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Page 1: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

C ’ S lConsumer’s Surplus

© 2010 W. W. Norton & Company, Inc.

Page 2: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Monetary Measures of Gains-to-T dTrade

You can buy as much gasoline as you wish at $1 per gallon once youyou wish at $1 per gallon once you enter the gasoline market. Q Wh i h ld Q: What is the most you would pay to enter the market?

© 2010 W. W. Norton & Company, Inc. 2

Page 3: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Monetary Measures of Gains-to-T dTrade

A: You would pay up to the dollar value of the gains-to-trade you wouldvalue of the gains to trade you would enjoy once in the market.H h i d b How can such gains-to-trade be measured?

© 2010 W. W. Norton & Company, Inc. 3

Page 4: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Monetary Measures of Gains-to-T dTrade

Three such measures are:Consumer’s Surplus– Consumer s Surplus

– Equivalent Variation, and– Compensating Variation.

Only in one special circumstance do Only in one special circumstance do these three measures coincide.

© 2010 W. W. Norton & Company, Inc. 4

Page 5: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y Suppose gasoline can be bought

l i l f llonly in lumps of one gallon. Use r1 to denote the most a single 1 g

consumer would pay for a 1st gallon -- call this her reservation price for-- call this her reservation price for the 1st gallon.

r1 is the dollar equivalent of the marginal utility of the 1st gallon.g y g

© 2010 W. W. Norton & Company, Inc. 5

Page 6: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

Now that she has one gallon, use r2to denote the most she would pay forto denote the most she would pay for a 2nd gallon -- this is her reservation price for the 2nd gallonprice for the 2nd gallon.

r2 is the dollar equivalent of the 2marginal utility of the 2nd gallon.

© 2010 W. W. Norton & Company, Inc. 6

Page 7: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

Generally, if she already has n-1 gallons of gasoline then r denotesgallons of gasoline then rn denotes the most she will pay for an nth gallongallon.

rn is the dollar equivalent of the nmarginal utility of the nth gallon.

© 2010 W. W. Norton & Company, Inc. 7

Page 8: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

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

So r1 + … + rn - pGn will be the 1 n Gdollar equivalent of the total change to utility from acquiring n gallons ofto utility from acquiring n gallons of gasoline at a price of $pG each.

© 2010 W. W. Norton & Company, Inc. 8

Page 9: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

A plot of r1, r2, … , rn, … against n is a reservation-price curve This is notreservation price curve. This is not quite the same as the consumer’s demand curve for gasolinedemand curve for gasoline.

© 2010 W. W. Norton & Company, Inc. 9

Page 10: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1

68r2

r3

24r4

r50

1 2 3 4 5 6r6

Gasoline (gallons)

© 2010 W. W. Norton & Company, Inc. 10

Page 11: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

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

© 2010 W. W. Norton & Company, Inc. 11

Page 12: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

The dollar equivalent net utility gain for the 1st gallon is $(r1 - pG)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 isgain to trade is$(r1 - pG) + $(r2 - pG) + …

for as long as r p > 0for as long as rn - pG > 0.

© 2010 W. W. Norton & Company, Inc. 12

Page 13: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1

68r2

r3

24r4

r5pG

01 2 3 4 5 6

r6

Gasoline (gallons)

© 2010 W. W. Norton & Company, Inc. 13

Page 14: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1

68r2

r3

24r4

r5pG

01 2 3 4 5 6

r6

Gasoline (gallons)

© 2010 W. W. Norton & Company, Inc. 14

Page 15: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1$ value of net utility gains-to-trade

68r2

r3

24r4

r5pG

01 2 3 4 5 6

r6

Gasoline (gallons)

© 2010 W. W. Norton & Company, Inc. 15

Page 16: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

Now suppose that gasoline is sold in half-gallon unitshalf gallon units.

r1, r2, … , rn, … denote the ’ i i fconsumer’s reservation prices for

successive half-gallons of gasoline.g g Our consumer’s new reservation

price curve isprice curve is

© 2010 W. W. Norton & Company, Inc. 16

Page 17: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1

68r3

r5

24r7

r90

1 2 3 4 5 6r11 7 8 9 10 11

Gasoline (half gallons)

© 2010 W. W. Norton & Company, Inc. 17

Page 18: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1

68r3

r5

24r7

r9pG

01 2 3 4 5 6

r11 7 8 9 10 11Gasoline (half gallons)

© 2010 W. W. Norton & Company, Inc. 18

Page 19: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline($) Res.

