Chapter Fourteen
Consumer’s Surplus
Monetary Measures of Gains-to-
Trade
?You can buy as much rice as you
wish at RMB1 per kilogram 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?
Monetary Measures of Gains-to-
Trade
?Three such measures are:
?Consumer’s Surplus
?Equivalent Variation,and
?Compensating Variation.
?Only in one special circumstance do
these three measures coincide.
Monetary Measures of Gains-to-
Trade
?Suppose rice can be bought only in
lumps of one kilogram.
?Use r1 to denote the most a single
consumer would pay for a 1st
kilogram -- call this her reservation
price for the 1st kilogram.
?r1 is the dollar equivalent of the
marginal utility of the 1st kilogram.
$ Equivalent Utility Gains
?Now that she has one kilogram,use
r2 to denote the most she would pay
for a 2nd kilogram -- this is her
reservation price for the 2nd
kilogram.
?r2 is the dollar equivalent of the
marginal utility of the 2nd gallon.
$ Equivalent Utility Gains
?Generally,if she already has n-1
kilograms of rice then rn denotes the
most she will pay for an nth
kilogram.
?rn is the dollar equivalent of the
marginal utility of the nth kilogram.
$ Equivalent Utility Gains
?r1 + … + r n will therefore be the dollar
equivalent of the total change to
utility from acquiring n kilograms of
rice at a price of $0.
?So r1 + … + r n - pGn will be the
dollar equivalent of the total change
to utility from acquiring n kilograms
of rice at a price of $pG each.
$ Equivalent Utility Gains
?A plot of r1,r2,…,r n,… against n is a
reservation-price curve,This is not
quite the same as the consumer’s
demand curve for rice.
$ Equivalent Utility Gains
$ Equivalent Utility Gains
R e s e r v a t i o n P r i c e C u r v e f o r R i c e
0
2
4
6
8
10
R i c e (k i l o g r a m 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
rice market at a price of $pG?
$ Equivalent Utility Gains
?The dollar equivalent net utility gain for
the 1st kilogram is $(r1 - pG)
?and is $(r2 - pG) for the 2nd kilogram,
?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
Re s e r v a t i o n P r i c e Cu r v e f o r G a s o l i 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,
V a l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ Equivalent Utility Gains
Re s e r v a t i o n P r i c e Cu r v e f o r G a s o l i 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,
V a l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ value of net utility gains-to-trade
Suppose rice can be purchased in any
continuous quantity then,..
$ Equivalent Utility Gains
$ Equivalent Utility Gains
Rice
Res.
Prices
pG
Reservation Price Curve for Rice
$ value of net utility gains-to-trade
?Unfortunately,estimating a
consumer’s reservation-price curve
is difficult,
?so,as an approximation,the
reservation-price curve is replaced
with the consumer’s ordinary
demand curve.
$ 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
?Approximating the net utility gain
area under the reservation-price
curve by the corresponding area
under the ordinary demand curve
gives the Consumer’s Surplus
measure of net utility gain.
Consumer’s Surplus
Consumer’s Surplus
Gasoline
($) Reservation price curve for gasoline
Ordinary demand curve for gasoline
Consumer’s Surplus
rice
Reservation price curve for rice
Ordinary demand curve for rice
pG
?The difference between the
consumer’s reservation-price and
ordinary demand curves is due to
income effects.
?But,if the consumer’s utility function
is quasilinear in income then there
are no income effects and
Consumer’s Surplus is an exact
measure of gains-to-trade,
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
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
?Consumer’s Surplus is an exact
dollar measure of utility gained from
consuming commodity 1 when the
consumer’s utility function is
quasilinear in commodity 2,
?Otherwise Consumer’s Surplus is an
approximation.
Consumer’s Surplus
?The change to a consumer’s total
utility due to a change to p1 is
approximately the change in her
Consumer’s Surplus,
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?
?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? ?' '" '"
EV = m1 - m2.
