)

324 OLIGOPOLY (Ch,27)

(f) Suppose that Simon makes his decisions in the same way that Carl

does,Notice that the problem is completely symmetric in the roles played

by Carl and Simon,Therefore without even calculating it,we can guess

that Simon’s reaction function is R

t

S

(q

t?1

C

)= 1;200?q

t?1

C

=2,(Of

course,if you don’t like to guess,you could work this out by following

similar steps to the ones you used to nd Carl’s reaction function.)

(g) Suppose that in year 1,Carl produced 200 pumpkins and Simon pro-

duced 1,000 pumpkins,In year 2,how many would Carl produce?

700,How many would Simon produce? 1,100,In year 3,how

many would Carl produce? 650,How many would Simon produce?

850,Use a calculator or pen and paper to work out several more

terms in this series,To what level of output does Carl’s output appear

to be converging? 800 How about Simon’s? 800.

(h) Write down two simultaneous equations that could be solved to nd

outputs q

S

and q

C

such that,if Carl is producing q

C

and Simon is produc-

ing q

S

,then they will both want to produce the same amount in the next

period,(Hint,Use the reaction functions.) q

s

=1;200?q

C

=2

and q

C

=1;200?q

S

=2.

(i) Solve the two equations you wrote down in the last part for an equi-

librium output for each farmer,Each farmer,in Cournot equilibrium,

produces 800 units of output,The total amount of pumpkins brought

to the Farmers’ Market in Lake Witchisit is 1,600,The price of

pumpkins in that market is $1,How much pro t does each farmer

make? $400.

27.2 (0) Suppose that the pumpkin market in Lake Witchisit is as

we described it in the last problem except for one detail,Every spring,

the snow thaws o of Carl’s pumpkin eld a week before it thaws o of

Simon’s,Therefore Carl can plant his pumpkins one week earlier than

Simon can,Now Simon lives just down the road from Carl,and he can

tell by looking at Carl’s elds how many pumpkins Carl planted and how

many Carl will harvest in the fall,(Suppose also that Carl will sell every

pumpkin that he produces.) Therefore instead of assuming that Carl will

sell the same amount of pumpkins that he did last year,Simon sees how

many Carl is actually going to sell this year,Simon has this information

before he makes his own decision about how many to plant.

NAME 325

(a) If Carl plants enough pumpkins to yield q

t

C

this year,then Simon

knows that the pro t-maximizing amount to produce this year is q

t

S

=

Hint,Remember the reaction functions you found in the last problem.

1;200?q

t

C

=2.

(b) When Carl plants his pumpkins,he understands how Simon will make

his decision,Therefore Carl knows that the amount that Simon will

produce this year will be determined by the amount that Carl produces.

In particular,if Carl’s output is q

t

C

,then Simon will produce and sell

1;200?q

t

C

=2 and the total output of the two producers will be

1;200 +q

t

C

=2,Therefore Carl knows that if his own output is q

C

,

the price of pumpkins in the market will be 1:25?q

t

C

=3;200.

(c) In the last part of the problem,you found how the price of pumpkins

this year in the Farmers’ Market is related to the number of pumpkins

that Carl produces this year,Now write an expression for Carl’s total

revenue in year t as a function of his own output,q

t

C

,1:25q

t

C

(q

t

C

)

2

=3;200,Write an expression for Carl’s marginal revenue in

year t as a function of q

t

C

,1:25?q

t

C

=1;600.

(d) Find the pro t-maximizing output for Carl,1,200,Find the

pro t-maximizing output for Simon,600,Find the equilibrium price

of pumpkins in the Lake Witchisit Farmers’ Market,$7/8,How

much pro t does Carl make? $450,How much pro t does Simon

make? $225,An equilibrium of the type we discuss here is known

as a Stackleberg equilibrium.

(e) If he wanted to,it would be possible for Carl to delay his plant-

ing until the same time that Simon planted so that neither of them

would know the other’s plans for this year when he planted,Would

it be in Carl’s interest to do this? Explain,(Hint,What are Carl’s

pro ts in the equilibrium above? How do they compare with his prof-

its in Cournot equilibrium?) No,Carl’s profits in

Stackleberg equilibrium are larger than

in Cournot equilibrium,So if the output

326 OLIGOPOLY (Ch,27)

when neither knows the other’s output this

year until after planting time is a Cournot

equilibrium,Carl will want Simon to know

his output.

27.3 (0) Suppose that Carl and Simon sign a marketing agreement.

They decide to determine their total output jointly and to each produce

the same number of pumpkins,To maximize their joint pro ts,how many

pumpkins should they produce in toto? 1,200,How much does each

one of them produce? 600,How much pro t does each one of them

make? 450.

27.4 (0) The inverse market demand curve for bean sprouts is given by

P(Y) = 100?2Y,and the total cost function for any rm in the industry

is given by TC(y)=4y.

(a) The marginal cost for any rm in the industry is equal to $4,The

change in price for a one-unit increase in output is equal to $?2.

(b) If the bean-sprout industry were perfectly competitive,the industry

output would be 48,and the industry price would be $4.

(c) Suppose that two Cournot rms operated in the market,The reaction

function for Firm 1 would be y

1

=24?y

2

=2,(Reminder,Unlike

the example in your textbook,the marginal cost is not zero here.) The

reaction function of Firm 2 would be y

2

=24?y

1

=2,If the rms

were operating at the Cournot equilibrium point,industry output would

be 32,each rm would produce 16,and the market price

would be $36.

(d) For the Cournot case,draw the two reaction curves and indicate the

equilibrium point on the graph below.

NAME 327

0 6 12 18 24

6

12

18

y1

y2

24

e

Firm 1's reaction

function

Firm 2's

reaction

function

(e) If the two rms decided to collude,industry output would be 24

and the market price would equal $52.

(f) Suppose both of the colluding rms are producing equal amounts of

output,If one of the colluding rms assumes that the other rm would

not react to a change in industry output,what would happen to a rm’s

own pro ts if it increased its output by one unit? Profits would

increase by $22.

(g) Suppose one rm acts as a Stackleberg leader and the other rm

behaves as a follower,The maximization problem for the leader can be

written as max

y

1

[100?2(y

1

+24?y

1

=2)]y

1

4y

1

.

Solving this problem results in the leader producing an output of

24 and the follower producing 12,This implies an industry

output of 36 and price of $28.

27.5 (0) Grinch is the sole owner of a mineral water spring that costlessly

burbles forth as much mineral water as Grinch cares to bottle,It costs

Grinch $2 per gallon to bottle this water,The inverse demand curve for

Grinch’s mineral water is p = $20?:20q,wherep is the price per gallon

and q is the number of gallons sold.

328 OLIGOPOLY (Ch,27)

(a) Write down an expression for pro ts as a function of q,(q)=

(20?:20q)q?2q,Find the pro t-maximizing choice of q for

Grinch,45.

(b) What price does Grinch get per gallon of mineral water if he produces

the pro t-maximizing quantity? $11,How much pro t does he make?

$405.

(c) Suppose,now,that Grinch’s neighbor,Grubb nds a mineral spring

that produces mineral water that is just as good as Grinch’s water,but

that it costs Grubb $6 a bottle to get his water out of the ground and

bottle it,Total market demand for mineral water remains as before.

Suppose that Grinch and Grubb each believe that the other’s quantity

decision is independent of his own,What is the Cournot equilibrium out-

put for Grubb? 50=3,What is the price in the Cournot equilibrium?

$9.33.

27.6 (1) Albatross Airlines has a monopoly on air travel between Peoria

and Dubuque,If Albatross makes one trip in each direction per day,the

demand schedule for round trips is q = 160?2p,whereq is the number of

passengers per day,(Assume that nobody makes one-way trips.) There

is an \overhead" xed cost of $2,000 per day that is necessary to fly the

airplane regardless of the number of passengers,In addition,there is a

marginal cost of $10 per passenger,Thus,total daily costs are $2;000+10q

if the plane flies at all.

(a) On the graph below,sketch and label the marginal revenue curve,and

the average and marginal cost curves.

020406080

20

40

60

Q

MR,MC

80

mc

mr

ac

NAME 329

(b) Calculate the pro t-maximizing price and quantity and total daily

pro ts for Albatross Airlines,p = 45,q = 70,=

$450 per day.

(c) If the interest rate is 10% per year,how much would someone be will-

ing to pay to own Albatross Airlines’s monopoly on the Dubuque-Peoria

route,(Assuming that demand and cost conditions remain unchanged

forever.) About $1.6 million.

(d) If another rm with the same costs as Albatross Airlines were to enter

the Dubuque-Peoria market and if the industry then became a Cournot

duopoly,would the new entrant make a pro t? No; losses

would be about $900 per day.

(e) Suppose that the throbbing night life in Peoria and Dubuque becomes

widely known and in consequence the population of both places doubles.

As a result,the demand for airplane trips between the two places dou-

bles to become q = 320?4p,Suppose that the original airplane had a

capacity of 80 passengers,If AA must stick with this single plane and if

no other airline enters the market,what price should it charge to maxi-

mize its output and how much pro t would it make? p = $60,=

$2,000.

(f) Let us assume that the overhead costs per plane are constant regardless

of the number of planes,If AA added a second plane with the same costs

and capacity as the rst plane,what price would it charge? $45.

How many tickets would it sell? 140,How much would its pro ts

be? $900,If AA could prevent entry by another competitor,would

it choose to add a second plane? No.

(g) Suppose that AA stuck with one plane and another rm entered the

market with a plane of its own,If the second rm has the same cost

function as the rst and if the two rms act as Cournot oligopolists,what

will be the price,$40,quantities,80,and pro ts? $400.

27.7 (0) Alex and Anna are the only sellers of kangaroos in Sydney,

Australia,Anna chooses her pro t-maximizing number of kangaroos to

sell,q

1

,based on the number of kangaroos that she expects Alex to sell.

Alex knows how Anna will react and chooses the number of kangaroos that

330 OLIGOPOLY (Ch,27)

she herself will sell,q

2

,after taking this information into account,The

inverse demand function for kangaroos is P(q

1

+q

2

)=2;000?2(q

1

+q

2

).

It costs $400 to raise a kangaroo to sell.

(a) Alex and Anna are Stackelberg competitors,Alex is the leader

and Anna is the follower.

(b) If Anna expects Alex to sell q

2

kangaroos,what will her own marginal

revenue be if she herself sells q

1

kangaroos? MR(q

1

+ q

2

)=

2;000?4q

1

2q

2

.

(c) What is Anna’s reaction function,R(q

2

)? R(q

2

) = 400?

1=2q

2

.

(d) Now if Alex sells q

2

kangaroos,what is the total number of kangaroos

that will be sold? 400 + 1=2q

2

,What will be the market price as

a function of q

2

only? P(q

2

)=1;200?q

2

.

(e) What is Alex’s marginal revenue as a function of q

2

only?

MR(q

2

)=1;200? 2q

2

,How many kangaroos will Alex

sell? 400,How many kangaroos will Anna sell? 200,What will

the industry price be? $800.

27.8 (0) Consider an industry with the following structure,There are

50 rms that behave in a competitive manner and have identical cost

functions given by c(y)=y

2

=2,There is one monopolist that has 0

marginal costs,The demand curve for the product is given by

D(p)=1;000?50p:

(a) What is the supply curve of one of the competitive rms? y = p.

The total supply from the competitive sector at price p is S(p)= 50p.

(b) If the monopolist sets a price p,the amount that it can sell is D

m

(p)=

1;000?100p.

NAME 331

(c) The monopolist’s pro t-maximizing output is y

m

= 500,What

is the monopolist’s pro t-maximizing price? p =5.

(d) How much output will the competitive sector provide at this price?

50 5 = 250,What will be the total amount of output sold in

this industry? y

m

+y

c

= 750.

27.9 (0) Consider a market with one large rm and many small rms.

The supply curve of the small rms taken together is

S(p) = 100 +p:

The demand curve for the product is

D(p) = 200?p:

The cost function for the one large rm is

c(y)=25y:

(a) Suppose that the large rm is forced to operate at a zero level of

output,What will be the equilibrium price? 50,What will be the

equilibrium quantity? 150.

(b) Suppose now that the large rm attempts to exploit its market power

and set a pro t-maximizing price,In order to model this we assume that

customers always go rst to the competitive rms and buy as much as

they are able to and then go to the large rm,In this situation,the

equilibrium price will be $37.50,The quantity supplied by the

large rm will be 25,and the equilibrium quantity supplied by the

competitive rms will be 137.5.

(c) What will be the large rm’s pro ts? $312.50.

(d) Finally suppose that the large rm could force the competitive rms

out of the business and behave as a real monopolist,What will be the

equilibrium price? 225=2,What will be the equilibrium quantity?

175=2,What will be the large rm’s pro ts? (175=2)

2

.

332 OLIGOPOLY (Ch,27)

Calculus 27.10 (2) In a remote area of the American Midwest before the railroads

arrived,cast iron cookstoves were much desired,but people lived far apart,

roads were poor,and heavy stoves were expensive to transport,Stoves

could be shipped by river boat to the town of Bouncing Springs,Missouri.

Ben Kinmore was the only stove dealer in Bouncing Springs,He could

buy as many stoves as he wished for $20 each,delivered to his store.

The only farmers who traded in Bouncing Springs lived along a road that

ran east and west through town,Along that road,there was one farm

every mile and the cost of hauling a stove was $1 per mile,There were

no other stove dealers on the road in either direction,The owners of

every farm along the road had a reservation price of $120 for a cast iron

cookstove,That is,any of them would be willing to pay up to $120 to

have a stove rather than to not have one,Nobody had use for more than

one stove,Ben Kinmore charged a base price of $p for stoves and added

to the price the cost of delivery,For example,if the base price of stoves

was $40 and you lived 45 miles west of Bouncing Springs,you would have

to pay $85 to get a stove,$40 base price plus a hauling charge of $45.

Since the reservation price of every farmer was $120,it follows that if the

base price were $40,any farmer who lived within 80 miles of Bouncing

Springs would be willing to pay $40 plus the price of delivery to have a

cookstove,Therefore at a base price of $40,Ben could sell 80 cookstoves

to the farmers living west of him,Similarly,if his base price is $40,he

could sell 80 cookstoves to the farmers living within 80 miles to his east,

for a total of 160 cookstoves.

(a) If Ben set a base price of $p for cookstoves where p<120,and if he

charged $1 a mile for delivering them,what would be the total number of

cookstoves he could sell? 2(120?p),(Remember to count the ones

he could sell to his east as well as to his west.) Assume that Ben has no

other costs than buying the stoves and delivering them,Then Ben would

make a pro t of p?20 per stove,Write Ben’s total pro t as a function

of the base price,$p,that he charges,2(120?p)(p?20) =

2(140p?p

2

2;400).

(b) Ben’s pro t-maximizing base price is $70,(Hint,You just wrote

pro ts as a function of prices,Now di erentiate this expression for pro ts

with respect to p.) Ben’s most distant customer would be located at a

distance of 50 miles from him,Ben would sell 100 cookstoves

and make a total pro t of $5,000.

(c) Suppose that instead of setting a single base price and making all

buyers pay for the cost of transportation,Ben o ers free delivery of cook-

stoves,He sets a price $p and promises to deliver for free to any farmer

who lives within p?20 miles of him,(He won’t deliver to anyone who lives

NAME 333

further than that,because it then costs him more than $p to buy a stove

and deliver it.) If he is going to price in this way,how high should he set

p? $120,How many cookstoves would Ben deliver? 200,How

much would his total revenue be? $24,000 How much would his

total costs be,including the cost of deliveries and the cost of buying the

stoves? $14,000,(Hint,What is the average distance that he has

to haul a cookstove?) How much pro t would he make? $10,000.

Can you explain why it is more pro table for Ben to use this pricing

scheme where he pays the cost of delivery himself rather than the scheme

where the farmers pay for their own deliveries? If Ben pays

for delivery,he can price-discriminate

between nearby farmers and faraway ones.

He charges a higher price,net of transport

cost,to nearby farmers and a lower net

price to faraway farmers,who are willing

to pay less net of transport cost.

Calculus 27.11 (2) Perhaps you wondered what Ben Kinmore,who lives o in

the woods quietly collecting his monopoly pro ts,is doing in this chapter

on oligopoly,Well,unfortunately for Ben,before he got around to selling

any stoves,the railroad built a track to the town of Deep Furrow,just 40

miles down the road,west of Bouncing Springs,The storekeeper in Deep

Furrow,Huey Sunshine,was also able to get cookstoves delivered by train

to his store for $20 each,Huey and Ben were the only stove dealers on

the road,Let us concentrate our attention on how they would compete

for the customers who lived between them,We can do this,because Ben

can charge di erent base prices for the cookstoves he ships east and the

cookstoves he ships west,So can Huey.

Suppose that Ben sets a base price,p

B

,for stoves he sends west

and adds a charge of $1 per mile for delivery,Suppose that Huey sets

a base price,p

H

,for stoves he sends east and adds a charge of $1 per

mile for delivery,Farmers who live between Ben and Huey would buy

from the seller who is willing to deliver most cheaply to them (so long as

the delivered price does not exceed $120),If Ben’s base price is p

B

and

Huey’s base price is p

H

,somebody who lives x miles west of Ben would

have to pay a total of p

B

+ x to have a stove delivered from Ben and

p

H

+(40?x) to have a stove delivered by Huey.

(a) If Ben’s base price is p

B

and Huey’s is p

H

,write down an equation that

could be solved for the distance x

to the west of Bouncing Springs that

334 OLIGOPOLY (Ch,27)

Ben’s market extends,p

B

+x

= p

H

+(40?x

),If Ben’s base

price is p

B

and Huey’s is p

H

,then Ben will sell 20 + (p

H

p

B

)=2

cookstoves and Huey will sell 20 + (p

B

p

H

)=2 cookstoves.

(b) Recalling that Ben makes a pro t of p

B

20 on every cookstove that

he sells,Ben’s pro ts can be expressed as the following function of p

B

and p

H

,(20 + (p

H

p

B

)=2)(p

B

20).

(c) If Ben thinks that Huey’s price will stay at p

H

,no matter what price

Ben chooses,what choice of p

B

will maximize Ben’s pro ts? p

B

=

30 + p

H

=2,(Hint,Set the derivative of Ben’s pro ts with respect

to his price equal to zero.) Suppose that Huey thinks that Ben’s price

will stay at p

B

,no matter what price Huey chooses,what choice of p

H

will maximize Huey’s pro ts? p

H

=30+p

B

=2,(Hint,Use the

symmetry of the problem and the answer to the last question.)

(d) Can you nd a base price for Ben and a base price for Huey such that

each is a pro t-maximizing choice given what the other guy is doing?

(Hint,Find prices p

B

and p

H

that simultaneously solve the last two

equations.) p

B

= p

H

=60,How many cookstoves does Ben sell

to farmers living west of him? 20,How much pro t does he make on

these sales? $800.

(e) Suppose that Ben and Huey decided to compete for the customers

who live between them by price discriminating,Suppose that Ben o ers

to deliver a stove to a farmer who lives x miles west of him for a price

equal to the maximum of Ben’s total cost of delivering a stove to that

farmer and Huey’s total cost of delivering to the same farmer less 1 penny.

Suppose that Huey o ers to deliver a stove to a farmer who lives x miles

west of Ben for a price equal to the maximum of Huey’s own total cost of

delivering to this farmer and Ben’s total cost of delivering to him less a

penny,For example,if a farmer lives 10 miles west of Ben,Ben’s total cost

of delivering to him is $30,$20 to get the stove and $10 for hauling it 10

miles west,Huey’s total cost of delivering it to him is $50,$20 to get the

stove and $30 to haul it 30 miles east,Ben will charge the maximum of

his own cost,which is $30,and Huey’s cost less a penny,which is $49.99.

The maximum of these two numbers is $49.99,Huey will charge the

maximum of his own total cost of delivering to this farmer,which is $50,

and Ben’s cost less a penny,which is $29.99,Therefore Huey will charge

$50.00 to deliver to this farmer,This farmer will buy from Ben

NAME 335

whose price to him is cheaper by one penny,When the two merchants

have this pricing policy,all farmers who live within 20 miles of

Ben will buy from Ben and all farmers who live within 20 miles

of Huey will buy from Huey,A farmer who lives x miles west of Ben

and buys from Ben must pay 59:99?x dollars to have a cookstove

delivered to him,A farmer who lives x miles east of Huey and buys from

Huey must pay 59:99?x for delivery of a stove,On the graph

below,use blue ink to graph the cost to Ben of delivering to a farmer who

lives x miles west of him,Use red ink to graph the total cost to Huey

of delivering a cookstove to a farmer who lives x miles west of Ben,Use

pencil to mark the lowest price available to a farmer as a function of how

far west he lives from Ben.

010203040

20

40

60

Miles west of Ben

Dollars

80

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Blue line

Red line

Pencil line

Ben's profit

Huey's profit

(f) With the pricing policies you just graphed,which farmers get stoves

delivered most cheaply,those who live closest to the merchants or those

who live midway between them? Those who live midway

between them,On the graph you made,shade in the area rep-

resenting each merchant’s pro ts,How much pro ts does each merchant

make? $400,If Ben and Huey are pricing in this way,is there any

way for either of them to increase his pro ts by changing the price he

charges to some farmers? No.

336 OLIGOPOLY (Ch,27)

Chapter 28 NAME

Game Theory

Introduction,In this introduction we o er three examples of two-person

games,The rst game has a dominant strategy equilibrium,The second

has a Nash equilibrium in pure strategies that is not a dominant strategy

equilibrium,The third has no pure strategy Nash equilibrium,but it does

have a mixed strategy equilibrium.

Example,Albert and Victoria are roommates,Each of them prefers a

clean room to a dirty room,but neither likes to clean the room,If both

clean the room,they each get a payo of 5,If one cleans and the other

doesn’t clean the room,the person who does the cleaning has a utility of

0,and the person who doesn’t clean the room has a utility of 8,If neither

cleans the room,the room stays a mess and each has a utility of 1,The

payo s from the strategies \Clean" and \Don’t Clean" are shown in the

box below.

Clean Room{Dirty Room

Albert

Victoria

Clean Don’t Clean

Clean 5;5 0;8

Don’t Clean 8;0 1;1

In this game,notice that if Victoria chooses to clean,then Albert

will be better o not cleaning than he would be if he chose to clean.

Likewise if Victoria chooses not to clean,Albert is better o not clean-

ing than cleaning,Therefore \Don’t Clean" is a dominant strategy for

Albert,Similar reasoning shows that no matter what Albert chooses to

do,Victoria is better o if she chooses \Don’t Clean." Therefore the out-

come where both roommates choose \Don’t Clean" is a dominant strategy

equilibrium,It is interesting to notice that this is true,even though both

persons would be better o if they both chose the strategy \Clean."

Example,This game is set in the South Paci c in 1943,Admiral Imamura

must transport Japanese troops from the port of Rabaul in New Britain,

across the Bismarck Sea to New Guinea,The Japanese fleet could either

travel north of New Britain,where it is likely to be foggy,or south of

New Britain,where the weather is likely to be clear,U.S,Admiral Ken-

ney hopes to bomb the troop ships,Kenney has to choose whether to

338 GAME THEORY (Ch,28)

concentrate his reconnaissance aircraft on the Northern or the Southern

route,Once he nds the convoy,he can bomb it until its arrival in New

Guinea,Kenney’s sta has estimated the number of days of bombing

time for each of the outcomes,The payo s to Kenney and Imamura from

each outcome are shown in the box below,The game is modeled as a

\zero-sum game." For each outcome,Imamura’s payo is the negative of

Kenney’s payo,

TheBattleoftheBismarckSea

Kenney

Imamura

North South

North 2;?2 2;?2

South 1;?1 3;?3

This game does not have a dominant strategy equilibrium,since there

is no dominant strategy for Kenney,His best choice depends on what Ima-

mura does,The only Nash equilibrium for this game is where Imamura

chooses the northern route and Kenney concentrates his search on the

northern route,To check this,notice that if Imamura goes North,then

Kenney gets an expected two days of bombing if he (Kenney) chooses

North and only one day if he (Kenney) chooses South,Furthermore,if

Kenney concentrates on the north,Imamura is indi erent between go-

ing north or south,since he can be expected to be bombed for two days

either way,Therefore if both choose \North," then neither has an incen-

tive to act di erently,You can verify that for any other combination of

choices,one admiral or the other would want to change,As things actually

worked out,Imamura chose the Northern route and Kenney concentrated

his search on the North,After about a day’s search the Americans found

the Japanese fleet and inflicted heavy damage on it.

Some two-player games do not have a \Nash equilibrium in pure

strategies." But every two-player game of the kind we look at has a

Nash equilibrium in mixed strategies,If a player is indi erent between

two strategies,then he is also willing to choose randomly between them.

Sometimes this is just what is needed to give an equilibrium.

Example,A soccer player has been awarded a free kick,The only player

allowed to defend against his kick is the opposing team’s goalie,The

kicker has two possible strategies,He can try to kick the ball into the

right side of the goal or he can try to kick the ball into the left side of the

This example is discussed in Luce and Rai a’s Games and Decisions,

John Wiley,1957 or Dover,1989,We recommend this book to anyone

interested in reading more about game theory.

NAME 339

goal,There is not time for the goalie to determine where the ball is going

before he must commit himself by jumping either to the left or to the

right side of the net,Let us suppose that if the goalie guesses correctly

where the kicker is going to kick,then the goalie always stops the ball.

The kicker has a very accurate shot to the right side of the net,but is not

so good at shooting left,If he kicks to the right side of the net and the

goalie jumps left,the kicker will always score,But the kicker kicks to the

left side of the net and the goalie jumps to the right,then the kicker will

score only half of the time,This story leads us to the following payo

matrix,where if the kicker makes the goal,the kicker gets a payo of 1

and the goalie a payo of 0 and if the kicker does not make the goal,the

goalie gets a payo of 1 and the kicker a payo of 0.

The Free Kick

Goalie

Kicker

Kick Left Kick Right

Jump Left 1;0 0;1

Jump Right,5;:5 1;0

This game has no Nash equilibrium in pure strategies,There is no

combination of actions taken with certainty such that each is making the

best response to the other’s action,The goalie always wants to be where

the kicker is kicking and the kicker always wants to kick where the goalie

isn’t,What we can nd is a pair of equilibrium mixed strategies.

In this mixed strategy equilibrium each player’s strategy is chosen at

random,The kicker will be willing to choose a random strategy only if

the expected payo is the same from kicking to either side,The payo s

from kicking to the right and to the left depend on what the goalie is

doing,Let

G

be the probability that the goalie will jump left and 1?

G

be the probability that he will jump right,The kicker realizes that if he

kicks to the right,he will score when the goalie goes left and he will not

score when the goalie goes right,The expected payo to the kicker if he

kicks to the right is therefore just

G

,If the kicker kicks to the left,then

the only way that he can score is if the goalie jumps right,This happens

with probability 1?

G

,Even then he will only score half the time,So

the expected payo to the kicker from kicking left is,5(1?

G

),These

two expected payo s are equalized when

G

=,5(1?

G

),If we solve

this equation,we nd

G

=1=3,This has to be the probability that the

goalie goes left in a mixed strategy equilibrium.

Now let us nd the probability that the kicker kicks left in a mixed

strategy equilibrium,In equilibrium,the kicker’s probability

K

of kick-

ing left must be such that the goalie gets the same expected payo from

jumping left as from jumping right,The expected payo to the goalie is

340 GAME THEORY (Ch,28)

the probability that the kicker does not score,If the goalie jumps left,

then the kicker will not score if he kicks left and will score if he kicks

right,so the expected payo to the goalie from going left is

K

.Ifthe

goalie jumps right,then with probability (1?

K

),the kicker will kick

right and the goalie will stop the ball,When the kicker is kicking to the

undefended left side of the net,he only makes it half the time,so if the

goalie jumps right,the probability that the kicker kicks left and makes

the kick is only,5

K

,Therefore the expected payo to the goalie from

jumping right is (1?

K

)+:5

K

=1?:5

K

,Equalizing the payo to the

goalie from jumping left or jumping right requires

K

=1?:5

K

.Solving

this equation we nd that in the equilibrium mixed strategy,

K

=2=3.

28.1 (0) Perhaps you have wondered what it could mean that \the meek

shall inherit the earth." While we don’t claim this is always the case,here

is an example where it is true,In a famous experiment,two psychologists

put two pigs|a little one and a big one|into a pen that had a lever at

one end and a trough at the other end,When the lever was pressed,a

serving of pigfeed would appear in a trough at the other end of the pen.

If the little pig would press the lever,then the big pig would eat all of

the pigfeed and keep the little pig from getting any,If the big pig pressed

the lever,there would be time for the little pig to get some of the pigfeed

before the big pig was able to run to the trough and push him away.

Let us represent this situation by a game,in which each pig has two

possible strategies,One strategy is Press the Lever,The other strategy

is Wait at the Trough,If both pigs wait at the trough,neither gets any

feed,If both pigs press the lever,the big pig gets all of the feed and the

little pig gets a poke in the ribs,If the little pig presses the lever and

the big pig waits at the trough,the big pig gets all of the feed and the

little pig has to watch in frustration,If the big pig presses the lever and

the little pig waits at the trough,then the little pig is able to eat 2=3

of the feed before the big pig is able to push him away,The payo s are

as follows,(These numbers are just made up,but their relative sizes are

consistent with the payo s in the Baldwin-Meese experiment.)

Big Pig{Little Pig

Little Pig

Big Pig

Press Wait

Press?1;9?1;10

Wait 6;4 0;0

Baldwin and Meese (1979),\Social Behavior in Pigs Studied by

Means of Operant Conditioning," Animal Behavior

NAME 341

(a) Is there a dominant strategy for the little pig? Yes,Wait,Is

there a dominant strategy for the big pig? No.

(b) Find a Nash equilibrium for this game,Does the game have more than

one Nash equilibrium? The only Nash equilibrium is

where little pig waits and big pig presses.

(Incidentally,while Baldwin and Meese did not interpret this experiment

as a game,the result they observed was the result that would be predicted

by Nash equilibrium.)

(c) Which pig gets more feed in Nash equilibrium? Little pig.

28.2 (0) Consider the following game matrix.

A Game Matrix

Player A

Player B

Left Right

Top a;b c;d

Bottom e;f g;h

(a) If (top,left) is a dominant strategy equilibrium,then we know that

a> e,b> d,c >g,and f >h.

(b) If (top,left) is a Nash equilibrium,then which of the above inequalities

must be satis ed? a>e; b>d.

(c) If (top,left) is a dominant strategy equilibrium must it be a Nash

equilibrium? Why? Yes,A dominant strategy

equilibrium is always a Nash equilibrium.

28.3 (1) This problem is based on an example developed by the biologist

John Maynard Smith to illustrate the uses of game theory in the theory

of evolution,Males of a certain species frequently come into conflict with

other males over the opportunity to mate with females,If a male runs

into a situation of conflict,he has two alternative \strategies." A male

342 GAME THEORY (Ch,28)

can play \Hawk" in which case he will ght the other male until he either

wins or is badly hurt,Or he can play \Dove," in which case he makes

a display of bravery but retreats if his opponent starts to ght,If an

animal plays Hawk and meets another male who is playing Hawk,they

both are seriously injured in battle,If he is playing Hawk and meets an

animal who is playing Dove,the Hawk gets to mate with the female and

the Dove slinks o to celibate contemplation,If an animal is playing Dove

and meets another Dove,they both strut around for a while,Eventually

the female either chooses one of them or gets bored and wanders o,The

expected payo s to each of two males in a single encounter depend on

which strategy each adopts,These payo s are depicted in the box below.

The Hawk-Dove Game

Animal A

Animal B

Hawk Dove

Hawk?5;?5 10;0

Dove 0;10 4;4

(a) Now while wandering through the forest,a male will encounter many

conflict situations of this type,Suppose that he cannot tell in advance

whether another animal that he meets will behave like a Hawk or like

a Dove,The payo to adopting either strategy oneself depends on the

proportion of the other guys that is Hawks and the proportion that is

Doves,For example,suppose all of the other males in the forest act

like Doves,Any male that acted like a Hawk would nd that his rival

always retreated and would therefore enjoy a payo of 10 on every

encounter,If a male acted like a Dove when all other males acted like

Doves,he would receive an average payo of 4.

(b) If strategies that are more pro table tend to be chosen over strategies

that are less pro table,explain why there cannot be an equilibrium in

which all males act like Doves,If you know that you

are meeting a Dove,it pays to be a Hawk.

(c) If all the other males acted like Hawks,then a male who adopted the

Hawk strategy would be sure to encounter another Hawk and would get

a payo of?5,If instead,this male adopted the Dove strategy,he

would again be sure to encounter a Hawk,but his payo would be 0.

NAME 343

(d) Explain why there could not be an equilibrium where all of the an-

imals acted like Hawks,If everyone plays Hawk,it

would be profitable to play Dove.

(e) Since there is not an equilibrium in which everybody chooses the same

strategy,we might ask whether there might be an equilibrium in which

some fraction of the males chose the Hawk strategy and the rest chose

the Dove strategy,Suppose that the fraction of a large male population

that chooses the Hawk strategy is p,Then if one acts like a Hawk,the

fraction of one’s encounters in which he meets another Hawk is about p

and the fraction of one’s encounters in which he meets a Dove is about

1?p,Therefore the average payo to being a Hawk when the fraction of

Hawks in the population is p,mustbep (?5) + (1?p) 10 = 10?15p.

Similarly,if one acts like a Dove,the probability of meeting a Hawk is

about p and the probability of meeting another Dove is about (1?p).

Therefore the average payo to being a Dove when the proportion of

Hawks in the population is p will be p 0+(1?p) 4.

(f) Write an equation that states that when the proportion of the popu-

lation that acts like Hawks is p,the payo to Hawks is the same as the

payo s to Doves,4?4p =10?15p.

(g) Solve this equation for the value of p such that at this value Hawks

do exactly as well as Doves,This requires that p = 6=11.

(h) On the axes below,use blue ink to graph the average payo to the

strategy Dove when the proportion of the male population who are Hawks

is p,Use red ink to graph the average payo to the strategy,Hawk,when

the proportion of the male population who are Hawks is p,Label the

equilibrium proportion in your diagram by E.