10Values

r1$ value of net utility gains-to-trade

68r3

r5

24r7

r9pG

01 2 3 4 5 6

r11 7 8 9 10 11Gasoline (half gallons)

© 2010 W. W. Norton & Company, Inc. 19

Page 20: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

And if gasoline is available in one-quarter gallon unitsquarter gallon units ...

© 2010 W. W. Norton & Company, Inc. 20

Page 21: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline

10

68($) Res.

Values

24

01 2 3 4 5 6 7 8 9 10 11

Gasoline (one-quarter gallons)

© 2010 W. W. Norton & Company, Inc. 21

Page 22: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline

10

68($) Res.

Values

24

pG

01 2 3 4 5 6 7 8 9 10 11

Gasoline (one-quarter gallons)

© 2010 W. W. Norton & Company, Inc. 22

Page 23: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline

10 $ value of net utility gains-to-trade

68($) Res.

Values

24

pG

0

Gasoline (one-quarter gallons)

© 2010 W. W. Norton & Company, Inc. 23

Page 24: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

Finally, if gasoline can be purchased in any quantity thenin any quantity then ...

© 2010 W. W. Norton & Company, Inc. 24

Page 25: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y($) Res. Reservation Price Curve for GasolinePrices

Gasoline

© 2010 W. W. Norton & Company, Inc. 25

Page 26: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y($) Res. Reservation Price Curve for GasolinePrices

pG

Gasoline

© 2010 W. W. Norton & Company, Inc. 26

Page 27: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y($) Res. Reservation Price Curve for GasolinePrices $ value of net utility gains-to-trade

pG

Gasoline

© 2010 W. W. Norton & Company, Inc. 27

Page 28: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

$ Equivalent Utility Gainsq y

Unfortunately, estimating a consumer’s reservation-price curveconsumer s reservation price curve is difficult,

i i h so, as an approximation, the reservation-price curve is replaced with the consumer’s ordinary demand curve.demand curve.

© 2010 W. W. Norton & Company, Inc. 28

Page 29: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusp A consumer’s reservation-price p

curve is not quite the same as her ordinary demand curve Why not?ordinary demand curve. Why not?

A reservation-price curve describes ti ll th l f isequentially the values of successive

single units of a commodity. An ordinary demand curve describes

the most that would be paid for qthe most that would be paid for q units of a commodity purchased i lt l

© 2010 W. W. Norton & Company, Inc. 29

simultaneously.

Page 30: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusp

Approximating the net utility gain area under the reservation-pricearea under the reservation price curve by the corresponding area under the ordinary demand curveunder the ordinary demand curve gives the Consumer’s Surplus measure of net utility gain.

© 2010 W. W. Norton & Company, Inc. 30

Page 31: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusp($) Reservation price curve for gasoline

Ordinary demand curve for gasoline

Gasoline

© 2010 W. W. Norton & Company, Inc. 31

Page 32: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspReservation price curve for gasoline($)Ordinary demand curve for gasoline

pG

Gasoline

© 2010 W. W. Norton & Company, Inc. 32

Page 33: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspReservation price curve for gasoline($)Ordinary demand curve for gasoline

$ value of net utility gains-to-trade

pG

Gasoline

© 2010 W. W. Norton & Company, Inc. 33

Page 34: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspReservation price curve for gasoline($)Ordinary demand curve for gasoline

$ value of net utility gains-to-tradeConsumer’s Surplus

pG

Gasoline

© 2010 W. W. Norton & Company, Inc. 34

Page 35: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspReservation price curve for gasoline($)Ordinary demand curve for gasoline

$ value of net utility gains-to-tradeConsumer’s Surplus

pG

Gasoline

© 2010 W. W. Norton & Company, Inc. 35

Page 36: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusp

The difference between the The difference between the consumer’s reservation-price and ordinary demand curves is due toordinary demand curves is due to income effects.

But, if the consumer’s utility function is quasilinear in income then thereis quasilinear in income then there are no income effects and Consumer’s Surplus is an exact $Consumer s Surplus is an exact $ measure of gains-to-trade.

