Consumption | Preferences 1
Goods and Bundles
Consumers make choices over bundles of goods,Consumer theory models the way in which these choices are made.
A good is simply a product | such as apples or bananas,A good may be specifled in terms of time | such as
apples today or apples tomorrow,or place | such as apples in Oxford or apples in London.
A bundle of goods is a collection of goods | such as (3 apples,2 bananas),The bundle can contain as many goods
as necessary,For much of economics,only 2 goods are required,This allows a graphical representation.
Number of Bananas (= x2)
Number of Apples (= x1)
￠x
￠y
￠z
0
1
2
3
4
5
0 1 2 3 4 5
Write a bundle of goods as x =(x1;x2) where x1 is the amount of good 1 and x2 the amount of good 2 in the
bundle,How do consumers choose between bundles such as x and difierent bundles like y =(y1;y2) and z =(z1;z2)?
Consumption | Preferences 2
Preferences
The basic premise of consumer theory is that consumers have preferences over bundles of goods.
Write x ′ y if and only if bundle x is strictly preferred to bundle y.
Write x? y if bundle x is not preferred to bundle y and bundle y is not preferred to bundle x,The consumer is
indifierent between the two bundles.
Write x,y if the consumer strictly prefers bundle x to bundle y or the consumer is indifierent,Equivalently,x is
weakly preferred to y.
Armed with such preferences the consumer can compare difierent bundles of goods.
However,preferences are di–cult things to work with | is there a simple alternative?
Consumption | Preferences 3
Representation
Economists make three assumptions about preferences,These are:
1,Completeness,Either x,y or y,x for all x and y,In other words,all the difierent bundles can be compared.
2,Re exivity,x,x for all x,In other words,all the bundles are at least as good as themselves.
3,Transitivity,If x,y and y,z then x,z for all x,y and z,In other words,if a consumer prefers bundle x to
bundle y and bundle y to bundle z then the consumer also prefers bundle x to bundle z.
At a flrst glance these do not seem unreasonable,Nevertheless they are assumptions and not facts.
Theorem,Any preferences that satisfy the above three assumptions can be represented by a utility function.
The utility function is written as u(￠),It takes each bundle of goods and assigns a number to it,The theorem states
that there is a utility function u(￠) such that x,y if and only if u(x)? u(y),That is,a consumer prefers bundle x
to bundle y if and only if they get bigger \utility" from bundle x,The ordering induced by a consumer’s preferences
can be very simply represented by numbers.
The numbers u(￠) themselves might have no meaning | it is the relative size that matters,Utility functions allow
the construction of indifierence curves.
Consumption | Preferences 4
Indifierence Curves 1
As the theorem says,\utility numbers" can be assigned to each bundle of goods,This can be graphically
represented,Suppose all the bundles with a utility number greater than 5 are found and a line is drawn to separate
them from the bundles with a utility number less than 5,This line is called an indifierence curve.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.......................................................................................................................................0
x2
x1
￠x
￠y
￠z u =5
u > 5
u < 5
......
......
......
......
......
.......
.......
.......
.....
.....
......
......
......
.......
........
.............
.................
............................
In the picture above u(x) > 5 and u(y) < 5 therefore x,y,Also x,z but more information is required to rank y
and z,More indifierence curves need to be drawn,Clearly they can be drawn for any utility number.
With the current assumptions indifierence curves can be of various shapes,However,they cannot cross,Why not?
Consumption | Preferences 5
Indifierence Curves 2
Suppose that two indifierence curves were to cross,as below.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.......................................................................................................................................0
x2
x1
x
y
z
u =5
u =6
.....
.....
......
......
......
......
......
.......
.......
.....
.....
......
......
.......
........
...........
.................
.............................
..
......
......
......
......
.......
.......
.......
......
.....
......
......
.......
.
.........
.................
.............................................
