Chapter Fourteen

Consumer’s Surplus消费者剩余

In previous chapters From underlying preference or utility

function to consumer’s demand function

In practice, From observed demand behavior to

estimate preferences or utility .

Monetary Measures of Gains-to-Trade

You can buy as much gasoline as you wish 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.

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( 等价效用所得 )

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( 等价效用所得 )

r1 + … + rn will be the dollar equivalent of the total change to utility from consuming n gallons of gasoline at a price of $0.

So r1 + … + rn - 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( 等价效用所得 )

What is the monetary value of our consumer’s gain-to-trading in the gasoline market at a price of $pG?

Equivalent Utility Gains( 等价效用所得 )

Equivalent Utility Gains( 等价效用所得 )

Reservation Price Curve for Gasoline

0

2

4

6

8

10

Gasoline (gallons)

($) Res.Values

1 2 3 4 5 6

r1

r2

r3

r4

r5

r6

pG

$ value of net utility gains-to-tradeConsumer’s surplus （消费者剩余） or

Net consumer’s surplus （净消费者剩余）

Gross consumer’s surplus （总消费者剩余） or gross benefit （总效益） Yellow area

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( 等价效用所得 )

If gasoline can be purchased in any quantity then ...

Equivalent Utility Gains( 等价效用所得 )

Equivalent Utility Gains( 等价效用所得 )

Gasoline

($) Res.Prices

pG

Reservation Price Curve for Gasoline

$ value of net utility gains-to-trade

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( 等价效用所得 )

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

The difference of ordinary demand curve and reservation price curve is induced by income effect.

If income effects are small, then the approximation is good.

But, if the consumer’s utility function is quasilinear, income effect is zero, two curves are exactly the same, why??

Consumer’s Surplus

Consumer’s Surplus

U x x v x x( , ) ( )1 2 1 2

The consumer’s utility function isquasilinear in x2.

Take p2 = 1. Then the consumer’schoice 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 isquasilinear in x2.

Take p2 = 1. Then the consumer’schoice problem is to maximize

U x x v x x( , ) ( )1 2 1 2 subject to

p x x m1 1 2 .

Consumer’s SurplusThat 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’sordinary demand for commodity 1.

Consumer’s SurplusOrdinary demand curve,p1

p v x1 1 '( )

x1*x1

'

CS

p1'

Consumer’s SurplusOrdinary demand curve,p1

p v x1 1 '( )

x1*x1

'

CS

CS v x dx p xx '( ) ' ''

1 1 1 101

p1'

Consumer’s SurplusOrdinary demand curve,p1

p v x1 1 '( )

x1*x1

'

CS

CS v x dx p xx '( ) ' ''

1 1 1 101

v x v p x( ) ( )' ' '1 1 10

p1'

Consumer’s SurplusOrdinary demand curve,p1

p v x1 1 '( )

x1*x1

'

p1'

CSis exactly the consumer’s utility gain from consuming x1’ units of commodity 1.

CS v x dx p xx '( ) ' ''

1 1 1 101

v x v p x( ) ( )' ' '1 1 10

Price changes??

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 Surplusp1

x1*x1

'x1"

Lost CS

p1"

p1'

"1'1

pp 11

*1 dp)p(xCS

measures the loss in Consumer’s Surplus.

Two additional dollar measures of the total utility change caused by a price change are Compensating Variation( 补偿变化 ) and Equivalent Variation （等价变化） .

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.

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.

使消费者回到初始无差异曲线上所必需的收入变化称作收入的补偿变化

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.

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.

EV

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.

使消费者回到初始无差异曲线上所必需的收入变化称作收入的补偿变化

CV

Compensating VariationVS Equivalent Variation

Both CV and EV measure “how far apart” two indifference curves are.

Depend on the slope of the tangent line CV does not equal EV in most cases. Only in quasilinear utility. Why?

The distance of two indifference curves is the same no matter where it is measured.

CV = EV = CS消费者剩余 .

Changes in a firm’s welfare can be measured in dollars much as for a consumer.

Producer’s Surplus

Producer’s Surplus

m (output units)

Output price (p)

Marginal Cost

Producer’s Surplus

m (output units)

Output price (p)

Marginal Cost

p'

'm

Producer’s Surplus

m (output units)

Output price (p)

Marginal Cost

p'

'm

Revenue= mp

Producer’s Surplus

m (output units)

Output price (p)

Marginal Cost

p'

Variable Cost of producingm’ units is the sum of themarginal costs

'm

Producer’s Surplus

y (output units)

Output price (p)

Marginal Cost

p'

y'

Variable Cost of producingy’ units is the sum of themarginal costs

Revenue less VCis the Producer’sSurplus.

How can such gains-to-trade be measured? Three such measures are:

Consumer’s Surplus （消费者剩余） Equivalent Variation （等价变换） , and Compensating Variation （补偿变换） .

Only in one special circumstance (quasilinear preference ) do these three measures coincide.

Producer’s surplus

Monetary Measures of Gains-to-Trade