© 2010 W. W. Norton & Company, Inc. 36

Page 37: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspThe consumer’s utility function is

ili i

Ux x v x x( ) ( )1 2 1 2 quasilinear in x2.

Ux x v x x( , ) ( )1 2 1 2 Take p2 = 1. Then the consumer’sp2choice problem is to maximize

Ux x v x x( ) ( ) Ux x v x x( , ) ( )1 2 1 2 subject toj

px x m1 1 2 .

© 2010 W. W. Norton & Company, Inc. 37

Page 38: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspThe consumer’s utility function is

ili i

Ux x v x x( ) ( )1 2 1 2 quasilinear in x2.

Ux x v x x( , ) ( )1 2 1 2 Take p2 = 1. Then the consumer’sp2choice problem is to maximize

Ux x v x x( ) ( ) Ux x v x x( , ) ( )1 2 1 2 subject toj

px x m1 1 2 .

© 2010 W. W. Norton & Company, Inc. 38

Page 39: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspThat is, choose x1 to maximize

v x m px( ) .1 1 1 The first-order condition is

v x p'( )1 1 0 v x p( )1 1 0

That is, p v x1 1 '( ).That is, p v x1 1( ).

This is the equation of the consumer’sqordinary demand for commodity 1.

© 2010 W. W. Norton & Company, Inc. 39

Page 40: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspOrdinary demand curve,p1

p v x1 1 '( )p1

CS'p1

x1*x1

'

© 2010 W. W. Norton & Company, Inc. 40

Page 41: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspOrdinary demand curve,p1

p v x1 1 '( )p1CS v x dx pxx '( ) ' ''

1 1 1 101

CS'p1

x1*x1

'

© 2010 W. W. Norton & Company, Inc. 41

Page 42: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspOrdinary demand curve,p1

p v x1 1 '( )p1CS v x dx pxx '( ) ' ''

1 1 1 101

v x v px( ) ( )' ' '1 1 10

CS'p1

x1*x1

'

© 2010 W. W. Norton & Company, Inc. 42

Page 43: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s SurpluspOrdinary demand curve,p1

p v x1 1 '( )p1CS v x dx pxx '( ) ' ''

1 1 1 101

is exactly the consumer’s utility v x v px( ) ( )' ' '

1 1 10

'CS

is exactly the consumer s utilitygain from consuming x1’

units of commodity 1p1units of commodity 1.

x1*x1

'

© 2010 W. W. Norton & Company, Inc. 43

Page 44: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusp

Consumer’s Surplus is an exact dollar measure of utility gained fromdollar measure of utility gained from consuming commodity 1 when the consumer’s utility function isconsumer’s utility function is quasilinear in commodity 2.

Otherwise Consumer’s Surplus is an approximation.approximation.

© 2010 W. W. Norton & Company, Inc. 44

Page 45: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusp

The change to a consumer’s total utility due to a change to p1 isutility due to a change to p1 is approximately the change in her Consumer’s SurplusConsumer’s Surplus.

© 2010 W. W. Norton & Company, Inc. 45

Page 46: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surpluspp1p1

p1(x1), the inverse ordinary demandf dit 1curve for commodity 1

'p1

x1*x1

'

© 2010 W. W. Norton & Company, Inc. 46

Page 47: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surpluspp1p1

p1(x1)

CS before' CS beforep1

x1*x1

'

© 2010 W. W. Norton & Company, Inc. 47

Page 48: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surpluspp1p1

p1(x1)

CS afterp" CS afterp1

'p1

x1*x1

'x1"

© 2010 W. W. Norton & Company, Inc. 48

Page 49: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surpluspp1p1

p1(x1), inverse ordinary demandf dit 1curve for commodity 1.

p"

Lost CSp1

'p1

x1*x1

'x1"

© 2010 W. W. Norton & Company, Inc. 49

Page 50: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplusx1* px1

x1*(p1), the consumer’s ordinaryx1'

"p *

1 1demand curve for commodity 1.

" 1

'1

pp 11

*1 dp)p(xCS

x1Lost

measures the loss inConsumer’s Surplus.

CSp

p1p1"p1

'

© 2010 W. W. Norton & Company, Inc. 50

Page 51: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variation and E i l t V i tiEquivalent Variation

Two additional dollar measures of the total utility change caused by athe total utility change caused by a price change are Compensating Variation and Equivalent VariationVariation and Equivalent Variation.