The consumer is indifierent between x and y,and between x and z,z is strictly preferred to y,But by transitivity
x? y and x? z implies z? y,A contradiction.
Even with the current \minimal" assumptions two things can be said,First,indifierence curves can be drawn and
second they do not cross,What do they look like?
Consumption | Preferences 6
Some Examples of Indifierence Curves
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
x2
x1
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......................................................................................................................................
Perfect Substitutes
x2
x1
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
..............................................................................................
......
......
......
......
......
......
......
......
......
......
......
..
...................................................................
......
......
......
......
......
......
.....
........................................
..............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
Perfect Complements
0 0
......
......
......
......
........
...
.........,....
......
......
......
...........
.
.........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
x2
x1.........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......................................................................................................................................
A \Bad" | x2
x2
x1..........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
..
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
..
..........................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
A \Neutral" | x2
0 0
......
......
......
......
...........,.....
...
.....................................,........
The arrows are in the direction of increasing utility,A great variety of other shapes are possible | economists
restrict the sorts of indifierence curves possible by making further technical assumptions on preferences.
Consumption | Preferences 7
Well-Behaved Preferences
Two further assumptions are made in order to obtain well-behaved preferences.
1,Monotonicity,If x1? y1 and x2 > y2 or if x1 > y1 and x2? y2 then x ′ y,In other words,more is better.
2,Convexity,If x? y then?x+(1)y,x,In other words,averages are better than extremes.
It is easiest to see what these assumptions entail by looking at the diagrams below.
........................................................................................................................................,.............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
x2
x1
Well-Behaved
x2
x1
Non-Convexity
0 0
.....
.....
......
......
......
......
......
.......
.......
.....
.....
......
......
.......
........
...........
.................
..............................
................
.............
............
........
.......
......
......
......
......
.....
.....
......
.......
.......
.......
......
......
......
......
......
......
......
........
...
.........
The second diagram is ruled out by the convexity assumption,It is these two assumptions which give indifierence
curves their nice regular shape that will become so familiar during the course,Monotonicity makes the curves
downward sloping and convexity makes them \bowed".
Consumption | Preferences 8
The Marginal Rate of Substitution
The marginal rate of substitution or MRS is the rate at which the consumer is just willing to give up a small
amount of good 1 in order to gain a small amount of good 2,It is the private rate of exchange.
In the diagrams this is represented by the slope of the indifierence curve.
........................................................................................................................................,.............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
x2
x1
The MRS
x2
x1
Diminishing MRS
0 0
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
........................
......
......
.,....
................
................
................
................
...........,
................
................
................
.............
.........................
......
......
..
......
.......................
￠x2
-￠x1
Slope = MRS =￠x2=￠x1..........................................
....
......
......
......
......
......
......
.......
.......
.....
.....
....
.......
.........
..............
..................
.................
......
......
......
......
......
......
.......
.......
......
.....
....
.......
.........
.
.......................
￠x1 and ￠x2 are both taken to be very small numbers | hence the word marginal,More mathematically
￠x2=￠x1 … dx2=dx1,the derivative,The concept of marginal is central to economics and is always associated with
derivatives,or difierence ratios involving ￠ (capital delta).
The second diagram shows how the MRS changes along the indifierence curve,Notice it is decreasing (in absolute
value) as x1 increases,This is called a diminishing marginal rate of substitution,It is a direct consequence of the
convexity assumption,The consumer is willing to give up more of good 1 in exchange for good 2 as the amount of
good 1 increases.
Consumption | Preferences 9
Utility Functions
A utility function assigns a number to each bundle of goods,x.
Early economists thought of utility as being a measure of happiness or satisfaction,In this case,the numbers
attached to various bundles of goods matter,It means something for a bundle to have twice the utility of another.
This is called cardinal utility.
Nowadays most economists think of utility as being ordinal,The numbers themselves only matter insofar as they
rank the difierent bundles,If u(x) > u(y) then x ′ y,Therefore preferences can be represented by many difierent
utility functions,All of the below utility functions are equivalent.