344 GAME THEORY (Ch,28)

0255075100

2

4

6

Percentage of hawks

Payoff

8

Blue

Line

Red Line

e

(i) If the proportion of Hawks is slightly greater than E,whichstrat-

egy does better? Dove,If the proportion of Hawks is slightly less

than E,which strategy does better? Hawk,If the more pro table

strategy tends to be adopted more frequently in future plays,then if the

strategy proportions are out of equilibrium,will changes tend to move the

proportions back toward equilibrium or further away from equilibrium?

Closer.

28.4 (1) Evangeline and Gabriel met at a freshman mixer,They want

desperately to meet each other again,but they forgot to exchange names

or phone numbers when they met the rst time,There are two possible

strategies available for each of them,These are Go to the Big Party or

Stay Home and Study,They will surely meet if they both go to the party,

and they will surely not otherwise,The payo to meeting is 1,000 for

each of them,The payo to not meeting is zero for both of them,The

payo s are described by the matrix below.

Close Encounters of the Second Kind

Evangeline

Gabriel

Go to Party Stay Home

Go to Party 1000;1000 0;0

Stay Home 0;0 0;0

NAME 345

(a) A strategy is said to be a weakly dominant strategy for a player if

the payo from using this strategy is at least as high as the payo from

using any other strategy,Is there any outcome in this game where both

players are using weakly dominant strategies? The only one is

(top,left).

(b) Find all of the pure-strategy Nash equilibria for this game,There

are two,(top,left) and (bottom,right).

(c) Do any of the pure Nash equilibria that you found seem more rea-

sonable than others? Why or why not? Although (bottom,

right) is a Nash equilibrium,it seems a

silly one,If either player believes that

there is any chance that the other will go

to the party,he or she will also go.

(d) Let us change the game a little bit,Evangeline and Gabriel are still

desperate to nd each other,But now there are two parties that they

might go to,There is a little party at which they would be sure to meet

if they both went there and a huge party at which they might never see

each other,The expected payo to each of them is 1,000 if they both go

to the little party,Since there is only a 50-50 chance that they would nd

each other at the huge party,the expected payo to each of them is only

500,If they go to di erent parties,the payo to both of them is zero.

The payo matrix for this game is:

More Close Encounters

Evangeline

Gabriel

Little Party Big Party

Little Party 1000;1000 0;0

Big Party 0;0 500;500

346 GAME THEORY (Ch,28)

(e) Does this game have a dominant strategy equilibrium? No,What

are the two Nash equilibria in pure strategies? (1) Both go

to the little party,(2) Both go to the big

party.

(f) One of the Nash equilibria is Pareto superior to the other,Suppose

that each person thought that there was some slight chance that the

other would go to the little party,Would that be enough to convince

them both to attend the little party? No,Can you think of any rea-

son why the Pareto superior equilibrium might emerge if both players

understand the game matrix,if both know that the other understands

it,and each knows that the other knows that he or she understands the

game matrix? If both know the game matrix and

each knows that the other knows it,then

each may predict the other will choose the

little party.

28.5 (1) This is a famous game,known to game theorists as \The Battle

of the Sexes." The story goes like this,Two people,let us call them

Michelle and Roger,although they greatly enjoy each other’s company,

have very di erent tastes in entertainment,Roger’s tastes run to ladies’

mud wrestling,while Michelle prefers Italian opera,They are planning

their entertainment activities for next Saturday night,For each of them,

there are two possible actions,go to the wrestling match or go to the

opera,Roger would be happiest if both of them went to see mud wrestling.

His second choice would be for both of them to go to the opera,Michelle

would prefer if both went to the opera,Her second choice would be that

they both went to see the mud wrestling,They both think that the worst

outcome would be that they didn’t agree on where to go,If this happened,

they would both stay home and sulk.

BattleoftheSexes

Roger

Michelle

Wrestling Opera

Wrestling 2;1 0;0

Opera 0;0 1;2

NAME 347

(a) Is the sum of the payo s to Michelle and Roger constant over all

outcomes? No,(If so,this is called a \zero-sum game." Otherwise it is

called a \nonzero sum game.") Does this game have a dominant strategy

equilibrium? No.

(b) Find two Nash equilibria in pure strategies for this game,Both

go to opera,Both go to mud wrestling.

(c) Find a Nash equilibrium in mixed strategies,Michele

chooses opera with probability 2=3 and

wrestling with probability 1=3,Roger chooses

opera with probability 1=3 and mud wrestling

with probability 2=3.

28.6 (1) This is another famous two-person game,known to game the-

orists as \Chicken." Two teenagers in souped-up cars drive toward each

other at great speed,The rst one to swerve out of the road is \chicken."

The best thing that can happen to you is that the other guy swerves and

you don’t,Then you are the hero and the other guy is the chicken,If you

both swerve,you are both chickens,If neither swerves,you both end up

in the hospital,A payo matrix for a chicken-type game is the following.

Chicken

Joe Bob

Leroy

Swerve Don’t Swerve

Swerve 1;1 1;2

Don’t Swerve 2;1 0;0

(a) Does this game have a dominant strategy? No,What are the two

Nash equilibria in pure strategies? The two outcomes where

one teenager swervesand the does not.

348 GAME THEORY (Ch,28)

(b) Find a Nash equilibrium in mixed strategies for this game,Play

each strategy with probability 1=2.

28.7 (0) I propose the following game,I flip a coin,and while it is in the

air,you call either heads or tails,If you call the coin correctly,you get

to keep the coin,Suppose that you know that the coin always comes up

heads,What is the best strategy for you to pursue? Always call

heads.

(a) Suppose that the coin is unbalanced and comes up heads 80% of

the time and tails 20% of the time,Now what is your best strategy?

Always call heads.

(b) What if the coin comes up heads 50% of the time and tails 50% of the

time? What is your best strategy? It doesn’t matter.

You can call heads always,tails always,or

randomize your calls.

(c) Now,suppose that I am able to choose the type of coin that I will toss

(where a coin’s type is the probability that it comes up heads),and that

you will know my choice,What type of coin should I choose to minimize

my losses? A fair coin.

(d) What is the Nash mixed strategy equilibrium for this game? (It may

help to recognize that a lot of symmetry exists in the game.) I

choose a fair coin,and you randomize with

50% heads and 50% tails.

28.8 (0) Ned and Ruth love to play \Hide and Seek." It is a simple

game,but it continues to amuse,It goes like this,Ruth hides upstairs or

downstairs,Ned can look upstairs or downstairs but not in both places.

If he nds Ruth,Ned gets one scoop of ice cream and Ruth gets none,If

he does not nd Ruth,Ruth gets one scoop of ice cream and Ned gets

none,Fill in the payo s in the matrix below.

NAME 349

Hide and Seek

Ned

Ruth

Upstairs Downstairs

Upstairs 1;0 0;1

Downstairs 0;1 1;0

(a) Is this a zero-sum game? Yes,What are the Nash equilibria in

pure strategies? There are none.

(b) Find a Nash equilibrium in mixed strategies for this game.

Ruth hides upstairs and Ned searches

upstairs with probability 1/2; Ruth hides

downstairs and Ned searches downstairs with

probability 1/2.

(c) After years of playing this game,Ned and Ruth think of a way to

liven it up a little,Now if Ned nds Ruth upstairs,he gets two scoops of

ice cream,but if he nds her downstairs,he gets one scoop,If Ned nds

Ruth,she gets no ice cream,but if he doesn’t nd her she gets one scoop.

Fill in the payo s in the graph below.

Advanced Hide and Seek

Ned

Ruth

Upstairs Downstairs

Upstairs 2;0 0;1

Downstairs 0;1 1;0

350 GAME THEORY (Ch,28)

(d) Are there any Nash equilibria in pure strategies? No,What mixed

strategy equilibrium can you nd? Ruth hides downstairs

2/3 of the time,Ned looks downstairs 1/2

the time,If both use equilibrium strategies,what fraction of the

time will Ned nd Ruth? 1/2.

28.9 (1) Let’s have another look at the soccer example that was discussed

in the introduction to this section,But this time,we will generalize the

payo matrix just a little bit,Suppose the payo matrix is as follows.

The Free Kick

Goalie

Kicker

Kick Left Kick Right

Jump Left 1;0 0;1

Jump Right 1-p,p 1;0

Now the probability that the kicker will score if he kicks to the left

and the goalie jumps to the right is p,Wewillwanttoseehowthe

equilibrium probabilities change as p changes.

(a) If the goalie jumps left with probability

G

,then if the kicker kicks

right,his probability of scoring is

G

.

(b) If the goalie jumps left with probability

G

,then if the kicker kicks

left,his probability of scoring is p(1?

G

).

(c) Find the probability

G

that makes kicking left and kicking right lead

to the same probability of scoring for the kicker,(Your answer will be a

function of p.)

G

=

p

1+p

.

(d) If the kicker kicks left with probability

K

,then if the goalie jumps

left,the probability that the kicker will not score is

K

.

NAME 351

(e) If the kicker kicks left with probability

K

,then if the goalie jumps

right,the probability that the kicker will not score is (1?p)

K

+

(1?

K

).

(f) Find the probability

K

that makes the payo to the goalie equal from

jumping left or jumping right.

1

1+p

.

(g) The variable p tells us how good the kicker is at kicking the ball

into the left side of the goal when it is undefended,As p increases,does

the equilibrium probability that the kicker kicks to the left increase or

decrease? Decreases,Explain why this happens in a way that

even a TV sports announcer might understand,The better

the kicker’s weak side gets,the less often

the goalie defends the kicker’s good side.

So kicker can kick to good side more often.

28.10 (0) Maynard’s Cross is a trendy bistro that specializes in carpac-

cio and other uncooked substances,Most people who come to Maynard’s

come to see and be seen by other people of the kind who come to May-

nard’s,There is,however,a hard core of 10 customers per evening who

come for the carpaccio and don’t care how many other people come,The

number of additional customers who appear at Maynard’s depends on

how many people they expect to see,In particular,if people expect that

the number of customers at Maynard’s in an evening will be X,then

the number of people who actually come to Maynard’s is Y =10+:8X:

In equilibrium,it must be true that the number of people who actually

attend the restaurant is equal to the number who are expected to attend.

(a) What two simultaneous equations must you solve to nd the equilib-

rium attendance at Maynard’s? y =10+:8x and x = y.

(b) What is the equilibrium nightly attendance? 50.

(c) On the following axes,draw the lines that represent each of the

two equations you mentioned in Part (a),Label the equilibrium atten-

352 GAME THEORY (Ch,28)

dance level.

020406080

20

40

60

x

y

80

X=Y

Y=10+.8X

e

Y=11+.8X

(d) Suppose that one additional carpaccio enthusiast moves to the area.

Like the other 10,he eats at Maynard’s every night no matter how many

others eat there,Write down the new equations determining attendance

at Maynard’s and solve for the new equilibrium number of customers.

y =11+:8x and y = x,sox = y =55.

(e) Use a di erent color ink to draw a new line representing the equa-

tion that changed,How many additional customers did the new steady

customer attract (besides himself)? 4.

(f) Suppose that everyone bases expectations about tonight’s attendance

on last night’s attendance and that last night’s attendance is public knowl-

edge,Then X

t

= Y

t?1

,whereX

t

is expected attendance on day t and

Y

t?1

is actual attendance on day t?1,At any time t,Y

t

=10+:8X

t

.

Suppose that on the rst night that Maynard’s is open,attendance is 20.

What will be attendance on the second night? 26.

(g) What will be the attendance on the third night? 30.8.

(h) Attendance will tend toward some limiting value,What is it? 50.

28.11 (0) Yogi’s Bar and Grill is frequented by unsociable types who

hate crowds,If Yogi’s regular customers expect that the crowd at Yogi’s

will beX,then the number of people who show up at Yogi’s,Y,will be the

larger of the two numbers,120?2X and 0,Thus Y =maxf120?2X;0g:

NAME 353

(a) Solve for the equilibrium attendance at Yogi’s,Draw a diagram de-

picting this equilibrium on the axes below.

020406080

20

40

60

x

y

80

e

X=Y

Y=120-2X

(b) Suppose that people expect the number of customers on any given

night to be the same as the number on the previous night,Suppose that

50 customers show up at Yogi’s on the rst day of business,How many

will show up on the second day? 20,The third day? 80,The

fourth day? 0,The fth day? 120,The ninety-ninth day?

120,The hundredth day? 0.

(c) What would you say is wrong with this model if at least some of Yogi’s

customers have memory spans of more than a day or two?

They’d notice that last night’s attendance

is not a good predictor of tonight’s,If

attendance is low on odd-numbered days and

high on even-numbered days,it would be

smart to adjust by coming on odd-numbered

days.

28.12 (2) Economic ideas and equilibrium analysis have many fascinat-

ing applications in biology,Popular discussions of natural selection and

biological tness often take it for granted that animal traits are selected

for the bene t of the species,Modern thinking in biology emphasizes that

individuals (or strictly speaking,genes) are the unit of selection,A mu-

tant gene that induces an animal to behave in such a way as to help the

354 GAME THEORY (Ch,28)

species at the expense of the individuals that carry that gene will soon

be eliminated,no matter how bene cial that behavior is to the species.

A good illustration is a paper in the Journal of Theoretical Biology,

1979,by H,J,Brockmann,A,Grafen,and R,Dawkins,called \Evo-

lutionarily Stable Nesting Strategy in a Digger Wasp." They maintain

that natural selection results in behavioral strategies that maximize an

individual animal’s expected rate of reproduction over the course of its

lifetime,According to the authors,\Time is the currency which an animal

spends."

Females of the digger wasp Sphex ichneumoneus nest in underground

burrows,Some of these wasps dig their own burrows,After she has dug

her burrow,a wasp goes out to the elds and hunts katydids,These

she stores in her burrow to be used as food for her o spring when they

hatch,When she has accumulated several katydids,she lays a single egg

in the burrow,closes o the food chamber,and starts the process over

again,But digging burrows and catching katydids is time-consuming,An

alternative strategy for a female wasp is to sneak into somebody else’s

burrow while she is out hunting katydids,This happens frequently in

digger wasp colonies,A wasp will enter a burrow that has been dug by

another wasp and partially stocked with katydids,The invader will start

catching katydids,herself,to add to the stock,When the founder and

the invader nally meet,they ght,The loser of the ght goes away and

never comes back,The winner gets to lay her egg in the nest.

Since some wasps dig their own burrows and some invade burrows

begun by others,it is likely that we are observing a biological equilibrium

in which each strategy is as e ective a way for a wasp to use its time for

producing o spring as the other,If one strategy were more e ective than

the other,then we would expect that a gene that led wasps to behave

in the more e ective way would prosper at the expense of genes that led

them to behave in a less e ective way.

Suppose the average nesting episode takes 5 days for a wasp that

digs its own burrow and tries to stock it with katydids,Suppose that the

average nesting episode takes only 4 days for invaders,Suppose that when

they meet,half the time the founder of the nest wins the ght and half

the time the invader wins,Let D be the number of wasps that dig their

own burrows and let I be the number of wasps that invade the burrows

of others,The fraction of the digging wasps that are invaded will be

about

5

4

I

D

,(Assume for the time being that

5

4

I

D

< 1.) Half of the diggers

who are invaded will win their ght and get to keep their burrows,The

fraction of digging wasps who lose their burrows to other wasps is then

1

2

5

4

I

D

=

5

8

I

D

,Assume also that all the wasps who are not invaded by other

wasps will successfully stock their burrows and lay their eggs.

(a) Then the fraction of the digging wasps who do not lose their burrows

is just 1?

5

8

I

D

.

Therefore over a period of 40 days,a wasp who dug her own bur-

row every time would have 8 nesting episodes,Her expected number of

successes would be 8?5

I

D

.

NAME 355

(b) In 40 days,a wasp who chose to invade every time she had a chance

would have time for 10 invasions,Assuming that she is successful half the

time on average,her expected number of successes would be 5.

Write an equation that expresses the condition that wasps who always dig

their own burrows do exactly as well as wasps who always invade burrows

dug by others,8?5

I

D

=5.

(c) The equation you have just written should contain the expression

I

D

.

Solve for the numerical value of

I

D

that just equates the expected number

of successes for diggers and invaders,The answer is

3

5

.

(d) But there is a problem here,the equilibrium we found doesn’t appear

to be stable,On the axes below,use blue ink to graph the expected num-

ber of successes in a 40-day period for wasps that dig their own burrows

every time where the number of successes is a function of

I

D

.Useblack

ink to graph the expected number of successes in a 40-day period for in-

vaders,Notice that this number is the same for all values of

I

D

,Label the

point where these two lines cross and notice that this is equilibrium,Just

to the right of the crossing,where

I

D

is just a little bit bigger than the

equilibrium value,which line is higher,the blue or the black? Black.

At this level of

I

D

,which is the more e ective strategy for any individ-

ual wasp? Invade,Suppose that if one strategy is more e ective

than the other,the proportion of wasps adopting the more e ective one

increases,If,after being in equilibrium,the population got joggled just

a little to the right of equilibrium,would the proportions of diggers and

invaders return toward equilibrium or move further away? Further

away.

Success

e

Blue line

8-5(I/D)

5

Black line

I_

D

356 GAME THEORY (Ch,28)

(e) The authors noticed this likely instability and cast around for possible

changes in the model that would lead to stability,They observed that

an invading wasp does help to stock the burrow with katydids,This may

save the founder some time,If founders win their battles often enough

and get enough help with katydids from invaders,it might be that the

expected number of eggs that a founder gets to lay is an increasing rather

than a decreasing function of the number of invaders,On the axes below,

show an equilibrium in which digging one’s own burrow is an increasingly

e ective strategy as

I

D

increases and in which the payo to invading is

constant over all ratios of

I

D

,Is this equilibrium stable? Yes.

Success

I_

D

e

Chapter 29 NAME

Exchange

Introduction,The Edgeworth box is a thing of beauty,An amazing

amount of information is displayed with a few lines,points and curves,In

fact one can use an Edgeworth box to tell just about everything there is to

say about the case of two traders dealing in two commodities,Economists

know that the real world has more than two people and more than two

commodities,But it turns out that the insights gained from this model

extend nicely to the case of many traders and many commodities,So

for the purpose of introducing the subject of exchange equilibrium,the

Edgeworth box is exactly the right tool,We will start you out with an

example of two gardeners engaged in trade.

Example,Alice and Byron consume two goods,camelias and dahlias.

Alice has 16 camelias and 4 dahlias,Byron has 8 camelias and 8 dahlias.

They consume no other goods,and they trade only with each other,To

describe the possible allocations of flowers,we rst draw a box whose

width is the total number of camelias and whose height is the total number

of dahlias that Alice and Byron have between them,The width of the

box is therefore 16 + 8 = 24 and the height of the box is 4 + 8 = 12.

Dahlias Byron

12

6

0 6 121824

Alice Camelias

Any feasible allocation of flowers between Alice and Byron is fully

described by a single point in the box,Consider,for example,the alloca-

tion where Alice gets the bundle (15;9) and Byron gets the bundle (9;3).

This allocation is represented by the point A =(15;9) in the Edgeworth

box,The distance 15 from A to the left side of the box is the number of

camelias for Alice and the distance 9 from A to the bottom of the box is

the number of dahlias for Alice,This point also determines Byron’s con-

sumption of camelias and dahlias,The distance 9 from A to the right side

of the box is the total number of camelias consumed by Byron,and the

distance from A to the top of the box is the number of dahlias consumed

by Byron,Since the width of the box is the total supply of camelias and

the height of the box is the total supply of dahlias,these conventions en-

sure that any point in the box represents a feasible allocation of the total

358 EXCHANGE (Ch,29)

supply of camelias and dahlias.

It is useful to mark the initial allocation on the Edgeworth box,In

this case,the initial allocation is represented by the point E =(16;4).

Now suppose that Alice’s utility function is U(c;d)=c+2d and Byron’s

utility funtion is U(c;d)=cd,Alice’s indi erence curves will be straight

lines with slope?1=2,The indi erence curve that passes through her

initial endowment,for example,will be a line that runs from the point

(24;0) to the point (0;12),Since Byron has Cobb-Douglas utility,his

indi erence curves will be rectangular hyperbolas,but since quantities

for Byron are measured from the upper right corner of the box,these

indi erence curves will be flipped over as in the diagram.

The Pareto set or contract curve is the set of points where Alice’s

indi erence curves are tangent to Byron’s,There will be tangency if the

slopes are the same,The slope of Alice’s indi erence curve at any point is

1=2,The slope of Byron’s indi erence curve depends on his consumption

of the two goods,When Byron is consuming the bundle (c

B;d

B

),the slope

of his indi erence curve is equal to his marginal rate of substitution,which

is?d

B

=c

B

,Therefore Alice’s and Byron’s indi erence curves will nuzzle

up in a nice tangency whenever?d

B

=c

B

=?1=2,So the Pareto set in

this example is just the diagonal of the Edgeworth box.

Some problems ask you to nd a competitive equilibrium,For an

economy with two goods,the following procedure is often a good way to

calculate equilibrium prices and quantities.

Since demand for either good depends only on the ratio of prices of

good 1 to good 2,it is convenient to set the price of good 1 equal to

1andletp

2

be the price of good 2.

With the price of good 1 held at 1,calculate each consumer’s demand

for good 2 as a function of p

2

.

Write an equation that sets the total amount of good 2 demanded by

all consumers equal to the total of all participants’ initial endowments

of good 2.

Solve this equation for the value of p

2

that makes the demand for

good 2 equal to the supply of good 2,(When the supply of good 2

equals the demand of good 2,it must also be true that the supply of

good 1 equals the demand for good 1.)

Plug this price into the demand functions to determine quantities.

Example,Frank’s utility function is U(x

1;x

2

)=x

1

x

2

and Maggie’s is

U(x

1;x

2

)=minfx

1;x

2

g,Frank’s initial endowment is 0 units of good 1

and 10 units of good 2,Maggie’s initial endowment is 20 units of good 1

and 5 units of good 2,Let us nd a competitive equilibrium for Maggie

and Frank.

Set p

1

= 1 and nd Frank’s and Maggie’s demand functions for good

2 as a function of p

2

,Using the techniques learned in Chapter 6,we

nd that Frank’s demand function for good 2 is m=2p

2

,wherem is his

income,Since Frank’s initial endowment is 0 units of good 1 and 10 units

of good 2,his income is 10p

2

,Therefore Frank’s demand for good 2 is

10p

2

=2p

2

= 5,Since goods 1 and 2 are perfect complements for Maggie,

she will choose to consume where x

1

= x

2

,This fact,together with her

budget constraint implies that Maggie’s demand function for good 2 is

NAME 359

m=(1 + p

2

),Since her endowment is 20 units of good 1 and 5 units of

good 2,her income is 20 + 5p

2

,Therefore at price p

2

,Maggie’s demand

is (20 + 5p

2

)=(1 +p

2

),Frank’s demand plus Maggie’s demand for good 2

adds up to 5 + (20 + 5p

2

)=(1 +p

2

),The total supply of good 2 is Frank’s

10 unit endowment plus Maggie’s 5 unit endowment,which adds to 15

units,Therefore demand equals supply when

5+

(20 + 5p

2

)

(1 +p

2

)

=15:

Solving this equation,one nds that the equilibrium price is p

2

=2,At

the equilibrium price,Frank will demand 5 units of good 2 and Maggie

will demand 10 units of good 2.

29.1 (0) Morris Zapp and Philip Swallow consume wine and books.

Morris has an initial endowment of 60 books and 10 bottles of wine,Philip

has an initial endowment of 20 books and 30 bottles of wine,They have

no other assets and make no trades with anyone other than each other.

For Morris,a book and a bottle of wine are perfect substitutes,His utility

function is U(b;w)=b+w,whereb is the number of books he consumes

and w is the number of bottles of wine he consumes,Philip’s preferences

are more subtle and convex,He has a Cobb-Douglas utility function,

U(b;w)=bw,In the Edgeworth box below,Morris’s consumption is

measured from the lower left,and Philip’s is measured from the upper

right corner of the box.

020406080

20

40

Books

PhilipWine

Morris

e

Blue curve

Red curve

Black

line

(a) On this diagram,mark the initial endowment and label it E.Usered

ink to draw Morris Zapp’s indi erence curve that passes through his initial

endowment,Use blue ink to draw in Philip Swallow’s indi erence curve

that passes through his initial endowment,(Remember that quantities

for Philip are measured from the upper right corner,so his indi erence

curves are \Phlipped over.")

360 EXCHANGE (Ch,29)

(b) At any Pareto optimum,where both people consume some of each

good,it must be that their marginal rates of substitution are equal,No

matter what he consumes,Morris’s marginal rate of substitution is equal

to -1,When Philip consumes the bundle,(b

P;w

P

),his MRS is

w

P

=b

P

,Therefore every Pareto optimal allocation where both

consume positive amounts of both goods satis es the equation w

P

=

b

P

,Use black ink on the diagram above to draw the locus of Pareto

optimal allocations.

(c) At a competitive equilibrium,it will have to be that Morris consumes

some books and some wine,But in order for him to do so,it must be that

the ratio of the price of wine to the price of books is 1,Therefore

we know that if we make books the numeraire,then the price of wine in

competitive equilibrium must be 1.

(d) At the equilibrium prices you found in the last part of the question,

what is the value of Philip Swallow’s initial endowment? 50,At these

prices,Philip will choose to consume 25 books and 25

bottles of wine,If Morris Zapp consumes all of the books and all of the

wine that Philip doesn’t consume,he will consume 55 books and

15 bottles of wine.

(e) At the competitive equilibrium prices that you found above,Morris’s

income is 70,Therefore at these prices,the cost to Morris of con-

suming all of the books and all of the wine that Philip doesn’t consume

is (the same as,more than,less than) the same as his income.

At these prices,can Morris a ord a bundle that he likes better than the

bundle (55;15)? No.

(f) Suppose that an economy consisted of 1,000 people just like Morris

and 1,000 people just like Philip,Each of the Morris types had the same

endowment and the same tastes as Morris,Each of the Philip types had

the same endowment and tastes as Philip,Would the prices that you

found to be equilibrium prices for Morris and Philip still be competitive

equilibrium prices? Yes,If each of the Morris types and each of the

Philip types behaved in the same way as Morris and Philip did above,

would supply equal demand for both wine and books? Yes.

NAME 361

29.2 (0) Consider a small exchange economy with two consumers,Astrid

and Birger,and two commodities,herring and cheese,Astrid’s initial

endowment is 4 units of herring and 1 unit of cheese,Birger’s initial en-

dowment has no herring and 7 units of cheese,Astrid’s utility function is

U(H

A;C

A

)=H

A

C

A

,Birger is a more inflexible person,His utility func-

tion is U(H

B;C

B

)=minfH

B;C

B

g.(HereH

A

and C

A

are the amounts

of herring and cheese for Astrid,and H

B

and C

B

are amounts of herring

and cheese for Birger.)

(a) Draw an Edgeworth box,showing the initial allocation and sketching

in a few indi erence curves,Measure Astrid’s consumption from the lower

left and Birger’s from the upper right,In your Edgeworth box,draw two

di erent indi erence curves for each person,using blue ink for Astrid’s

and red ink for Birger’s.

02468

2

4

Cheese

BirgerHerring

Astrid

e

Blue curves

Red curves

Black

line

(b) Use black ink to show the locus of Pareto optimal allocations,(Hint:

Since Birger is kinky,calculus won’t help much here,But notice that

because of the rigidity of the proportions in which he demands the two

goods,it would be ine cient to give Birger a positive amount of either

good if he had less than that amount of the other good,What does that

tell you about where the Pareto e cient locus has to be?) Pareto

efficient allocations lie on the line with

slope 1 extending from Birger’s corner of

the box.

29.3 (0) Dean Foster Z,Interface and Professor J,Fetid Nightsoil ex-

change bromides and platitudes,Dean Interface’s utility function is

U

I

(B

I;P

I

)=B

I

+2

p

P

I

:

Professor Nightsoil’s utility function is

U

N

(B

N;P

N

)=B

N

+4

p

P

N

:

362 EXCHANGE (Ch,29)

Dean Interface’s initial endowment is 8 bromides and 12 platitudes,Pro-

fessor Nightsoil’s initial endowment is 8 bromides and 4 platitudes.

0481216

4

8

12

Bromides

Platitudes

16

Nightsoil

Interface

e

Red curve

Pencil curve

Blue line

3.2

(a) If Dean Interface consumes P

I

platitudes and B

I

bromides,his mar-

ginal rate of substitution will be?P

1=2

I

,If Professor Nightsoil

consumes P

N

platitudes and B

N

bromides,his marginal rate of substitu-

tion will be?2P

1=2

N

.

(b) On the contract curve,Dean Interface’s marginal rate of substitution

equals Professor Nightsoil’s,Write an equation that states this condition.

p

P

I

=

p

P

N

=2,This equation is especially simple because each

person’s marginal rate of substitution depends only on his consumption

of platitudes and not on his consumption of bromides.

(c) From this equation we see that P

I

=P

N

= 1=4 at all points on the

contract curve,This gives us one equation in the two unknowns P

I

and

P

N

.

(d) But we also know that along the contract curve it must be that P

I

+

P

N

= 16,since the total consumption of platitudes must equal

the total endowment of platitudes.

(e) Solving these two equations in two unknowns,we nd that everywhere

on the contract curve,P

I

and P

N

are constant and equal to P

I

=3:2

and P

N

=12:8.

NAME 363

(f) In the Edgeworth box,label the initial endowment with the letter

E,Dean Interface has thick gray penciled indi erence curves,Profes-

sor Nightsoil has red indi erence curves,Draw a few of these in the

Edgeworth box you made,Use blue ink to show the locus of Pareto op-

timal points,The contract curve is a (vertical,horizontal,diagonal)

horizontal line in the Edgeworth box.

(g) Find the competitive equilibrium prices and quantities,You know

what the prices have to be at competitive equilibrium because you know

what the marginal rates of substitution have to be at every Pareto

optimum,P

I

=3:2,P

N

=12:8,platitude

price/bromide price =

1

p

3:2

.

29.4 (0) A little exchange economy contains just two consumers,named

Ken and Barbie,and two commodities,quiche and wine,Ken’s initial

endowment is 3 units of quiche and 2 units of wine,Barbie’s initial en-

dowment is 1 unit of quiche and 6 units of wine,Ken and Barbie have iden-

tical utility functions,We write Ken’s utility function as,U(Q

K;W

K

)=

Q

K

W

K

and Barbie’s utility function as U(Q

B;W

B

)=Q

B

W

B

,whereQ

K

and W

K

are the amounts of quiche and wine for Ken and Q

B

and W

B

are amounts of quiche and wine for Barbie.

(a) Draw an Edgeworth box below,to illustrate this situation,Put quiche

on the horizontal axis and wine on the vertical axis,Measure goods for

Ken from the lower left corner of the box and goods for Barbie from the

upper right corner of the box,(Be sure that you make the length of the

box equal to the total supply of quiche and the height equal to the total

supply of wine.) Locate the initial allocation in your box,and label it W.

On the sides of the box,label the quantities of quiche and wine for each

of the two consumers in the initial endowment.

364 EXCHANGE (Ch,29)

24

2

4

6

8

0

Ken Quiche

Wine Barbie

w

ce

Black line

Red

curve

Blue

curve

Pareto

efficient

points

(b) Use blue ink to draw an indi erence curve for Ken that shows alloca-

tions in which his utility is 6,Use red ink to draw an indi erence curve

for Barbie that shows allocations in which her utility is 6.

(c) At any Pareto optimal allocation where both consume some of each

good,Ken’s marginal rate of substitution between quiche and wine must

equal Barbie’s,Write an equation that states this condition in terms

of the consumptions of each good by each person,W

B

=Q

B

=

W

K

=Q

K

.

(d) On your graph,show the locus of points that are Pareto e cient.

(Hint,If two people must each consume two goods in the same proportions

as each other,and if together they must consume twice as much wine as

quiche,what must those proportions be?)

(e) In this example,at any Pareto e cient allocation,where both persons

consume both goods,the slope of Ken’s indi erence curve will be?2.

Therefore,since we know that competitive equilibrium must be Pareto

e cient,we know that at a competitive equilibrium,p

Q

=p

W

= 2.

(f) What must be Ken’s consumption bundle in competitive equilibrium?

2 quiche,4 wine,How about Barbie’s consumption bundle?

2 quiche,4 wine,(Hint,You found competitive equilib-

rium prices above,You know Ken’s initial endowment and you know the

NAME 365

equilibrium prices,In equilibrium Ken’s income will be the value of his

endowment at competitive prices,Knowing his income and the prices,

you can compute his demand in competitive equilibrium,Having solved

for Ken’s consumption and knowing that total consumption by Ken and

Barbie equals the sum of their endowments,it should be easy to nd

Barbie’s consumption.)

(g) On the Edgeworth box for Ken and Barbie,draw in the competitive

equilibrium allocation and draw Ken’s competitive budget line (with black

ink).

29.5 (0) Linus Straight’s utility function is U(a;b)=a +2b,wherea

is his consumption of apples and b is his consumption of bananas,Lucy

Kink’s utility function isU(a;b)=minfa;2bg,Lucy initially has 12 apples

and no bananas,Linus initially has 12 bananas and no apples,In the

Edgeworth box below,goods for Lucy are measured from the upper right

corner of the box and goods for Linus are measured from the lower left

corner,Label the initial endowment point on the graph with the letter

E,Draw two of Lucy’s indi erence curves in red ink and two of Linus’s

indi erence curves in blue ink,Use black ink to draw a line through all

of the Pareto optimal allocations.

612

6

12

0

Linus Apples

Bananas Lucy

e

Red curves

Blue

curves

Black line

(a) In this economy,in competitive equilibrium,the ratio of the price of

apples to the price of bananas must be 1/2.

(b) Let a

S

be Linus’s consumption of apples and let b

S

be his consumption

of bananas,At competititive equilibrium,Linus’s consumption will have

to satisfy the budget constraint,a

s

+ 2 b

S

= 24,This gives us

one equation in two unknowns,To nd a second equation,consider Lucy’s

366 EXCHANGE (Ch,29)

consumption,In competitive equilibrium,total consumption of apples

equals the total supply of apples and total consumption of bananas equals

the total supply of bananas,Therefore Lucy will consume 12?a

s

apples

and 12?b

s

bananas,At a competitive equilibrium,Lucy will be

consuming at one of her kink points,The kinks occur at bundles where

Lucy consumes 2 apples for every banana that she consumes.

Therefore we know that

12?a

s

12?b

s

= 2.

