Chapter Fourteen
Consumer’s Surplus
消费者剩余
Structure
Money equivalent of utility gains to
trade
Consumer’s surplus
Changes in consumer’s surplus
Compensating and equivalent
variations
Producer’s surplus
Monetary Measures of Gains-to-
Trade
You can buy as much gasoline as
you wish at $1 per gallon once you
enter the gasoline market,
Q,What is the most you would pay to
enter the market?
A,You would pay up to the dollar
value of the gains-to-trade you would
enjoy once in the market.
How can such gains-to-trade be
measured?
Three such measures are:
Consumer’s Surplus
Equivalent Variation （ 等价变换）,and
Compensating Variation （ 补偿变换）,
Only in one special circumstance do these
three measures coincide.
Monetary Measures of Gains-to-
Trade
Use r1 to denote the most a consumer
would pay for a 1st gallon -- reservation
price （ 保留价格） for the 1st gallon.
r1 is the dollar equivalent of the marginal
utility of the 1st gallon.
use r2 to denote the most she would pay
for a 2nd gallon -- this is her reservation
price for the 2nd gallon.
r2 is the dollar equivalent of the marginal
utility of the 2nd gallon.
$ Equivalent Utility Gains
Generally,if she already has n-1
gallons of gasoline then rn denotes
the most she will pay for an nth
gallon.
rn is the dollar equivalent of the
marginal utility of the nth gallon.
$ Equivalent Utility Gains
r1 + … + r n will be the dollar
equivalent of the total change to
utility from consuming n gallons of
gasoline at a price of $0.
So r1 + … + r n - pGn will be the
dollar equivalent of the total change
to utility from consuming n gallons
of gasoline at a price of $pG each.
$ Equivalent Utility Gains
$ Equivalent Utility Gains
R es er v at io n P r ic e C u r v e f o r G as o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
What is the monetary value of our
consumer’s gain-to-trading in the
gasoline market at a price of $pG?
$ Equivalent Utility Gains
The dollar equivalent net utility gain for
the 1st gallon is $(r1 - pG)
and is $(r2 - pG) for the 2nd gallon,
and so on,so the dollar value of the
gain-to-trade is
$(r1 - pG) + $(r2 - pG) + …
for as long as rn - pG > 0.
$ Equivalent Utility Gains
$ Equivalent Utility Gains
R es er v at io n P r ic e C u r v e f o r G as o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ Equivalent Utility Gains
R es er v at io n P r ic e C u r v e f o r G as o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ Equivalent Utility Gains
Res e r v a t io n P r ic e Cu r v e f o r G a s o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ value of net utility gains-to-trade
If gasoline can be purchased in any
quantity then,..
$ Equivalent Utility Gains
$ Equivalent Utility Gains
Gasoline
($) Res.
Prices
Reservation Price Curve for Gasoline
$ Equivalent Utility Gains
Gasoline
($) Res.
Prices
pG
Reservation Price Curve for Gasoline
$ Equivalent Utility Gains
Gasoline
($) Res.
Prices
pG
Reservation Price Curve for Gasoline
$ value of net utility gains-to-trade
Quasi-Linear Utility Example
U(x,y) = v(x) + y
Reservation prices for the first 3 units:
r1= v(1) - v(0)
r2= v(2) - v(1)
r3= v(3) - v(2)
r1+r2+r3=v(3)-v(0)=v(3)
This is the gross benefit （ 毛收益） of
consuming 3 units of good x.
Quasi-Linear Utility
G r o s s b e n e f it = v ( 3 )
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
Quasi-Linear Utility
When x=3,y=m-3p,
U(3,m-3p)= v(3) + m-3p
If consume 0 good x,then y=m
U(0,m) = m
The net benefit of consuming x (gains-to-
trade)
U(3,m-3p) – U(0,m) = v(3) - 3p
If n units of x then utility gain is
v(n) – pn
This is the net benefit （ 净收益）,
Quasi-Linear Utility
Net b e n e f it = v ( 3 ) - 3 p
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
p
Unfortunately,estimating a
consumer’s reservation-price curve
is difficult,
so,as an approximation,we use the
consumer’s ordinary demand curve.
This approximation gives the
Consumer’s Surplus measure of net
utility gain.
$ Equivalent Utility Gains
A consumer’s reservation-price
curve is not quite the same as her
ordinary demand curve,Why not?
A reservation-price curve describes
sequentially the values of successive
single units of a commodity.
An ordinary demand curve describes
the most that would be paid for q
units of a commodity purchased
simultaneously.
Consumer’s Surplus
But,if the consumer’s utility function
is quasilinear in income then
Consumer’s Surplus is an exact $
measure of gains-to-trade,
If income effects are small,then the
approximation is good.