Bundle u1 (￠) u2 (￠) u3 (￠)
x 1 0.01 -10
y 2 98 0
z 3 101 65
Any monotonic transformation of a utility function yields another utility function which represents the same
preferences,A monotonic transformation is simply one which leaves order unchanged | like multiplying by 10.
Utility functions can be constructed from indifierence curves and vice-versa,In fact,indifierence curves are like
cross sections of the utility function,Some examples will clarify this point.
Consumption | Preferences 10
Some Examples
1,Perfect Substitutes,Consider u(x1;x2)= x1 +x2,To draw indifierence curves from such a function set utility
equal to a constant,c,It must be the case that the consumer is indifierent between any combination of goods which
results in the same level of utility,c,x1 +x2 = c deflnes a straight line (x2 = c?x1) with a slope of?1,The
intercept of this line is at c,The below graph can be drawn.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
x2
x1
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......................................................................................................................................
Perfect Substitutes
u =2
u =3
u =4
x2
x1
u =1
u =2
u =3
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
..............................................................................................
......
......
......
......
......
......
......
......
......
......
......
..
...................................................................
......
......
......
......
......
......
.....
........................................
..............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
Perfect Complements
0 0
2,Perfect Complements,The same operation can be done for u(x1;x2)=minfx1;x2g,See above.
3,Quasilinear Preferences,u(x1;x2)= v(x1)+x2 where v(￠) is an increasing function,In this case each
indifierence curve is an identical shape,simply shifted upward.
4,Cobb-Douglas Preferences,u(x1;x2)= x1fix2fl,Another useful way to write this is to take logarithms (a
monotonic transformation) yielding the equivalent v(x1;x2)= filnx1 +fllnx2.
Consumption | Preferences 11
Marginal Utility
How much extra \utility" would a consumer obtain if given a little more of good 1? The answer is found by
calculating the marginal utility,This is the change in utility due to a small change in x1:
MU1 = ￠u￠x
1
= u(x1 +￠x1;x2)?u(x1;x2)￠x
1
… @u@x
1
Notice x2 is flxed in this calculation,To flnd how much extra \utility" is generated when a consumer obtains a little
more of good 1 the change in utility is required,￠u = MU1￠x1.
The same can be done for good 2 of course,This time x1 is flxed.
MU2 = ￠u￠x
2
= u(x1;x2 +￠x2)?u(x1;x2)￠x
2
… @u@x
2
Again it follows that ￠u = MU2￠x2.
Ordinal marginal utility has no behavioural content however,How could it be calculated from observed behaviour?
A choice might reveal a ranking over bundles but never marginal utility,However,marginal utility can be used to
calculate something that does have behavioural content.
Consumption | Preferences 12
Marginal Utility and the MRS
Marginal utility can be used to calculate the marginal rate of substitution.
Recall that the MRS is the slope of the indifierence curve,An indifierence curve,by deflnition,maps the set of
bundles of goods for which utility is equal,Hence,when moving along an indifierence curve by increasing
consumption of good 1 and decreasing consumption of good 2 utility does not change.
From the last slide the change in utility from a small change in the amount of good 1 is MU1￠x1 and from good 2
is MU2￠x2,The total change is therefore:
MU1￠x1 +MU2￠x2 =￠u =0
Rearranging this equation yields an important fact:
MRS= ￠x2￠x
1
=?MU1MU
2
Notice MRS is negative,a consumer is willing to give up one good in order to gain another,Economists often refer
to MRS by its absolute value,MRS can be deduced from actual behaviour,By ofiering a consumer difierent rates of
exchange between the two goods an economist can work out a consumer’s MRS,How?
The above material is more simply understood with the aid of calculus,Difierences become derivatives and the
mathematics becomes trivial,The appendix to Varian (2002),Chapter 4 contains a useful summary.
