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
Reservation Price
?Maximum willingness to pay for an
additional unit of a good
?Two goods,good 1 (x1) and
expenditure on others (x2); p2=1
?By budget constraint (p1x1+x2=m):
If x1=0,then x2=m
If x1=1,then x2=m-p1
If x1=2,then x2=m-2p1
If x1=3,then x2=m-3p1
Reservation Price
?Reservation price for the 1st unit (r1)
u(0,m) = u(1,m-r1)
? r1 is the dollar equivalent of the
marginal utility of the 1st unit.
?Reservation price for the 2nd unit (r2)
u(1,m-r2) = u(2,m-2r2)
? r2 is the dollar equivalent of the
marginal utility of the 2nd unit.
?Reservation price for the 3rd unit (r3)
u(2,m-2r3) = u(2,m-3r3)
?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.
$ 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
?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
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
? U(x,y) = v(x) + y
? Reservation prices for the first 3 units:
v(0)+m=v(1)+m- r1 ? r1= v(1) - v(0)
v(1)+m- r2=v(2)+m- 2r2 ? r2= v(2) - v(1)
v(2)+m- 2r3=v(3)+m- 3r3 ? 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
? The reservation price curve is
exactly the demand curve
? 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'
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:
?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
?A,The 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?
Equivalent Variation
?A,The 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
When the consumer has quasilinear
utility,
CV = EV = ?CS.
But,otherwise,we have:
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
Reservation Price
?Maximum willingness to pay for an
additional unit of a good
?Two goods,good 1 (x1) and
expenditure on others (x2); p2=1
?By budget constraint (p1x1+x2=m):
If x1=0,then x2=m
If x1=1,then x2=m-p1
If x1=2,then x2=m-2p1
If x1=3,then x2=m-3p1
Reservation Price
?Reservation price for the 1st unit (r1)
u(0,m) = u(1,m-r1)
? r1 is the dollar equivalent of the
marginal utility of the 1st unit.
?Reservation price for the 2nd unit (r2)
u(1,m-r2) = u(2,m-2r2)
? r2 is the dollar equivalent of the
marginal utility of the 2nd unit.
?Reservation price for the 3rd unit (r3)
u(2,m-2r3) = u(2,m-3r3)
?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.
$ 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
?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
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
? U(x,y) = v(x) + y
? Reservation prices for the first 3 units:
v(0)+m=v(1)+m- r1 ? r1= v(1) - v(0)
v(1)+m- r2=v(2)+m- 2r2 ? r2= v(2) - v(1)
v(2)+m- 2r3=v(3)+m- 3r3 ? 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
? The reservation price curve is
exactly the demand curve
? 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'
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:
?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
?A,The 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?
Equivalent Variation
?A,The 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
When the consumer has quasilinear
utility,
CV = EV = ?CS.
But,otherwise,we have:
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.