© 2010 W. W. Norton & Company, Inc. 51

Page 52: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variationp g

p1 rises. Q: What is the least extra income Q: What is the least extra income

that, at the new prices, just restores h ’ i i l ili l l?the consumer’s original utility level?

© 2010 W. W. Norton & Company, Inc. 52

Page 53: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variationp g

p1 rises. Q: What is the least extra income Q: What is the least extra income

that, at the new prices, just restores h ’ i i l ili l l?the consumer’s original utility level?

A: The Compensating Variation. A: The Compensating Variation.

© 2010 W. W. Norton & Company, Inc. 53

Page 54: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variationp gp1=p1’ p2 is fixed.

x2m px p x1 1 1 2 2 ' ' '

'

u1

x2'

1

x1x1'

© 2010 W. W. Norton & Company, Inc. 54

Page 55: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variationp gp1=p1’ p2 is fixed.

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

'x2" p x p x1 1 2 2

" " "

x2'

u11

u2

x1x1'x1

"

© 2010 W. W. Norton & Company, Inc. 55

Page 56: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variationp gp1=p1’ p2 is fixed.

x2

x2'"

p1=p1” m px p x1 1 1 2 2 ' ' '

x2"

'

x2 p x p x1 1 2 2

" " "

'"'"" xpxpm

u1

x2' 22112 xpxpm

1

u2

x1x1'x1

" x1'"

© 2010 W. W. Norton & Company, Inc. 56

Page 57: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Compensating Variationp gp1=p1’ p2 is fixed.

x2

x2'"

p1=p1” m px p x1 1 1 2 2 ' ' '

x2"

'

x2 p x p x1 1 2 2

" " "

'"'"" xpxpm

u1

x2' 22112 xpxpm

1

u2 CV = m2 - m1.

x1x1'x1

" x1'"

© 2010 W. W. Norton & Company, Inc. 57

Page 58: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Equivalent Variationq

p1 rises. Q: What is the least extra income Q: What is the least extra income

that, at the original prices, just h ’ i i lrestores the consumer’s original

utility level?y A: The Equivalent Variation.

© 2010 W. W. Norton & Company, Inc. 58

Page 59: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Equivalent Variationqp1=p1’ p2 is fixed.

x2m px p x1 1 1 2 2 ' ' '

'

u1

x2'

1

x1x1'

© 2010 W. W. Norton & Company, Inc. 59

Page 60: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Equivalent Variationqp1=p1’ p2 is fixed.

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

'x2" p x p x1 1 2 2

" " "

x2'

u11

u2

x1x1'x1

"

© 2010 W. W. Norton & Company, Inc. 60

Page 61: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Equivalent Variationqp1=p1’ p2 is fixed.

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

x2"

'

p x p x1 1 2 2" " "

m px p x' '" '"

u1

x2'

x'"

m px p x2 1 1 2 2

1

u2

x2

x1x1'x1

" x1'"

© 2010 W. W. Norton & Company, Inc. 61

Page 62: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Equivalent Variationqp1=p1’ p2 is fixed.

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

x2"

'

p x p x1 1 2 2" " "

m px p x' '" '"

u1

x2'

x'"

m px p x2 1 1 2 2

1

u2

x2EV = m1 - m2.

x1x1'x1

" x1'"

© 2010 W. W. Norton & Company, Inc. 62

Page 63: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

Relationship 1: When the consumer’s preferences areconsumer s preferences are quasilinear, all three measures are the samethe same.

© 2010 W. W. Norton & Company, Inc. 63

Page 64: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

Consider first the change in Consumer’s Surplus when p1 risesConsumer s Surplus when p1 rises from p1’ to p1”.

© 2010 W. W. Norton & Company, Inc. 64

Page 65: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

Ux x v x x( ) ( )If thUx x v x x( , ) ( )1 2 1 2 If then

CSp v x v px( ) ( ) ( )' ' ' '0CSp v x v px( ) ( ) ( )1 1 1 10

© 2010 W. W. Norton & Company, Inc. 65

Page 66: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

Ux x v x x( ) ( )If thUx x v x x( , ) ( )1 2 1 2 If then

CSp v x v px( ) ( ) ( )' ' ' '0CSp v x v px( ) ( ) ( )1 1 1 10

and so the change in CS when p1 risesand so the change in CS when p1 risesfrom p1’ to p1” is