Goods and Bundles
Consumers make choices over bundles of goods,Consumer theory models the way in which these choices are made.
A good is simply a product | such as apples or bananas,A good may be specifled in terms of time | such as
apples today or apples tomorrow,or place | such as apples in Oxford or apples in London.
A bundle of goods is a collection of goods | such as (3 apples,2 bananas),The bundle can contain as many goods
as necessary,For much of economics,only 2 goods are required,This allows a graphical representation.
Number of Bananas (= x2)
Number of Apples (= x1)
￠x
￠y
￠z
0
1
2
3
4
5
0 1 2 3 4 5
Write a bundle of goods as x =(x1;x2) where x1 is the amount of good 1 and x2 the amount of good 2 in the
bundle,How do consumers choose between bundles such as x and difierent bundles like y =(y1;y2) and z =(z1;z2)?
Consumption | Preferences 2
Preferences
The basic premise of consumer theory is that consumers have preferences over bundles of goods.
Write x ′ y if and only if bundle x is strictly preferred to bundle y.
Write x? y if bundle x is not preferred to bundle y and bundle y is not preferred to bundle x,The consumer is
indifierent between the two bundles.
Write x,y if the consumer strictly prefers bundle x to bundle y or the consumer is indifierent,Equivalently,x is
weakly preferred to y.
Armed with such preferences the consumer can compare difierent bundles of goods.
However,preferences are di–cult things to work with | is there a simple alternative?
Consumption | Preferences 3
Representation
Economists make three assumptions about preferences,These are:
1,Completeness,Either x,y or y,x for all x and y,In other words,all the difierent bundles can be compared.
2,Re exivity,x,x for all x,In other words,all the bundles are at least as good as themselves.
3,Transitivity,If x,y and y,z then x,z for all x,y and z,In other words,if a consumer prefers bundle x to
bundle y and bundle y to bundle z then the consumer also prefers bundle x to bundle z.
At a flrst glance these do not seem unreasonable,Nevertheless they are assumptions and not facts.
Theorem,Any preferences that satisfy the above three assumptions can be represented by a utility function.
The utility function is written as u(￠),It takes each bundle of goods and assigns a number to it,The theorem states
that there is a utility function u(￠) such that x,y if and only if u(x)? u(y),That is,a consumer prefers bundle x
to bundle y if and only if they get bigger \utility" from bundle x,The ordering induced by a consumer’s preferences
can be very simply represented by numbers.
The numbers u(￠) themselves might have no meaning | it is the relative size that matters,Utility functions allow
the construction of indifierence curves.
Consumption | Preferences 4
Indifierence Curves 1
As the theorem says,\utility numbers" can be assigned to each bundle of goods,This can be graphically
represented,Suppose all the bundles with a utility number greater than 5 are found and a line is drawn to separate
them from the bundles with a utility number less than 5,This line is called an indifierence curve.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.......................................................................................................................................0
x2
x1
￠x
￠y
￠z u =5
u > 5
u < 5
......
......
......
......
......
.......
.......
.......
.....
.....
......
......
......
.......
........
.............
.................
............................
In the picture above u(x) > 5 and u(y) < 5 therefore x,y,Also x,z but more information is required to rank y
and z,More indifierence curves need to be drawn,Clearly they can be drawn for any utility number.
With the current assumptions indifierence curves can be of various shapes,However,they cannot cross,Why not?
Consumption | Preferences 5
Indifierence Curves 2
Suppose that two indifierence curves were to cross,as below.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.......................................................................................................................................0
x2
x1
x
y
z
u =5
u =6
.....
.....
......
......
......
......
......
.......
.......
.....
.....
......
......
.......
........
...........
.................
.............................
..
......
......
......
......
.......
.......
.......
......
.....
......
......
.......
.
.........
.................
.............................................
The consumer is indifierent between x and y,and between x and z,z is strictly preferred to y,But by transitivity
x? y and x? z implies z? y,A contradiction.