?Relationship 1,When the
consumer’s preferences are
quasilinear,all three measures are
the same.
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
?Consider first the change in
Consumer’s Surplus when p1 rises
from p1’ to p1”.
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
U x x v x x(,) ( )1 2 1 2? ?If then
CS p v x v p x( ) ( ) ( )' ' ' '1 1 1 10? ? ?
and so the change in CS when p1 rises
from p1’ to p1” is
? CS CS p CS p? ?( ) ( )' "1 1
? ?? ? ? ? ? ?v x v p x v x v p x( ) ( ) ( ) ( )' ' ' " " "1 1 1 1 1 10 0
? ? ? ?v x v x p x p x( ) ( ) ( ).' " ' ' " "1 1 1 1 1 1
?Now consider the change in CV when
p1 rises from p1’ to p1”.
?The consumer’s utility for given p1 is
and CV is the extra income which,at
the new prices,makes the
consumer’s utility the same as at the
old prices,That is,...
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
v x p m p x p( ( )) ( )* *1 1 1 1 1? ?
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
v x m p x( )' ' '1 1 1? ?
? ? ? ?v x m CV p x( )," " "1 1 1
So
CV v x v x p x p x? ? ? ?( ) ( ) ( )' " ' ' " "1 1 1 1 1 1
? ? CS,
?Now consider the change in EV when
p1 rises from p1’ to p1”.
?The consumer’s utility for given p1 is
and EV is the extra income which,at
the old prices,makes the consumer’s
utility the same as at the new prices,
That is,...
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
v x p m p x p( ( )) ( )* *1 1 1 1 1? ?
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
That is,
EV v x v x p x p x? ? ? ?( ) ( ) ( )' " ' ' " "1 1 1 1 1 1
? ? CS,
v x m p x( )' ' '1 1 1? ?
? ? ? ?v x m EV p x( )," " "1 1 1
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
Monetary Measures of Gains-to-
Trade
?You can buy as much rice as you
wish at RMB1 per kilogram 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?
Monetary Measures of Gains-to-
Trade
?Three such measures are:
?Consumer’s Surplus
?Equivalent Variation,and
?Compensating Variation.
?Only in one special circumstance do
these three measures coincide.
Monetary Measures of Gains-to-
Trade
?Suppose rice can be bought only in
lumps of one kilogram.
?Use r1 to denote the most a single
consumer would pay for a 1st
kilogram -- call this her reservation
price for the 1st kilogram.
?r1 is the dollar equivalent of the
marginal utility of the 1st kilogram.
$ Equivalent Utility Gains
?Now that she has one kilogram,use
r2 to denote the most she would pay
for a 2nd kilogram -- this is her
reservation price for the 2nd
kilogram.
?r2 is the dollar equivalent of the
marginal utility of the 2nd gallon.
$ Equivalent Utility Gains
?Generally,if she already has n-1
kilograms of rice then rn denotes the
most she will pay for an nth
kilogram.
?rn is the dollar equivalent of the
marginal utility of the nth kilogram.
$ Equivalent Utility Gains
?r1 + … + r n will therefore be the dollar
equivalent of the total change to
utility from acquiring n kilograms of
rice at a price of $0.
?So r1 + … + r n - pGn will be the
dollar equivalent of the total change
to utility from acquiring n kilograms
of rice at a price of $pG each.
$ Equivalent Utility Gains
?A plot of r1,r2,…,r n,… against n is a
reservation-price curve,This is not
quite the same as the consumer’s
demand curve for rice.
$ Equivalent Utility Gains
$ Equivalent Utility Gains
R e s e r v a t i o n P r i c e C u r v e f o r R i c e
0
2
4
6
8
10
R i c e (k i l o g r a m 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
rice market at a price of $pG?