(c) You can solve the two equations that you found above to nd the

quantities of apples and bananas consumed in competitive equilibrium

by Linus and Lucy,Linus will consume 6 units of apples and

9 units of bananas,Lucy will consume 6 units of apples

and 3 units of bananas.

29.6 (0) Consider a pure exchange economy with two consumers and

two goods,At some given Pareto e cient allocation it is known that both

consumers are consuming both goods and that consumer A has a marginal

rate of substitution between the two goods of 2,What is consumer B’s

marginal rate of substitution between these two goods? 2.

29.7 (0) Charlotte loves apples and hates bananas,Her utility function

is U(a;b)=a?

1

4

b

2

,wherea is the number of apples she consumes and

b is the number of bananas she consumes,Wilbur likes both apples and

bananas,His utility function is U(a;b)=a+2

p

b,Charlotte has an initial

endowment of no apples and 8 bananas,Wilbur has an initial endowment

of 16 apples and 8 bananas.

(a) On the graph below,mark the initial endowment and label it E.Use

red ink to draw the indi erence curve for Charlotte that passes through

this point,Use blue ink to draw the indi erence curve for Wilbur that

passes through this point.

NAME 367

0481216

4

8

12

Bananas

Apples

16

Wilbur

Charlotte

e

Red

line

Blue line

Black line

(b) If Charlotte hates bananas and Wilbur likes them,how many bananas

can Charlotte be consuming at a Pareto optimal allocation? 0.

On the graph above,use black ink to mark the locus of Pareto optimal

allocations of apples and bananas between Charlotte and Wilbur.

(c) We know that a competitive equilibrium allocation must be Pareto

optimal and the total consumption of each good must equal the total

supply,so we know that at a competitive equilibrium,Wilbur must be

consuming 16 bananas,If Wilbur is consuming this number of

bananas,his marginal utility for bananas will be 1/4 and his marginal

utility of apples will be 1,If apples are the numeraire,then

the only price of bananas at which he will want to consume exactly 16

bananas is 1/4,In competitive equilibrium,for the Charlotte-Wilbur

economy,Wilbur will consume 16 bananas and 14 apples

and Charlotte will consume 0 bananas and 2 apples.

29.8 (0) Mutt and Je have 8 cups of milk and 8 cups of juice to

divide between themselves,Each has the same utility function given by

u(m;j)=maxfm;jg,wherem is the amount of milk and j is the amount

of juice that each has,That is,each of them cares only about the larger

of the two amounts of liquid that he has and is indi erent to the liquid

of which he has the smaller amount.

368 EXCHANGE (Ch,29)

(a) Sketch an Edgeworth box for Mutt and Je,Use blue ink to show a

couple of indi erence curves for each,Use red ink to show the locus of

Pareto optimal allocations,(Hint,Look for boundary solutions.)

02468

2

4

6

Milk

Juice

8

Jeff

Mutt

Red

point

Red

point

Blue

curves

(Jeff)

Blue

curves

(Mutt)

29.9 (1) Remember Tommy Twit from Chapter 3,Tommy is happiest

when he has 8 cookies and 4 glasses of milk per day and his indi erence

curves are concentric circles centered around (8,4),Tommy’s mother,

Mrs,Twit,has strong views on nutrition,She believes that too much

of anything is as bad as too little,She believes that the perfect diet for

Tommy would be 7 glasses of milk and 2 cookies per day,In her view,

a diet is healthier the smaller is the sum of the absolute values of the

di erences between the amounts of each food consumed and the ideal

amounts,For example,if Tommy eats 6 cookies and drinks 6 glasses of

milk,Mrs,Twit believes that he has 4 too many cookies and 1 too few

glasses of milk,so the sum of the absolute values of the di erences from

her ideal amounts is 5,On the axes below,use blue ink to draw the locus

of combinations that Mrs,Twit thinks are exactly as good for Tommy

as (6;6),Also,use red ink to draw the locus of combinations that she

thinks is just as good as (8;4),On the same graph,use red ink to draw an

indi erence \curve" representing the locus of combinations that Tommy

likes just as well as 7 cookies and 8 glasses of milk.

NAME 369

2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Cookies

Milk

0

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

,,,,,,,,,,,,

Black

line

Red curve

Blue curve

Tommy's red

curve

(a) On the graph,shade in the area consisting of combinations of cookies

and milk that both Tommy and his mother agree are better than 7 cookies

and 8 glasses of milk,where \better" for Mrs,Twit means she thinks it

is healthier,and where \better" for Tommy means he likes it better.

(b) Use black ink to sketch the locus of \Pareto optimal" bundles of

cookies and milk for Tommy,In this situation,a bundle is Pareto optimal

if any bundle that Tommy prefers to this bundle is a bundle that Mrs.

Twit thinks is worse for him,The locus of Pareto optimal points that you

just drew should consist of two line segments,These run from the point

(8,4) to the point 5,7 and from that point to the point 2,7.

29.10 (2) This problem combines equilibrium analysis with some of the

things you learned in the chapter on intertemporal choice,It concerns the

economics of saving and the life cycle on an imaginary planet where life

is short and simple,In advanced courses in macroeconomics,you would

study more-complicated versions of this model that build in more earthly

realism,For the present,this simple model gives you a good idea of how

the analysis must go.

370 EXCHANGE (Ch,29)

On the planet Drongo there is just one commodity,cake,and two

time periods,There are two kinds of creatures,\old" and \young." Old

creatures have an income of I units of cake in period 1 and no income in

period 2,Young creatures have no income in period 1 and an income of I

units of cake in period 2,There are N

1

old creatures and N

2

young crea-

tures,The consumption bundles of interest to creatures are pairs (c

1;c

2

),

where c

1

is cake in period 1 and c

2

is cake in period 2,All creatures,old

and young,have identical utility functions,representing preferences over

cake in the two periods,This utility function is U(c

1;c

2

)=c

a

1

c

1?a

2

,where

a is a number such that 0 a 1.

(a) If current cake is taken to be the numeraire,(that is,its price is

set at 1),write an expression for the present value of a consumption

bundle (c

1;c

2

),c

1

+ c

2

=(1 + r),Write down the present value

of income for old creatures I and for young creatures

I

=(1 + r),The budget line for any creature is determined by the

condition that the present value of its consumption bundle equals the

present value of its income,Write down this budget equation for old

creatures,c

1

+ c

2

=(1 + r)=I and for young creatures:

c

1

+c

2

=(1 +r)=I

=(1 +r).

(b) If the interest rate is r,write down an expression for an old creature’s

demand for cake in period 1 c

1

= aI andinperiod2 c

2

=

(1?a)I(1+r),Write an expression for a young creature’s demand

for cake in period 1 c

1

= aI

=(1 + r) andinperiod2 c

2

=

(1?a)I

,(Hint,If its budget line is p

1

c

1

+p

2

c

2

= W and its utility

function is of the form proposed above,then a creature’s demand function

for good 1 is c

1

= aW=p and demand for good 2 is c

2

=(1?a)W=p.) If

the interest rate is zero,how much cake would a young creature choose in

period 1? aI

,For what value of a would it choose the same amount

in each period if the interest rate is zero? a =1=2,If a =,55,

what would r have to be in order that young creatures would want to

consume the same amount in each period?,22.

(c) The total supply of cake in period 1 equals the total cake earnings of

all old creatures,since young creatures earn no cake in this period,There

are N

1

old creatures and each earns I units of cake,so this total is N

1

I.

Similarly,the total supply of cake in period 2 equals the total amount

earned by young creatures,This amount is N

2

I

.

NAME 371

(d) At the equilibrium interest rate,the total demand of creatures for

period-1 cake must equal total supply of period-1 cake,and similarly the

demand for period-2 cake must equal supply,If the interest rate is r,then

the demand for period-1 cake by each old creature is aI and the

demand for period-1 cake by each young creature is aI

=(1 + r).

Since there areN

1

old creatures and N

2

young creatures,the total demand

for period-1 cake at interest rate r is N

1

aI +N

2

aI

=(1 +r).

(e) Using the results of the last section,write an equation that sets the

demand for period-1 cake equal to the supply,N

1

aI+N

2

aI

=(1+

r)=N

1

I,Write a general expression for the equilibrium value of r,

given N

1

,N

2

,I,andI

,r =

N

2

I

a

N

1

I(1?a)

1,Solve this equation

for the special case when N

1

= N

2

and I = I

and a =11=21.

r = 10%.

(f) In the special case at the end of the last section,show that the interest

rate that equalizes supply and demand for period-1 cake will also equalize

supply and demand for period-2 cake,(This illustrates Walras’s law.)

Supply = demand for period 2 if N

1

(1?

a)I(1 + r)+N

2

(1?a)I

= N

2

I

.IfN

1

= N

2

and I = I

,then (1?a)(1 +r)+(1?a)=1.

If a =11=21,then r = 10%.

372 EXCHANGE (Ch,29)

Chapter 30 NAME

Production

Introduction,In this section we explore economywide production pos-

sibility sets,We pay special attention to the principle of comparative

advantage,The principle is simply that e ciency suggests that people

should specialize according to their relative abilities in di erent activities

rather than absolute abilities.

Example,For simplicity,let us imagine an island with only two people

on it,both of them farmers,They do not trade with the outside world.

Farmer A has 100 acres and is able to grow two crops,wheat and hay.

Each acre of his land that he plants to wheat will give him 50 bushels

of wheat,Each acre of his land that he plants to hay will give him 2

tons of hay,Farmer B also has 100 acres,but his land is not so good.

Each acre of his land yields only 20 bushels of wheat and only 1 ton of

hay,Notice that,although Farmer A’s land is better for both wheat and

hay,Farmer B’s land has comparative advantage in the production of hay.

This is true because the ratio of tons of hay to bushels of wheat per acre

2=50 =,04 for Farmer A and 1=20 =,05 for Farmer B,Farmer A,on the

other hand,has comparative advantage in the production of wheat,since

the ratio of bushels of wheat to tons of hay is 50=2 = 25 for Farmer A

and 20=1 = 20 for Farmer B,The e cient way to arrange production is to

have Farmer A \specialize" in wheat and farmer B \specialize" in hay,If

Farmer A devotes all of his land to wheat and Farmer B devotes all of his

land to hay,then total wheat production will be 5,000 bushels and total

hay production will be 100 tons,Suppose that they decide to produce

only 4,000 bushels of wheat,Given that they are going to produce 4,000

bushels of wheat,the most hay they can possibly produce together will

be obtained if Farmer A devotes 80 acres to wheat and 20 acres to hay

while Farmer B devotes all of his land to hay,Suppose that they decide to

produce 6,000 bushels of wheat,Then they will get the most hay possible

given that they are producing 6,000 bushels of wheat if Farmer A puts

all of his land into wheat and Farmer B puts 50 acres into wheat and the

remaining 50 acres into hay.

30.1 (0) Tip and Spot nally got into college,Tip can write term papers

at the rate of 10 pages per hour and solve workbook problems at the rate

of 3 per hour,Spot can write term papers at the rate of 6 pages per

hour and solve workbook problems at the rate of 2 per hour,Which of

these two has comparative advantage in solving workbook problems?

Spot.

374 PRODUCTION (Ch,30)

0 2040608010

20

40

60

80

Problems

Pages

120

Spot

Tip

Joint

30

18

12

36 96

(a) Tip and Spot each work 6 hours a day,They decide to work together

and to produce a combination of term papers and workbook problems

that lies on their joint production possibility frontier,On the above graph

plot their joint production possibility frontier,If they produce less than

60 pages of term papers,then Tip will write all of the term papers.

If they produce more than 60 pages of term papers,then Tip

will continue to specialize in writing term papers and Spot will also

write some term papers.

30.2 (0) Robinson Crusoe has decided that he will spend exactly 8

hours a day gathering food,He can either spend this time gathering

coconuts or catching sh,He can catch 1 sh per hour and he can gather

2 coconuts per hour,On the graph below,show Robinson’s production

possibility frontier between sh and coconuts per day,Write an equation

for the line segment that is Robinson’s production possibility frontier.

F +C=2=8.

NAME 375

0481216

4

8

12

Fish

Coconuts

16

Utility of 4

Utility of 8

Production

possibility frontier

(a) Robinson’s utility function is U(F;C)=FC,whereF is his daily

sh consumption and C is his daily coconut consumption,On the graph

above,sketch the indi erence curve that gives Robinson a utility of 4,

and also sketch the indi erence curve that gives him a utility of 8,How

many sh will Robinson choose to catch per day? 4,How many

coconuts will he collect? 8,(Hint,Robinson will choose a bundle

that maximizes his utility subject to the constraint that the bundle lies

in his production possibility set,But for this technology,his production

possibility set looks just like a budget set.)

(b) Suppose Robinson is not isolated on an island in the Paci c,but is

retired and lives next to a grocery store where he can buy either sh or

coconuts,If sh cost $1 per sh,how much would coconuts have to cost in

order that he would choose to consume twice as many coconuts as sh?

$.50,Suppose that a social planner decided that he wanted Robinson

to consume 4 sh and 8 coconuts per day,He could do this by setting

the price of sh equal to $1,the price of coconuts equal to $.50 and

giving Robinson a daily income of $ 8,

(c) Back on his island,Robinson has little else to do,so he pretends that

he is running a competitive rm that produces sh and coconuts,He

wonders,\What would the price have to be to make me do just what I

am actually doing? Let’s assume that sh are the numeraire and have a

price of $1,And let’s pretend that I have access to a competitive labor

market where I can hire as much labor as I want at some given wage.

There is a constant returns to scale technology,An hour’s labor produces

376 PRODUCTION (Ch,30)

one sh or 2 coconuts,At wages above $ 1 per hour,I wouldn’t

produce any sh at all,because it would cost me more than $1 to produce

a sh,At wages below $ 1 per hour,I would want to produce

in nitely many sh since I would make a pro t on every one,So the

only possible wage rate that would make me choose to produce a positive

nite amount of sh is $ 1 per hour,Now what would the price

of coconuts have to be to induce me to produce a positive number of

coconuts,At the wage rate I just found,the cost of producing a coconut

is $.50,At this price and only at this price,would I be willing to

produce a nite positive number of coconuts."

30.3 (0) We continue the story of Robinson Crusoe from the previous

problem,One day,while walking along the beach,Robinson Crusoe saw

a canoe in the water,In the canoe was a native of a nearby island,The

native told Robinson that on his island there were 100 people and that

they all lived on sh and coconuts,The native said that on his island,it

takes 2 hours to catch a sh and 1 hour to nd a coconut,The native said

that there was a competitive economy on his island and that sh were

the numeraire,The price of coconuts on the neighboring island must

have been $.50,The native o ered to trade with Crusoe at these

prices,\I will trade you either sh for coconuts or coconuts for sh at

the exchange rate of 2 coconuts for a sh," said he,\But you

will have to give me 1 sh as payment for rowing over to your island."

Would Robinson gain by trading with him? No,If so,would he buy

sh and sell coconuts or vice versa? Neither,Since their

prices are the same as the rate at which

he can transform the two goods,he can gain

nothing by trading.

(a) Several days later,Robinson saw another canoe in the water on the

other side of his island,In this canoe was a native who came from a

di erent island,The native reported that on his island,one could catch

only 1 sh for every 4 hours of shing and that it takes 1 hour to nd a

coconut,This island also had a competitive economy,The native o ered

to trade with Robinson at the same exchange rate that prevailed on his

own island,but said that he would have to have 2 sh in return for rowing

between the islands,If Robinson decides to trade with this island,he

chooses to produce only fish and will get his coconuts from

the other island,On the graph above,use black ink to draw Robinson’s

production possibility frontier if he doesn’t trade and use blue ink to

NAME 377

show the bundles he can a ord if he chooses to trade and specializes

appropriately,Remember to take away 2 sh to pay the trader.

0481216

4

8

12

Fish

Coconuts

16

Utility of 4

Utility of 8

Production

possibility frontier

(b) Write an equation for Crusoe’s \budget line" if he specializes appro-

priately and trades with the second trader,If he does this,what bundle

will he choose to consume? 3 fish,12 coconuts,Does he

like this bundle better than the bundle he would have if he didn’t trade?

Yes.

30.4 (0) The Isle of Veritas has made it illegal to trade with the outside

world,Only two commodities are consumed on this island,milk and

wheat,On the north side of the island are 40 farms,Each of these

farms can produce any combination of non-negative amounts of milk and

wheat that satis es the equation m =60?6w,On the south side of the

island are 60 farms,Each of these farms can produce any combination

of non-negative amounts of milk and wheat that satis es the equation

m =40?2w,The economy is in competitive equilibrium and 1 unit of

wheat exchanges for 4 units of milk.

(a) On the diagram below,use black ink to draw the production possibility

set for a typical farmer from the north side of the island,Given the

equilibrium prices,will this farmer specialize in milk,specialize in wheat,

or produce both goods? Specialize in milk,Use blue ink

to draw the budget that he faces in his role as a consumer if he makes

the optimal choice of what to produce.

378 PRODUCTION (Ch,30)

020406080

20

40

60

Wheat

Milk

80

Black line

Blue line

Red line

Pencil line

15

10

(b) On the diagram below,use black ink to draw the production possibility

set for a typical farmer from the south side of the island,Given the

equilibrium prices,will this farmer specialize in milk,specialize in wheat,

or produce both goods? Specialize in wheat,Use blue

ink to draw the budget that he faces in his role as a consumer if he makes

the optimal choice of what to produce.

020406080

20

40

60

Wheat

Milk

80

Black line

Blue line

Red line

Pencil line

(c) Suppose that peaceful Viking traders discover Veritas and o er to

exchange either wheat for milk or milk for wheat at an exchange rate of

NAME 379

1 unit of wheat for 3 units of milk,If the Isle of Veritas allows free trade

with the Vikings,then this will be the new price ratio on the island,At

this price ratio,would either type of farmer change his output? No.

(d) On the rst of the two graphs above,use red ink to draw the budget

for northern farmers if free trade is allowed and the farmers make the

right choice of what to produce,On the second of the two graphs,use

red ink to draw the budget for southern farmers if free trade is allowed

and the farmers make the right choice of what to produce.

(e) The council of elders of Veritas will meet to vote on whether to accept

the Viking o er,The elders from the north end of the island get 40

votes and the elders from the south end get 60 votes,Assuming that

everyone votes in the sel sh interest of his end of the island,how will

the northerners vote? In favor,How will the southerners vote?

Against,How is it that you can make a de nite answer to the last

two questions without knowing anything about the farmers’s consumption

preferences? The change strictly enlarges the

budget set for northerners and strictly

shrinks it for southerners.

(f) Suppose that instead of o ering to make exchanges at the rate of 1 unit

of wheat for 3 units of milk,the Vikings had o ered to trade at the price

of 1 unit of wheat for 1 unit of milk and vice versa,Would either type

of farmer change his output? Yes,Southerners would

now switch to specializing in milk,Use pencil

to sketch the budget line for each kind of farmer at these prices if he

makes the right production decision,How will the northerners vote now?

In favor,How will the southerners vote now? Depends

on their preferences about consumption.

Explain why it is that your answer to one of the last two questions has

to be \it depends." The two alternative budget

lines for southerners are not nested.

30.5 (0) Recall our friends the Mungoans of Chapter 2,They have a

strange two-currency system consisting of Blue Money and Red Money.

Originally,there were two prices for everything,a blue-money price and

a red-money price,The blue-money prices are 1 bcu per unit of ambrosia

and 1 bcu per unit of bubble gum,The red-money prices are 2 rcu’s per

unit of ambrosia and 4 rcu’s per unit of bubble gum.

380 PRODUCTION (Ch,30)

(a) Harold has a blue income of 9 and a red income of 24,If it has to

pay in both currencies for any purchase,draw its budget set in the graph

below,(Hint,You answered this question a few months ago.)

0 5 10 15 20

5

10

15

Ambrosia

Bubble gum

20

Part j budget set

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

,,,,,,,,,,,

(12,9)

9

12

6

9

Part a budget set

(b) The Free Choice party campaigns on a platform that Mungoans should

be allowed to purchase goods at either the blue-money price or the red-

money price,whichever they prefer,We want to construct Harold’s bud-

get set if this reform is instituted,To begin with,how much bubble gum

could Harold consume if it spent all of its blue money and its red money

on bubble gum? 15 units of bubble gum.

(c) How much ambrosia could it consume if it spent all of its blue

money and all of its red money on ambrosia? 21 units of

ambrosia.

(d) If Harold were spending all of its money of both colors on bubble gum

and it decided to purchase a little bit of ambrosia,which currency would

it use? The red currency.

(e) How much ambrosia could it buy before it ran out of that color money?

12 units of ambrosia.

(f) What would be the slope of this budget line before it ran out of that

kind of money? The slope would be?

1

2

.

NAME 381

(g) If Harold were spending all of its money of both colors on ambrosia

and it decided to purchase a little bit of bubble gum,which currency

would it use? The blue currency.

(h) How much bubble gum could it buy before it ran out of that color

money? He could buy 9 units of bubble gum.

(i) What would be the slope of this budget line before it ran out of that

kind of money? The slope would be?1.

(j) Use your answers to the above questions to draw Harold’s budget set

in the above graph if it could purchase bubble gum and ambrosia using

either currency.

382 PRODUCTION (Ch,30)

Chapter 31 NAME

Welfare

Introduction,Here you will look at various ways of determining social

preferences,You will check to see which of the Arrow axioms for ag-

gregating individual preferences are satis ed by these welfare relations.

You will also try to nd optimal allocations for some given social welfare

functions,The method for solving these last problems is analogous to

solving for a consumer’s optimal bundle given preferences and a budget

constraint,Two hints,Remember that for a Pareto optimal allocation

inside the Edgeworth box,the consumers’ marginal rates of substitution

will be equal,Also,in a \fair allocation," neither consumer prefers the

other consumer’s bundle to his own.

Example,A social planner has decided that she wants to allocate income

between 2 people so as to maximize

p

Y

1

+

p

Y

2

where Y

i

is the amount of

income that person i gets,Suppose that the planner has a xed amount

of money to allocate and that she can enforce any income distribution

such that Y

1

+ Y

2

= W,whereW is some xed amount,This planner

would have ordinary convex indi erence curves between Y

1

and Y

2

and

a \budget constraint" where the \price" of income for each person is 1.

Therefore the planner would set her marginal rate of substitution between

income for the two people equal to the relative price which is 1,When you

solve this,you will nd that she sets Y

1

= Y

2

= W=2,Suppose instead

that it is \more expensive" for the planner to give money to person 1 than

to person 2,(Perhaps person 1 is forgetful and loses money,or perhaps

person 1 is frequently robbed.) For example,suppose that the planner’s

budget is 2Y

1

+Y

2

= W,Then the planner maximizes

p

Y

1

+

p

Y

2

subject

to 2Y

1

+Y

2

= W,Setting her MRS equal to the price ratio,we nd that

p

Y

2

p

Y

1

=2,SoY

2

=4Y

1

,Therefore the planner makes Y

1

= W=5and

Y

2

=4W=5.

31.1 (2) One possible method of determining a social preference relation

is the Borda count,also known as rank-order voting,Each voter is asked

to rank all of the alternatives,If there are 10 alternatives,you give your

rst choice a 1,your second choice a 2,and so on,The voters’ scores for

each alternative are then added over all voters,The total score for an

alternative is called its Borda count,For any two alternatives,x and y,

if the Borda count of x is smaller than or the same as the Borda count

for y,thenx is \socially at least as good as" y,Suppose that there are

a nite number of alternatives to choose from and that every individual

has complete,reflexive,and transitive preferences,For the time being,

let us also suppose that individuals are never indi erent between any two

di erent alternatives but always prefer one to the other.

384 WELFARE (Ch,31)

(a) Is the social preference ordering de ned in this way complete? Yes.

Reflexive? Yes,Transitive? Yes.

(b) If everyone prefers x to y,will the Borda count rank x as socially

preferred to y? Explain your answer,Yes,If everybody

ranks x ahead of y,then everyone must give

x a higher rank than y,Then the sum of

the ranks of x must be larger than the sum

of the ranks of y.

(c) Suppose that there are two voters and three candidates,x,y,and

z,Suppose that Voter 1 ranks the candidates,x rst,z second,and y

third,Suppose that Voter 2 ranks the candidates,y rst,x second,and z

third,What is the Borda count for x? 3,For y? 4,For

z? 5,Now suppose that it is discovered that candidate z once

lifted a beagle by the ears,Voter 1,who has rather large ears himself,

is appalled and changes his ranking to x rst,y second,z third,Voter

2,who picks up his own children by the ears,is favorably impressed and

changes his ranking to y rst,z second,x third,Now what is the Borda

count for x? 4,For y? 3,For z? 5.

(d) Does the social preference relation de ned by the Borda count have

the property that social preferences between x and y depend only on how

people rank x versus y and not on how they rank other alternatives? Ex-

plain,No,In the above example,the ranking

of z changed,but nobody changed his mind

about whether x was better than y or vice

versa,Before the change x beat y,and after

the change y beat x.

31.2 (2) Suppose the utility possibility frontier for two individuals is

given by U

A

+2U

B

= 200,On the graph below,plot the utility frontier.

NAME 385

0 50 100 150 200

50

100

150

UA

UB

200

Blue line

Black line

Red line

Utility frontier

(a) In order to maximize a \Nietzschean social welfare function,"

W(U

A;U

B

)=maxfU

A;U

B

g,on the utility possibility frontier shown

above,one would set U

A

equal to 200 and U

B

equal to 0.

(b) If instead we use a Rawlsian criterion,W(U

A;U

B

)=minfU

A;U

B

g,

then the social welfare function is maximized on the above utility possi-

bility frontier where U

A

equals 66.66 and U

B

equals 66.66.

(c) Suppose that social welfare is given by W(U

A;U

B

)=U

1=2

A

U

1=2

B

.In

this case,with the above utility possibility frontier,social welfare is max-

imized where U

A

equals 100 and U

B

is 50,(Hint,You might

want to think about the similarities between this maximization problem

and the consumer’s maximization problem with a Cobb-Douglas utility

function.)

(d) Show the three social maxima on the above graph,Use black ink

to draw a Nietzschean isowelfare line through the Nietzschean maximum.

Use red ink to draw a Rawlsian isowelfare line through the Rawlsian

maximum,Use blue ink to draw a Cobb-Douglas isowelfare line through

the Cobb-Douglas maximum.

31.3 (2) A parent has two children named A and B and she loves both

of them equally,She has a total of $1,000 to give to them.

386 WELFARE (Ch,31)

(a) The parent’s utility function is U(a;b)=

p

a +

p

b,wherea is the

amount of money she gives to A and b istheamountofmoneyshegives

to B,How will she choose to divide the money? a = b = $500.

(b) Suppose that her utility function is U(a;b)=?

1

a

1

b

,How will she

choose to divide the money? a = b = $500.

(c) Suppose that her utility function is U(a;b)=loga +logb,How will

she choose to divide the money? a = b = $500.

(d) Suppose that her utility function is U(a;b)=minfa;bg,How will she

choose to divide the money? a = b = $500.

(e) Suppose that her utility function is U(a;b)=maxfa;bg,How will she

choose to divide the money? a =$1;000,b =0,or vice

versa.

(Hint,In each of the above cases,we notice that the parent’s problem is

to maximize U(a;b) subject to the constraint that a+b =1;000,This is

just like the consumer problems we studied earlier,It must be that the

parent sets her marginal rate of substitution between a and b equal to 1

since it costs the same to give money to each child.)

(f) Suppose that her utility function is U(a;b)=a

2

+ b

2

,How will

she choose to divide the money between her children? Explain why she

doesn’t set her marginal rate of substitution equal to 1 in this case.

She gives everything to one child,Her

preferences are not convex,indifference

curves are quarter circles.

31.4 (2) In the previous problem,suppose that A is a much more e cient

shopper than B so that A is able to get twice as much consumption

goods as B can for every dollar that he spends,Let a be the amount of

consumption goods that A gets and b the amount that B gets,We will

measure consumption goods so that one unit of consumption goods costs

$1 for A and $2 for B,Thus the parent’s budget constraint is a +2b =

1;000.

(a) If the mother’s utility function is U(a;b)=a+b,which child will get

more money? A,Which child will consume more goods? A.

NAME 387

(b) If the mother’s utility function is U(a;b)=a b,which child will get

more money? They get the same amount of money.

Which child will get to consume more? A consumes more.

(c) If the mother’s utility function is U(a;b)=?

1

a

1

b

,which child will

get more money? B gets more money,Which child will get

to consume more? They consume the same amount.

(d) If the mother’s utility function is U(a;b)=maxfa;bg,which child

will get more money? A,Which child will get to consume more?

A.

(e) If the mother’s utility function is U(a;b)=minfa;bg,which child will

get more money? B,Which child will get to consume more?

They consume the same amount.

Calculus 31.5 (1) Norton and Ralph have a utility possibility frontier that is given

by the following equation,U

R

+U

2

N

= 100 (where R and N signify Ralph

and Norton respectively).

(a) If we set Norton’s utility to zero,what is the highest possible utility

Ralph can achieve? 100,If we set Ralph’s utility to zero,what is

the best Norton can do? 10.

(b) Plot the utility possibility frontier on the graph below.

0 5 10 15 20

25

50

75

Norton's utility

Ralph's utility

100

388 WELFARE (Ch,31)

(c) Derive an equation for the slope of the above utility possibility curve.

dU

R

dU

N

=?2U

N

.

(d) Both Ralph and Norton believe that the ideal allocation is given by

maximizing an appropriate social welfare function,Ralph thinks that

U

R

= 75,U

N

= 5 is the best distribution of welfare,and presents the

maximization solution to a weighted-sum-of-the-utilities social welfare

function that con rms this observation,What was Ralph’s social welfare

function? (Hint,What is the slope of Ralph’s social welfare function?)

W = U

R

+10U

N

.

(e) Norton,on the other hand,believes that U

R

= 19,U

N

=9isthe

best distribution,What is the social welfare function Norton presents?

W = U

R

+18U

N

.

31.6 (2) Roger and Gordon have identical utility functions,U(x;y)=

x

2

+y

2

,There are 10 units of x and 10 units of y to be divided between

them,Roger has blue indi erence curves,Gordon has red ones.

(a) Draw an Edgeworth box showing some of their indi erence curves and

mark the Pareto optimal allocations with black ink,(Hint,Notice that

the indi erence curves are nonconvex.)

010

10

Roger

Gordon

Black lines

Black lines

Red curves

Blue

curves

Fair

Fair

y

x

(b) What are the fair allocations in this case? See diagram.

31.7 (2) Paul and David consume apples and oranges,Paul’s util-

ity function is U

P

(A

P;O

P

)=2A

P

+ O

P

and David’s utility function is

NAME 389

U

D

(A

D;O

D

)=A

D

+2O

D

,whereA

P

and A

D

are apple consumptions for

Paul and David,and O

P

and O

D

are orange consumptions for Paul and

David,There are a total of 12 apples and 12 oranges to divide between

Paul and David,Paul has blue indi erence curves,David has red ones.

Draw an Edgeworth box showing some of their indi erence curves,Mark

the Pareto optimal allocations on your graph.

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

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,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,

12

Apples

Oranges

0

Red curves

Blue

curves

Blue shading

Red shading

Pareto

optimal

Pareto optimal

Fair

Paul

David

12

(a) Write one inequality that says that Paul likes his own bundle as well

as he likes David’s and write another inequality that says that David likes

his own bundle as well as he likes Paul’s,2A

P

+O

P

2A

D

+O

D

and A

D

+2O

D

A

P

+2O

P

.

(b) Use the fact that at feasible allocations,A

P

+A

D

=12andO

P

+O

D

=

12 to eliminate A

D

and O

D

from the rst of these equations,Write the

resulting inequality involving only the variables A

P

and O

P

.Nowinyour

Edgeworth box,use blue ink to shade in all of the allocations such that

Paul prefers his own allocation to David’s,2A

P

+O

P

18.

(c) Use a procedure similar to that you used above to nd the allocations

where David prefers his own bundle to Paul’s,Describe these points

with an inequality and shade them in on your diagram with red ink.

A

D

+2O

D

18.

(d) On your Edgeworth box,mark the fair allocations.

31.8 (3) Romeo loves Juliet and Juliet loves Romeo,Besides love,

they consume only one good,spaghetti,Romeo likes spaghetti,but he

390 WELFARE (Ch,31)

also likes Juliet to be happy and he knows that spaghetti makes her

happy,Juliet likes spaghetti,but she also likes Romeo to be happy and

she knows that spaghetti makes Romeo happy,Romeo’s utility function

is U

R

(S

R;S

J

)=S

a

R

S

1?a

J

and Juliet’s utility function is U

J

(S

J;S

R

)=

S

a

J

S

1?a

R

,whereS

J

and S

R

are the amount of spaghetti for Romeo and

the amount of spaghetti for Juliet respectively,There is a total of 24 units

of spaghetti to be divided between Romeo and Juliet.

(a) Suppose that a =2=3,If Romeo got to allocate the 24 units of

spaghetti exactly as he wanted to,how much would he give himself?

16,How much would he give Juliet? 8,(Hint,Notice that this

problem is formally just like the choice problem for a consumer with a

Cobb-Douglas utility function choosing between two goods with a budget

constraint,What is the budget constraint?)

(b) If Juliet got to allocate the spaghetti exactly as she wanted to,how

much would she take for herself? 16,How much would she give

Romeo? 8.

(c) What are the Pareto optimal allocations? (Hint,An allocation

will not be Pareto optimal if both persons’ utility will be increased by

a gift from one to the other.) The Pareto optimal

allocations are all of the allocations in

which each person gets at least 8 units of

spaghetti.

(d) When we had to allocate two goods between two people,we drew an

Edgeworth box with indi erence curves in it,When we have just one

good to allocate between two people,all we need is an \Edgeworth line"

and instead of indi erence curves,we will just have indi erence dots.

Consider the Edgeworth line below,Let the distance from left to right

denote spaghetti for Romeo and the distance from right to left denote

spaghetti for Juliet.

(e) On the Edgeworth line you drew above,show Romeo’s favorite point

and Juliet’s favorite point.

NAME 391

(f) Suppose that a =1=3,If Romeo got to allocate the spaghetti,how

much would he choose for himself? 8,If Juliet got to allocate

the spaghetti,how much would she choose for herself? 8,Label

the Edgeworth line below,showing the two people’s favorite points and

the locus of Pareto optimal points.