' "CS CSp CSp ( ) ( )' "1 1

© 2010 W. W. Norton & Company, Inc. 66

Page 67: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

Ux x v x x( ) ( )If thUx x v x x( , ) ( )1 2 1 2 If then

CSp v x v px( ) ( ) ( )' ' ' '0CSp v x v px( ) ( ) ( )1 1 1 10

and so the change in CS when p1 risesand so the change in CS when p1 risesfrom p1’ to p1” is

' "CS CSp CSp ( ) ( )' "1 1

( ) ( ) ( ) ( )' ' ' " " "0 0 v x v px v x v p x( ) ( ) ( ) ( )1 1 1 1 1 10 0

© 2010 W. W. Norton & Company, Inc. 67

Page 68: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

Ux x v x x( ) ( )If thUx x v x x( , ) ( )1 2 1 2 If then

CSp v x v px( ) ( ) ( )' ' ' '0CSp v x v px( ) ( ) ( )1 1 1 10

and so the change in CS when p1 risesand so the change in CS when p1 risesfrom p1’ to p1” is

' "CS CSp CSp ( ) ( )' "1 1

( ) ( ) ( ) ( )' ' ' " " "0 0 v x v px v x v p x( ) ( ) ( ) ( )1 1 1 1 1 10 0

v x v x px p x( ) ( ) ( )' " ' ' " "

© 2010 W. W. Norton & Company, Inc. 68

v x v x px p x( ) ( ) ( ).1 1 1 1 1 1

Page 69: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation Now consider the change in CV when Now consider the change in CV when

p1 rises from p1’ to p1”. The consumer’s utility for given p1 is

v x p m px p( ( )) ( )* *1 1 1 1 1

and CV is the extra income which, at

v x p m px p( ( )) ( )1 1 1 1 1

and CV is the extra income which, at the new prices, makes the consumer’s utility the same as at theconsumer s utility the same as at the old prices. That is, ...

© 2010 W. W. Norton & Company, Inc. 69

Page 70: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

' ' 'v x m px( )' ' '1 1 1

CV" " " v x m CV p x( ) .1 1 1

© 2010 W. W. Norton & Company, Inc. 70

Page 71: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

' ' 'v x m px( )' ' '1 1 1

CV" " " v x m CV p x( ) .1 1 1SoSoCV v x v x px p x ( ) ( ) ( )' " ' ' " "

1 1 1 1 1 1p p( ) ( ) ( )1 1 1 1 1 1CS.

© 2010 W. W. Norton & Company, Inc. 71

Page 72: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation Now consider the change in EV when Now consider the change in EV when

p1 rises from p1’ to p1”. The consumer’s utility for given p1 is

v x p m px p( ( )) ( )* *

and EV is the extra income which at

v x p m px p( ( )) ( )1 1 1 1 1

and EV is the extra income which, at the old prices, makes the consumer’s

tilit th t th iutility the same as at the new prices. That is, ...

© 2010 W. W. Norton & Company, Inc. 72

Page 73: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

' ' 'v x m px( )1 1 1

EV( )" " " v x m EV p x( ) .1 1 1

© 2010 W. W. Norton & Company, Inc. 73

Page 74: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent Variation

' ' 'v x m px( )1 1 1

EV( )" " "

That is, v x m EV p x( ) .1 1 1

,EV v x v x px p x ( ) ( ) ( )' " ' ' " "

1 1 1 1 1 1

CS.

© 2010 W. W. Norton & Company, Inc. 74

Page 75: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Consumer’s Surplus, Compensating V i ti d E i l t V i tiVariation and Equivalent VariationS h th h iliSo when the consumer has quasilinearutility,

CV = EV = CS.

But, otherwise, we have:

Relationship 2: In size, EV < CS < CV.