Even with the current \minimal" assumptions two things can be said,First,indifierence curves can be drawn and
second they do not cross,What do they look like?
Consumption | Preferences 6
Some Examples of Indifierence Curves
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
x2
x1
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......................................................................................................................................
Perfect Substitutes
x2
x1
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
..............................................................................................
......
......
......
......
......
......
......
......
......
......
......
..
...................................................................
......
......
......
......
......
......
.....
........................................
..............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
Perfect Complements
0 0
......
......
......
......
........
...
.........,....
......
......
......
...........
.
.........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
x2
x1.........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......................................................................................................................................
A \Bad" | x2
x2
x1..........
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
..
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
..
..........................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
A \Neutral" | x2
0 0
......
......
......
......
...........,.....
...
.....................................,........
The arrows are in the direction of increasing utility,A great variety of other shapes are possible | economists
restrict the sorts of indifierence curves possible by making further technical assumptions on preferences.
Consumption | Preferences 7
Well-Behaved Preferences
Two further assumptions are made in order to obtain well-behaved preferences.
1,Monotonicity,If x1? y1 and x2 > y2 or if x1 > y1 and x2? y2 then x ′ y,In other words,more is better.
2,Convexity,If x? y then?x+(1)y,x,In other words,averages are better than extremes.
It is easiest to see what these assumptions entail by looking at the diagrams below.
........................................................................................................................................,.............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
x2
x1
Well-Behaved
x2
x1
Non-Convexity
0 0
.....
.....
......
......
......
......
......
.......
.......
.....
.....
......
......
.......
........
...........
.................
..............................
................
.............
............
........
.......
......
......
......
......
.....
.....
......
.......
.......
.......
......
......
......
......
......
......
......
........
...
.........
The second diagram is ruled out by the convexity assumption,It is these two assumptions which give indifierence
curves their nice regular shape that will become so familiar during the course,Monotonicity makes the curves
downward sloping and convexity makes them \bowed".
Consumption | Preferences 8
The Marginal Rate of Substitution
The marginal rate of substitution or MRS is the rate at which the consumer is just willing to give up a small
amount of good 1 in order to gain a small amount of good 2,It is the private rate of exchange.
In the diagrams this is represented by the slope of the indifierence curve.
........................................................................................................................................,.............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
...
x2
x1
The MRS
x2
x1
Diminishing MRS
0 0
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
........................
......
......
.,....
................
................
................
................
...........,
................
................
................
.............
.........................
......
......
..
......
.......................
￠x2
-￠x1
Slope = MRS =￠x2=￠x1..........................................
....
......
......
......
......
......
......
.......
.......
.....
.....
....
.......
.........
..............
..................
.................
......
......
......
......
......
......
.......
.......
......
.....
....
.......
.........
.
.......................
￠x1 and ￠x2 are both taken to be very small numbers | hence the word marginal,More mathematically
￠x2=￠x1 … dx2=dx1,the derivative,The concept of marginal is central to economics and is always associated with
derivatives,or difierence ratios involving ￠ (capital delta).
The second diagram shows how the MRS changes along the indifierence curve,Notice it is decreasing (in absolute
value) as x1 increases,This is called a diminishing marginal rate of substitution,It is a direct consequence of the
convexity assumption,The consumer is willing to give up more of good 1 in exchange for good 2 as the amount of
good 1 increases.
Consumption | Preferences 9
Utility Functions
A utility function assigns a number to each bundle of goods,x.
Early economists thought of utility as being a measure of happiness or satisfaction,In this case,the numbers
attached to various bundles of goods matter,It means something for a bundle to have twice the utility of another.
This is called cardinal utility.
Nowadays most economists think of utility as being ordinal,The numbers themselves only matter insofar as they
rank the difierent bundles,If u(x) > u(y) then x ′ y,Therefore preferences can be represented by many difierent
utility functions,All of the below utility functions are equivalent.