(g) When a =1=3,at the Pareto optimal allocations what do Romeo and

Juliet disagree about? Romeo wants to give spaghetti

to Juliet,but she doesn’t want to take it.

Juliet wants to give spaghetti to Romeo,

but he doesn’t want to take it,Both like

spaghetti for themselves,but would rather

the other had it.

31.9 (2) Hat eld and McCoy hate each other but love corn whiskey.

Because they hate for each other to be happy,each wants the other to

have less whiskey,Hat eld’s utility function isU

H

(W

H;W

M

)=W

H

W

2

M

and McCoy’s utility function is U

M

(W

M;W

H

)=W

M

W

2

H

,whereW

M

is McCoy’s daily whiskey consumption and W

H

is Hat eld’s daily whiskey

consumption (both measured in quarts),There are 4 quarts of whiskey

to be allocated.

(a) If McCoy got to allocate all of the whiskey,how would he allocate it?

All for himself,If Hat eld got to allocate all of the whiskey,

how would he allocate it? All for himself.

(b) If each of them gets 2 quarts of whiskey,what will the utility of each

of them be2,If a bear spilled 2 quarts of their whiskey and they

divided the remaining 2 quarts equally between them,what would the

utility of each of them be? 0,If it is possible to throw away some

of the whiskey,is it Pareto optimal for them each to consume 2 quarts of

whiskey? No.

392 WELFARE (Ch,31)

(c) If it is possible to throw away some whiskey and they must consume

equal amounts of whiskey,how much should they throw away? 3

quarts.

Chapter 32 NAME

Externalities

Introduction,When there are externalities,the outcome from indepen-

dently chosen actions is typically not Pareto e cient,In these exercises,

you explore the consequences of alternative mechanisms and institutional

arrangements for dealing with externalities.

Example,A large factory pumps its waste into a nearby lake,The lake

is also used for recreation by 1,000 people,Let X betheamountofwaste

that the rm pumps into the lake,Let Y

i

be the number of hours per day

that person i spends swimming and boating in the lake,and let C

i

be the

number of dollars that person i spends on consumption goods,If the rm

pumps X units of waste into the lake,its pro ts will be 1;200X?100X

2

.

Consumers have identical utility functions,U(Y

i;C

i;X)=C

i

+9Y

i

Y

2

i

XY

i

,and identical incomes,Suppose that there are no restrictions

on pumping waste into the lake and there is no charge to consumers for

using the lake,Also,suppose that the factory and the consumers make

their decisions independently,The factory will maximize its pro ts by

choosing X = 6,(Set the derivative of pro ts with respect to X equal

to zero.) When X = 6,each consumer maximizes utility by choosing

Y

i

=1:5,(Set the derivative of utility with respect to Y

i

equal to zero.)

Notice from the utility functions that when each person is spending 1.5

hours a day in the lake,she will be willing to pay 1.5 dollars to reduce

X by 1 unit,Since there are 1,000 people,the total amount that people

will be willing to pay to reduce the amount of waste by 1 unit is $1,500.

If the amount of waste is reduced from 6 to 5 units,the factory’s pro ts

will fall from $3,600 to $3,500,Evidently the consumers could a ord to

bribe the factory to reduce its waste production by 1 unit.

32.1 (2) The picturesque village of Horsehead,Massachusetts,lies on a

bay that is inhabited by the delectable crustacean,homarus americanus,

also known as the lobster,The town council of Horsehead issues permits

to trap lobsters and is trying to determine how many permits to issue.

The economics of the situation is this:

1,It costs $2,000 dollars a month to operate a lobster boat.

2,If there are x boats operating in Horsehead Bay,the total revenue

from the lobster catch per month will be f(x)=$1;000(10x?x

2

).

(a) In the graph below,plot the curves for the average product,AP(x)=

f(x)=x,and the marginal product,MP(x)=10;000?2;000x.Inthe

same graph,plot the line indicating the cost of operating a boat.

394 EXTERNALITIES (Ch,32)

2 4 6 8 10 12

2

4

6

8

10

12

x

AP,MP

0

AP

Cost

MP

(b) If the permits are free of charge,how many boats will trap lobsters

in Horsehead,Massachusetts? (Hint,How many boats must enter before

there are zero pro ts?) 8 boats.

(c) What number of boats maximizes total pro ts? Set MP

equal to cost to give 10?2x =2,orx =4

boats.

(d) If Horsehead,Massachusetts,wants to restrict the number of boats to

the number that maximizes total pro ts,how much should it charge per

month for a lobstering permit? (Hint,With a license fee of F thousand

dollars per month,the marginal cost of operating a boat for a month

would be (2 + F) thousand dollars per month.) $4,000 per

month.

32.2 (2) Suppose that a honey farm is located next to an apple orchard

and each acts as a competitive rm,Let the amount of apples produced

be measured by A and the amount of honey produced be measured by H.

The cost functions of the two rms are c

H

(H)=H

2

=100 and c

A

(A)=

A

2

=100?H,The price of honey is $2 and the price of apples is $3.

(a) If the rms each operate independently,the equilibrium amount of

honey produced will be 100 and the equilibrium amount of apples

produced will be 150.

NAME 395

(b) Suppose that the honey and apple rms merged,What would be

the pro t-maximizing output of honey for the combined rm? 150.

What would be the pro t-maximizing amount of apples? 150.

(c) What is the socially e cient output of honey? 150,If the rms

stayed separate,how much would honey production have to be subsidized

to induce an e cient supply? $1 per unit.

32.3 (2) In El Carburetor,California,population 1,001,there is not

much to do except to drive your car around town,Everybody in town

is just like everybody else,While everybody likes to drive,everybody

complains about the congestion,noise,and pollution caused by tra c,A

typical resident’s utility function is U(m;d;h)=m+16d?d

2

6h=1;000,

where m is the resident’s daily consumption of Big Macs,d is the number

of hours per day that he,himself,drives,and h is the total amount of

driving (measured in person-hours per day) done by all other residents

of El Carburetor,The price of Big Macs is $1 each,Every person in El

Carburetor has an income of $40 per day,To keep calculations simple,

suppose it costs nothing to drive a car.

(a) If an individual believes that the amount of driving he does won’t af-

fect the amount that others drive,how many hours per day will he choose

to drive? 8,(Hint,What value of d maximizes U(m;d;h)?)

(b) If everybody chooses his best d,then what is the total amount h of

driving by other persons? 8,000.

(c) What will be the utility of each resident? 56.

(d) If everybody drives 6 hours a day,what will be the utility level of a

typical resident of El Carburetor? 64.

(e) Suppose that the residents decided to pass a law restricting the total

number of hours that anyone is allowed to drive,How much driving

should everyone be allowed if the objective is to maximize the utility of

the typical resident? (Hint,Rewrite the utility function,substituting

1;000d for h,and maximize with respect to d.) 5 hours per

day.

396 EXTERNALITIES (Ch,32)

(f) The same objective could be achieved with a tax on driving,How

much would the tax have to be per hour of driving? (Hint,This price

would have to equal an individual’s marginal rate of substitution between

driving and Big Macs when he is driving the \right" amount.) $6.

32.4 (3) Tom and Jerry are roommates,They spend a total of 80 hours

a week together in their room,Tom likes loud music,even when he sleeps.

His utility function is U

T

(C

T;M)=C

T

+ M,whereC

T

is the number

of cookies he eats per week and M is the number of hours of loud music

per week that is played while he is in their room,Jerry hates all kinds

of music,His utility function is U(C

J;M)=C

J

M

2

=12,Every week,

Tom and Jerry each get two dozen chocolate chip cookies sent from home.

They have no other source of cookies,We can describe this situation with

a box that looks like an Edgeworth box,The box has cookies on the

horizontal axis and hours of music on the vertical axis,Since cookies are

private goods,the number of cookies that Tom consumes per week plus

the number that Jerry consumes per week must equal 48,But music in

their room is a public good,Each must consume the same number of

hours of music,whether he likes it or not,In the box,let the height of a

point represent the total number of hours of music played in their room

per week,Let the distance of the point from the left side of the box be

\cookies for Tom" and the distance of the point from the right side of the

box be \cookies for Jerry."

012243648

20

40

60

Cookies

Music

80

Blue Line

Red Line

Blue Shading

a

b

Blue Line

Red Line

Tom

Jerry

(a) Suppose the dorm’s policy is that you must have your roommate’s

permission to play music,The initial endowment in this case denotes the

situation if Tom and Jerry make no deals,There would be no music,and

each person would consume 2 dozen cookies a week,Mark this initial

endowment on the box above with the label A,Use red ink to sketch

the indi erence curve for Tom that passes through this point,and use

NAME 397

blue ink to sketch the indi erence curve for Jerry that passes through

this point,[Hint,When you draw Jerry’s indi erence curve,remember

two things,(1) He hates music,so he prefers lower points on the graph

to higher ones,(2) Cookies for Jerry are measured from the right side

of the box,so he prefers points that are toward the left side of the box

to points that are toward the right.] Use blue ink to shade in the points

representing situations that would make both roommates better o than

they are at point A.

(b) Suppose,alternatively,that the dorm’s policy is \rock-n-roll is good

for the soul." You don’t need your roommate’s permission to play music.

Then the initial endowment is one in which Tom plays music for all of

the 80 hours per week that they are in the room together and where each

consumes 2 dozen cookies per week,Mark this endowment point in the

box above and label it B,Use red ink to sketch the indi erence curve

for Tom that passes through this point,and use blue ink to sketch the

indi erence curve for Jerry that passes through this point,Given the

available resources,can both Tom and Jerry be made better o than they

areatpointB? Yes.

Calculus 32.5 (0) A clothing store and a jewelry store are located side by side

in a small shopping mall,The number of customers who come to the

shopping mall intending to shop at either store depends on the amount

of money that the store spends on advertising per day,Each store also

attracts some customers who came to shop at the neighboring store,If

the clothing store spends $x

C

per day on advertising,and the jeweler

spends $x

J

on advertising per day,then the total pro ts per day of the

clothing store are

C

(x

C;x

J

)=(60+x

J

)x

C

2x

2

C

,and the total pro ts

per day of the jewelry store are

J

(x

C;x

J

) = (105 + x

C

)x

J

2x

2

J

.(In

each case,these are pro ts net of all costs,including advertising.)

(a) If each store believes that the other store’s amount of advertising

is independent of its own advertising expenditure,then we can nd the

equilibrium amount of advertising for each store by solving two equations

in two unknowns,One of these equations says that the derivative of the

clothing store’s pro ts with respect to its own advertising is zero,The

other equation requires that the derivative of the jeweler’s pro ts with

respect to its own advertising is zero,These two equations are written as

60 +x

J

4x

C

=0 and 105 +x

C

4x

J

=0,The

equilibrium amounts of advertising are x

C

=23 and x

J

=32.

Pro ts of the clothing store are $1,058 and pro ts of the jeweler

are 2,048.

398 EXTERNALITIES (Ch,32)

(b) The extra pro t that the jeweler would get from an extra dollar’s

worth of advertising by the clothing store is approximately equal to the

derivative of the jeweler’s pro ts with respect to the clothing store’s ad-

vertising expenditure,When the two stores are doing the equilibrium

amount of advertising that you calculated above,a dollar’s worth of ad-

vertising by the clothing store would give the jeweler an extra pro t of

about $32 and an extra dollar’s worth of advertising by the jeweler

would give the clothing store an extra pro t of about $23.

(c) Suppose that the owner of the clothing store knows the pro t functions

of both stores,She reasons to herself as follows,Suppose that I can decide

how much advertising I will do before the jeweler decides what he is going

to do,When I tell him what I am doing,he will have to adjust his behavior

accordingly,I can calculate his reaction function to my choice of x

C

,by

setting the derivative of his pro ts with respect to his own advertising

equal to zero and solving for his amount of advertising as a function of

my own advertising,When I do this,I nd thatx

J

= 105=4+x

C

=4.

If I substitute this value of x

J

into my pro t function and then choose x

C

to maximize my own pro ts,I will choose x

C

= 24.64 and he will

choose x

J

= 32.41,In this case my pro ts will be $1,062.72

and his pro ts will be $2,100.82.

(d) Suppose that the clothing store and the jewelry store have the same

pro t functions as before but are owned by a single rm that chooses

the amounts of advertising so as to maximize the sum of the two stores’

pro ts,The single rm would choose x

C

= $37.50 and x

J

=

$45,Without calculating actual pro ts,can you determine whether

total pro ts will be higher,lower,or the same as total pro ts would be

when they made their decisions independently? Yes,they

would be higher,How much would the total pro ts be?

$3,487.50.

32.6 (2) The cottagers on the shores of Lake Invidious are an unsavory

bunch,There are 100 of them,and they live in a circle around the lake.

Each cottager has two neighbors,one on his right and one on his left.

There is only one commodity,and they all consume it on their front

lawns in full view of their two neighbors,Each cottager likes to consume

the commodity but is very envious of consumption by the neighbor on

his left,Curiously,nobody cares what the neighbor on his right is doing.

In fact every consumer has a utility function U(c;l)=c?l

2

,wherec is

NAME 399

his own consumption and l is consumption by his neighbor on the left.

Suppose that each consumer owns 1 unit of the consumption good and

consumes it.

(a) Calculate his utility level,0.

(b) Suppose that each consumer consumes only 3=4 of a unit,Will all

individuals be better o or worse o? Better off.

(c) What is the best possible consumption if all are to consume the same

amount? 1=2.

(d) Suppose that everybody around the lake is consuming 1 unit,Can

any two people make themselves both better o either by redistributing

consumption between them or by throwing something away? No.

(e) How about a group of three people? No.

(f) How large is the smallest group that could cooperate to bene t all its

members? 100.

32.7 (0) Jim and Tammy are partners in Business and in Life,As

is all too common in this imperfect world,each has a little habit that

annoys the other,Jim’s habit,we will call activity X,and Tammy’s

habit,activity Y.Letx be the amount of activity X that Jim pursues

and y be the amount of activity Y that Tammy pursues,Due to a series

of unfortunate reverses,Jim and Tammy have a total of only $1,000,000

a year to spend,Jim’s utility function is U

J

= c

J

+ 500 lnx?10y,where

c

J

is the money he spends per year on goods other than his habit,x is

the number of units of activity X that he consumes per year,and y is the

number of units of activity Y that Tammy consumes per year,Tammy’s

utility function is U

T

= c

T

+ 500 lny?10x,wherec

T

istheamountof

money she spends on goods other than activity Y,y is the number of

units of activity Y that she consumes,and x is the number of units of

activity X that Jim consumes,Activity X costs $20 per unit,Activity

Y costs $100 per unit.

(a) Suppose that Jim has a right to half their joint income and Tammy

has a right to the other half,Suppose further that they make no bargains

with each other about how much activity X and Y they will consume.

How much of activity X will Jim choose to consume? 25 units.

How much of activity Y will Tammy consume? 5 units.

400 EXTERNALITIES (Ch,32)

(b) Because Jim and Tammy have quasilinear utility functions,their util-

ity possibility frontier includes a straight line segment,Furthermore,this

segment can be found by maximizing the sum of their utilities,Notice

that

U

J

(c

J;x;y)+U

T

(c

T;x;y)

= c

J

+ 500 lnx?20y +c

T

+ 500 lny?10x

= c

J

+c

T

+ 500 lnx?10x+ 500 lny?10y:

But we know from the family budget constraint that c

J

+c

T

=1;000;000?

20x?100y,Therefore we can write

U

J

(c

J;x;y)+U

T

(c

T;x;y)=1;000;000?20x?100y + 500 lnx?10x

+ 500 lny?10y

=1;000;000 + 500 lnx+ 500 lny?30x?110y:

Let us now choose x and y so as to maximize U

J

(c

J;x;y)+U

T

(c

T;x;y).

Setting the partial derivatives with respect to x and y equal to zero,we

nd the maximum where x = 16.67 and y = 4.54,Ifweplug

these numbers into the equation U

J

(c

J;x;y)+U

T

(c

T;x;y)=1;000;000+

500 lnx+500 lny?30x?110y,we nd that the utility possibility frontier is

described by the equation U

J

+U

C

= 1,001,163.86,(You need

a calculator or a log table to nd this answer.) Along this frontier,the

total expenditure on the annoying habits X and Y by Jim and Tammy is

787:34,The rest of the $1,000,000 is spent on c

J

and c

T

,Each possible

way of dividing this expenditure corresponds to a di erent point on the

utility possibility frontier,The slope of the utility possibility frontier

constructed in this way is -1.

32.8 (0) An airport is located next to a large tract of land owned by a

housing developer,The developer would like to build houses on this land,

but noise from the airport reduces the value of the land,The more planes

that fly,the lower is the amount of pro ts that the developer makes,Let

X be the number of planes that fly per day and let Y be the number of

houses that the developer builds,The airport’s total pro ts are 48X?X

2

,

and the developer’s total pro ts are 60Y?Y

2

XY,Let us consider the

outcome under various assumptions about institutional rules and about

bargaining between the airport and the developer.

(a) \Free to Choose with No Bargaining",Suppose that no bargains can

be struck between the airport and the developer and that each can decide

on its own level of activity,No matter how many houses the developer

builds,the number of planes per day that maximizes pro ts for the airport

is 24,Given that the airport is landing this number of planes,the

NAME 401

number of houses that maximizes the developer’s pro ts is 18,Total

pro ts of the airport will be 576 and total pro ts of the developer

will be 324,The sum of their pro ts will be 900.

(b) \Strict Prohibition",Suppose that a local ordinance makes it illegal

to land planes at the airport because they impose an externality on the

developer,Then no planes will fly,The developer will build 30

houses and will have total pro ts of 900.

(c) \Lawyer’s Paradise",Suppose that a law is passed that makes the

airport liable for all damages to the developer’s property values,Since the

developer’s pro ts are 60Y?Y

2

XY and his pro ts would be 60Y?Y

2

if no planes were flown,the total amount of damages awarded to the

developer will be XY,Therefore if the airport flies X planes and the

developer builds Y houses,then the airport’s pro ts after it has paid

damages will be 48X?X

2

XY,The developer’s pro ts including the

amount he receives in payment of damages will be 60Y?Y

2

XY+XY =

60Y?Y

2

,To maximize his net pro ts,the developer will choose to build

30 houses no matter how many planes are flown,To maximize its

pro ts,net of damages,the airport will choose to land 9 planes.

Total pro ts of the developer will be 900 and total pro ts of the

airport will be 81,The sum of their pro ts will be 981.

Calculus 32.9 (1) This problem concerns the airport and the developer from the

previous problem.

(a) \The Conglomerate",Suppose that a single rm bought the de-

veloper’s land and the airport and managed both to maximize joint

pro ts,Total pro ts,expressed as a function of X and Y would be

48X?X

2

+60Y?Y

2

XY Total pro ts are maximized

when X = 12 and Y = 24,Total pro ts are then equal to

1,008.

(b) \Dealing",Suppose that the airport and the developer remain in-

dependent,If the original situation was one of \free to choose," could

the developer increase his net pro ts by bribing the airport to cut back

one flight per day if the developer has to pay for all of the airport’s lost

pro ts? Yes,The developer decides to get the airport to reduce its

402 EXTERNALITIES (Ch,32)

flights by paying for all lost pro ts coming from the reduction of flights.

To maximize his own net pro ts,how many flights per day should he pay

the airport to eliminate? 12.

32.10 (1) Every morning,6,000 commuters must travel from East Potato

to West Potato,Commuters all try to minimize the time it takes to get to

work,There are two ways to make the trip,One way is to drive straight

across town,throught the heart of Middle Potato,The other way is to take

the Beltline Freeway that circles the Potatoes,The Beltline Freeway is

entirely uncongested,but the drive is roundabout and it takes 45 minutes

to get from East Potato to West Potato by this means,The road through

Middle Potato is much shorter,and if it were uncongested,it would take

only 20 minutes to travel from East Potato to West Potato by this means.

But this road can get congested,In fact,if the number of commuters who

use this road is N,then the number of minutes that it takes to drive from

East Potato to West Potato through Middle Potato is 20 +N=100.

(a) Assuming that no tolls are charged for using either road,in equilib-

rium how many commuters will use the road through Middle Potato?

2,500,What will be the total number of person-minutes per

day spent by commuters traveling from East Potato to West Potato?

45 6;000 = 270;000.

(b) Suppose that a social planner controlled access to the road through

Middle Potato and set the number of persons permitted to travel this

way so as to minimize the total number of person-minutes per day spent

by commuters traveling from East Potato to West Potato,Write an

expression for the total number of person-minutes per day spent by

commuters traveling from East Potato to West Potato as a function of

the number N of commuters permitted to travel on the Middle Potato

road,N(20 +

N

100

)+(6;000?N)45,How many com-

muters per day would the social planner allow to use the road through

Middle Potato? 1,250,In this case,how long would it take com-

muters who drove through Middle Potato to get to work? 32.5

minutes,What would be the total number of person-minutes per

day spent by commuters traveling from East Potato to West Potato?

1;250 32:5+4;750 45 = 254;375

NAME 403

(c) Suppose that commuters value time saved from commuting at $w per

minute and that the Greater Potato metropolitan government charges a

toll for using the Middle Potato road and divides the revenue from this

toll equally among all 6,000 commuters,If the government chooses the

toll in such a way as to minimize the total amount of time that people

spend commuting from East Potato to West Potato,how high should it

set the toll? $12:5w,How much revenue will it collect per day from

this toll? $15;625w,Show that with this policy every commuter is

better o than he or she was without the tolls and evaluate the gain per

consumer in dollars,Before the toll was in place,

all commuters spent 45 minutes traveling

to work,With the toll in place,commuters

who travel on the Beltline still spend

45 minutes traveling to work and commuters

who travel through Middle Potato are

indifferent between spending 45 minutes

traveling on the Beltline and paying the

toll to go through Middle Potato,Thus

nobody would be worse off even if toll

revenue were wasted,But everybody gets

back about $2:6w per day from the toll

revenue,so all are better off.

32.11 (2) Suppose that the Greater Potato metropolitan government

rejects the idea of imposing tra c tolls and decides instead to rebuild the

Middle Potato highway so as to double its capacity,With the doubled

capacity,the amount of time it takes to travel from East Potato to West

Potato on the Middle Potato highway is given by 20 + N=200,where

N is the number of commuters who use the Middle Potato highway,In

the new equilibrium,with expanded capacity and no tolls,how many

commuters will use the Middle Potato highway? 5,000 How long

will it take users of the Middle Potato highway to get to work? 45

minutes How many person-minutes of commuting time will be saved

404 EXTERNALITIES (Ch,32)

by expanding the capacity of the Middle Potato highway? 0 Do

you think people will think that this capacity expansion will be a good

use of their tax dollars? No.

Chapter 33 NAME

Law

Introduction,These problems are based on the survey of law and eco-

nomics found in your text,We hope that you will be pleased to see that

the techniques you learned in earlier chapters can provide useful insights

into issues that arise in law.

33.1 (2) Madame Norrell makes her living in Florida by stealing gold

buttons from designer jackets in expensive boutiques,She can sell each

button to a fence for $10,The maximum number of buttons she can steal

in a day is 50,Florida has a law against button theft,There is a ne of F

dollars if someone is caught stealing any number of buttons,The police

catch about 10 percent of all button thieves,and these must pay the ne

and forfeit any buttons they have stolen.

(a) Suppose that the only thing that Madame Norrell cares about is her

expected pro ts,What is the smallest ne that will discourage Madame

Norrell from stealing buttons? 4,500.

(b) Due to an oversupply of buttons,Madame Norrell’s fence announces

that he will no longer pay her a flat price for buttons,If Madame Norrell

delivers x buttons,she will be paid 5 lnx,(Assume that Madame Norrell

will take at least 1 button if she takes any at all.) Initially Madame

Norrell has $100,and the ne if she is caught stealing x buttons is $3

per button,However,she only has to pay the ne if she is caught,in

which case all her buttons are con scated and she collects zero from the

fence,How many buttons will Madame Norrell try to take,assuming she

maximizes her expected pro t? 15.

(c) What does the ne per button have to be to induce Madame Norrell

to limit herself to taking 10 buttons? 4.50.

(d) Now assume that Madame Norrell is an expected utility maximizer.

With probability,10,she is caught with x buttons and pays a ne of 3x.

With probability,90,she gets away with x buttons,which she can sell for

$10 each,She cares about the expected utility of her wealth,with von

Neumann-Morgenstern utility function lnx,Initially her wealth is $100.

How many buttons will she take? 29.

33.2 (2) Jim Levson rides his bike through the forest with reckless

abandon,while Dick Stout likes to hike in the woods,Let s be the speed

in miles per hour that James rides and w the speed with which Dick walks.

406 LAW (Ch,33)

Jim’s utility depends on how fast he rides and how many dollars he has,

while Dick’s utility depends on how fast he walks and how much money

he has.

U

Jim

=6

p

s?s+m

U

Dick

=4

p

w?w +m:

(a) How fast will Dick walk? 4 miles per hour,How fast will

Jim ride? 9 miles per hour.

(b) Alas,since Jim and Dick are both moving in the same forest,there

is some chance that Jim will run into Dick,Suppose that the expected

cost to Dick of such an accident depends on the speed that each moves:

c(s;w)=

s

2

16

+

w

2

2

,(Assume that Jim is tter than Dick and will incur

negligable costs in an accident.) If Dick has to pay the entire cost of an

accident,how fast will he walk? 1 mile per hour,How fast

will Jim ride? 9 miles per hour.

(c) Suppose that Jim now has full liability and must pay any costs that he

imposes on Dick,How fast will Dick walk? 4 miles per hour.

How fast will Jim ride? 4 miles per hour.

(d) What are the socially optimal speeds for Jim and Dick to move? Dick

should walk 1 mile per hour and Jim should ride 4 miles

per hour.

33.3 (2) Derri Bottled Water of Christchurch,New Zealand,sells bottled

water from \the bottom of the world." Due to a number of fortuitous

circumstances,Derri has a monopoly on bottled water in the South Island.

The demand for bottled water in the South Island is p(x)=10?x=200,

and the cost of producing x bottles of water is c(x)=x

2

=200,Here the

price is measured in New Zealand dollars and the quantity is measured

in 1;000 cases per month.

(a) Draw the demand curve,the marginal revenue curve,and the marginal

cost curve in the graph below,The pro t-maximizing quantity is 500

cases of water,and the pro t maximizing price is 7.5 dollars per case.

NAME 407

200 400 600 800 1000 1200

2

4

6

8

10

12

Quantity

Price

0

5

500

mc

Demand

mr

7.5

(b) The New Zealand antitrust authorities now bring action against Derri

waters for monopolizing the bottled water industry,They announce that

during the coming year they will con scate 50 percent of Derri’s prof-

its,Part of these con scated pro ts will be used to distribute rebates

to the consumers of bottled water,In particular,each purchaser of bot-

tled water will receive $2 per case from the government,How does this

rebate influence the demand for bottled water? Shifts it up

by $2 What is the equation for the new inverse demand curve?

p(x)=12?x=200.

(c) Solve for the new levels of output and price,Draw the marginal

revenue curve,marginal cost curve,and inverse demand curve in the

following graph.

408 LAW (Ch,33)

200 400 600 800 1000 1200

2

4

6

8

10

12

Quantity

Price

0

mc

Demand

mr

625

8

1_

8

Chapter 34 NAME

Information Technology

Introduction,We all recognize that information technology has revolu-

tionized the way we produce and consume,Some think that it is necessary

to have a \new economics" to understand this New Economy,We think

not,The economic tools that you have learned in this course can o er

very powerful insights into the economics of information technology,as

we illustrate in this set of problems.

34.1 (2) Bill Barriers,the president of MightySoft software company is

about to introduce a new computer operating system called DoorKnobs.

Because it is easier to swap les with people who have the same operating

system,the amount people are willing to pay to have DoorKnobs on their

computers is greater the larger they believe DoorKnobs’s market share to

be.

The perceived market share for DoorKnobs is the fraction of all com-

puters that the public believes is using DoorKnobs,When the price of

DoorKnobs is p,thenitsactual market share is the fraction of all com-

puter owners that would be willing to pay at least $p to have DoorKnobs

installed on their computers,Market researchers have discovered that if

DoorKnobs’s perceived market share is s and the price of DoorKnobs is

$p,then its actual market share will be x,wherex is related to the price

p and perceived market share s by the formula

p = 256s(1?x),(1)

In the short run,MightySoft can influence the perceived market share

of DoorKnobs by publicity,advertising,giving liquor and gifts to friendly

journalists,and giving away copies in conspicuous ways,In the long run,

the truth will emerge,and DoorKnobs’s perceived market share s must

equal its actual market share x.

(a) If the perceived market share is s,then the demand curve for Door-

Knobs is given by Equation 1,On the graph below,draw the demand

curve relating price to actual market share in the case in which Door-

Knobs’s perceived market share is s =1=2,Label this curve s =1=2.

(b) On the demand curve that you just drew with s =1=2,mark a

red dot on the point at which the actual market share of DoorKnobs is

1/2,(This is the point on the demand curve directly above x =1=2.)

What is the price at which half of the computer owners actually want to

buy DoorKnobs,given that everybody believes that half of all computer

owners want to buy DoorKnobs? $64

410 INFORMATION TECHNOLOGY (Ch,34)

(c) On the same graph,draw and label a separate demand curve for the

case where DoorKnobs’s perceived market share s takes on each of the

following values,s =1/8,1/4,3/4,7/8,1.

2 4 6 8 10 12 14 16

32

64

96

128

160

192

224

256

Actual Market Share (in sixteenths)

Willingness to Pay

0

S=1/8

S=1/4

S=1/2

S=3/4

S=7/8

S=1

(d) On the demand curve for a perceived market share of s =1=4,put

a red dot on the point at which the actual market share of DoorKnobs

is 1/4,(This is the point on this demand curve directly above x =1=4.)

If the perceived market share of DoorKnobs is 1/4,at what price is the

actual market share of DoorKnobs also 1/4? $48

(e) Just as you did for s =1=2ands =1=4,make red marks on the

demand curves corresponding to s = 1/8,3/4,7/8,and 1,showing the

price at which the actual market share is s,given that the perceived

market share is s.

(f) Let us now draw the long-run demand curve for DoorKnobs,where we

assume that computer owners’ perceived market shares s are the same as

the actual market shares x,If this is the case,it must be that s = x,so

the demand curve is given by p = 256x(1?x),On the graph above,plot

a few points on this curve and sketch in an approximation of the curve.

(Hint,Note that the curve you draw must go through all the red points

that you have already plotted.)

(g) Suppose that MightySoft sets a price of $48 for DoorKnobs and sticks

with that price,There are three di erent perceived market shares such

that the fraction of consumers who would actually want to buy Door-

Knobs for $48 is equal to the perceived market share,One such perceived

NAME 411

market share is 0,What are the other two possibilities? s =1=4

and s =3=4

(h) Suppose that by using its advertising and media influence,MightySoft

can temporarily set its perceived market share at any number between

0 and 1,If DoorKnobs’s perceived market share is x and if MightySoft

charges a price p = 256x(1?x),the actual market fraction will also be x

and the earlier perceptions will be reinforced and maintained,Assuming

that MightySoft chooses a perceived market share x and a price that

makes the actual market share equal to the perceived market share,what

market share x should MightySoft choose in order to maximize its revenue

and what price should it charge in order to maintain this market share?

(Hint,Revenue is px = 256x

2

(1?x).) Use calculus and show your

work,x =2=3,The first-order condition is

d

dx

256(x

2

x

3

)=0,This implies 2x =3x

2

,

which implies that x =2=3 or x =0,The

second order condition is satisfied only

when x =2=3,Price should be $256 1=3

2=3 = $56:89.

34.2 (1) Suppose that demand for DoorKnobs is as given in the previous

problem,and assume that the perceived market share in any period is

equal to the actual market share in the previous period,Then where x

t

is the actual market share in period t,the equation p = 256x

t?1

(1?x

t

)

is satis ed,Rearranging this equation,we nd that x

t

=1?(p=256x

t?1

)

whenever p=256x

t?1

1,If p=256x

t?1

0,then x

t

=0,Withthis

formula,if we know actual market share for any time period,we can

calculate market share for the next period.

Let us assume that DoorKnobs sets the price at p = $32 and never

changes this price,(To answer the following questions,you will nd a

calculator useful.)

(a) If the actual market share in the rst period was 1/2,nd the actual

market share in the second period,75,the third period,833.

Write down the actual market shares for the next few periods,8529,

8534,Do they seem to be approaching a limit? If so,what?

.853553.

412 INFORMATION TECHNOLOGY (Ch,34)

(b) Notice that when price is held constant at p,if DoorKnobs’s mar-

ket share converges to a constant x,itmustbethat x =1?(p=256 x).

Solve this equation for x in the case where p = $32,What do you make

of the fact that there are two solutions? This equation

implies x

2

x +1=8=0,Solutions are

x =0:85355 and x =0:14645,Both are

equilibrium market shares with a price of

$32.

34.3 (1) A group of 13 consumers are considering whether to connect to a

new computer network,Consumer 1 has an initial value of $1 for hooking

up to the network,consumer 2 has an initial value of $2,consumer 3 has

an iinitial value of $3,and so on up to consumer 13,Each consumer’s

willingness to pay to connect to the network depends on the total number

of persons who are connected to it,In fact,for each i,consumeri’s

willingness to pay to connect to the network is i times the total number

of persons connected,Thus if 5 people are connected to the network,

consumer 1’s willingness to pay is $5,consumer 2’s willingness to pay is

$10 and so on.

(a) What is the highest price at which 9 customers could hook up to the

market and all of them either make a pro t or break even? $45

(b) Suppose that the industry that supplies the computer network is com-

petitive and that the cost of hooking up each consumer to the network is

$45,Suppose that consumers are very conservative and nobody will sign

up for the network unless her buyer value will be at least as high as the

price she paid as soon as she signs up,How many people will sign up if

the price is $45? 0

(c) Suppose that the government o ers to subsidize \pioneer users" of the

system,The rst two users are allowed to connect for $10 each,After

the rst two users are hooked up,the government allows the next two

to connect for $25,After that,everyone who signs up will have to pay

the full cost of $45,Assume that users remain so conservative that will

sign up only if their buyer values will be at least equal to the price they

are charged when they connect,With the subsidy in place,how many

consumers in toto will sign up for the network? 9

34.4 (2) Professor Kremepu has written a new,highly simpli ed eco-

nomics text,Microeconomics for the Muddleheaded,which will be pub-

lished by East Frisian Press,The rst edition of this book will be in print

for two years,at which time it will be replaced by a new edition,East

NAME 413

Frisian Press has already made all its xed cost investments in the book

and must pay a constant marginal cost of $c for each copy that it sells.