Consumer’s Surplus
Consumer’s Surplus
U x x v x x(,) ( )1 2 1 2
The consumer’s utility function is
quasilinear in x2.
Take p2 = 1,Then the consumer’s
choice problem is to maximize
U x x v x x(,) ( )1 2 1 2
subject to
p x x m1 1 2,
Consumer’s Surplus
U x x v x x(,) ( )1 2 1 2
The consumer’s utility function is
quasilinear in x2.
Take p2 = 1,Then the consumer’s
choice problem is to maximize
U x x v x x(,) ( )1 2 1 2
subject to
p x x m1 1 2,
Consumer’s Surplus
That is,choose x1 to maximize
v x m p x( ),1 1 1
The first-order condition is
v x p' ( )1 1 0
That is,p v x1 1? ' ( ).
This is the equation of the consumer’s
ordinary demand for commodity 1.
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
CS
p1'
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
CS
CS v x dx p xx ' ( ) ' '' 1 1 1 10 1
p1'
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
CS
CS v x dx p xx ' ( ) ' '' 1 1 1 10 1
v x v p x( ) ( )' ' '1 1 10
p1'
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
p1'
CS is exactly the consumer’s utilitygain from consuming x1’
units of commodity 1.
CS v x dx p xx ' ( ) ' '' 1 1 1 10 1
v x v p x( ) ( )' ' '1 1 10
Interpreting Consumer’s Surplus
Net benefit of consuming n units of
the good:
Utility minus expenditure
Compensation needed to give up
consuming the product.
The change to a consumer’s total
utility due to a change to p1 is
approximately the change in her
Consumer’s Surplus,
Change in Consumer’s Surplus
Consumer’s Surplus
p1
x1*x1'
p1'
p1(x1),the inverse ordinary demand
curve for commodity 1
Consumer’s Surplus
p1
x1*x1'
CS before
p1(x1)
p1'
Consumer’s Surplus
p1
x1*x1'
CS afterp1"
x1"
p1(x1)
p1'
Consumer’s Surplus
p1
x1*x1'x1"
Lost CS
p1(x1),inverse ordinary demand
curve for commodity 1.
p1"
p1'
Consumer’s Surplus
p1
x1*
x1'
x1"
Lost
CS
"
1
'
1
p
p 11
*
1 dp)p(xCS
x1*(p1),the consumer’s ordinary
demand curve for commodity 1.
p1"p1'
measures the loss in
Consumer’s Surplus.
Two additional dollar measures of
the total utility change caused by a
price change are Compensating
Variation and Equivalent Variation.
Compensating Variation and
Equivalent Variation
p1 rises.
Q,What is the least extra income
that,at the new prices,just restores
the consumer’s original utility level?
Compensating Variation
p1 rises.
Q,What is the least extra income
that,at the new prices,just restores
the consumer’s original utility level?
A,The Compensating Variation.
Compensating Variation
Compensating Variation
x2
x1x1'
u1
x2'
p1=p1’ p2 is fixed.
m p x p x1 1 1 2 2' ' '
Compensating Variation
x2
x1x1'
x2'
x1"
x2"
u1
u2
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
Compensating Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " " '"
22'"1"12 xpxpm
Compensating Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " " '"
22'"1"12 xpxpm
CV = m2 - m1.
p1 rises.
Q,What is the least extra income
that,at the original prices,just
restores the consumer’s original
utility level?
A,The Equivalent Variation.
Equivalent Variation
Equivalent Variation
x2
x1x1'
u1
x2'
p1=p1’ p2 is fixed.
m p x p x1 1 1 2 2' ' '
Equivalent Variation
x2
x1x1'
x2'
x1"
x2"
u1
u2
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
Equivalent Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
m p x p x2 1 1 2 2' '" '"
Equivalent Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
m p x p x2 1 1 2 2' '" '"
EV = m1 - m2.
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
So when the consumer has quasilinear
utility,
CV = EV =?CS.
But,otherwise,we have:
Relationship 2,In size,EV <?CS < CV.
Changes in a firm’s welfare can be
measured in dollars much as for a
consumer.
Producer’s Surplus
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Revenue
= py' '
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Variable Cost of producing
y’ units is the sum of the
marginal costs
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Variable Cost of producing
y’ units is the sum of the
marginal costs
Revenue less VC
is the Producer’s
Surplus.
Consumer’s Surplus
消费者剩余
Structure
Money equivalent of utility gains to
trade
Consumer’s surplus
Changes in consumer’s surplus
Compensating and equivalent
variations
Producer’s surplus
Monetary Measures of Gains-to-
Trade
You can buy as much gasoline as
you wish at $1 per gallon once you
enter the gasoline market,
Q,What is the most you would pay to
enter the market?