$ Equivalent Utility Gains
?The dollar equivalent net utility gain for
the 1st kilogram is $(r1 - pG)
?and is $(r2 - pG) for the 2nd kilogram,
?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
Re s e r v a t i o n P r i c e Cu r v e f o r G a s o l i 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,
V a l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ Equivalent Utility Gains
Re s e r v a t i o n P r i c e Cu r v e f o r G a s o l i 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,
V a l u e s
1 2 3 4 5 6
r1
r2
r3
r4
r5
r6
pG
$ value of net utility gains-to-trade
Suppose rice can be purchased in any
continuous quantity then,..
$ Equivalent Utility Gains
$ Equivalent Utility Gains
Rice
Res.
Prices
pG
Reservation Price Curve for Rice
$ value of net utility gains-to-trade
?Unfortunately,estimating a
consumer’s reservation-price curve
is difficult,
?so,as an approximation,the
reservation-price curve is replaced
with the consumer’s ordinary
demand curve.
$ 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
?Approximating the net utility gain
area under the reservation-price
curve by the corresponding area
under the ordinary demand curve
gives the Consumer’s Surplus
measure of net utility gain.
Consumer’s Surplus
Consumer’s Surplus
Gasoline
($) Reservation price curve for gasoline
Ordinary demand curve for gasoline
Consumer’s Surplus
rice
Reservation price curve for rice
Ordinary demand curve for rice
pG
?The difference between the
consumer’s reservation-price and
ordinary demand curves is due to
income effects.
?But,if the consumer’s utility function
is quasilinear in income then there
are no income effects and
Consumer’s Surplus is an exact
measure of gains-to-trade,
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
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
?Consumer’s Surplus is an exact
dollar measure of utility gained from
consuming commodity 1 when the
consumer’s utility function is
quasilinear in commodity 2,
?Otherwise Consumer’s Surplus is an
approximation.
Consumer’s Surplus
?The change to a consumer’s total
utility due to a change to p1 is
approximately the change in her
Consumer’s Surplus,
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?
?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? ?' '" '"
EV = m1 - m2.
?Relationship 1,When the
consumer’s preferences are
quasilinear,all three measures are
the same.
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
?Consider first the change in
Consumer’s Surplus when p1 rises
from p1’ to p1”.
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
U x x v x x(,) ( )1 2 1 2? ?If then
CS p v x v p x( ) ( ) ( )' ' ' '1 1 1 10? ? ?
and so the change in CS when p1 rises
from p1’ to p1” is
? CS CS p CS p? ?( ) ( )' "1 1
? ?? ? ? ? ? ?v x v p x v x v p x( ) ( ) ( ) ( )' ' ' " " "1 1 1 1 1 10 0
? ? ? ?v x v x p x p x( ) ( ) ( ).' " ' ' " "1 1 1 1 1 1
?Now consider the change in CV when
p1 rises from p1’ to p1”.
?The consumer’s utility for given p1 is
and CV is the extra income which,at
the new prices,makes the
consumer’s utility the same as at the
old prices,That is,...
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
v x p m p x p( ( )) ( )* *1 1 1 1 1? ?
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
v x m p x( )' ' '1 1 1? ?
? ? ? ?v x m CV p x( )," " "1 1 1
So
CV v x v x p x p x? ? ? ?( ) ( ) ( )' " ' ' " "1 1 1 1 1 1
? ? CS,
?Now consider the change in EV when
p1 rises from p1’ to p1”.
?The consumer’s utility for given p1 is
and EV is the extra income which,at
the old prices,makes the consumer’s
utility the same as at the new prices,
That is,...
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
v x p m p x p( ( )) ( )* *1 1 1 1 1? ?
Consumer’s Surplus,Compensating
Variation and Equivalent Variation
That is,
EV v x v x p x p x? ? ? ?( ) ( ) ( )' " ' ' " "1 1 1 1 1 1
? ? CS,
v x m p x( )' ' '1 1 1? ?
? ? ? ?v x m EV p x( )," " "1 1 1
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.