Bundle u1 (￠) u2 (￠) u3 (￠)
x 1 0.01 -10
y 2 98 0
z 3 101 65
Any monotonic transformation of a utility function yields another utility function which represents the same
preferences,A monotonic transformation is simply one which leaves order unchanged | like multiplying by 10.
Utility functions can be constructed from indifierence curves and vice-versa,In fact,indifierence curves are like
cross sections of the utility function,Some examples will clarify this point.
Consumption | Preferences 10
Some Examples
1,Perfect Substitutes,Consider u(x1;x2)= x1 +x2,To draw indifierence curves from such a function set utility
equal to a constant,c,It must be the case that the consumer is indifierent between any combination of goods which
results in the same level of utility,c,x1 +x2 = c deflnes a straight line (x2 = c?x1) with a slope of?1,The
intercept of this line is at c,The below graph can be drawn.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
x2
x1
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
......
......
......
......
......
......
......
......
......................................................................................................................................
Perfect Substitutes
u =2
u =3
u =4
x2
x1
u =1
u =2
u =3
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.....
..............................................................................................
......
......
......
......
......
......
......
......
......
......
......
..
...................................................................
......
......
......
......
......
......
.....
........................................
..............................................................................................................................................
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
Perfect Complements
0 0
2,Perfect Complements,The same operation can be done for u(x1;x2)=minfx1;x2g,See above.
3,Quasilinear Preferences,u(x1;x2)= v(x1)+x2 where v(￠) is an increasing function,In this case each
indifierence curve is an identical shape,simply shifted upward.
4,Cobb-Douglas Preferences,u(x1;x2)= x1fix2fl,Another useful way to write this is to take logarithms (a
monotonic transformation) yielding the equivalent v(x1;x2)= filnx1 +fllnx2.
Consumption | Preferences 11
Marginal Utility
How much extra \utility" would a consumer obtain if given a little more of good 1? The answer is found by
calculating the marginal utility,This is the change in utility due to a small change in x1:
MU1 = ￠u￠x
1
= u(x1 +￠x1;x2)?u(x1;x2)￠x
1
… @u@x
1
Notice x2 is flxed in this calculation,To flnd how much extra \utility" is generated when a consumer obtains a little
more of good 1 the change in utility is required,￠u = MU1￠x1.
The same can be done for good 2 of course,This time x1 is flxed.
MU2 = ￠u￠x
2
= u(x1;x2 +￠x2)?u(x1;x2)￠x
2
… @u@x
2
Again it follows that ￠u = MU2￠x2.
Ordinal marginal utility has no behavioural content however,How could it be calculated from observed behaviour?
A choice might reveal a ranking over bundles but never marginal utility,However,marginal utility can be used to
calculate something that does have behavioural content.
Consumption | Preferences 12
Marginal Utility and the MRS
Marginal utility can be used to calculate the marginal rate of substitution.
Recall that the MRS is the slope of the indifierence curve,An indifierence curve,by deflnition,maps the set of
bundles of goods for which utility is equal,Hence,when moving along an indifierence curve by increasing
consumption of good 1 and decreasing consumption of good 2 utility does not change.
From the last slide the change in utility from a small change in the amount of good 1 is MU1￠x1 and from good 2
is MU2￠x2,The total change is therefore:
MU1￠x1 +MU2￠x2 =￠u =0
Rearranging this equation yields an important fact:
MRS= ￠x2￠x
1
=?MU1MU
2
Notice MRS is negative,a consumer is willing to give up one good in order to gain another,Economists often refer
to MRS by its absolute value,MRS can be deduced from actual behaviour,By ofiering a consumer difierent rates of
exchange between the two goods an economist can work out a consumer’s MRS,How?
The above material is more simply understood with the aid of calculus,Difierences become derivatives and the
mathematics becomes trivial,The appendix to Varian (2002),Chapter 4 contains a useful summary.