A,You would pay up to the dollar
value of the gains-to-trade you would
enjoy once in the market.
How can such gains-to-trade be
measured?
Three such measures are:
Consumer’s Surplus
Equivalent Variation （ 等价变换）,and
Compensating Variation （ 补偿变换）,
Only in one special circumstance do these
three measures coincide.
Monetary Measures of Gains-to-
Trade
Use r1 to denote the most a consumer
would pay for a 1st gallon -- reservation
price （ 保留价格） for the 1st gallon.
r1 is the dollar equivalent of the marginal
utility of the 1st gallon.
use r2 to denote the most she would pay
for a 2nd gallon -- this is her reservation
price for the 2nd gallon.
r2 is the dollar equivalent of the marginal
utility of the 2nd gallon.
$ Equivalent Utility Gains
Generally,if she already has n-1
gallons of gasoline then rn denotes
the most she will pay for an nth
gallon.
rn is the dollar equivalent of the
marginal utility of the nth gallon.
$ Equivalent Utility Gains
r1 + … + r n will be the dollar
equivalent of the total change to
utility from consuming n gallons of
gasoline at a price of $0.
So r1 + … + r n - pGn will be the
dollar equivalent of the total change
to utility from consuming n gallons
of gasoline at a price of $pG each.
$ Equivalent Utility Gains
$ Equivalent Utility Gains
R es er v at io n P r ic e C u r v e f o r G as o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
What is the monetary value of our
consumer’s gain-to-trading in the
gasoline market at a price of $pG?
$ Equivalent Utility Gains
The dollar equivalent net utility gain for
the 1st gallon is $(r1 - pG)
and is $(r2 - pG) for the 2nd gallon,
and so on,so the dollar value of the
gain-to-trade is
$(r1 - pG) + $(r2 - pG) + …
for as long as rn - pG > 0.
$ Equivalent Utility Gains
$ Equivalent Utility Gains
R es er v at io n P r ic e C u r v e f o r G as o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ Equivalent Utility Gains
R es er v at io n P r ic e C u r v e f o r G as o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ Equivalent Utility Gains
Res e r v a t io n P r ic e Cu r v e f o r G a s o li n e
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ value of net utility gains-to-trade
If gasoline can be purchased in any
quantity then,..
$ Equivalent Utility Gains
$ Equivalent Utility Gains
Gasoline
($) Res.
Prices
Reservation Price Curve for Gasoline
$ Equivalent Utility Gains
Gasoline
($) Res.
Prices
pG
Reservation Price Curve for Gasoline
$ Equivalent Utility Gains
Gasoline
($) Res.
Prices
pG
Reservation Price Curve for Gasoline
$ value of net utility gains-to-trade
Quasi-Linear Utility Example
U(x,y) = v(x) + y
Reservation prices for the first 3 units:
r1= v(1) - v(0)
r2= v(2) - v(1)
r3= v(3) - v(2)
r1+r2+r3=v(3)-v(0)=v(3)
This is the gross benefit （ 毛收益） of
consuming 3 units of good x.
Quasi-Linear Utility
G r o s s b e n e f it = v ( 3 )
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
Quasi-Linear Utility
When x=3,y=m-3p,
U(3,m-3p)= v(3) + m-3p
If consume 0 good x,then y=m
U(0,m) = m
The net benefit of consuming x (gains-to-
trade)
U(3,m-3p) – U(0,m) = v(3) - 3p
If n units of x then utility gain is
v(n) – pn
This is the net benefit （ 净收益）,
Quasi-Linear Utility
Net b e n e f it = v ( 3 ) - 3 p
0
2
4
6
8
10
G a s o l i n e (g a l l o n s )
($ ) R e s,
Va l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
p
Unfortunately,estimating a
consumer’s reservation-price curve
is difficult,
so,as an approximation,we use the
consumer’s ordinary demand curve.
This approximation gives the
Consumer’s Surplus measure of net
utility gain.
$ Equivalent Utility Gains
A consumer’s reservation-price
curve is not quite the same as her
ordinary demand curve,Why not?
A reservation-price curve describes
sequentially the values of successive
single units of a commodity.
An ordinary demand curve describes
the most that would be paid for q
units of a commodity purchased
simultaneously.
Consumer’s Surplus
But,if the consumer’s utility function
is quasilinear in income then
Consumer’s Surplus is an exact $
measure of gains-to-trade,
If income effects are small,then the
approximation is good.
Consumer’s Surplus
Consumer’s Surplus
U x x v x x(,) ( )1 2 1 2
The consumer’s utility function is
quasilinear in x2.