Let p

1

be the price charged for new copies sold in the rst year of

publication and let p

2

be the price charged for new copies sold in the

second year of publication,The publisher and the students who buy

the book are aware that there will be an active market for used copies of

Microeconomics for the Muddleheaded one year after publication and that

used copies of the rst edition will have zero resale value two years after

publication,At the end of the rst year of publication,students can resell

their used textbooks to bookstores for 40% of the second-year price,p

2

.

The net cost to a student of buying the book in the rst year,using it

for class,and reselling it at the end of the year is p

1

0:4p

2

.Thenumber

of copies demanded in the rst year of publication is given by a demand

function,q

1

= D

1

(p

1

0:4p

2

).

Some of the students who use the book in the rst year of publication

will want to keep their copies for future reference,and some will damage

their books so that they cannot be resold,The cost of keeping one’s old

copy or of damaging it is the resale price 0:4p

2

,The number of books that

are either damaged or kept for reference is given by a \keepers" demand

function,D

k

(0:4p

2

),It follows that the number of used copies available

at the end of the rst year will be D

1

(p

1

0:4p

2

)?D

k

(0:4p

2

).

Students who buy Microeconomics for the Muddleheaded in the sec-

ond year of publication will not be able to resell their used copies,since

a new edition will then be available,These students can,however,buy

either a new copy or a used copy of the book,For simplicity of calcula-

tions,let us assume that students are indi erent between buying a new

copy or a used copy and that used copies cost the same as new copies in

the book store,(The results would be the same if students preferred new

to used copies,but bookstores priced used copies so that students were

indi erent between buying new and used copies.) The total number of

copies,new and used,that are purchased in the second year of publication

is q

2

= D

2

(p

2

).

(a) Write an expression for the number of new copies that East Frisian

Press can sell in the second year after publication if it sets prices p

1

in

year 1 and p

2

in year 2,D

2

(p

2

)?D

1

(p

1

:4p

2

)+D

k

(:4p

2

).

(b) Write an expression for the total number of new copies of Microeco-

nomics for the Muddleheaded that East Frisian can sell over two years at

prices p

1

and p

2

in years 1 and 2,D

1

(p

1

:4p

2

)+D

2

(p

2

)?

D

1

(p

1

:4p

2

)+D

k

(:4p

2

)=D

2

(p

2

)+D

k

(:4p

2

).

(c) Would the total number of copies sold over two years increase,de-

crease,or remain constant if p

1

were increased and p

2

remained constant?

It would remain constant.

414 INFORMATION TECHNOLOGY (Ch,34)

(d) Write an expression for the total revenue that East Frisian Press will

receive over the next two years if it sets prices p

1

and p

2

,p

1

D

1

(p

1

:4p

2

)+p

2

(D

2

(p

2

)?D

1

(p

1

:4p

2

)+D

k

(:4p

2

)) = (p

1

p

2

)D

1

(p

1

:4p

2

)+p

2

(D

2

(p

2

)+D

k

(:4p

2

)):

(e) To maximize its total pro ts over the next two years,East Frisian

must maximize the di erence between its total revenue and its variable

costs,Show that this di erence can be written as

(p

1

p

2

)D

1

(p

1

:4p

2

)+(p

2

c)

D

2

(p

2

)+D

k

(:4p

2

)

:

Variable cost is c(D

2

(p

2

)+D

k

(:4p

2

)).

Subtract this from previous answer.

(f) Suppose that East Frisian has decided that it must charge the same

price for the rst edition in both years that it is sold,Thus it must

set p = p

1

= p

2

,Write an expression for East Frisian’s revenue net

of variable costs over the next two years as a function of p.

(p?c)(D

2

(p)+D

k

(:4p))

34.5 (2) Suppose that East Frisian Press,discussed in the previous

problem,has a constant marginal cost of c = $10 for each copy of Micro-

economics for the Muddleheaded that it sells and let the demand functions

be

D

1

(p

1

0:4p

2

) = 100 (90?p

1

+0:4p

2

)

D

2

(p

2

) = 100(90?p

2

):

The number of books that people either damage or keep for reference

after the rst year is

D

k

(0:4p

2

) = 100(90?0:8p

2

):

(This assumption is consistent with the assumption that everyone’s will-

ingness to pay for keeping the book is half as great as her willingness to

pay to have the book while she is taking the course.) Assume that East

Frisian Press is determined to charge the same price in both years,so that

p

1

= p

2

= p.

NAME 415

(a) If East Frisian Press charges the same price p for Microeconomics for

the Muddleheaded in the rst and second years,show that the total sales

of new copies over the two years are equal to

18;000?180p:

Total sales are D

2

(p)+D

k

(:4p

2

) = 100(90?

p) + 100(90?:8p)) = 18;000?180p

(b) Write an expression for East Frisian’s total revenue,net of variable

costs,over the rst two years as a function of the price p,(p?

10)(18;000?180p)=19;800p?180p

2

180;000

(c) Solve for the price p that maximizes its total revenue net of variable

costs over the rst two years,p = $55,At this price,the net cost

to students in the rst year of buying the text and reselling it is $33.

The total number of copies sold in the rst year will be 5,700,The

total number of copies that are resold as used books is 1,100,The

total number of copies purchased by students in the second year will be

3,500,(Remember students in the second year know that they

cannot resell the book,so they have to pay the full price p for using it.)

The total number of new copies purchased by students in the second year

will be 2,400,Total revenue net of variable costs over the two years

will be $364,500.

34.6 (2) East Frisian Press is trying to decide whether it would be prof-

itable to produce a new edition of Microeconomics for the Muddleheaded

after one year rather than after two years,If it produces a new edition

after one year,it will destroy the used book market and all copies that

are purchased will be new copies,In this case,the number of new copies

that will be demanded in each of the two years will be 100(90?p),where

p is the price charged,The variable cost of each copy sold remains $10.

(a) Write an expression for the total number of copies sold over the course

of two years if the price is p in each year 200(90-p),Also,write

an expression for total revenue net of variable costs as a function of p.

200(p?10)(90?p).

416 INFORMATION TECHNOLOGY (Ch,34)

(b) Find the price that maximizes total revenue net of variable costs.

$50.

(c) The total number of new books sold in the rst year would be

4,000,and the total number of books sold in the second year would

be 4,000.

(d) East Frisian’s total revenue net of variable costs,if it markets a new

edition after one year,will be $320,000.

(e) Would it be more pro table for East Frisian Press to produce a new

edition after one year or after two years? After two years.

Which would be better for students? (Hint,The answer is not the same

for all students.) After two years is better for

students who take the course in the first

year of publication and plan to sell.

After one year is better for the other

students.

34.7 (3) Suppose that East Frisian Press publishes a new edition only af-

ter two years and that demands and costs are as in the previous problems.

Suppose that it sets two di erent prices p

1

and p

2

in the two periods.

(a) Write an expression for the total number of new copies sold at prices

p

1

and p

2

and show that this number depends on p

2

but not on p

1

.

100 ((70?p

1

+:4p

2

) + (140?1:8p

2

)?(70?p

1

+:4p

2

)) =

100(140?1:8p

2

)

(b) Show that at prices p

1

and p

2

,the di erence between revenues and

variablecostsisequalto

100

parenleftbig

90p

1

+ 108p

2

+1:4p

1

p

2

p

2

1

2:2p

2

2

1;800

:

This difference is 100(p

1

p

2

)(90?p

1

+

:4p

2

)+(p

2

10)(180?1:8p

2

),Expand this

expression.

NAME 417

(c) Calculate the prices p

1

and p

2

that maximize the di erence between

total revenue and variable costs and hence maximize pro ts,p

1

=

$80,p

2

= $50

(d) If East Frisian Press chooses its pro t-maximizing p

1

and p

2

,compare

the cost of using Microeconomics for the Muddleheaded for a student who

buys the book when it is rst published and resells it at the end of the

rst year with the cost for a student who buys the book at the beginning

of the second year and then discards it,The former has a

net cost of $80?:4 50 = $60 and the latter

has a cost of $50.

34.8 (2) The Silicon Valley company Intoot produces checkwriting soft-

ware,The program itself,Fasten,sells for $50 and includes a package of

checks,Check re ll packets for Fasten cost $20 to produce and Intoot sells

the checks at cost,Suppose that a consumer purchases Fasten for $50 in

period 1 and spends $20 on checks in each subsequent period,Assume

for simplicity that the consumer uses the program for an in nite number

of periods.

(a) If the interest rate is r =,10 per period,what is the present value

of the stream of payments made by the consumer? (Hint,a stream of

payments of x starting next period has a present value of x=r.) The total

cost of ownership of Fasten is 50+20/.10 = $250.

(b) Fasten’s competitor produces an equally e ective product called

Czechwriter,Czechwriter can do everything Fasten can do and vice versa

except that Fasten cannot use check re ll packets that are sold by anyone

other than Fasten,Czechwriter also sells for $50 and sells its checks for

$20 per period,A Fasten customer can switch to Czechwriter simply by

purchasing the program,This means his switching costs are $50

(c) Fasten is contemplating raising the price of checks to $30 per period.

If so,will its customers switch to Czechwriter? Explain,Yes,the

present value of continuing to use Fasten

are $300 while the costs of switching to

Czechwriter are $250.

418 INFORMATION TECHNOLOGY (Ch,34)

(d) Fasten contemplates raising the price of checks to $22 per period,Will

its customers switch? No,The present value of

continuing to use Fasten are $220 while the

present value of switching to Czechwriter

is $250.

(e) At what price for checks will Fasten’s customers just be indi erent to

switching? (Hint,Let x be this amount,Compare the present value of

staying with Fasten with the present value of switching to Czechwriter.)

Solve the equation x=:10 = 250 to find

x =25.

(f) If it charges the highest price that it can without making its customers

switch,what pro t does Fasten make on checks from each of its customers

per period? $5,What is the present value of the pro t per customer

that Fasten gets if it sets the price of checks equal to the number deter-

minedinthelastquestion? PV =5=:10 = 50,How does this

compare to the customer switching cost? It is the same.

(g) Suppose now that the cost of switching also involves several hours

of data conversion that the consumer values at $100,The total cost of

switching is the cost of the new program plus the data conversion cost

which is $150.

(h) Making allowances for the cost of data conversion,what is the highest

pricethatIntootcanchargeforitschecks? Solve x=:10 =

250 + 100 for x =35,What is the present value of pro t

from this price? $150,How does this compare to total switching

costs? It is the same.

(i) Suppose that someone writes a computer program that eliminates

the cost of converting data and makes this program available for free.

Suppose that Intoot continues to price its check re ll packages at $25,A

new customer is contemplating buying Fasten at a price of $50 and paying

$25 per period for checks,versus paying $50 for Czechwriter and paying

$20 for checks,If the functionality of the software is identical,which will

the consumer buy? Czechwriter.

NAME 419

(j) Intoot decides to distribute a coupon that o ers a discount of $50

o of the regular purchase price,What price would it have to set to

make consumers indi erent between purchasing Fasten and Czechwriter?

Solve 50 + 25=:10?d =50+20=:10 to find

that the discount should be $50.

(k) Suppose that consumers are shortsighted and only look at the cost

of the software itself,neglecting the cost of the checks,Which program

would they buy if Intoot o ered this coupon? Fasten,How might

Czechwriter respond to the Fasten o er? Issue its own

coupon for $50 and raise the price of its

checks to $35.

34.9 (2) Sol Microsystems has recently invented a new language,Guava,

which runs on a proprietary chip,the Guavachip,The chip can only be

used to run Guava,and Guava can only run on the Guavachip,Sol

estimates that if it sells the chip for a price p

c

and the language for a

price p

g

,the demand for the chip-language system will be

x = 120?(p

c

+p

g

):

(a) Sol initially sets up two independent subsidiaries,one to produce the

chip and one to produce the language,Each of the subsidiaries will price

its product so as to maximize its pro ts,while assuming that a change in

its own price will not a ect the pricing decision of the other subsidiary.

Assume that marginal costs are negligible for each company,If the price

of the language is set at p

g

,the chip company’s pro t function (neglecting

xed costs) is [120?p

c

p

g

]p

c

.

(b) Di erentiate this pro t function with respect to p

c

and set the result

equal to zero to calculate the optimal choice of p

c

as a function of p

g

.

p

c

= 120?2p

g

:

(c) Now consider the language subsidiary’s pricing decision,The optimal

choice of p

g

as a function of p

c

is p

g

= 120?2p

c

:

(d) Solving these two equations in two unknowns,we nd that p

c

=

40 and p

g

= 40,sothatp

c

+p

g

= 80

420 INFORMATION TECHNOLOGY (Ch,34)

(e) Sol Microsystems decides that the independent subsidiary system is

cumbersome,so it sets up Guava Computing which sells a bundled system

consisting of the chip and the language,Let p be the price of the bundle.

Guava Computing’s pro t function is [120?p]p.

(f) Di erentiate this pro t with respect to p and set the resulting expres-

sion to zero to determine p = 60.

(g) Compare the prices charged by the integrated system and the separate

subsidiaries,Which is lower? Integrated system,Which

is better for consumers? Integrated system,Which makes

more pro t? Integrated system.

34.10 (2) South Belgium Press produces the academic journal Nano-

economics,which has a loyal following among short microeconomists,and

Gigaeconomics,a journal for tall macroeconomists,It o ers a license for

the electronic version of each journal to university libraries at a subscrip-

tion cost per journal of $1,000 per year,The 200 top universities all

subscribe to both journals,each paying $2,000 per year to South Bel-

gium,By revealed preference,their willingness to pay for each journal is

at least $1,000.

(a) In an attempt to lower costs,universities decide to form pairs,with

one member of each pair subscribing to Nanoeconomics and one member

of each pair subscribing to Gigaeconomics,They agree to use interlibrary

loan to share the other journal,Since the copies are electronic,there is

no incremental cost to doing this,Under this pairing scheme,how many

subscriptions of each journal will South Belgium sell? 100

.

(b) In order to stem the revenue hemorrhage,South Belgium raises the

price of each journal,Assuming library preferences and budgets haven’t

changed,how high can they set this price? They can raise

the price to $2,000,since libraries have

already indicated that they are willing

to pay this much for the pair of journals.

(c) How does library expenditure and South Belgium’s revenue compare

to those of the previous regime? They remain the same.

NAME 421

(d) If there were a cost of interlibrary loan,how would your an-

swer change? Assuming they still bought

both journals,libraries would be worse

off since they would have to pay the

transactions cost for interlibrary loan.

422 INFORMATION TECHNOLOGY (Ch,34)

Chapter 35 NAME

Public Goods

Introduction,In previous chapters we studied sel sh consumers con-

suming private goods,A unit of private goods consumed by one person

cannot be simultaneously consumed by another,If you eat a ham sand-

wich,Icannoteatthesamehamsandwich,(Ofcoursewecanbotheat

ham sandwiches,but we must eat di erent ones.) Public goods are a dif-

ferent matter,They can be jointly consumed,You and I can both enjoy

looking at a beautiful garden or watching reworks at the same time,The

conditions for e cient allocation of public goods are di erent from those

for private goods,With private goods,e ciency demands that if you and

I both consume ham sandwiches and bananas,then our marginal rates of

substitution must be equal,If our tastes di er,however,we may consume

di erent amounts of the two private goods.

If you and I live in the same town,then when the local reworks

show is held,there will be the same amount of reworks for each of us.

E ciency does not require that my marginal rate of substitution between

reworks and ham sandwiches equal yours,Instead,e ciency requires

that the sum of the amount that viewers are willing to pay for a marginal

increase in the amount of reworks equal the marginal cost of reworks.

This means that the sum of the absolute values of viewers’ marginal rates

of substitution between reworks and private goods must equal the mar-

ginal cost of public goods in terms of private goods.

Example,A quiet midwestern town has 5,000 people,all of whom are in-

terested only in private consumption and in the quality of the city streets.

The utility function of person i is U(X

i;G)=X

i

+A

i

G?B

i

G

2

,whereX

i

is the amount of money that person i has to spend on private goods and

G is the amount of money that the town spends on xing its streets,To

nd the Pareto optimal amount of money for this town to spend on xing

its streets,we must set the sum of the absolute values of marginal rates of

substitution between public and private goods equal to the relative prices

of public and private goods,In this example we measure both goods in

dollar values,so the price ratio is 1,The absolute value of person i’s

marginal rate of substitution between public goods and private goods is

the ratio of the marginal utility of public goods to the marginal utility of

private goods,The marginal utility of private goods is 1 and the marginal

utility of public goods for person i is A

i

B

i

G,Therefore the absolute

value of person i’s MRS is A

i

B

i

G and the sum of absolute values

of marginal rates of substitution is

P

i

(A

i

B

i

G)=

P

i

A

i

(

P

B

i

)G.

Therefore Pareto e ciency requires that

P

i

A

i

(

P

i

B

i

)G =1,Solving

this for G,wehaveG =(

P

i

A

i

1)=

P

i

B

i

.

35.1 (0) Muskrat,Ontario,has 1,000 people,Citizens of Muskrat con-

sume only one private good,Labatt’s ale,There is one public good,the

town skating rink,Although they may di er in other respects,inhabitants

424 PUBLIC GOODS (Ch,35)

have the same utility function,This function is U(X

i;G)=X

i

100=G,

where X

i

is the number of bottles of Labatt’s consumed by citizen i and

G is the size of the town skating rink,measured in square meters,The

price of Labatt’s ale is $1 per bottle and the price of the skating rink is

$10 per square meter,Everyone who lives in Muskrat has an income of

$1,000 per year.

(a) Write down an expression for the absolute value of the marginal rate

of substitution between skating rink and Labatt’s ale for a typical citizen.

100=G

2

What is the marginal cost of an extra square meter of skating

rink (measured in terms of Labatt’s ale)? 10.

(b) Since there are 1,000 people in town,all with the same marginal

rate of substitution,you should now be able to write an equation that

states the condition that the sum of absolute values of marginal rates of

substitution equals marginal cost,Write this equation and solve it for the

Pareto e cient amount of G,1;000

100

G

2

=10.SoG = 100.

(c) Suppose that everyone in town pays an equal share of the cost of

the skating rink,Total expenditure by the town on its skating rink will

be $10G,Then the tax bill paid by an individual citizen to pay for the

skating rink is $10G=1;000 = $G=100,Every year the citizens of Muskrat

vote on how big the skating rink should be,Citizens realize that they will

have to pay their share of the cost of the skating rink,Knowing this,a

citizen realizes that if the size of the skating rink is G,then the amount

of Labatt’s ale that he will be able to a ord is 1;000?G=100.

(d) Therefore we can write a voter’s budget constraint as X

i

+G=100 =

1;000,In order to decide how big a skating rink to vote for,a voter simply

solves for the combination of X

i

and G that maximizes his utility subject

to his budget constraint and votes for that amount of G.HowmuchG is

that in our example? G = 100.

(e) If the town supplies a skating rink that is the size demanded by the

voters will it be larger than,smaller than,or the same size as the Pareto

optimal rink? The same.

(f) Suppose that the Ontario cultural commission decides to promote

Canadian culture by subsidizing local skating rinks,The provincial gov-

ernment will pay 50% of the cost of skating rinks in all towns,The costs

of this subsidy will be shared by all citizens of the province of Ontario.

There are hundreds of towns like Muskrat in Ontario,It is true that to

pay for this subsidy,taxes paid to the provincial government will have

to be increased,But there are hundreds of towns from which this tax

NAME 425

is collected,so that the e ect of an increase in expenditures in Muskrat

on the taxes its citizens have to pay to the state can be safely neglected.

Now,approximately how large a skating rink would citizens of Muskrat

vote for? G = 100

p

2,(Hint,Rewrite the budget constraint for

individuals observing that local taxes will be only half as large as before

and the cost of increasing the size of the rink only half as much as before.

Then solve for the utility-maximizing combination.)

(g) Does this subsidy promote economic e ciency? No.

35.2 (0) Ten people have dinner together at an expensive restaurant

and agree that the total bill will be divided equally among them.

(a) What is the additional cost to any one of them of ordering an appetizer

that costs $20? $2.

(b) Explain why this may be an ine cient system,Each pays

less than full cost of own meal,so all

overindulge.

35.3 (0) Cowflop,Wisconsin,has 1,000 people,Every year they have

a reworks show on the Fourth of July,The citizens are interested in

only two things|drinking milk and watching reworks,Fireworks cost 1

gallon of milk per unit,People in Cowflop are all pretty much the same.

In fact,they have identical utility functions,The utility function of each

citizen i is U

i

(x

i;g)=x

i

+

p

g=20,where x

i

is the number of gallons

of milk per year consumed by citizen i and g is the number of units of

reworks exploded in the town’s Fourth of July extravaganza,(Private

use of reworks is outlawed.)

(a) Solve for the absolute value of each citizen’s marginal rate of substi-

tution between reworks and milk,1=(40

p

g).

(b) Find the Pareto optimal amount of reworks for Cowflop,625.

35.4 (0) Bob and Ray are two hungry economics majors who are sharing

an apartment for the year,In a flea market they spot a 25-year-old sofa

that would look great in their living room.

Bob’s utility function is u

B

(S;M

B

)=(1+S)M

B

,and Ray’s utility

function is u

R

(S;M

R

)=(2+S)M

R

,In these expressions M

B

and M

R

are

the amounts of money that Bob and Ray have to spend on other goods,

S = 1 if they get the sofa,and S = 0 if they don’t get the sofa,Bob has

W

B

dollars to spend,and Ray has W

R

dollars.

426 PUBLIC GOODS (Ch,35)

(a) What is Bob’s reservation price for the sofa? Solve W

B

=

2(W

B

p

B

) to get p

B

= W

B

=2.

(b) What is Ray’s reservation price for the sofa? Solve 2W

R

=

3(W

R

p

R

),which gives p

R

= W

R

=3.

(c) If Bob has a total of W

B

= $100 and Ray has a total of W

R

= $75

to spend on sofas and other stu,they could buy the sofa and have a

Pareto improvement over not buying it so long as the cost of the sofa is

no greater than $75.

35.5 (0) Bonnie and Clyde are business partners,Whenever they work,

they have to work together,Their only source of income is pro t from

their partnership,Their total pro t per year is 50H,whereH is the

number of hours that they work per year,Since they must work together,

they both must work the same number of hours,so the variable \hours of

labor" is like a public \bad" for the two person community consisting of

Bonnie and Clyde,Bonnie’s utility function is U

B

(C

B;H)=C

B

:02H

2

and Clyde’s utility function is U

C

(C

C;H)=C

C

:005H

2

,whereC

B

and

C

C

are the annual amounts of money spent on consumption for Bonnie

and for Clyde.

(a) If the number of hours that they both work is H,what is the ratio

of Bonnie’s marginal utility of hours of work to her marginal utility of

private goods:04H,What is the ratio of Clyde’s marginal utility

of hours of work to his marginal utility of private goods:01H.

(b) If Bonnie and Clyde are both working H hours,then the total amount

of money that would be needed to compensate them both for having to

work an extra hour is the sum of what is needed to compensate Bonnie

and the amount that is needed to compensate Clyde,This amount is

approximately equal to the sum of the absolute values of their marginal

rates of substitution between work and money,Write an expression for

this amount as a function of H.,05H,How much extra money will

they make if they work an extra hour? $50.

(c) Write an equation that can be solved for the Pareto optimal number

of hours for Bonnie and Clyde to work.,05H =50.

Find the Pareto optimal H,H =1;000,(Hint,Notice that

this model is formally the same as a model with one public good H and

one private good,income.)

NAME 427

35.6 (0) Lucy and Melvin share an apartment,They spend some of

their income on private goods like food and clothing that they consume

separately and some of their income on public goods like the refrigerator,

the household heating,and the rent,which they share,Lucy’s utility

function is 2X

L

+G and Melvin’s utility function is X

M

G,whereX

L

and

X

M

are the amounts of money spent on private goods for Lucy and for

Melvin and where G is the amount of money that they spend on public

goods,Lucy and Melvin have a total of $8,000 per year between them to

spend on private goods for each of them and on public goods.

(a) What is the absolute value of Lucy’s marginal rate of substitution

between public and private goods? 1=2,What is the absolute value

of Melvin’s? X

M

=G.

(b) Write an equation that expresses the condition for provision of the

Pareto e cient quantity of the public good,1=2+X

M

=G =1.

(c) Suppose that Melvin and Lucy each spend $2,000 on private goods

for themselves and they spend the remaining $4,000 on public goods,Is

this a Pareto e cient outcome? Yes.

(d) Give an example of another Pareto optimal outcome in which Melvin

gets more than $2,000 and Lucy gets less than $2,000 worth of private

goods,One example,Melvin gets $2,500; Lucy

gets $500 and G =$5;000.

(e) Give an example of another Pareto optimum in which Lucy gets

more than $2,000,Lucy gets $5,000; Melvin gets

$1;000 and G =$2;000.

(f) Describe the set of Pareto optimal allocations,The allocations

that satisfy the equations X

M

=G =1=2 and

X

L

+X

M

+G =$8;000.

(g) The Pareto optima that treat Lucy better and Melvin worse will have

(more of,less of,the same amount of) public good as the Pareto optimum

that treats them equally,Less of.

428 PUBLIC GOODS (Ch,35)

35.7 (0) This problem is set in a fanciful location,but it deals with a

very practical issue that concerns residents of this earth,The question

is,\In a Democracy,when can we expect that a majority of citizens will

favor having the government supply pure private goods publicly?" This

problem also deals with the e ciency issues raised by public provision

of private goods,We leave it to you to see whether you can think of

important examples of publicly supplied private goods in modern Western

economies.

On the planet Jumpo there are two goods,aerobics lessons and

bread,The citizens all have Cobb-Douglas utility functions of the form

U

i

(A

i;B

i

)=A

1=2

i

B

1=2

i

,whereA

i

and B

i

are i’s consumptions of aerobics

lessons and bread,Although tastes are all the same,there are two di er-

ent income groups,the rich and the poor,Each rich creature on Jumpo

has an income of 100 fondas and every poor creature has an income of

50 fondas (the currency unit on Jumpo),There are two million poor

creatures and one million rich creatures on Jumpo,Bread is sold in the

usual way,but aerobics lessons are provided by the state despite the fact

that they are private goods,The state gives the same amount of aerobics

lessons to every creature on Jumpo,The price of bread is 1 fonda per

loaf,The cost to the state of aerobics lessons is 2 fondas per lesson,This

cost of the state-provided lessons is paid for by taxes collected from the

citizens of Jumpo,The government has no other expenses than providing

aerobics lessons and collects no more or less taxes than the amount needed

to pay for them,Jumpo is a democracy,and the amount of aerobics to

be supplied will be determined by majority vote.

(a) Suppose that the cost of the aerobics lessons provided by the state

is paid for by making every creature on Jumpo pay an equal amount of

taxes,On planets,such as Jumpo,where every creature has exactly one

head,such a tax is known as a \head tax." If every citizen of Jumpo gets

20 lessons,how much will be total government expenditures on lessons?

120 million fondas,How much taxes will every citizen

have to pay? 40 fondas,If 20 lessons are given,how much will a

rich creature have left to spend on bread after it has paid its taxes? 60

fondas,How much will a poor creature have left to spend on bread

after it has paid its taxes? 10 fondas.

(b) More generally,when everybody pays the same amount of taxes,if x

lessons are provided by the government to each creature,the total cost

to the government is 6 million times x and the taxes that one

creature has to pay is 2 times x.

NAME 429

(c) Since aerobics lessons are going to be publicly provided with every-

body getting the same amount and nobody able to get more lessons from

another source,each creature faces a choice problem that is formally the

same as that faced by a consumer,i,who is trying to maximize a Cobb-

Douglas utility function subject to the budget constraint 2A + B = I,

whereI is its income,Explain why this is the case,If A lessons

are provided,your taxes are 2A fondas.

After taxes,you have I?2A fondas to

spend on B.

(d) Suppose that the aerobics lessons are paid for by a head tax and all

lessons are provided by the government in equal amounts to everyone.

How many lessons would the rich people prefer to have supplied? 25.

How many would the poor people prefer to have supplied? 12.5.

(Hint,In each case you just have to solve for the Cobb-Douglas demand

with an appropriate budget.)

(e) If the outcome is determined by majority rule,how many aerobics

lessons will be provided? 12.5,How much bread will the rich get?

75,How much bread will the poor get? 25.

(f) Suppose that aerobics lessons are \privatized," so that no lessons are

supplied publicly and no taxes are collected,Every creature is allowed to

buy as many lessons as it likes and as much bread as it likes,Suppose

that the price of bread stays at 1 fonda per unit and the price of lessons

stays at 2 fondas per unit,How many aerobics lessons will the rich get?

25,How many will the poor get? 12.5,How much bread will the

rich get? 50,How much bread will the poor get? 25.

(g) Suppose that aerobics lessons remain publicly supplied but are paid

for by a proportional income tax,The tax rate is set so that tax rev-

enue pays for the lessons,If A aerobics lessons are o ered to each

creature on Jumpo,the tax bill for a rich person will be 3A fondas

and the tax bill for a poor person will be 1:5A fondas,If A lessons

are given to each creature,show that total tax revenue collected will

be the total cost of A lessons,There are 2,000,000

poor and 1,000,000 rich,total revenue is

2;000;000 1:5A+1;000;000 3A =6;000;000A.

430 PUBLIC GOODS (Ch,35)

There are 3,000,000 people in all,If each

gets A lessons and lessons cost 2 fondas,

total cost is 6;000;000A.

(h) With the proportional income tax scheme discussed above,what bud-

get constraint would a rich person consider in deciding how many aerobics

lessons to vote for? 3A + B = 100,What is the relevant bud-

get constraint for a poor creature? 1:5A + B =50,With these

tax rates,how many aerobics lessons per creature would the rich favor?

50=3,How many would the poor favor? 50=3,What quantity of

aerobics lessons per capita would be chosen under majority rule? 50=3.

How much bread would the rich get? 50,How much bread would the

poor get? 25.

(i) Calculate the utility of a rich creature under a head tax.

p

937:5

Under privatization.

p

1;250,Under a proportional income tax.

p

833:3,(Hint,In each case,solve for the consumption of bread and

the consumption of aerobics lessons that a rich person gets,and plug these

into the utility function.) Now calculate the utility of each poor creature

under the head tax.

p

312:5,Under privatization.

p

312:5,Un-

der the proportional income tax.

p

416:67,(Express these utilities

as square roots rather than calculating out the roots.)

(j) Is privatization Pareto superior to the head tax? Yes,Is a propor-

tional income tax Pareto superior to the head tax? No,Is privatization

Pareto superior to the proportional income tax? No,Explain the last

two answers,Rich prefer privatization,poor

prefer proportional income tax.

Chapter 36 NAME

Information

Introduction,The economics of information and incentives is a rela-

tively new branch of microeconomics,in which much intriguing work is

going on,This chapter shows you a sample of these problems and the

way that economists think about them.

36.1 (0) There are two types of electric pencil-sharpener producers.

\High-quality" manufacturers produce very good sharpeners that con-

sumers value at $14,\Low-quality" manufacturers produce less good ones

that are valued at $8,At the time of purchase,customers cannot distin-

guish between a high-quality product and a low-quality product; nor can

they identify the manufacturer,However,they can determine the quality

of the product after purchase,The consumers are risk neutral; if they

have probability q of getting a high-quality product and 1?q of getting

a low-quality product,then they value this prospect at 14q +8(1?q).

Each type of manufacturer can manufacture the product at a constant

unit cost of $11.50,All manufacturers behave competitively.

(a) Suppose that the sale of low-quality electric pencil-sharpeners is ille-

gal,so that the only items allowed to appear on the market are of high

quality,What will be the equilibrium price? $11.50.

(b) Suppose that there were no high-quality sellers,How many low-quality

sharpeners would you expect to be sold in equilibrium? Sellers

won’t sell for less than $11.50,consumers

won’t pay that much for low-quality product.

So in equilibrium there would be no sales.

(c) Could there be an equilibrium in which equal (positive) quantities

of the two types of pencil sharpeners appear in the market? No.

Average willingness to pay would be $11,

which is less than the cost of production.

So there would be zero trade.

432 INFORMATION (Ch,36)

(d) Now we change our assumptions about the technology,Suppose

that each producer can choose to manufacture either a high-quality or

a low-quality pencil-sharpener,with a unit cost of $11.50 for the for-

mer and $11 for the latter,what would we expect to happen in equilib-

rium? No trade,Producers would produce the

low-quality product since it has a lower

production cost,If all producers produce

low-quality output,costs will be $11 and

the willingness-to-pay for low quality is

$8.

(e) Assuming that each producer is able to make the production choice

described in the last question,what good would it do if the government

banned production of low-quality electric pencil-sharpeners? If

there is no ban,there will be no output

and no consumers’ surplus,If low-quality

products are banned,then in equilibrium

there is output and positive consumers’

surplus.

36.2 (0) In West Bend,Indiana,there are exactly two kinds of workers.

One kind has a (constant) marginal product worth $10 and the other kind

has a (constant) marginal product worth $15,There are equal numbers

of workers of each kind,A rm cannot directly tell the di erence between

the two kinds of workers,Even after it has hired them,it won’t be able

to monitor their work closely enough to determine which workers are of

which type.

(a) If the labor market is competitive,workers will be paid the average

value of their marginal product,This amount is $12.50.

(b) Suppose that the local community college o ers a microeconomics

course in night school,taught by Professor M,De Sade,The high-

productivity workers think that taking this course is just as bad as a

$3 wage cut,and the low-productivity workers think it is just as bad as

a $6 wage cut,The rm can observe whether or not an individual takes

the microeconomics course,Suppose that the high-productivity workers

all choose to take the microeconomics course and the low-productivity

NAME 433

workers all choose not to,The competitive wage for people who take the

microeconomics course will be $15 and the wage for people who don’t

take the microeconomics course will be $10.

(c) If there is a separating equilibrium,with high-productivity workers

taking the course and low-productivity workers not taking it,then the

net bene ts from taking the microeconomics course will be $2

for the high-productivity workers and $?1 for the low-productivity

workers,Therefore there (will be,won’t be) will be a separating

equilibrium of this type.