© 2010 W. W. Norton & Company, Inc. 75

Page 76: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Producer’s Surplusp

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

© 2010 W. W. Norton & Company, Inc. 76

Page 77: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Producer’s Surplusp

Output price (p)Output price (p)

Marginal Cost

y (output units)

© 2010 W. W. Norton & Company, Inc. 77

Page 78: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Producer’s Surplusp

Output price (p)Output price (p)

Marginal Cost

'p'

y (output units)y'

© 2010 W. W. Norton & Company, Inc. 78

Page 79: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Producer’s Surplusp

Output price (p)Output price (p)

Marginal Cost

'p'

Revenue' '= py

y (output units)y'

© 2010 W. W. Norton & Company, Inc. 79

Page 80: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Producer’s Surplusp

Output price (p)Output price (p)

Marginal Cost

'p'Variable Cost of producing

fy’ units is the sum of themarginal costs

y (output units)y'

© 2010 W. W. Norton & Company, Inc. 80

Page 81: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Producer’s Surplusp

Output price (p)Output price (p)Revenue less VCis the Producer’s

Marginal Cost

'

is the Producer sSurplus.

p'Variable Cost of producing

fy’ units is the sum of themarginal costs

y (output units)y'

© 2010 W. W. Norton & Company, Inc. 81

Page 82: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit Cost AnalysisBenefit-Cost Analysis

Can we measure in money units the net gain or loss caused by a marketnet gain, or loss, caused by a market intervention; e.g., the imposition or the removal of a market regulation?the removal of a market regulation?

Yes, by using measures such as the y gConsumer’s Surplus and the Producer’s Surplus.Producer s Surplus.

© 2010 W. W. Norton & Company, Inc. 82

Page 83: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice The free-market equilibrium

Supply

p0

D d

QD QS

Demand

q© 2010 W. W. Norton & Company, Inc. 83

QD, QSq0

Page 84: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice The free-market equilibrium

and the gains from tradeSupply

and the gains from tradegenerated by it.

CSCSp0

PS

D d

PS

QD QS

Demand

q© 2010 W. W. Norton & Company, Inc. 84

QD, QSq0

Page 85: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice The gain from freely

trading the q th unitSupply

Consumer’s

trading the q1th unit.

CS

Consumer’sgain

CSp0

PS

D d

PS Producer’sgain

QD QS

Demand

qq

g

© 2010 W. W. Norton & Company, Inc. 85

QD, QSq0q1

Page 86: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice The gains from freely

trading the units fromSupply

trading the units fromq1 to q0.

Consumer’s

CS

Consumer’sgains

CSp0

PS

D d

PS Producer’sgains

QD QS

Demand

qq

g

© 2010 W. W. Norton & Company, Inc. 86

QD, QSq0q1

Page 87: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice The gains from freely

trading the units fromSupply

trading the units fromq1 to q0.

Consumer’s

CS

Consumer’sgains

CSp0

PS

D d

PS Producer’sgains

QD QS

Demand

qq

g

© 2010 W. W. Norton & Company, Inc. 87

QD, QSq0q1

Page 88: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice

Consumer’sAny regulation that

CS

Consumer’sgains

causes the unitsfrom q1 to q0 to be

CSp0

PS

not traded destroysthese gains. This

PS Producer’sgains

loss is the net costof the regulation.

QD QSqq

g

© 2010 W. W. Norton & Company, Inc. 88

QD, QSq0q1

Page 89: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice An excise tax imposed at a rate of $t

t d d it d t th iper traded unit destroys these gains.

Deadweight

pbCS

gLoss

TaxRevenuet

PSps

QD QSqq© 2010 W. W. Norton & Company, Inc. 89

QD, QSq0q1

Page 90: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice An excise tax imposed at a rate of $t

t d d it d t th iper traded unit destroys these gains.

Deadweight So does a floor

pfCS

gLoss price set at pf

PS

QD QSqq© 2010 W. W. Norton & Company, Inc. 90

QD, QSq0q1

Page 91: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice An excise tax imposed at a rate of $t

t d d it d t th iper traded unit destroys these gains.

Deadweight So does a floorgLoss price set at pf,

a ceiling price setgat pcCS

pc PS

QD QSqq© 2010 W. W. Norton & Company, Inc. 91

QD, QSq0q1

Page 92: Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s reservation-price curve is not quite the same as her ordinary demand curve Why not?ordinary

Benefit-Cost AnalysisyPrice An excise tax imposed at a rate of $t

t d d it d t th iper traded unit destroys these gains.

Deadweight So does a floorgLoss price set at pf,

a ceiling price setpeCS

gat pc, and a rationscheme that

pc allows only q1units to be traded.PS

QD QSqq© 2010 W. W. Norton & Company, Inc. 92

QD, QSq0q1Revenue received by holders of ration coupons.