Take p2 = 1,Then the consumer’s
choice problem is to maximize
U x x v x x(,) ( )1 2 1 2
subject to
p x x m1 1 2,
Consumer’s Surplus
U x x v x x(,) ( )1 2 1 2
The consumer’s utility function is
quasilinear in x2.
Take p2 = 1,Then the consumer’s
choice problem is to maximize
U x x v x x(,) ( )1 2 1 2
subject to
p x x m1 1 2,
Consumer’s Surplus
That is,choose x1 to maximize
v x m p x( ),1 1 1
The first-order condition is
v x p' ( )1 1 0
That is,p v x1 1? ' ( ).
This is the equation of the consumer’s
ordinary demand for commodity 1.
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
CS
p1'
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
CS
CS v x dx p xx ' ( ) ' '' 1 1 1 10 1
p1'
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
CS
CS v x dx p xx ' ( ) ' '' 1 1 1 10 1
v x v p x( ) ( )' ' '1 1 10
p1'
Consumer’s Surplus
Ordinary demand curve,p1 p v x1 1? ' ( )
x1*x1'
p1'
CS is exactly the consumer’s utilitygain from consuming x1’
units of commodity 1.
CS v x dx p xx ' ( ) ' '' 1 1 1 10 1
v x v p x( ) ( )' ' '1 1 10
Interpreting Consumer’s Surplus
Net benefit of consuming n units of
the good:
Utility minus expenditure
Compensation needed to give up
consuming the product.
The change to a consumer’s total
utility due to a change to p1 is
approximately the change in her
Consumer’s Surplus,
Change in Consumer’s Surplus
Consumer’s Surplus
p1
x1*x1'
p1'
p1(x1),the inverse ordinary demand
curve for commodity 1
Consumer’s Surplus
p1
x1*x1'
CS before
p1(x1)
p1'
Consumer’s Surplus
p1
x1*x1'
CS afterp1"
x1"
p1(x1)
p1'
Consumer’s Surplus
p1
x1*x1'x1"
Lost CS
p1(x1),inverse ordinary demand
curve for commodity 1.
p1"
p1'
Consumer’s Surplus
p1
x1*
x1'
x1"
Lost
CS
"
1
'
1
p
p 11
*
1 dp)p(xCS
x1*(p1),the consumer’s ordinary
demand curve for commodity 1.
p1"p1'
measures the loss in
Consumer’s Surplus.
Two additional dollar measures of
the total utility change caused by a
price change are Compensating
Variation and Equivalent Variation.
Compensating Variation and
Equivalent Variation
p1 rises.
Q,What is the least extra income
that,at the new prices,just restores
the consumer’s original utility level?
Compensating Variation
p1 rises.
Q,What is the least extra income
that,at the new prices,just restores
the consumer’s original utility level?
A,The Compensating Variation.
Compensating Variation
Compensating Variation
x2
x1x1'
u1
x2'
p1=p1’ p2 is fixed.
m p x p x1 1 1 2 2' ' '
Compensating Variation
x2
x1x1'
x2'
x1"
x2"
u1
u2
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
Compensating Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " " '"
22'"1"12 xpxpm
Compensating Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " " '"
22'"1"12 xpxpm
CV = m2 - m1.
p1 rises.
Q,What is the least extra income
that,at the original prices,just
restores the consumer’s original
utility level?
A,The Equivalent Variation.
Equivalent Variation
Equivalent Variation
x2
x1x1'
u1
x2'
p1=p1’ p2 is fixed.
m p x p x1 1 1 2 2' ' '
Equivalent Variation
x2
x1x1'
x2'
x1"
x2"
u1
u2
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
Equivalent Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
m p x p x2 1 1 2 2' '" '"
Equivalent Variation
x2
x1x1'
u1
u2
x1"
x2"
x2'
x2'"
x1'"
p1=p1’
p1=p1”
p2 is fixed.
m p x p x1 1 1 2 2' ' '
p x p x1 1 2 2" " "
m p x p x2 1 1 2 2' '" '"
EV = m1 - m2.
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
So when the consumer has quasilinear
utility,
CV = EV =?CS.
But,otherwise,we have:
Relationship 2,In size,EV <?CS < CV.
Changes in a firm’s welfare can be
measured in dollars much as for a
consumer.
Producer’s Surplus
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Revenue
= py' '
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Variable Cost of producing
y’ units is the sum of the
marginal costs
Producer’s Surplus
y (output units)
Output price (p)
Marginal Cost
p'
y'
Variable Cost of producing
y’ units is the sum of the
marginal costs
Revenue less VC
is the Producer’s
Surplus.