(d) Suppose that Professor De Sade is called o to Washington,to lec-

ture wayward representaatives on the economics of family values,His

replacement is Professor Morton Kremepu,Kremepu prides himself on

his ability to make economics \as easy as political science and as fun as

the soaps on TV." Professor Kremepu ’s claims are exaggerated,but at

least students like him better than De Sade,High-productivity workers

think that taking Kremepu ’s course is as bad as a $1 wage cut,and

low-productivity workers think that taking Kremepu ’s course is as bad

as a $4 wage cut,If the high-productivity workers all choose to take the

microeconomics course and the low-productivity workers all choose not to,

the competitive wage for people who take the microeconomics course will

be $15 and the wage for people who don’t take the microeconomics

course will be $10.

(e) If there is a separating equilibrium with high-productivity workers

taking the course and low-productivity workers not taking it,then the net

bene ts from taking Kremepu ’s microeconomics course will be $4

for the high-productivity workers and $1 for the low-productivity

workers,Therefore there (will be,won’t be) won’t be a separating

equilibrium of this type.

36.3 (1) In Enigma,Ohio,there are two kinds of workers,Klutzes

whose labor is worth $1,000 per month and Kandos,whose labor is worth

$2,500 per month,Enigma has exactly twice as many Klutzes as Kandos.

Klutzes look just like Kandos and are accomplished liars,If you ask,

they will claim to be Kandos,Kandos always tell the truth,Monitoring

individual work accomplishments is too expensive to be worthwhile,In

the old days,there was no way to distinguish the two types of labor,so

everyone was paid the same wage,If labor markets were competitive,

what was this wage? $1,500

434 INFORMATION (Ch,36)

(a) A professor who loves to talk o ered to give a free monthly lecture

on macroeconomics and personal hygiene to the employees of one small

rm,These lectures had no e ect on productivity,but both Klutzes and

Kandos found them to be excruciatingly dull,To a Klutz,each hour’s

lecture was as bad as losing $100,To a Kando,each hour’s lecture was as

bad as losing $50,Suppose that the rm gave each of its employees a pay

raise of $55 a month but insisted that he attend the professor’s lectures.

What would happen to the rm’s labor force? All Klutzes

would leave,Kandos would stay on,More

Kandos could be hired at these terms.

Klutzes would not accept job,What would happen

to the average productivity of the rm’s employees? Rise by

$1,000--from $1,500 to $2,500.

(b) Other rms noticed that those who had listened to the professor’s

lectures were more productive than those who had not,So they tried to

bid them away from their original employer,Since all those who agreed

to listen to the original lecture series were Kandos,their wage was bid up

to $2,500.

(c) After observing the \e ect of his lectures on labor productivity," the

professor decided to expand his e orts,He found a huge auditorium where

he could lecture to all the laborers in Enigma who would listen to him.

If employers believed that listening to the professor’s lectures improved

productivity by the improvement in productivity in the rst small rm

and o ered bonuses for attending the lectures accordingly,who would

attend the lectures? Everybody,Having observed this outcome,

how much of a wage premium would rms pay for those who had attended

the professor’s lectures? 0.

(d) The professor was disappointed by the results of his big lecture and

decided that if he gave more lectures per month,his pupils might \learn

more." So he decided to give a course of lectures for 20 hours a month.

Would there now be an equilibrium in which the Kandos all took his

course and none of the Klutzes took it and where those who took the

course were paid according to their true productivity? Yes,If

those who take the course get $2,500 and

people who do not get $1,000 a month,then

Kandos would take the course,since the

NAME 435

pain of 20 hours of lecture costs $1,000,

but the wage premium is $1,500,Klutzes

would not take the course,since the pain

of lectures costs $2,000 a month and the

wage premium is $1,500.

(e) What is the smallest number of hours the professor could lecture and

still maintain a separating equilibrium? 15 hours

36.4 (1) Old MacDonald produces hay,He has a single employee,Jack.

If Jack works for x hours he can produce x bales of hay,Each bale of hay

sells for $1,The cost to Jack of working x hours is c(x)=x

2

=10.

(a) What is the e cient number of bales of hay for Jack to cut? 5.

(b) If the most that Jack could earn elsewhere is zero,how much would

MacDonald have to pay him to get him to work the e cient amount?

5

2

=10 = $2:50.

(c) What is MacDonald’s net pro t? 5?2:50 = $2:50.

(d) Suppose that Jack would receive $1 for passing out leaflets,an activity

that involves no e ort whatsoever,How much would he have to receive

from MacDonald for producing the e cient number of bales of hay?

$3.50.

(e) Suppose now that the opportunity for passing out leaflets is no longer

available,but that MacDonald decides to rent his hay eld out to Jack for

a flat fee,How much would he rent it for? $2.50.

36.5 (0) In Rustbucket,Michigan,there are 200 people who want to sell

their used cars,Everybody knows that 100 of these cars are \lemons"

and 100 of these cars are \good." The problem is that nobody except the

original owners know which are which,Owners of lemons will be happy

to get rid of their cars for any price greater than $200,Owners of good

used cars will be willing to sell them for any price greater than $1,500,

but will keep them if they can’t get $1,500,There are a large number of

buyers who would be willing to pay $2,500 for a good used car,but would

pay only $300 for a lemon,When these buyers are not sure of the quality

of the car they buy,they are willing to pay the expected value of the car,

given the knowledge they have.

436 INFORMATION (Ch,36)

(a) If all 200 used cars in Rustbucket were for sale,how much would

buyers be willing to pay for a used car? $1,400,Would owners

of good used cars be willing to sell their used cars at this price? No.

Would there be an equilibrium in which all used cars are sold? No.

Describe the equilibrium that would take place in Rustbucket,Good

car owners won’t sell,Lemon owners will

sell,Price of a used car will be $300.

(b) Suppose that instead of there being 100 cars of each kind,everyone

in town is aware that there are 120 good cars and 80 lemons,How much

would buyers be willing to pay for a used car? $1,620,Would

owners of good used cars be willing to sell their used cars at this price?

Yes,Would there be an equilibrium in which all used cars are sold?

Yes,Would there be an equilibrium in which only the lemons were

sold? Yes,Describe the possible equilibrium or equilibria that would

take place in Rustbucket,One equilibrium has all

cars sold at a price of $1,620,There is

also an equilibrium where only the lemons

are sold.

36.6 (1) Each year,1,000 citizens of New Crankshaft,Pennsylvania,sell

their used cars and buy new cars,The original owners of the old cars

have no place to keep second cars and must sell them,These used cars

vary a great deal in quality,Their original owners know exactly what is

good and what is bad about their cars,but potential buyers can’t tell

them apart by looking at them,Lamentably,though they are in other

respects model citizens,the used-car owners in New Crankshaft have no

scruples about lying about their old jalopies,Each car has a value,V,

which a buyer who knew all about its qualities would be willing to pay.

There is a very large number of potential buyers,any one of which would

be willing to pay $V foracarofvalue$V:

The distribution of values of used cars on the market is quite simply

described,In any year,for any V between 0 and $2,000,the number of

used cars available for sale that are worth less than $V is V=2,Potential

used-car buyers are all risk-neutral,That is if they don’t know the value of

a car for certain,they value it at its expected value,given the information

they have.

NAME 437

Rod’s Garage in New Crankshaft will test out any used car and nd

its true value V,Rod’s Garage is known to be perfectly accurate and

perfectly honest in its appraisals,The only problem is that getting an

accurate appraisal costs $200,People with terrible cars are not going to

want to pay $200 to have Rod tell the world how bad their cars are,But

people with very good cars will be willing to pay Rod the $200 to get

their cars appraised,so they can sell them for their true values.

Let’s try to gure our exactly how the equilibrium works,which cars

get appraised,and what the unappraised cars sell for.

(a) If nobody had their car appraised,what would the market price

for used cars in North Crankshaft be and what would be the total

revenue received by used-car owners for their cars? They’d

all sell for $1,000 for total revenue of

$1,000,000.

(b) If all the cars that are worth more than $X are appraised and all

the cars that are worth less than $X are sold without appraisal,what

will the market price of unappraised used cars be? (Hint,What is the

expected value of a random draw from the set of cars worth less than

$X?) $X=2.

(c) If all the cars that are worth more than $X are appraised and all

thecarsthatareworthlessthan$X are sold without appraisal,then if

your car is worth $X,how much money would you have left if you had

it appraised and then sold it for its true value? $X?200,How

much money would you get if you sold it without having it appraised?

$X=2.

(d) In equilibrium,there will be a car of marginal quality such that all

cars better than this car will be appraised and all cars worse than this car

will be sold without being appraised,The owner of this car will be just

indi erent between selling his car unappraised and having it appraised.

What will be the value of this marginal car? Solve X=2=

X?200 to get X = $400.

(e) In equilibrium,how many cars will be sold unappraised and what

will they sell for? The worst 200 cars will be

unappraised and will sell for $200.

438 INFORMATION (Ch,36)

(f) In equilibrium,what will be the total net revenue of all owners

of used cars,after Rod’s Garage has been paid for its appraisals?

$1;000;000?800 200 = 840;000.

36.7 (2) In Pot Hole,Georgia,1,000 people want to sell their used cars.

These cars vary in quality,Original owners know exactly what their cars

are worth,All used cars look the same to potential buyers until they have

bought them; then they nd out the truth,For any number X between

0 and 2,000,the number of cars of quality lower than X is X=2,If a car

is of quality X,its original owner will be willing to sell it for any price

greater than X,If a buyer knew that a car was of quality X,she would

be willing to pay X + 500 for it,When buyers are not sure of the quality

of a car,they are willing to pay its expected value,given their knowledge

of the distribution of qualities on the market.

(a) Suppose that everybody knows that all the used cars in Pot Hole are

for sale,What would used cars sell for? $1,500,Would every

used car owner be willing to sell at this price? No,Which used

cars would appear on the market? Those worth less than

$1,500.

(b) Let X

be some number between 0 and 2,000 and suppose that all

cars of quality lower than X

are sold,but original owners keep all cars

of quality higher than X

,What would buyers be willing to pay for a

used car? X

=2 + 500,At this price,which used cars would be

for sale? Cars worth less than X

=2 + 500.

(c) Write an equation for the equilibrium value of X

,atwhichtheprice

that buyers are willing to pay is exactly enough to induce all cars of

quality less than X

into the market,X

=2 + 500 = X

,Solve

this equation for the equilibrium value of X

,X

=$1;000.

QUIZZES

This section contains short multiple-choice quizzes based on the workbook

problems in each chapter,Typically the questions are slight variations on

the workbook problems,so that if you have worked and understood the

corresponding workbook problem,the quiz question will be pretty easy.

Instructors who have adopted Workouts for their course can make use

of the test-item le o ered with the textbook,The test-item le contains

alternative versions of each quiz question in the back of Workouts,The

questions in these quizzes use di erent numerical values but the same in-

ternal logic,They can be used to provide additional problems for student

practice or for in-class quizzes.

When we teach this course we tell the students to work through all

the quiz questions in Workouts for each chapter,either by themselves

or with a study group,During the term we have a short in-class quiz

every other week or so,using the alternative versions from the test-item

le,These are essentially the Workouts quizzes with di erent numbers.

Hence,students who have done their homework nd it easy to do well on

the quizzes.

440 QUIZZES (Ch,36)

Quiz 2 NAME

The Budget Set

2.1 In Problem 2.1,if you have an income of $12 to spend,if commodity 1

costs $2 per unit,and if commodity 2 costs $6 per unit,then the equation

for your budget line can be written as

(a) x

1

=2+x

2

=6 = 12.

(b) (x

1

+x

2

)=(8) = 12.

(c) x

1

+3x

2

=6.

(d) 3x

1

+7x

2

= 13.

(e) 8(x

1

+x

2

) = 12.

2.2 In Problem 2.3,if you could exactly a ord either 6 units of x and 14

units of y,or 10 units of x and 6 units of y,then if you spent all of your

income on y,how many units of y could you buy?

(a) 26.

(b) 18.

(c) 34.

(d) 16.

(e) None of the other options are correct.

2.3 In Problem 2.4,Murphy used to consume 100 units of x and 50 units

of y when the price of x was 2 and the price of y was 4,If the price of x

rose to 5 and the price of y rose to 8,how much would Murphy’s income

have to rise so that he could still a ord his original bundle?

(a) 700.

(b) 500.

(c) 350.

(d) 1,050.

442 THE BUDGET SET (Ch,2)

(e) None of the other options are correct.

2.4 In Problem 2.7,Edmund must pay $6 each for punk rock video

casettes,If Edmund is paid $48 per sack for accepting garbage and if

his relatives send him an allowance of $384,then his budget line is de-

scribed by the equation:

(a) 6V =48G.

(b) 6V +48G = 384.

(c) 6V?48G = 384.

(d) 6V = 384?G.

(e) None of the other options are correct.

2.5 InProblem2.10,ifinthesameamountoftimethatittakesher

to read 40 pages of economics and 30 pages of sociology,Martha could

read 30 pages of economics and 50 pages of sociology,then which of these

equations describes combinations of pages of economics,E,and sociology,

S,that she could read in the time it takes to read 40 pages of economics

and 30 pages of sociology?

(a) E +S = 70.

(b) E=2+S = 50.

(c) 2E +S = 110.

(d) E +S = 80.

(e) All of the above.

2.6 In Problem 2.11,ads in the boring business magazine are read by

300 lawyers and 1,000 M.B.As,Ads in the consumer publication are

read by 250 lawyers and 300 M.B.A.’s,If Harry had $3,000 to spend

on advertising,if the price of ads in the boring business magazine were

$600 and the price of ads in the consumer magazine were $300,then the

combinations of recent M.B.A’s and lawyers with hot tubs whom he could

reach with his advertising budget would be represented by the integer

values along a line segment that runs between the two points

(a) (2,500,3,000) and (1,500,5,000).

(b) (3,000,3,500) and (1,500,6,000).

(c) (0,3,000) and (1,500,0).

NAME 443

(d) (3,000,0) and (0,6,000).

(e) (2,000,0) and (0,5,000).

2.7 In the economy of Mungo,discussed in Problem 2.12,there is a third

creature called Ike,Ike has a red income of 40 and a blue income of

10,(Recall that blue prices are 1 bcu (blue currency unit) per unit of

ambrosia and 1 bcu per unit of bubble gum,Red prices are 2 rcus (red

currency units) per unit of ambrosia and 6 rcus per unit of bubble gum.

You have to pay twice for what you buy,once in red currency,once in

blue currency.) If Ike spends all of its blue income,but not all of its red

income,then it must be that

(a) it consumes at least 5 units of bubble gum.

(b) it consumes at least 5 units of ambrosia.

(c) it consumes exactly twice as much bubblegum as ambrosia.

(d) it consumes at least 15 units of bubble gum.

(e) it consumes equal amounts of ambrosia and bubble gum.

444 THE BUDGET SET (Ch,2)

Quiz 3 NAME

Preferences

3.1 In Problem 3.1,Charlie’s indi erence curves have the equation

x

B

= constant=x

A

,where larger constants correspond to better indif-

ference curves,Charlie strictly prefers the bundle (7,15) to the following

bundle:

(a) (15,7).

(b) (8,14).

(c) (11,11).

(d) all three of these bundles.

(e) none of these bundles.

3.2 In Problem 3.2,Ambrose has indi erence curves with the equation

x

2

= constant?4x

1=2

1

,where larger constants correspond to higher indif-

ference curves,If good 1 is drawn on the horizontal axis and good 2 on

the vertical axis,what is the slope of Ambrose’s indi erence curve when

his consumption bundle is (1,6)?

(a)?1=6

(b)?6=1

(c)?2

(d)?7

(e)?1

3.3 In Problem 3.8,Nancy Lerner is taking a course from Professor Good-

heart who will count only her best midterm grade and from Professor

Stern who will count only her worst midterm grade,In one of her classes,

Nancy has scores of 50 on her rst midterm and 30 on her second midterm.

When the rst midterm score is measured on the horizontal axis and her

second midterm score on the vertical,her indi erence curve has a slope

of zero at the point (50,30),Therefore it must be that

(a) this class could be Professor Goodheart’s but couldn’t be Professor

Stern’s.

446 PREFERENCES (Ch,3)

(b) this class could be Professor Stern’s but couldn’t be Professor Good-

heart’s.

(c) this class couldn’t be either Goodheart’s or Stern’s.

(d) this class could be either Goodheart’s or Stern’s.

3.4 In Problem 3.9,if we graph Mary Granola’s indi erence curves with

avocados on the horizontal axis and grapefruits on the vertical axis,then

whenever she has more grapefruits than avocados,the slope of her indif-

ference curve is?2,Whenever she has more avocados than grapefruits,

the slope is?1=2,Mary would be indi erent between a bundle with 24

avocados and 36 grapefruits and another bundle that has 34 avocados and

(a) 28 grapefruits.

(b) 32 grapefruits.

(c) 22 grapefruits.

(d) 25 grapefruits.

(e) 26.50 grapefruits.

3.5 In Problem 3.12,recall that Tommy Twit’s mother measures the de-

parture of any bundle from her favorite bundle for Tommy by the sum

of the absolute values of the di erences,Her favorite bundle for Tommy

is (2,7){that is,2 cookies and 7 glasses of milk,Tommy’s mother’s in-

di erence curve that passes through the point (c;m)=(3;6) also passes

through

(a) (4,5).

(b) the points (2,5),(4,7),and (3,8).

(c) (2,7).

(d) the points (3,7),(2,6),and (2,8).

(e) None of the other options are correct.

3.6 In Problem 3.1,Charlie’s indi erence curves have the equation

x

B

= constant=x

A

,where larger constants correspond to better indif-

ference curves,Charlie strictly prefers the bundle (9,19) to the following

bundle:

(a) (19,9).

(b) (10,18).

(c) (15,17).

(d) More than one of these options are correct.

(e) None of the above are correct.

Quiz 4 NAME

Utility

4.1 In Problem 4.1,Charlie has the utility function U(x

A;x

B

)=x

A

x

B

.

His indi erence curve passing through 10 apples and 30 bananas will also

pass through the point where he consumes 2 apples and

(a) 25 bananas.

(b) 50 bananas.

(c) 152 bananas.

(d) 158 bananas.

(e) 150 bananas.

4.2 In Problem 4.1,Charlie’s utility function is U(A;B)=AB where

A and B are the numbers of apples and bananas,respectively,that he

consumes,When Charlie is consuming 20 apples and 100 bananas,then

if we put apples on the horizontal axis and bananas on the vertical axis,

the slope of his indi erence curve at his current consumption is

(a)?20.

(b)?5.

(c)?10.

(d)?1=5.

(e)?1=10.

4.3 In Problem 4.2,Ambrose has the utility function U(x

1;x

2

)=4x

1=2

1

+

x

2

,If Ambrose is initially consuming 81 units of nuts and 14 units of

berries,then what is the largest number of berries that he would be

willing to give up in return for an additional 40 units of nuts?

(a) 11

(b) 25

(c) 8

448 UTILITY (Ch,4)

(d) 4

(e) 2

4.4 Joe Bob,from Problem 4.12 has a cousin Jonas who consume goods

1 and 2,Jonas thinks that 2 units of good 1 is always a perfect substitute

for 3 units of good 2,Which of the following utility functions is the only

one that would NOT represent Jonas’s preferences?

(a) U(x

1;x

2

)=3x

1

+2x

2

+1;000.

(b) U(x

1;x

2

)=9x

2

1

+12x

1

x

2

+4x

2

2

.

(c) U(x

1;x

2

)=minf3x

1;2x

2

g.

(d) U(x

1;x

2

)=30x

1

+20x

2

10;000.

(e) More than one of the above does NOT represent Jonas’s preferences.

4.5 In Problem 4.7,Harry Mazzola has the utility function U(x

1;x

2

)=

minfx

1

+2x

2;2x

1

+ x

2

g,He has $40 to spend on corn chips and french

fries,If the price of corn chips is 5 dollar(s) per unit and the price of

french fries is 5 dollars per unit,then Harry will

(a) de nitely spend all of his income on corn chips.

(b) de nitely spend all of his income on french fries.

(c) consume at least as much corn chips as french fries,but might consume

both.

(d) consume at least as much french fries as corn chips,but might consume

both.

(e) consume equal amounts of french fries and corn chips.

4.6 Phil Rupp’s sister Ethel has the utility function U(x;y)=minf2x+

y;3yg.Wherex is measured on the horizontal axis and y on the vertical

axis,her indi erence curves

(a) consist of a vertical line segment and a horizontal line segment which

meet in a kink along the line y =2x.

(b) consist of a vertical line segment and a horizontal line segment which

meet in a kink along the line x =2y.

(c) consist of a horizontal line segment and a negatively sloped line seg-

ment which meet in a kink along the line x = y.

(d) consist of a positively sloped line segment and a negatively sloped line

segment which meet along the line x = y.

(e) consist of a horizontal line segment and a positively sloped line seg-

ment which meet in a kink along the line x =2y.

Quiz 5 NAME

Choice

5.1 In Problem 5.1,Charlie has a utility function U(x

A;x

B

)=x

A

x

B

,

the price of apples is 1 and the price of bananas is 2,If Charlie’s income

were 240,how many units of bananas would he consume if he chooses the

bundle that maximizes his utility subject to his budget constraint?

(a) 60

(b) 30

(c) 120

(d) 12

(e) 180

5.2 In Problem 5.1,if Charlie’s income is 40,the price of apples is 5

and the price of bananas is 6,how many apples are contained in the best

bundle that Charlie can a ord?

(a) 8

(b) 15

(c) 10

(d) 11

(e) 4

5.3 In Problem 5.2,Clara’s utility function is U(X;Y)=(X +2)(Y +1).

If Clara’s marginal rate of substitution is?2 and she is consuming 10

units of good X,how many units of good Y is she consuming?

(a) 2

(b) 24

(c) 12

(d) 23

450 CHOICE (Ch,5)

(e) 5

5.4 In Problem 5.3,Ambrose’s utility function is U(x

1;x

2

)=4x

1=2

1

+x

2

.

If the price of nuts is 1,the price of berries is 4,and his income is 72,how

many units of nuts will Ambrose choose?

(a) 2

(b) 64

(c) 128

(d) 67

(e) 32

5.5 Ambrose’s utility function is 4x

1=2

1

+x

2

,If the price of nuts is 1,the

price of berries is 4,and his income is 100,how many units of berries will

Ambrose choose?

(a) 65

(b) 9

(c) 18

(d) 8

(e) 12

5.6 In Problem 5.6,Elmer’s utility function is U(x;y)=minfx;y

2

g.If

the price of x is 15,the price of y is 10,and Elmer chooses to consume 7

units of y,what must Elmer’s income be?

(a) 1,610

(b) 175

(c) 905

(d) 805

(e) There is not enough information to tell.

Quiz 6 NAME

Demand

6.1 (See Problem 6.1,) If Charlie’s utility function is X

4

A

X

B

,apples

cost 90 cents each,and bananas cost 10 cents each,then Charlie’s budget

line is tangent to one of his indi erence curves whenever the following

equation is satis ed:

(a) 4X

B

=9X

A

.

(b) X

B

= X

A

.

(c) X

A

=4X

B

.

(d) X

B

=4X

A

.

(e) 90X

A

+10X

B

= M.

6.2 (See Problem 6.1.) If Charlie’s utility function is X

4

A

X

B

,the price

of apples is p

A

,the price of bananas is p

B

,and his income is m,then

Charlie’s demand for apples is

(a) m=(2p

A

).

(b) 0:25p

A

m.

(c) m=(p

A

+p

B

).

(d) 0:80m=p

A

.

(e) 1:25p

B

m=p

A

.

6.3 Ambrose’s brother Bartholomew has a utility function U(x

1;x

2

)=

24x

1=2

1

+ x

2

,His income is 51,the price of good 1 (nuts) is 4,and the

price of good 2 (berries) is 1,How many units of nuts will Bartholomew

demand?

(a) 19

(b) 5

(c) 7

(d) 9

452 DEMAND (Ch,6)

(e) 16

6.4 Ambrose’s brother Bartholomew has a utility function U(x

1;x

2

)=

8x

1=2

1

+x

2

,His income is 23,the price of nuts is 2,and the price of berries

is 1,How many units of berries will Bartholomew demand?

(a) 15

(b) 4

(c) 30

(d) 10

(e) There is not enough information to determine the answer.

6.5 In Problem 6.6,recall that Miss Mu et insists on consuming 2 units

of whey per unit of curds,If the price of curds is 3 and the price of whey

is 6,then if Miss Mu ett’s income is m,her demand for curds will be

(a) m=3.

(b) 6m=3.

(c) 3C +6W = m.

(d) 3m.

(e) m=15.

6.6 In Problem 6.8,recall that Casper’s utility function is 3x+y,where

x is his consumption of cocoa and y is his consumption of cheese,If the

total cost of x units of cocoa is x

2

,the price of cheese is 8,and Casper’s

income is $174,how many units of cocoa will he consume?

(a) 9

(b) 12

(c) 23

(d) 11

(e) 24

6.7 (See Problem 6.13.) Kinko’s utility function is U(w;j)=

minf7w;3w +12jg,wherew is the number of whips that he owns and j

is the number of leather jackets,If the price of whips is $20 and the price

of leather jackets is $60,Kinko will demand:

NAME 453

(a) 6 times as many whips as leather jackets.

(b) 5 times as many leather jackets as whips.

(c) 3 times as many whips as leather jackets.

(d) 4 times as many leather jackets as whips.

(e) only leather jackets.

454 DEMAND (Ch,6)

Quiz 7 NAME

Revealed Preference

7.1 In Problem 7.1,if the only information we had about Goldie were

that she chooses the bundle (6,6) when prices are (6,3) and she chooses

the bundle (10,0) when prices are (5,5),then we could conclude that

(a) the bundle (6,6) is revealed preferred to (10,0) but there is no evidence

that she violates WARP.

(b) neither bundle is revealed preferred to the other.

(c) Goldie violates WARP.

(d) the bundle (10,0) is revealed preferred to (6,6) and she violates WARP.

(e) the bundle (10,0) is revealed preferred to (6,6) and there is no evidence

that she violates WARP.

7.2 In Problem 7.3,Pierre’s friend Henri lives in a town where he has

to pay 3 francs per glass of wine and 6 francs per loaf of bread,Henri

consumes 6 glasses of wine and 4 loaves of bread per day,Recall that Bob

has an income of $15 per day and pays $.50 per loaf of bread and $2 per

glass of wine,If Bob has the same tastes as Henri,and if the only thing

that either of them cares about is consumption of bread and wine,we can

deduce

(a) nothing about whether one is better than the other.

(b) that Henri is better o than Bob.

(c) that Bob is better o than Henri.

(d) that both of them violate the weak axiom of revealed preferences.

(e) that Bob and Henri are equally well o,

7.3 Let us reconsider the case of Ronald in Problem 7.4,Let the prices

and consumptions in the base year be as in Situation D,where p

1

=3,

p

2

=1,x

1

=5,andx

2

= 15,If in the current year,the price of good 1 is

1 and the price of good 2 is 3,and his current consumptions of good 1 and

good 2 are 25 and 10 respectively,what is the Laspeyres price index of

current prices relative to base-year prices? (Pick the most nearly correct

answer.)

456 REVEALED PREFERENCE (Ch,7)

(a) 1.67

(b) 1.83

(c) 1

(d) 0.75

(e) 2.50

7.4 On the planet Homogenia,every consumer who has ever lived con-

sumes only two goods x and y and has the utility function U(x;y)=xy.

The currency in Homogenia is the fragel,On this planet in 1900,the

price of good 1 was 1 fragel and the price of good 2 was 2 fragels,Per

capita income was 120 fragels,In 1990,the price of good 1 was 5 fragels

and the price of good 2 was 5 fragels,The Laspeyres price index for the

price level in 1990 relative to the price level in 1900 is

(a) 3.75.

(b) 5.

(c) 3.33.

(d) 6.25.

(e) not possible to determine from this information.

7.5 On the planet Hyperion,every consumer who has ever lived has a

utility function U(x;y)=minfx;2yg,The currency of Hyperion is the

doggerel,In 1850 the price of x was 1 doggerel per unit,and the price of

y was 2 doggerels per unit,In 1990,the price of x was 10 doggerels per

unit and the price of y was 4 doggerels per unit,Paasche price index of

prices in 1990 relative to prices in 1850 is

(a) 6.

(b) 4.67.

(c) 2.50.

(d) 3.50.

(e) not possible to determine without further information.

Quiz 8 NAME

Slutsky Equation

8.1 In Problem 8.1,Charlie’s utility function is x

A

x

B

,The price of

apples used to be $1 per unit and the price of bananas was $2 per unit.

His income was $40 per day,If the price of apples increased to $1.25 and

the price of bananas fell to $1.25,then in order to be able to just a ord

his old bundle,Charlie would have to have a daily income of

(a) $37.50.

(b) $76.

(c) $18.75.

(d) $56.25.

(e) $150.

8.2 In Problem 8.1,Charlie’s utility function isx

A

x

B

,The price of apples

used to be $1 and the price of bananas used to be $2,and his income used

to be $40,If the price of apples increased to 8 and the price of bananas

stayed constant,the substitution e ect on Charlie’s apple consumption

reduces his consumption by

(a) 17.50 apples.

(b) 7 apples.

(c) 8.75 apples.

(d) 13.75 apples.

(e) None of the other options are correct.

8.3 Neville,in Problem 8.2,has a friend named Colin,Colin has the same

demand function for claret as Neville,namely q =,02m?2p,wherem

is income and p is price,Colin’s income is 6,000 and he initially had to

pay a price of 30 per bottle of claret,The price of claret rose to 40,The

substitution e ect of the price change

(a) reduced his demand by 20.

(b) increased his demand by 20.

458 SLUTSKY EQUATION (Ch,8)

(c) reduced his demand by 8.

(d) reduced his demand by 32.

(e) reduced his demand by 18.

8.4 Goods 1 and 2 are perfect complements and a consumer always con-

sumes them in the ratio of 2 units of Good 2 per unit of Good 1,If a

consumer has income 120 and if the price of good 2 changes from 3 to 4,

while the price of good 1 stays at 1,then the income e ect of the price

change

(a) is 4 times as strong as the substitution e ect.

(b) does not change demand for good 1.

(c) accounts for the entire change in demand.

(d) is exactly twice as strong as the substitution e ect.

(e) is 3 times as strong as the substitution e ect.

8.5 Suppose that Agatha in Problem 8.10 had $570 to spend on tickets

for her trip,She needs to travel a total of 1,500 miles,Suppose that the

price of rst-class tickets is $0.50 per mile and the price of second-class

tickets is $0.30 per mile,How many miles will she travel by second class?

(a) 900

(b) 1,050

(c) 450

(d) 1,000

(e) 300

8.6 In Problem 8.4,Maude thinks delphiniums and hollyhocks are perfect

substitutes,one for one,If delphiniums currently cost $5 per unit and

hollyhocks cost $6 per unit,and if the price of delphiniums rises to $9 per

unit,

(a) the income e ect of the change in demand for delphiniums will be

bigger than the substitution e ect.

(b) there will be no change in the demand for hollyhocks.

(c) the entire change in demand for delphiniums will be due to the sub-

stitution e ect.

(d) the fraction 1=4 of the change will be due to the income e ect.

(e) the fraction 3=4 of the change will be due to the income e ect.

Quiz 9 NAME

Buying and Selling

9.1 In Problem 9.1,if Abishag owned 9 quinces and 10 kumquats,and if

the price of kumquats is 3 times the price of quinces,how many kumquats

could she a ord if she spent all of her money on kumquats?

(a) 26

(b) 19

(c) 10

(d) 13

(e) 10

9.2 Suppose that Mario in Problem 9.2 consumes eggplant and tomatoes

in the ratio of one bushel of eggplant per bushel of tomatoes,His garden

yields 30 bushels of eggplant and 10 bushels of tomatoes,He initially faced

prices of $10 per bushel for each vegetable,but the price of eggplant rose

to $30 per bushel,while the price of tomatoes stayed unchanged,After

the price change,he would

(a) increase his eggplant consumption by 5 bushels.

(b) decrease his eggplant consumption by at least 5 bushels.

(c) increase his consumption of eggplant by 7 bushels.

(d) decrease his consumption of eggplant by 7 bushels.

(e) decrease his tomato consumption by at least 1 bushel.

9.3 (See Problem 9.9(b).) Dr,Johnson earns $5 per hour for his labor

and has 80 hours to allocate between labor and leisure,His only other

income besides his earnings from labor is a lump sum payment of $50 per

week,Suppose that the rst $200 per week of his labor income is untaxed,

but all labor income above $200 is taxed at a rate of 40 percent.

(a) Dr,J.’s budget line has a kink in it at the point where he takes 50

units of leisure.

(b) Dr,J.’s budget line has a kink where his income is 250 and his leisure

is 40.

460 BUYING AND SELLING (Ch,9)

(c) The slope of Dr,J.’s budget line is everywhere?3.

(d) Dr,J.’s budget line has no kinks in the part of it that corresponds to

a positive labor supply.

(e) Dr,J.’s budget line has a piece that is a horizontal straight line.

9.4 Dudley,in Problem 9.15,has a utility function U(C;R)=C?(12?

R)

2

,whereR is leisure and C is consumption per day,He has 16 hours

per day to divide between work and leisure,If Dudley has a nonlabor

income of $40 per day and is paid a wage of $6 per hour,how many hours

of leisure will he choose per day?

(a) 6

(b) 7

(c) 8

(d) 10

(e) 9

9.5 Mr,Cog in Problem 9.7 has 18 hours a day to divide between labor

and leisure,His utility function is U(C;R)=CR where C is the number

of dollars per day that he spends on consumption and R is the number of

hours per day that he spends at leisure,If he has 16 dollars of nonlabor

income per day and gets a wage rate of 13 dollars per hour when he works,

his budget equation,expressing combinations of consumption and leisure

that he can a ord to have,can be written as:

(a) 13R +C = 16.

(b) 13R +C = 250.

(c) R+C=13 = 328.

(d) C = 250 + 13R.

(e) C = 298 + 13R.

9.6 Mr,Cog in Problem 9.7 has 18 hours per day to divide between labor

and leisure,His utility function is U(C;R)=CR where C is the number

of dollars per day that he spends on consumption and R is the number of

hours per day that he spends at leisure,If he has a nonlabor income of

42 dollars per day and a wage rate of 13 dollars per hour,he will choose

a combination of labor and leisure that allows him to spend

(a) 276 dollars per day on consumption.

NAME 461

(b) 128 dollars per day on consumption.

(c) 159 dollars per day on consumption.

(d) 138 dollars per day on consumption.

(e) 207 dollars per day on consumption.

462 BUYING AND SELLING (Ch,9)

Quiz 10 NAME

Intertemporal Choice

10.1 If Peregrine in Problem 10.1 consumes (1,000,1,155) and earns

(800,1365) and if the interest rate is 0.05,the present value of his endow-

ment is

(a) 2,165.

(b) 2,100.

(c) 2,155.

(d) 4,305.

(e) 5,105.

10.2 Suppose that Molly from Problem 10.2 had income $400 in period 1

and income 550 in period 2,Suppose that her utility function werec

a

1

c

1?a

2

,

where a =0:40 and the interest rate were 0:10,If her income in period 1

doubled and her income in period 2 stayed the same,her consumption in

period 1 would

(a) double.

(b) increase by 160.

(c) increase by 80

(d) stay constant.

(e) increase by 400.

10.3 Mr,O,B,Kandle,of Problem 10.8 has a utility function c

1

c

2

where

c

1

is his consumption in period 1 and c

2

is his consumption in period 2.

He will have no income in period 2,If he had an income of 30,000 in

period 1 and the interest rate increased from 10% to 12%,

(a) his savings would increase by 2% and his consumption in period 2

would also increase.

(b) his savings would not change,but his consumption in period 2 would

increase by 300.

(c) his consumption in both periods would increase.

464 INTERTEMPORAL CHOICE (Ch,10)

(d) his consumption in both periods would decrease.

(e) his consumption in period 1 would decrease by 12% and his consump-

tion in period 2 would also decrease.

10.4 Harvey Habit in Problem 10.9 has a utility function U(c

1;c

2

)=

minfc

1;c

2

g,If he had an income of 1,025 in period 1,and 410 in period

2,and if the interest rate were 0.05,how much would Harvey choose to

spend on bread in period 1?

(a) 1,087.50

(b) 241.67

(c) 362.50

(d) 1,450

(e) 725

10.5 In the village in Problem 10.10,if the harvest this year is 3,000 and

the harvest next year will be 1,100,and if rats eat 50% of any grain that

is stored for a year,how much grain could the villagers consume next year

if they consume 1,000 bushels of grain this year?

(a) 2,100.

(b) 1,000.

(c) 4,100.

(d) 3,150.

(e) 1,200.

10.6 Patience has a utility function U(c

1;c

2

)=c

1=2

1

+0:83c

1=2

2

,c

1

is her

consumption in period 1 and c

2

is her consumption in period 2,Her

income in period 1 is 2 times as large as her income in period 2,At what

interestratewillshechoosetoconsumethesameamountinperiod1as

in period 2?

(a) 0.40

(b) 0.10

(c) 0.20

(d) 0

(e) 0.30

Quiz 11 NAME

Asset Markets

11.1 Ashley,in Problem 11.6,has discovered another wine,Wine D,Wine

drinkers are willing to pay 40 dollars to drink it right now,The amount

that wine drinkers are willing to pay will rise by 10 dollars each year

that the wine ages,The interest rate is 10%,How much would Ashley

be willing to pay for the wine if he buys it as an investment? (Pick the

closest answer.)

(a) $56

(b) $40

(c) $100

(d) $440

(e) $61

11.2 Chillingsworth,from Problem 11.10 has a neighbor,Shivers,who

faces the same options for insulating his house as Chillingsworth,But

Shivers has a larger house,Shivers’s annual fuel bill for home heating is

1,000 dollars per year,Plan A will reduce his annual fuel bill by 15%,plan

B will reduce it by 20%,and plan C will eliminate his need for heating fuel

altogether,The Plan A insulation job would cost Shivers 1,000 dollars,

Plan B would cost him 1,900 dollars,and Plan C would cost him 11,000

dollars,If the interest rate is 10% and his house and the insulation job

last forever,which plan is the best for Shivers?

(a) Plan A.

(b) Plan B.

(c) Plan C.

(d) Plans A and B are equally good.

(e) He is best o using none of the plans.

11.3 The price of an antique is expected to rise by 2% during the next

year,The interest rate is 6%,You are thinking of buying an antique

and selling it a year from now,You would be willing to pay a total of

200 dollars for the pleasure of owning the antique for a year,How much

would you be willing to pay to buy this antique? (See Problem 11.5.)

466 ASSET MARKETS (Ch,11)

(a) $3,333.33

(b) $4,200

(c) $200

(d) $5,000

(e) $2,000

11.4 A bond has a face value of 9,000 dollars,It will pay 900 dollars in

interest at the end of every year for the next 46 years,At the time of the

nal interest payment,46 years from now,the company that issued the

bond will \redeem the bond at face value." That is,the company buys

back the bond from its owner at a price equal to the face value of the

bond,If the interest rate is 10% and is expected to remain at 10%,how

much would a rational investor pay for this bond right now?

(a) $9,000

(b) $50,400

(c) $41,400

(d) More than any of the above numbers.

(e) Less than any of the above numbers.

11.5 The sum of the in nite geometric series 1;0:86;0:86

2;0:86

3;::,is

closest to which of the following numbers?

(a) in nity.

(b) 1.86.

(c) 7.14.

(d) 0.54.

(e) 116.28.

11.6 If the interest rate is 11%,and will remain 11% forever,how much

would a rational investor be willing to pay for an asset that will pay him

5,550 dollars one year from now,1,232 dollars two years from now,and

nothing at any other time?

(a) $6,000

(b) $5,000

(c) $54,545.45

(d) $72,000

(e) $7,000

Quiz 12 NAME

Uncertainty

12.1 In Problem 12.9,Billy has a von Neumann-Morgenstern utility func-

tion U(c)=c

1=2

,If Billy is not injured this season,he will receive an

income of 25 million dollars,If he is injured,his income will be only

$10,000,The probability that he will be injured is,1 and the probability

that he will not be injured is,9,His expected utility is

(a) 4,510.

(b) between 24 million and 25 million dollars.

(c) 100,000.

(d) 9,020.

(e) 18,040.

12.2 (See Problem 12.2.) Willy’s only source of wealth is his chocolate

factory,He has the utility function pc

1=2

f

+(1?p)c

1=2

nf

where p is the

probability of a flood and 1?p is the probability of no flood,Let c

f

and

c

n

f be his wealth contingent on a flood and on no flood,respectively,The

probability of a flood is p =1=15,The value of Willy’s factory is $600,000

if there is no flood and 0 if there is a flood,Willy can buy insurance where

if he buys $x worth of insurance,he must pay the insurance company

$3x=17 whether there is a flood or not,but he gets back $x from the

company if there is a flood,Willy should buy

(a) no insurance since the cost per dollar of insurance exceeds the prob-

ability of a flood.

(b) enough insurance so that if there is a flood,after he collects his insur-

ance his wealth will be 1/9 of what it would be if there is no flood.

(c) enough insurance so that if there is a flood,after he collects his insur-

ance,his wealth will be the same whether there is a flood or not.

(d) enough insurance so that if there is a flood,after he collects his in-

surance,his wealth will be 1/4 of what it would be if there is no flood.

(e) enough insurance so that if there is a flood,after he collects his insur-

ance his wealth will be 1/7 of what it would be if there is no flood.

12.3 Sally Kink is an expected utility maximizer with utility function

pu(c

1

)+(1?p)u(c

2

)whereforanyx<4;000,u(x)=2x and where

u(x)=8;000 +x for x greater than or equal to 4,000.

468 UNCERTAINTY (Ch,12)

(a) Sally will be risk averse if her income is less than 4,000 but risk loving

if her income is more than 4,000.

(b) Sally will be risk neutral if her income is less than 4,000 and risk

averse if her income is more than 4,000.

(c) For bets that involve no chance of her wealth exceeding 4,000,Sally

will take any bet that has a positive expected net payo,

(d) Sally will never take a bet if there is a chance that it leaves her with

wealth less than 8,000.

(e) None of the above are true.

12.4 (See Problem 12.11.) Martin’s expected utility function is pc

1=2

1

+

(1?p)c

1=2

2

where p is the probability that he consumes c

1

and 1?p is

the probability that he consumes c

2

,Wilbur is o ered a choice between

getting a sure payment of $Z or a lottery in which he receives $2,500 with

probability 0.40 and he receives $900 with probability 0.60,Wilbur will

choose the sure payment if

(a) Z>1;444 and the lottery if Z<1;444.

(b) Z>1;972 and the lottery if Z<1;972.

(c) Z>900 and the lottery if Z<900.

(d) Z>1;172 and the lottery if Z<1;172.

(e) Z>1;540 and the lottery if Z<1;540.

12.5 Clancy has $4,800,He plans to bet on a boxing match between

Sullivan and Flanagan,He nds that he can buy coupons for $6 that

will pay o $10 each if Sullivan wins,He also nds in another store some

coupons that will pay o $10 if Flanagan wins,The Flanagan tickets cost

$4 each,Clancy believes that the two ghters each have a probability

of 1/2 of winning,Clancy is a risk averter who tries to maximize the

expected value of the natural log of his wealth,Which of the following

strategies would maximize his expected utility?

(a) Don’t gamble at all.

(b) Buy 400 Sullivan tickets and 600 Flanagan tickets.

(c) Buy exactly as many Flanagan tickets as Sullivan tickets.

(d) Buy 200 Sullivan tickets and 300 Flanagan tickets.

(e) Buy 200 Sullivan tickets and 600 Flanagan tickets.

Quiz 13 NAME

Risky Assets

13.1 Suppose that Ms,Lynch in Problem 13.1 can make up her portfolio

using a risk-free asset that o ers a sure- re rate of return of 15% and a

risky asset with expected rate of return 30%,with standard deviation 5.

If she chooses a portfolio with expected rate of return 18.75%,then the

standard deviation of her return on this portfolio will be:

(a) 0.63%.

(b) 4.25%.

(c) 1.25%.

(d) 2.50%,

(e) None of the other options are correct.

13.2 Suppose that Fenner Smith of Problem 13.2 must divide his portfolio

between two assets,one of which gives him an expected rate of return of

15 with zero standard deviation and one of which gives him an expected

rate of return of 30 and has a standard deviation of 5,He can alter the

expected rate of return and the variance of his portfolio by changing the

proportions in which he holds the two assets,If we draw a \budget line"

with expected return on the vertical axis and standard deviation on the

horizontal axis,depicting the combinations that Smith can obtain,the

slope of this budget line is

(a) 3.

(b)?3.

(c) 1.50.

(d)?1:50.

(e) 4.50.

470 RISKY ASSETS (Ch,13)

Quiz 14 NAME

Consumer’s Surplus

14.1 In Problem 14.1,Sir Plus has a demand function for mead that is

given by the equation D(p) = 100?p,If the price of mead is 75,how

much is Sir Plus’s net consumer surplus?

(a) 312.50

(b) 25

(c) 625

(d) 156.25

(e) 6,000

14.2 Ms,Quasimodo in Problem 14.3 has the utility function U(x;m)=

100x?x

2

=2+m where x is his consumption of earplugs and m is money

left over to spend on other stu,If she has $10,000 to spend on earplugs

and other stu,and if the price of earplugs rises from $50 to $95,then

her net consumer’s surplus

(a) falls by 1,237.50.

(b) falls by 3237.50.

(c) falls by 225.

(d) increases by 618.75.

(e) increases by 2,475.

14.3 Bernice in Problem 14.5 has the utility function u(x;y)=minfx;yg

where x is the number of pairs of earrings she buys per week and y is the

number of dollars per week she has left to spend on other things,(We

allow the possibility that she buys fractional numbers of pairs of earrings

per week.) If she originally had an income of $13 per week and was paying

a price of $2 per pair of earrings,then if the price of earrings rose to $4,

the compensating variation of that price change (measured in dollars per

week) would be closest to

(a) $5.20.

(b) $8.67.

472 CONSUMER’S SURPLUS (Ch,14)

(c) $18.33.

(d) $17.33.

(e) $16.33.

14.4 If Bernice (whose utility function is minfx;yg where x is her con-

sumption of earrings and y is money left for other stu ) had an income

of $16 and was paying a price of $1 for earrings when the price of earrings

went up to $8,then the equivalent variation of the price change was

(a) $12.44.

(b) $56.

(c) $112.

(d) $6.22.

(e) $34.22.

14.5 In Problem 14.7,Lolita’s utility function is U(x;y)=x?x

2

=2+y

where x is her consumption of cow feed and y is her consumption of hay.

If the price of cow feed is 0.40,the price of hay is 1,and her income is 4,

and if Lolita chooses the combination of hay and cow feed that she likes

best from among those combinations she can a ord,her utility will be

(a) 4.18.

(b) 3.60.

(c) 0.18.

(d) 6.18.

(e) 2.18.

Quiz 15 NAME

Market Demand

15.1 In Gas Pump,South Dakota,every Buick owner’s demand for gaso-

line is 20?5p for p less than or equal to 4 and 0 for p>4,Every Dodge

owner’sdemandis15?3p for p less than or equal to 5 and 0 for p>5.

Suppose that Gas Pump,S.D.,has 100 Buick owners and 50 Dodge own-

ers,If the price of gasoline is 4,what is the total amount of gasoline

demanded in Gas Pump?

(a) 300

(b) 75

(c) 225

(d) 150

(e) None of the other options are correct.

15.2 In Problem 15.5,the demand function for drangles is given by

D(p)=(p +1)

2

,If the price of drangles is 10,then the price elasticity

of demand is

(a)?7:27.

(b)?3:64.

(c)?5:45.

(d)?0:91.

(e)?1:82.

15.3 In Problem 15.6,the only quantities of good 1 that Barbie can buy

are 1 unit or zero units,For x

1

equal to zero or 1 and for all positive values

of x

2

,suppose that Barbie’s preferences were represented by the utility

function (x

1

+4)(x

2

+ 2),Then if her income were 28,her reservation

price for good 1 would be

(a) 12.

(b) 1.50.

(c) 6.

474 MARKET DEMAND (Ch,15)

(d) 2.

(e) 0.40.

15.4 In the same football conference as the university in Problem 15.9

is another university where the demand for football tickets at each game

is 80;000?12;000p,If the capacity of the stadium at that university is

50,000 seats,what is the revenue-maximizing price for this university to

charge per ticket?

(a) 3.33

(b) 2.50

(c) 6.67

(d) 1.67

(e) 10

15.5 In Problem 15.9,the demand for tickets is given byD(p) = 200;000?

10;000p,wherep is the price of tickets,If the price of tickets is 4,then

the price elasticity of demand for tickets is

(a)?0:50.

(b)?0:38.

(c)?0:75.

(d)?0:13.

(e)?0:25.

Quiz 16 NAME

Equilibrium

16.1 This problem will be easier if you have done Problem 16.3.The

inverse demand function for grapefruit is de ned by the equation p =

296?7q,whereq is the number of units sold,The inverse supply function

is de ned by p =17+2q,A tax of 27 is imposed on suppliers for each

unit of grapefruit that they sell,When the tax is imposed,the quantity

of grapefruit sold falls to

(a) 31.

(b) 17.50.

(c) 26.

(d) 28.

(e) 29.50.

16.2 In a crowded city far away,the civic authorities decided that rents

were too high,The long-run supply function of two-room rental apart-

ments was given by q =18+2p and the long run demand function was

given by q = 114?4p where p is the rental rate in crowns per week.

The authorities made it illegal to rent an apartment for more than 10

crowns per week,To avoid a housing shortage,the authorities agreed to

pay landlords enough of a subsidy to make supply equal to demand,How

much would the weekly subsidy per apartment have to be to eliminate

excess demand at the ceiling price?

(a) 9

(b) 15

(c) 18

(d) 36

(e) 27

16.3 Suppose that King Kanuta from Problem 16.11 demands that each

of his subjects gives him 4 coconuts for every coconut that the subject

consumes,The king puts all of the coconuts that he collects in a large

pile and burns them,The supply of coconuts is given by S(p

s

) = 100p

s

,

where p

s

is the price received by suppliers,The demand for coconuts by

the king’s subjects is given by D(p

d

)=8;320?100p

d

,wherep

d

is the

price paid by consumers,In equilibrium,the price received by suppliers

will be

476 EQUILIBRIUM (Ch,16)

(a) 16.

(b) 24.

(c) 41.60.

(d) 208.

(e) None of the other options are correct.

16.4 In Problem 16.6,the demand function for Schrecklichs is 200?4P

S

2P

L

and the demand function for LaMerdes is 200?3P

L

P

S

,whereP

S

and P

L

are respectively the price of Schrecklichs and LaMerdes,If the

world supply of Schrecklichs is 100 and the world supply of Lamerdes is

90,then the equilibrium price of Schrecklichs is

(a) 8.

(b) 25.

(c) 42.

(d) 34.

(e) 16.

Quiz 17 NAME

Auctions

17.1 First Fiddler’s Bank has foreclosed on a home mortgage and is selling

the house at auction,There are three bidders for the house,Jesse,Sheila,

and Elsie,First Fiddler’s does not know the willingness to pay of any

of these bidders but on the basis of its previous experience believes that

each of them has a probability of 1/3 of valuing the house at $700,000,

a probability of 1/3 of valuing it at $500,000,and a probability of 1/3 of

valuing it at $200,000,First Fiddlers believes that these probabilities are

independent between buyers,If First Fiddler’s sells the house by means

of a second-bidder sealed-bid auction (Vickrey auction),what will be the

bank’s expected revenue from the sale? (Choose the closest answer.)

(a) $500,000

(b) $474,074

(c) $466,667

(d) $666,667

(e) $266,667

17.2 An antique cabinet is being sold by means of an English auction.

There are four bidders,Natalie,Heidi,Linda,and Eva,These bidders

are unacquainted with each other and do not collude,Natalie values the

cabinet at $1,200,Heidi values it at $950,Linda values it at $1,700,and

Eva values it at $700,If the bidders bid in their rational self-interest,the

cabinet will be sold to

(a) Linda for about $1,700.

(b) Natalie for about $1,200.

(c) either Linda or Natalie for about $1,200,Which of these two buyers

gets it is randomly determined.

(d) Linda for slightly more than $1,200.

478 AUCTIONS (Ch,17)

(e) either Linda or Natalie for about $950,Which of these two buyers

gets it is randomly determined.

17.3 A dealer decides to sell an antique automobile by means of an English

auction with a reservation price of $900,There are two bidders,The

dealer believes that there are only three possible values that each bidder’s

willingness to pay might take,$6,300,$2,700,and $900,Each bidder has

a probability of 1/3 of having each of these willingnesses to pay,and the

probabilities of the two bidders are independent of the other’s valuation.

Assuming that the two bidders bid rationally and do not collude,the

dealer’s expected revenue from selling the automobile is

(a) $4,500.

(b) $3,300.

(c) $2,700.

(d) $2,100.

(e) $6,300.

17.4 A dealer decides to sell an oil painting by means of an English

auction with a reservation price of slightly below $81,000,If he fails to

get a bid as high as his reservation price,he will burn the painting,There

are two bidders,The dealer believes that each bidder’s willingness to

pay will take one of the three values,$90,000,$81,000,and $45,000,The

dealer believes that each bidder has a probability of 1/3 of having each

of these three values,The probability distribution of each buyer’s value

is independent of that of the other’s,Assuming that the two bidders bid

rationally and do not collude,the dealer’s expected revenue from selling

the painting is slightly less than

(a) $73,000.

(b) $81,000.

(c) $45,000.

(d) $63,000.

NAME 479

(e) $72,000.

17.5 Jerry’s Auction House in Purloined Hubcap,Oregon,holds sealed-

bid used car auctions every Wednesday,Each car is sold to the highest

bidder at the second-highest bidder’s bid,On average,2/3 of the cars that

are auctioned are lemons and 1/3 are good used cars,A good used car is

worth $1,500 to any buyer,A lemon is worth $150 to any buyer,Most

buyers can do no better than picking at random from among these used

cars,The only exception is Al Crankcase,Recall that Al can sometimes

detect lemons by tasting the oil on the car’s dipstick,A good car never

fails Al’s test,but half of the lemons fail his test,Al attends every auction,

licks every dipstick,and bids his expected value of every car given the

results of his test,Al will bid:

(a) $825 for cars that pass his test and $150 for cars that fail his test.

Normal bidders will get only lemons.

(b) $750 for cars that pass his test and $500 for cars that fail his test.

Normal bidders will get only lemons.

(c) $500 for cars that pass his test and $150 for cars that fail his test.

Normal bidders will get good cars only 1/6 of the time.

(d) $600 for cars that pass his test and $250 for cars that fail his test.

Normal bidders will get good cars only 1/6 of the time.

(e) $300 for cars that pass his test and $150 for cars that fail his test.

Normal bidders will get good cars only 1/12 of the time.

480 AUCTIONS (Ch,17)

Quiz 18 NAME

Technology

18.1 This problem will be easier if you have done Problem 18.1,A rm

has the production functionf(x

1;x

2

)=x

0:90

1

x

0:30

2

,The isoquant on which

output is 40

3=10

has the equation

(a) x

2

=40x

3

1

.

(b) x

2

=40x

3:33

1

.

(c) x

1

=x

2

=3.

(d) x

2

=40x

0:30

1

.

(e) x

1

=0:30x

0:70

2

.

18.2 A rm has the production function f(x;y)=x

0:70

y

0:30

.This rm

has

(a) decreasing returns to scale and dimininishing marginal product for

factor x.

(b) increasing returns to scale and decreasing marginal product of factor

x.

(c) decreasing returns to scale and increasing marginal product for factor

x.

(d) constant returns to scale.

(e) None of the other options are correct.

18.3 A rm uses 3 factors of production,Its production function is

f(x;y;z)=minfx

5

=y;y

4;(z

6

x

6

)=y

2

g,If the amount of each input is

multiplied by 6,its output will be multiplied by

(a) 7,776.

(b) 1,296.

(c) 216.

(d) 0.

482 TECHNOLOGY (Ch,18)

(e) The answer depends on the original choice of x,y,andz.

18.4 A rm has a production function f(x;y)=1:20(x

0:10

+y

0:10

)

1

when-

ever x>0andy>0,When the amounts of both inputs are positive,

this rm has

(a) increasing returns to scale.

(b) decreasing returns to scale.

(c) constant returns to scale.

(d) increasing returns to scale if x+y>1 and decreasing returns to scale

otherwise.

(e) increasing returns to scale if output is less than 1 and decreasing

returns to scale if output is greater than 1.

Quiz 19 NAME

Profit Maximization

19.1 In Problem 19.1,the production function is F(L)=6L

2=3

,Suppose

that the cost per unit of labor is 8 and the price of output is 8,how many

units of labor will the rm hire?

(a) 128

(b) 64

(c) 32

(d) 192

(e) None of the other options are correct.

19.2 In Problem 19.2,the production function is given by f(x)=4x

1=2

.

If the price of the commodity produced is 70 per unit and the cost of the

input is 35 per unit,how much pro ts will the rm make if it maximizes

pro ts?

(a) 560

(b) 278

(c) 1,124

(d) 545

(e) 283

19.3 In Problem 19.11,the production function is f(x

1;x

2

)=x

1=2

1

x

1=2

2

.If

the price of factor 1 is 8 and the price of factor 2 is 16,in what proportions

should the rm use factors 1 and 2 if it wants to maximize pro ts?

(a) x

1

= x

2

.

(b) x

1

=0:50x

2

.

(c) x

1

=2x

2

.

(d) We can’t tell without knowing the price of output.

484 PROFIT MAXIMIZATION (Ch,19)

(e) x

1

=16x

2

.

19.4 In Problem 19.9,when Farmer Hoglund applies N pounds of fer-

tilizer per acre,the marginal product of fertilizer is 1?(N=200) bushels

of corn,If the price of corn is $4 per bushel and the price of fertilizer

is $1.20 per pound,then how many pounds of fertilizer per acre should

Farmer Hoglund use in order to maximize his pro ts?

(a) 140

(b) 280

(c) 74

(d) 288

(e) 200

Quiz 20 NAME

Cost Minimization

20.1 Suppose that Nadine in Problem 20.1 has a production function

3x

1

+ x

2

,If the factor prices are 9 for factor 1 and 4 for factor 2,how

much will it cost her to produce 50 units of output?

(a) 1,550

(b) 150

(c) 200

(d) 875

(e) 175

20.2 In Problem 20.2,suppose that a new alloy is invented which uses

copper and zinc in xed proportions,where one unit of output requires 3

units of copper and 3 units of zinc for each unit of alloy produced,If no

other inputs are needed,if the price of copper is 2 and the price of zinc

is 2,what is the average cost per unit when 4,000 units of the alloy are

produced?

(a) 6.33

(b) 666.67

(c) 0.67

(d) 12

(e) 6,333.33

20.3 In Problem 20.3,the production function is f(L;M)=4L

1=2

M

1=2

,

where L is the number of units of labor and M is the number of machines

used,If the cost of labor is $25 per unit and the cost of machines is $64

per unit,then the total cost of producing 6 units of output will be

(a) $120.

(b) $267.

(c) $150.

486 COST MINIMIZATION (Ch,20)

(d) $240.

(e) None of the other options are correct.

20.4 Suppose that in the short run,the rm in Problem 20.3 which has

production function F(L;M)=4L

1=2

M

1=2

must use 25 machines,If the

cost of labor is 8 per unit and the cost of machines is 7 per unit,the

short-run total cost of producing 200 units of output is

(a) 1,500.

(b) 1,400.

(c) 1,600.

(d) 1,950.

(e) 975.

20.5 In Problem 20.12,Al’s production function for deer is f(x

1;x

2

)=

(2x

1

+ x

2

)

1=2

where x

1

is the amount of plastic and x

2

is the amount of

wood used,If the cost of plastic is $2 per unit and the cost of wood is $4

per unit,then the cost of producing 8 deer is

(a) $64.

(b) $70.

(c) $256.

(d) $8.

(e) $32.

20.6 Two rms,Wickedly E cient Widgets and Wildly Nepotistic Wid-

gets,produce widgets with the same production function y = K

1=2

L

1=2

where K is the input of capital and L is the input of labor,Each company

can hire labor at $1 per unit and capital at $1 per unit,WEW produces

10 widgets per week,choosing its input combination so as to produce

these 10 widgets in the cheapest way possible,WNW also produces 10

widgets per week,but its dotty CEO requires it to use twice as much

labor as WEW uses,Given that it must use twice as many laborers as

WEW does,and must produce the same output,how much more larger

are WNW’s total costs than WEW’s?

(a) $10 per week

(b) $20 per week

(c) $15 per week

(d) $5 per week

(e) $2 per week

Quiz 21 NAME

Cost Curves

21.1 In Problem 21.2,if Mr,Dent Carr’s total costs are 4s

2

+75s + 60,

then if he repairs 15 cars,his average variable costs will be

(a) 135.

(b) 139.

(c) 195.

(d) 270.

(e) 97.50.

21.2 In Problem 21.3,Rex Carr could pay $10 for a shovel that lasts one

year and pay $5 a car to his brother Scoop to bury the cars,or he could

buy a low-quality car smasher that costs $200 a year to own and that

smashes cars at a marginal cost of $1 per car,If it is also possible for

Rex to buy a high-quality hydraulic car smasher that cost $300 per year

to own and if with this smasher he could dispose of cars at a cost of $0.80

per car,it would be worthwhile for him to buy this high-quality smasher

smasher if

(a) he plans to dispose of at least 500 cars per year.

(b) he plans to dispose of no more than 250 cars per year.

(c) he plans to dispose of at least 510 cars per year.

(d) he plans to dispose of no more than 500 cars per year.

(e) he plans to dispose of at least 250 cars per year.

21.3 Mary Magnolia in Problem 21.4 has variable costs equal to y

2

=F

where y is the number of bouquets she sells per month and where F is the

number of square feet of space in her shop,If Mary has signed a lease for

a shop with 1,600 square feet and if she is not able to get out of the lease

or to expand her store in the short run,and if the price of a bouquet is

$3 per unit,how many bouquets per month should she sell in the short

run?

(a) 1,600

488 COST CURVES (Ch,21)

(b) 800

(c) 2,400

(d) 3,600

(e) 2,640

21.4 Touchie MacFeelie’s production function is,1J

1=2

L

3=4

,whereJ is

the number of old jokes used and L is the number of hours of cartoonists’

labor,Touchie is stuck with 900 old jokes for which he paid 6 dollars each.

If the wage rate for cartoonists is 5,then the total cost of producing 24

comics books is

(a) 5,480.

(b) 2,740.

(c) 8,220.

(d) 5,504.

(e) 1,370.

21.5 Recall that Touchie McFeelie’s production function for comic books

is,1J

1=2

L

3=4

,Suppose that Touchie can vary both jokes and cartoonists’

labor,If old jokes cost $2 each and cartoonists’ labor costs $18 per hour,

then the cheapest way to produce comics books requires using jokes and

labor in the ratio J=L =

(a) 9.

(b) 12.

(c) 3.

(d) 2/3.

(e) 6.

Quiz 22 NAME

Firm Supply

22.1 Suppose that Dent Carr’s long-run total cost of repairing s cars per

week is c(s)=3s

2

+ 192,If the price he receives for repairing a car is 36,

then in the long run,how many cars will he x per week if he maximizes

pro ts?

(a) 6

(b) 0

(c) 12

(d) 9

(e) 18

22.2 In Problem 22.9,suppose that Irma’s production function is

f(x

1;x

2

)=(minfx

1;2x

2

g)

1=2

,If the price of factor 1 is w

1

=6and

the price of factor 2 is w

2

= 4,then her supply function is given by the

equation:

(a) S(p)=p=16.

(b) S(p)=pmaxfw

1;2w

2

g

2

.

(c) S(p)=pminfw

1;2w

2

g

2

.

(d) S(p)=8p.

(e) S(p)=minf6p;8pg.

22.3 A rm has the long-run cost function C(q)=2q

2

+ 8,In the long

run,it will supply a positive amount of output,so long as the price is

greater than

(a) 16.

(b) 24.

(c) 4.

(d) 8.

(e) 13.

490 FIRM SUPPLY (Ch,22)

Quiz 23 NAME

Industry Supply

23.1 In Problem 23.1,if the cost of plaster and labor is $9 per gnome and

everything else is as in the problem,what is the lowest price of gnomes

at which there would be a positive supply in the long run?

(a) $9

(b) $18

(c) $11.20

(d) $9.90

(e) $10.80

23.2 Suppose that the garden gnome industry was in long-run equilib-

rium given the circumstances described in Problem 23.1,Suppose,as in

Problem 23.2,that it was discovered to everyone’s surprise,on January 1,

1993 after it was to late to change orders for gnome molds,that the cost

of the plaster and labor needed to make a gnome had changed to 8,If

the demand curve does not change,what will happen to the equilibrium

price of gnomes?

(a) It rises by 1.

(b) It falls by 1.

(c) It stays constant.

(d) It rises by 8.

(e) It falls by 4.

23.3 Suppose that the garden gnome industry was in long run equilib-

rium as described in Problem 23.1 and that on January 1,1993,the cost

of plaster and labor remained at $7 per gnome,and the government in-

troduced a tax of $10 on every garden gnome sold,Then the equilibrium

price of garden gnomes in 1993 would be

(a) $17.

(b) $9.20.

492 INDUSTRY SUPPLY (Ch,23)

(c) $7.

(d) $10.

(e) $27.

23.4 Suppose that the cost of capturing a cockatoo and transporting him

to the U,S,is about $40 per bird,Cockatoos are drugged and smuggled

in suitcases to the U,S,Half of the smuggled cockatoos die in transit.

Each smuggled cockatoo has a 10% probability of being discovered,in

which case the smuggler is ned,If the ne imposed for each smuggled

cockatoo is increased to $900,then the equilibrium price of cockatoos in

theU.S.willbe

(a) $288.89.

(b) $130.

(c) $85.

(d) $67.

(e) $200.

23.5 In Problem 23.13,in the absence of government interference,there

is a constant marginal cost of $5 per ounce for growing marijuana and

delivering it to buyers,If the probability that any shipment of marijuana

is seized is 0.20 and the ne if a shipper is caught is $20 per ounce,then

the equilibrium price of marijuana per ounce is

(a) $11.25.

(b) $9.

(c) $25.

(d) $4.

(e) $6.

23.6 In Problem 23.8,the supply curve of any rm is S

i

(p)=p=2,If a

rm produces 3 units of output,what are its total variable costs?

(a) $18

(b) $7

(c) $13.50

(d) $9

(e) There is not enough information given to determine total variable

costs.

Quiz 24 NAME

Monopoly

24.1 In Problem 24.1,if the demand schedule for Bong’s book is Q =

3;000? 100p,the cost of having the book typeset is 10,000,and the

marginal cost of printing an extra book is $4,he would maximize his

pro ts by

(a) having it typeset and selling 1,300 copies.

(b) having it typeset and selling 1,500 copies.

(c) not having it typeset and not selling any copies.

(d) having it typeset and selling 2,600 copies.

(e) having it typeset and selling 650 copies.

24.2 In Problem 24.2,if the demand for pigeon pies is p(y)=70?y=2,

then what level of output will maximize Peter’s pro ts?

(a) 74

(b) 14

(c) 140

(d) 210

(e) None of the above

24.3 A pro t-maximizing monopoly faces an inverse demand function

described by the equation p(y)=70?y and its total costs are c(y)=5y,

where prices and costs are measured in dollars,In the past it was not

taxed,but now it must pay a tax of 8 dollars per unit of output,After

the tax,the monopoly will

(a) increase its price by $8.

(b) increase its price by $12.

(c) increase its price by $4.

(d) leave its price constant.

494 MONOPOLY (Ch,24)

(e) None of the other options are correct.

24.4 A rm has invented a new beverage called Slops,It doesn’t taste

very good,but it gives people a craving for Lawrence Welk’s music and

Professor Johnson’s jokes,Some people are willing to pay money for this

e ect,so the demand for Slops is given by the equation q =14?p.Slops

can be made at zero marginal cost from old-fashioned macroeconomics

books dissolved in bathwater,But before any Slops can be produced,the

rm must undertake a xed cost of 54,Since the inventor has a patent

on Slops,it can be a monopolist in this new industry.

(a) The rm will produce 7 units of Slops.

(b) A Pareto improvement could be achieved by having the government

pay the rm a subsidy of 59 and insisting that the rm o er Slops at zero

price.

(c) From the point of view of social e ciency,it is best that no Slops be

produced.

(d) The rm will produce 14 units of Slops.

(e) None of the other options are correct.

Quiz 25 NAME

Monopoly Behavior

25.1 (See Problem 25.1.) If demand in the U.S,is given by Q

1

=23;400?

900p

1

,wherep

1

is the price in the U.S,and if the demand in England

is given by 2;800? 200p

2

where p

2

is the price in England,then the

di erence between the price charged in England and the price charged in

the U.S,will be

(a) 6.

(b) 12.

(c) 0.

(d) 14.

(e) 18.

25.2 (See Problem 25.2.) A monopolist faces a demand curve described

by p(y) = 100?2y and has constant marginal costs of 16 and zero xed

costs,If this monopolist is able to practice perfect price discrimination,

its total pro ts will be

(a) 1,764.

(b) 21.

(c) 882.

(d) 2,646.

(e) 441.

25.3 A price-discriminating monopolist sells in two separate markets such

that goods sold in one market are never resold in the other,It charges 4 in

one market and 8 in the other market,At these prices,the price elasticity

inthe rstmarketis?1:50 and the price elasticity in the second market

is?0:10,Which of the following actions is sure to raise the monopolists

pro ts?

(a) Lower p

2

.

(b) Raise p

2

.

496 MONOPOLY BEHAVIOR (Ch,25)

(c) Raise p

1

and lower p

2

.

(d) Raise both p

1

and p

2

.

(e) Raise p

2

and lower p

1

.

25.4 The demand for Professor Bongmore’s new book is given by the

function Q =2;000?100p,If the cost of having the book typeset is

8,000,if the marginal cost of printing an extra copy is 4,and if he has no

other costs,then he would maximize his pro ts by

(a) having it typeset and selling 800 copies.

(b) having it typeset and selling 1,000 copies.

(c) not having it typeset and not selling any copies.

(d) having it typeset and selling 1,600 copies.

(e) having it typeset and selling 400 copies.

Quiz 26 NAME

Factor Markets

26.1 Suppose that in Problem 26.2,the demand curve for mineral water

is given by p =30?12q,wherep is the price per bottle paid by consumers

and q is the number of bottles purchased by consumers,Mineral water

is supplied to consumers by a monopolistic distributor,who buys from a

monopolist producer who is able to produce mineral water at zero cost.

The producer charges the distributor a price of c per bottle,where the

price c maximizes the producer’s total revenue,Given his marginal cost

of c,the distributor chooses an output to maximize pro ts,The price

paid by consumers under this arrangement is

(a) 15.

(b) 22.50.

(c) 2.50.

(d) 1.25.

(e) 7.50.

26.2 Suppose that the labor supply curve for a large university in a small

town is given by w =60+0:08L where L is number of units of labor per

week and w is the weekly wage paid per unit of labor,If the university

is currently hiring 1,000 units of labor per week,the marginal cost of an

additional unit of labor

(a) equals the wage rate.

(b) is twice the wage rate.

(c) equals the wage rate plus 160.

(d) equals the wage rate plus 80.

(e) equals the wage rate plus 240

26.3 Rabelaisian Restaurants has a monopoly in the town of Upper Duo-

denum,Its production function is Q =40L,whereL istheamountof

labor it uses and Q is the number of meals produced,Rabelaisian Restau-

rants nds that in order to hire L units of labor,it must pay a wage of

40 +,1L per unit of labor,The demand curve for meals at Rabelaisian

Restaurants is given by P =30:75?Q=1;000,The pro t-maximizing

output for Rabelasian Restaurants is

498 FACTOR MARKETS (Ch,26)

(a) 14,000.

(b) 28,000.

(c) 3,500.

(d) 3,000.

(e) 1,750.

Quiz 27 NAME

Oligopoly

27.1 Suppose that the duopolists Carl and Simon in Problem 27.1 face

a demand function for pumpkins of Q =13;200?800P,whereQ is the

total number of pumpkins that reach the market and P is the price of

pumpkins,Suppose further that each farmer has a constant marginal

cost of $0.50 for each pumpkin produced,If Carl believes that Simon is

going to produce Q

s

pumpkins this year,then the reaction function tells

us how many pumpkins Carl should produce in order to maximize his

pro ts,Carl’s reaction function is R

C

(Q

s

)=

(a) 6;400?Q

s

=2.

(b) 13;200?800Q

s

.

(c) 13;200?1;600Q

s

.

(d) 3;200?Q

s

=2.

(e) 9;600?Q

s

.

27.2 If in Problem 27.4,the inverse demand for bean sprouts were given

by P(Y ) = 290?4Y and the total cost of producing y units for any

rm were TC(Y)=50Y,and if the industry consisted of two Cournot

duopolists,then in equilibrium each rm’s production would be

(a) 30 units.

(b) 15 units.

(c) 10 units.

(d) 20 units.

(e) 18.13 units.

27.3 In Problem 27.5,suppose that Grinch and Grubb go into the wine

business in a small country where wine is di cult to grow,The demand

for wine is given by p = $360?:2Q where p is the price and Q is the total

quantity sold,The industry consists of just the two Cournot duopolists,

Grinch and Grubb,Imports are prohibited,Grinch has constant marginal

costs of $15 and Grubb has marginal costs of $75,How much is Grinch’s

output in equilibrium?

500 OLIGOPOLY (Ch,27)

(a) 675

(b) 1,350

(c) 337.50

(d) 1,012.50

(e) 2,025

27.4 In Problem 27.6,suppose that two Cournot duopolists serve the

Peoria-Dubuque route,and the demand curve for tickets per day is Q =

200?2p (so p = 100?Q=2),Total costs of running a flight on this route

are 700+40q whereq is the number of passengers on the flight,Each flight

has a capacity of 80 passengers,In Cournot equilibrium,each duopolist

will run one flight per day and will make a daily pro t of

(a) 100.

(b) 350.

(c) 200.

(d) 200.

(e) 2,400.

27.5 In Problem 27.4,suppose that the market demand curve for bean

sproutsisgivenbyP = 880?2Q,whereP is the price and Q is total

industry output,Suppose that the industry has two rms,a Stackleberg

leader,and a follower,Each rm has a constant marginal cost of $80 per

unit of output,In equilibrium,total output by the two rms will be

(a) 200.

(b) 100.

(c) 300.

(d) 400.

(e) 50.

27.6 There are two rms in the blastopheme industry,The demand

curve for blastophemes is given by p =2;100?3q,Each rm has one

manufacturing plant and each rm i has a cost function C(q

i

)=q

2

i

where

q

i

is the output of rm i,The two rms form a cartel and arrange to

split total industry pro ts equally,Under this cartel arrangement,they

will maximize joint pro ts if

NAME 501

(a) and only if each rm produces 150 units in its plant.

(b) they produce a total of 300 units,no matter which rm produces

them.

(c) and only if they each produce a total of 350 units.

(d) they produce a total of 233.33 units,no matter which rm produces

them.

(e) they shut down one of the two plants,having the other operate as a

monopoly and splitting the pro ts.

502 OLIGOPOLY (Ch,27)

Quiz 28 NAME

Game Theory

28.1 (See Problem 28.1.) Big Pig and Little Pig have two possible strate-

gies,Press the Button,and Wait at the Trough,If both pigs choose Wait,

both get 4,If both pigs press the button then Big Pig gets 5 and Little

Pig gets 5,If Little Pig presses the button and Big Pig waits,then Big

Pig gets 10 and Little Pig gets 0,Finally,if Big Pig presses and Little

Pig waits,then Big Pig gets 4 and Little Pig gets 2,In Nash equilibrium,

(a) Little Pig will get a payo of 2 and Big Pig will get a payo of 4.

(b) Little Pig will get a payo of 5 and Big Pig will get a payo of 5.

(c) both pigs will wait at the trough.

(d) Little Pig will get a payo of zero.

(e) the pigs must be using mixed strategies.

28.2 (See Problem 28.6.) Two players are engaged in a game of \chicken."

There are two possible strategies,Swerve and Drive Straight,A player

who chooses to Swerve is called \Chicken" and gets a payo of zero,

regardless of what the other player does,A player who chooses to Drive

Straight gets a payo of 32 if the other player swerves and a payo of

48 if the other player also chooses to Drive Straight,This game has two

pure strategy equilibria and

(a) a mixed strategy equilibrium in which each player swerves with prob-

ability 0.60 and drives straight with probability 0.40.

(b) two mixed strategies in which players alternate between swerving and

driving straight.

(c) a mixed strategy equilibrium in which one player swerves with prob-

ability 0.60 and the other swerves with probability 0.40.

(d) a mixed strategy in which each player swerves with probability 0.30

and drives straight with probability 0.70.

504 GAME THEORY (Ch,28)

(e) no mixed strategies.

28.3 The old Michigan football coach had only two strategies,run the ball

to the left side of the line,and run the ball to the right side,The defense

can concentrate either on the left side or the right side of Michigan’s

line,If the opponent concentrates on the wrong side,Michigan is sure to

gain at least 5 yards,If the defense defended the left side and Michigan

ran left,Michigan would be stopped for no gain,But if the opponent

defended the right side when Michigan ran right,Michigan would still

gain at least 5 yards with probability 0.40,It is the last play of the

game and Michigan needs to gain 5 yards to win,Both sides choose Nash

equilibrium strategies,In Nash equilibrium,Michigan would

(a) be sure to run to the right side.

(b) run to the right side with probability 0.63.

(c) run to the right side with probability 0.77.

(d) run with equal probability to one side or the other.

(e) run to the right side with probability 0.60.

28.4 Suppose that in the Hawk-Dove game discussed in Problem 28.3,

the payo to each player is?4 if both play Hawk,If both play Dove,

the payo to each player is 1 and if one plays Hawk and the other plays

Dove,the one that plays Hawk gets a payo of 3 and the one that plays

Dove gets 0,In equilibrium,we would expect Hawks and Doves to do

equally well,This happens when the proportion of the total population

that plays Hawk is

(a) 0.33.

(b) 0.17.

(c) 0.08.

(d) 0.67.

(e) 1.

28.5 (See Problem 28.11.) If the number of persons who attend the club

meeting this week is X,then the number of people who will attend next

week is 27 + 0:70X,What is a long-run equilibrium attendance for this

club?

(a) 27

(b) 38.57

(c) 54

(d) 90

(e) 63

Quiz 29 NAME

Exchange

29.1 An economy has two people Charlie and Doris,There are two goods,

apples and bananas,Charlie has an initial endowment of 3 apples and

12 bananas,Doris has an initial endowment of 6 apples and 6 bananas.

Charlie’s utility function is U(A

C;B

C

)=A

C

B

C

where A

C

is his apple

consumption and B

C

is his banana consumption,Doris’s utility function

is U(A

D;B

D

)=A

D

B

D

where A

D

and B

D

are her apple and banana

consumptions,At every Pareto optimal allocation,

(a) Charlie consumes the same number of apples as Doris.

(b) Charlie consumes 9 apples for every 18 bananas that he consumes.

(c) Doris consumes equal numbers of apples and bananas.

(d) Charlie consumes more bananas per apple than Doris does.

(e) Doris consumes apples and bananas in the ratio of 6 apples for every

6 bananas that she consumes.

29.2 In Problem 29.4,Ken’s utility function is U(Q

K;W

K

)=Q

K

W

K

and Barbie’s utility function is U(Q

B;W

B

)=Q

B

W

B

,If Ken’s initial

endowment were 3 units of quiche and 10 units of wine and Barbie’s

endowment were 6 units of quiche and 10 units of wine,then at any Pareto

optimal allocation where both persons consume some of each good,

(a) Ken would consume 3 units of quiche for every 10 units of wine.

(b) Barbie would consume twice as much quiche as Ken.

(c) Ken would consume 9 units of quiche for every 20 units of wine that

he consumed.

(d) Barbie would consume 6 units of quiche for every 10 units of wine

that she consumed.

(e) None of the other options are correct.

29.3 In Problem 29.1,suppose that Morris has the utility function

U(b;w)=6b +24w and Philip has the utility function U(b;w)=bw.

If we draw an Edgeworth box with books on the horizontal axis and wine

on the vertical axis and if we measure Morris’s consumptions from the

lower left corner of the box,then the contract curve contains

506 EXCHANGE (Ch,29)

(a) a straight line running from the upper right corner of the box to the

lower left.

(b) a curve that gets steeper as you move from left to right.

(c) a straight line with slope 1=4 passing through the lower left corner of

the box.

(d) a straight line with slope 1=4 passing through the upper right corner

of the box.

(e) a curve that gets flatter as you move from left to right.

29.4 In Problem 29.2,Astrid’s utility function is U(H

a;C

A

)=H

A

C

A

.

Birger’s utility function is minfH

B;C

B

g,Astrid’s initial endowment is no

cheese and 4 units of herring,and Birger’s initial endowments are 6 units

of cheese and no herring,Where p is a competitive equilibrium price of

herring and cheese is the numeraire,it must be that demand equals supply

in the herring market,This implies that

(a) 6=(p +1)+2=4.

(b) 6=4=p.

(c) 4=6=p.

(d) 6=p +4=2p =6.

(e) minf4;6g= p.

29.5 Suppose that in Problem 29.8,Mutt’s utility function is U(m;j)=

maxf3m;jg and Je ’s utility function is U(m;j)=2m + j.Mutis

initially endowed with 4 units of milk and 2 units of juice,Je is initially

endowed with 4 units of milk and 6 units of juice,If we draw an Edgeworth

box with milk on the horizontal axis and juice on the vertical axis and if

we measure goods for Mutt by the distance from the lower left corner of

the box,then the set of Pareto optimal allocations includes the

(a) left edge of the Edgeworth box but no other edges.

(b) bottom edge of the Edgeworth box but no other edges.

(c) left edge and bottom edge of the Edgeworth box.

(d) right edge of the Edgeworth box but no other edges.

NAME 507

(e) right edge and top edge of the Edgeworth box.

29.6 In Problem 29.3,Professor Nightsoil’s utility function,U

N

(B

N;P

N

),

is B

N

+4P

1=2

N

and Dean Interface’s utility function is U

I

(B

I;P

I

)=B

I

+

2P

1=2

I

,If Nightsoil’s initial endowment is 7 bromides and 15 platitudes

and if Interface’s initial endowment is 7 bromides and 25 platitudes,then

at any Pareto e cient allocation where both persons consume positive

amounts of both goods,it must be that

(a) Nightsoil consumes the same ratio of bromides to platitudes as Inter-

face.

(b) Interface consumes 8 platitudes.

(c) Interface consumes 7 bromides.

(d) Interface consumes 3 bromides.

(e) Interface consumes 5 platitudes.

508 EXCHANGE (Ch,29)

Quiz 30 NAME

Production

30.1 Suppose that in Problem 30.1,Tip can write 5 pages of term papers

or solve 20 workbook problems in an hour,while Spot can write 2 pages

of term papers or solve 6 workbook problems in an hour,If they each

decide to work a total of 7 hours,and to share their output then if they

produce as many pages of term paper as possible given that they produce

30 workbook problems,

(a) Spot will spend all of his time writing term papers and Tip will spend

some time at each task.

(b) Tip will spend all of his time writing term papers and Spot will spend

some time at each task.

(c) bothstudentswillspendsometimeateachtask.

(d) Spot will write term papers only and Tip will do workbook problems

only.

(e) Tip will write term papers only and Spot will do workbook problems

only.

30.2 Al and Bill are the only workers in a small factory which makes

geegaws and doodads,Al can make 3 geegaws per hour or 15 doodads per

hour,Bill can make 2 geegaws per hour or 6 doodads per hour,Assuming

that neither of them nds one task more odious than the other,

(a) Al has comparative advantage in producing geegaws,and Bill has

comparative advantage in producing doodads.

(b) Bill has comparative advantage in producing geegaws,and Al has

comparative advantage in producing doodads.

(c) Al has comparative advantage in producing both geegaws and doo-

dads.

(d) Bill has comparative advantage in producing both geegaws and doo-

dads.

510 PRODUCTION (Ch,30)

(e) both persons have comparative advantage in producing doodads.

30.3 (See Problem 30.5.) Every consumer has a red-money income and

a blue-money income and each commodity has a red price and a blue

price,You can buy a good by paying for it either with blue money at

the blue price,or with red money at the red price,Harold has 10 units

of red money to spend and 18 units of blue money to spend,The red

price of ambrosia is 1 and the blue price of ambrosia is 2,The red price

of bubblegum is 1 and the blue price of bubblegum is 1,If ambrosia is on

the horizontal axis,and bubblegum on the vertical,axis,then Harold’s

budget set is bounded

(a) by two line segments,one running from (0,28) to (10,18) and another

running from (10,18) to (19,0).

(b) by two line segments one running from (0,28) to (9,10) and the other

running from (9,10) to (19,0).

(c) by two line segments,one running from (0,27)to (10,18) and the other

running from (10,18) to (20,0).

(d) a vertical line segment and a horizontal line segement,intersecting at

(10,18).

(e) a vertical line segment and a horizontal line segment,intersecting at

(9,10).

30.4 (See Problem 30.2.) Robinson Crusoe has exactly 12 hours per day

to spend gathering coconuts or catching sh,He can catch 4 sh per hour

or he can pick 16 coconuts per hour,His utility function is U(F;C)=FC

where F is his consumption of sh and C is his consumption of coconuts.

If he allocates his time in the best possible way between catching sh and

picking coconuts,his consumption will be the same as it would be if he

could buy sh and coconuts in a competitive market where the price of

coconuts is 1,and where

(a) his income is 192 and the price of sh is 4.

(b) his income is 48 and the price of sh is 4.

(c) his income is 240 and the price of sh is 4.

(d) his income is 192 and the price of sh is 0.25.

(e) his income is 120 and the price of sh is 0.25.

30.5 On a certain island there are only two goods,wheat and milk,The

only scarce resource is land,There are 1,000 acres of land,An acre of land

will produce either 16 units of milk or 37 units of wheat,Some citizens

have lots of land,some have just a little bit,The citizens of the island

all have utility functions of the form U(M;W)=MW,At every Pareto

optimal allocation,

NAME 511

(a) the number of units of milk produced equals the number of units of

wheat produced.

(b) total milk production is 8,000.

(c) all citizens consume the same commodity bundle.

(d) every consumer’s marginal rate of substitution between milk and

wheat is?1.

(e) None of the above is true at every Pareto optimal allocation.

512 PRODUCTION (Ch,30)

Quiz 31 NAME

Welfare

31.1 A Borda count is used to decide an election between 3 candidates,

x,y,and z where a score of 1 is awarded to a rst choice,2 to a second

choice and 3 to a third choice,There are 25 voters,7 voters rank the

candidates x rst,y second,z third; 4 voters rank the candidates x rst,z

second,y third; 6 rank the candidates,z rst,y second,x third; 8 voters

rank the candidates,y rst,z second,x third,Which candidate wins?

(a) Candidate x.

(b) Candidate y.

(c) Candidate z.

(d) There is a tie between x and y,with z coming in third.

(e) There is a tie between y and z,with x coming in third.

31.2 A parent has two children living in cities with di erent costs of

living,The cost of living in city B is 3 times the cost of living in city A.

The child in city A has an income of 3,000 and the child in city B has an

income of $9,000,The parent wants to give a total of $4,000 to her two

children,Her utility function is U(C

A;C

B

)=C

A

C

B

,whereC

A

and C

B

are the consumptions of the children living in cities A and B respectively.

She will choose to

(a) give each child $2,000,even though this will buy less goods for the

child in city B.

(b) give the child in city B 3 times as much money as the child in city A.

(c) give the child in city A 3 times as much money as the child in city B.

(d) give the child in city B 1.50 times as much money as the child in

city A.

(e) give the child in city A 1.50 times as much money as the child in

city B.

31.3 Suppose that Paul and David from Problem 31.7 have utility func-

tions U =5A

P

+ O

P

and U = A

D

+5O

D

,respectively,where A

P

and

O

P

are Paul’s consumptions of apples and oranges and A

D

and O

D

are

David’s consumptions of apples and oranges,The total supply of apples

and oranges to be divided between them is 8 apples and 8 oranges,The

\fair" allocations consist of all allocations satisfying the following condi-

tions.

514 WELFARE (Ch,31)

(a) A

D

= A

P

and O

D

= O

P

.

(b) 10A

P

+2O

P

is at least 48,and 2A

D

+10O

D

is at least 48.

(c) 5A

P

+O

P

is at least 48,and 2A

D

+5O

D

is at least 48.

(d) A

D

+O

D

is at least 8,and A

S

+O

S

is at least 8.

(e) 5A

P

+O

P

is at least A

D

+5O

D

,andA

D

+5O

D

is at least 5A

P

+O

P

.

31.4 Suppose that Romeo in Problem 31.8 has the utility function U =

S

8

R

S

4

J

and Juliet has the utility function U = S

4

R

S

8

J

,whereS

R

is Romeo’s

spaghetti consumption and S

J

is Juliet’s,They have 96 units of spaghetti

to divide between them.

(a) Romeo would want to give Juliet some spaghetti if he had more than

48 units of spaghetti.

(b) Juliet would want to give Romeo some spaghetti if she has more than

62 units.

(c) Romeo and Juliet would never disagree about how to divide the

spaghetti.

(d) Romeo would want to give Juliet some spaghetti if he has more than

60 units of spaghetti.

(e) Juliet would want to give Romeo some spaghetti if she has more than

64 units of spaghetti.

31.5 Hat eld and McCoy burn with hatred for each other,They both

consume corn whisky,Hat eld’s utility function is U = W

H

W

2=8

M

and

McCoy’s utility is U = W

M

W

2=8

H

,whereW

H

is Hat eld’s whisky con-

sumption and W

M

is McCoy’s whisky consumption,measured in gallons.

The sheri has a total of 28 units of con scated whisky that he could

give back to them,For some reason,the sheri wants them both to be as

happy as possible,and he wants to treat them equally,The sheri should

give them each

(a) 14 gallons.

(b) 4 gallons and spill 20 gallons in the creek.

(c) 2 gallons and spill 24 gallons in the creek.

(d) 8 gallons and spill the rest in the creek.

(e) 1 gallon and spill the rest in the creek.

Quiz 32 NAME

Externalities

32.1 Suppose that in Horsehead,Massachusetts,the cost of operating

a lobster boat is $3,000 per month,Suppose that if X lobster boats

operate in the bay,the total monthly revenue from lobster boats in the

bay is $1;000(23x?x

2

),If there are no restrictions on entry and new

boats come into the bay until there is no pro t to be made by a new

entrant,then the number of boats that enter will be X1,If the number

of boats that operate in the bay is regulated to maximize total pro ts,

the number of boats in the bay will be X2.

(a) X1 = 20 and X2 = 20.

(b) X1 = 10 and X2=8.

(c) X1 = 20 and X2 = 10.

(d) X1 = 24 and X2 = 14.

(e) None of the other options are correct.

32.2 An apiary is located next to an apple orchard,The apiary produces

honey and the apple orchard produces apples,The cost function of the

apiary is C

H

(H;A)=H

2

=100?1A and the cost function of the apple

orchard is C

A

(H;A)=A

2

=100,where H and A are the number of units

of honey and apples produced respectively,The price of honey is 8 and

the price of apples is 7 per unit,Let A1 be the output of apples if the

rms operate independently,and let A2 be the output of apples if the

rms are operated by a single owner,It follows that

(a) A1 = 175 and A2 = 350.

(b) A1=A2 = 350.

(c) A1 = 200 and A2 = 350.

(d) A1 = 350 and A2 = 400.

516 EXTERNALITIES (Ch,32)

(e) A1 = 400 and A2 = 350.

32.3 Martin’s utility is U(c;d;h)=2c +5d?d

2

2h,whered is the

number of hours per day that he spends driving around,h is the number

of hours per day spent driving around by other people in his home town

and c is the amount of money he has left to spend on other stu besides

gasoline and auto repairs,Gas and auto repairs cost $.50 per hour of

driving,All the people in Martin’s home town have the same tastes,If

each citizen believes that his own driving will not a ect the amount of

driving done by others,they will all drive D1hoursperday,Iftheyall

drive the same amount,they would all be best o if each drove D2hours

per day,where

(a) D1=2andD2=1.

(b) D1=D2=2.

(c) D1=4andD2=2.

(d) D1=5andD2=0.

(e) D1 = 24 and D2=0.

32.4 (See Problems 32.8,32.9.) An airport is located next to a housing

development,Where X is the number of planes that land per day and Y

is the number of houses in the housing development,pro ts of the airport

are 22X?X

2

and pro ts of the developer are 32Y?Y

2

XY.Let

H1 be the number of houses built if a single pro t-maximizing company

owns the airport and the housing development,Let H2bethenumberof

houses built if the airport and the housing development are operated in-

dependently and the airport has to pay the developer the total \damages"

XY done by the planes to developer’s pro ts,Then

(a) H1=H2 = 14.

(b) H1 = 14 and H2 = 16.

(c) H1 = 16 and H2 = 14.

(d) H1 = 16 and H2 = 15.

(e) H1 = 15 and H2 = 19.

32.5 (See Problem 32.5.) A clothing store and a jeweler are located

side by side in a shopping mall,If the clothing store spends C dollars

on advertising and the jeweler spends J dollars on advertising,then the

pro ts of the clothing store will be (48 + J)C?2C

2

and the pro ts of

the jeweler will be (42 +C)J?2J

2

,The clothing store gets to choose his

amount of advertising rst,knowing that the jeweler will nd out how

much the clothing store advertised before deciding how much to spend.

The amount spent by the clothing store will be

NAME 517

(a) 16.71.

(b) 46.

(c) 69.

(d) 11.50.

(e) 34.50.

518 EXTERNALITIES (Ch,32)

Quiz 33 NAME

Law

33.1 Consider Madame Norrell,in Problem 33.1,She gets 5 logx if she

delivers x buttons to her fence,She has to pay a ne Fxif she is caught,

and she has a 10 percent chance of getting caught,If she is caught,she

cannot collect anything from her fence,How big should the ne be if we

want to limit Madam Norrell to taking 5 buttons?

(a) 4.5

(b) 5.5

(c) 9

(d) 11

(e) 12

33.2 Consider Jim and Dick,described in Problem 33.2,Jim rides at

speed s and has money m; his utility function is 10s +m,Dick walks at

speed w and has money m; his utility function is 10w + m.Thecostof

an accident to Jim is c

J

(s;w)=s

2

+ w

2

,and the cost of an accident to

Dick is also c

D

(s;w)=s

2

+w

2

,If there is no liability,how fast will Dick

and Jim move?

(a) s =10andw = 10.

(b) s =5andw =5.

(c) s =5andw = 10.

(d) s =10andw =5.

(e) s =15andw = 15.

520 LAW (Ch,33)

Quiz 34 NAME

Information Technology

34.1 If the demand function for the DoorKnobs operating system is re-

lated to perceived market share s and actual market share t by the equa-

tion p = 512s(1?x),then in the long run,the highest price at which

DoorKnobs could sustain a market share of 3/4 is

(a) $156.

(b) $64.

(c) $96.

(d) $128.

(e) $256.

34.2 Eleven consumers are trying to decide whether to connect to a new

communications network,Consumer 1 is of type 1,consumer 2 is of type

2,consumer 3 is of type 3,and so on,Where k is the number of consumers

connected to the network (including oneself),a consumer of type n has

willingness to pay to belong to this network equal to k times n.Whatis

the highest price at which 7 consumers could all connect to the network

and either make a pro t or at least break even?

(a) $40

(b) $33

(c) $25

(d) $40

(e) $35

34.3 Professor Kremepu ’s new,user-friendly textbook has just been

published,This book will be used in classes for two years,after which it

will be replaced by a new edition,The publisher charges a price of p

1

in

the rst year and p

2

in the second year,After the rst year,bookstores

buy back used copies for p

2

=2 and resell them to students in the second

year for p

2

,(Students are indi erent between new and used copies.) The

cost to a student of owning the book during the rst year is therefore

p

1

(p

2

=2),In the rst year of publication,the number of students

willing to pay $v to own a copy of the book for a year is 60;000?1;000v.

The number of students taking the course in the rst year who are willing

522 INFORMATION TECHNOLOGY (Ch,34)

to pay $w to keep the book for reference rather than sell it at the end of

the year is 60;000?5;000w,The number of persons who are taking the

course in the second year and are willing to pay at least $p for a copy of

the book is 50;000?1;000p,If the publisher sets a price of p

1

in the rst

year and p

2

p

1

in the second year,then the total number of copies of

the book that the publisher sells over the two years will be

(a) 120;000?1;000p

1

1;000p

2

.

(b) 120;000?1;000(p

1

p

2

=2).

(c) 120;000?3;000p

2

.

(d) 110;000?1;000(p

1

+p

2

=2).

(e) 110;000?1;500p

2

.

Quiz 35 NAME

Public Goods

35.1 Just north of the town of Muskrat,Ontario,is the town of Brass

Monkey,population 500,Brass Monkey,like Muskrat,has a single pub-

lic good,the town skating rink and a single private good,Labatt’s ale.

Everyone’s utility function is U

i

(X

i;Y)=X

i

64=Y,whereX

i

is the

number of bottles of ale consumed by i and Y is the size of the skating

rink in square meters,The price of ale is $1 per bottle,The cost of the

skating rink to the city is $5 per square meter,Everyone has an income

of at least $5,000,What is the Pareto e cient size for the town skating

rink?

(a) 80 square meters

(b) 200 square meters

(c) 100 square meters

(d) 165 square meters

(e) None of the other options are correct.

35.2 Recall Bob and Ray in Problem 35.4,They are thinking of buying a

sofa,Bob’s utility function is U

B

(S;M

B

)=(1+S)M

B

,and Ray’s utility

function is U

R

(S;M

R

)=(4+S)M

R

,whereS = 0 if they don’t get the sofa

and S = 1 if they do and where M

B

and M

R

are the amounts of money

they have respectively to spend on their private consumptions,Bob has

a total of $800 to spend on the sofa and other stu,Ray has a total of

$2,000 to spend on the sofa and other stu,The maximum amount that

they could pay for the sofa and still arrange to both be better o than

without it is

(a) $1,200.

(b) $500.

(c) $450.

(d) $800.

524 PUBLIC GOODS (Ch,35)

(e) $1,600.

35.3 Recall Bonnie and Clyde from Problem 35.5,Suppose that their

total pro ts are 48H,whereH is the number of hours they work per

year,Their utility functions are,respectively,U

B

(C

B;H)=C

B

0:01H

2

and U

C

(C

C;H)=C

C

0:01H

2

,whereC

B

and C

C

are their private goods

consumptions and H is the number of hours they work per year,If they

nd a Pareto optimal choice of hours of work and income distribution,it

must be that the number of hours they work per year is

(a) 1,300.

(b) 1,800.

(c) 1,200.

(d) 550.

(e) 650.

35.4 Recall Lucy and Melvin from Problem 35.6,Lucy’s utility function

is 2X

L

+ G,and Melvin’s utility function is X

M

G,where G is their ex-

penditures on the public goods they share in their apartment and where

X

L

and X

M

are their respective private consumption expenditures,The

total amount they have to spend on private goods and public goods is

32,000,They agree on a Pareto optimal pattern of expenditures in which

the amount that is spent on Lucy’s private consumption is 8,000,How

much do they spent on public goods?

(a) 8,000

(b) 16,000

(c) 8,050

(d) 4,000

(e) There is not enough information here to be able to determine the

answer.

Quiz 36 NAME

Information

36.1 As in Problem 36.2,suppose that low-productivity workers have

marginal products of 10 and high-productivity workers have marginal

products of 16,The community has equal numbers of each type of worker.

The local community college o ers a course in microeconomics,High-

productivity workers think taking this course is as bad as a wage cut of

4,and low-productivity workers think it is as bad as a wage cut of 7.

(a) There is a separating equilibrium in which high-productivity workers

take the course and are paid 16 and low-productivity workers do not take

the course and are paid 10.

(b) There is no separating equilibrium and no pooling equilibrium.

(c) There is no separating equilibrium,but there is a pooling equilibrium

in which everybody is paid 13.

(d) There is a separating equilibrium in which high-productivity workers

take the course and are paid 20 and low-productivity workers do not take

the course and are paid 10.

(e) There is a separating equilibrium in which high-productivity workers

take the course and are paid 16 and low-productivity workers are paid 13.

36.2 Suppose that in Enigma,Ohio,Klutzes have productivity of $1,000

and Kandos have productivity of $5,000 per month,You can’t tell Klutzes

from Kandos by looking at them or asking them,and it is too expensive

to monitor individual productivity,Kandos,however,have more patience

than Klutzes,Listening to an hour of dull lectures is as bad as losing $200

for a Klutz and $100 for a Kando,There will be a separating equilibrium

in which anybody who attends a course of H hours of lectures is paid

$5,000 per month and anybody who does not is paid $1,000 per month

(a) if H<40 and H>20.

(b) if H<80 and H>20.

(c) for all positive values of H.

(d) only in the limit as H approaches in nity.

526 INFORMATION (Ch,36)

(e) if H<35 and H>17:50.

36.3 In Rustbucket,Michigan,there are 200 used cars for sale,Half of

them are good,and half of them are lemons,Owners of lemons are willing

to sell them for $300,Owners of good used cars are willing to sell them for

prices above $1,100 but will keep them if the price is lower than $1,100.

There is a large number of potential buyers who are willing to pay $400

for a lemon and $2,100 for a good car,Buyers can’t tell good cars from

bad,but original owners know.

(a) There will be an equilibrium in which all used cars sell for $1,250.

(b) The only equilibrium is one in which all used cars on the market are

lemons and they sell for 400.

(c) There will be an equilibrium in which lemons sell for 300 and good

used cars sell for 1,100.

(d) There will be an equilibrium in which all used cars sell for 700.

(e) There will be an equilibrium in which lemons sell for 400 and good

used cars sell for 2,100.

36.4 Suppose that in Burnt Clutch,Pa.,the quality distribution of the

1000 used cars on the market is such that the number of used cars of value

less than V is V=2,Original owners must sell their used cars,Original

owners know what their cars are worth,but buyers can’t determine a car’s

quality until they buy it,An owner can either take his car to an appraiser

and pay the appraiser $100 to appraise the car (accurately and credibly),

or he can sell the car unappraised,In equilibrium,car owners will have

their cars appraised if and only if their value is at least

(a) $100.

(b) $500.

(c) $300.

(d) $200.

(e) $400.