Varian Chapter14 Consumer's Surplus.ppts_Surplus.pdf · Consumer’s Surplus A consumer’s...

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C ’ S lConsumer’s Surplus

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?

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?

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.

$ 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

$ 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.

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.

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.

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.

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

10Values

1 2 3 4 5 6r6

Gasoline (gallons)

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?

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.

10Values

01 2 3 4 5 6

Gasoline (gallons)

10Values

01 2 3 4 5 6

Gasoline (gallons)

10Values

r1$ value of net utility gains-to-trade

01 2 3 4 5 6

Gasoline (gallons)

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

10Values

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

Gasoline (half gallons)

10Values

01 2 3 4 5 6

r11 7 8 9 10 11Gasoline (half gallons)

10Values

r1$ value of net utility gains-to-trade

01 2 3 4 5 6

r11 7 8 9 10 11Gasoline (half gallons)

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

$ Equivalent Utility Gainsq yReservation Price Curve for Gasoline

68($) Res.

Values

01 2 3 4 5 6 7 8 9 10 11

Gasoline (one-quarter gallons)

68($) Res.

Values

01 2 3 4 5 6 7 8 9 10 11

10 $ value of net utility gains-to-trade

68($) Res.

Values

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

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

Gasoline

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

Gasoline

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

Gasoline

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.

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

simultaneously.

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.

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

Ordinary demand curve for gasoline

Gasoline

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

Gasoline

$ value of net utility gains-to-trade

Gasoline

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

Gasoline

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

Gasoline

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.

Consumer’s SurpluspThe consumer’s utility function is

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 .

Consumer’s SurpluspThe consumer’s utility function is

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 .

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.

Consumer’s SurpluspOrdinary demand curve,p1

p v x1 1 '( )p1

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

1 1 1 101

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

1 1 1 101

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

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

is exactly the consumer s utilitygain from consuming x1’

units of commodity 1p1units of commodity 1.

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.

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.

Consumer’s Surpluspp1p1

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

p1(x1)

CS before' CS beforep1

p1(x1)

CS afterp" CS afterp1

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

Lost CSp1

Consumer’s Surplusx1* px1

x1*(p1), the consumer’s ordinaryx1'

1 1demand curve for commodity 1.

*1 dp)p(xCS

x1Lost

measures the loss inConsumer’s Surplus.

p1p1"p1

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.

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?

Compensating Variationp g

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.

Compensating Variationp gp1=p1’ p2 is fixed.

x2m px p x1 1 1 2 2 ' ' '

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

'x2" p x p x1 1 2 2

x1x1'x1

p1=p1” m px p x1 1 1 2 2 ' ' '

x2 p x p x1 1 2 2

'"'"" xpxpm

x2' 22112 xpxpm

x1x1'x1

" x1'"

p1=p1” m px p x1 1 1 2 2 ' ' '

x2 p x p x1 1 2 2

'"'"" xpxpm

x2' 22112 xpxpm

u2 CV = m2 - m1.

x1x1'x1

" x1'"

Equivalent Variationq

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

utility level?y A: The Equivalent Variation.

Equivalent Variationqp1=p1’ p2 is fixed.

x2m px p x1 1 1 2 2 ' ' '

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

'x2" p x p x1 1 2 2

x1x1'x1

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

p x p x1 1 2 2" " "

m px p x' '" '"

m px p x2 1 1 2 2

x1x1'x1

" x1'"

x2 p1=p1” m px p x1 1 1 2 2 ' ' '

p x p x1 1 2 2" " "

m px p x' '" '"

m px p x2 1 1 2 2

x2EV = m1 - m2.

x1x1'x1

" x1'"

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.

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

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

' "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( ) ( ) ( )' " ' ' " "

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

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, ...

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

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

' ' '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.

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, ...

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

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

' ' '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

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.

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.

Output price (p)Output price (p)

Marginal Cost

y (output units)

Marginal Cost

y (output units)y'

Marginal Cost

Revenue' '= py

y (output units)y'

Marginal Cost

'p'Variable Cost of producing

fy’ units is the sum of themarginal costs

y (output units)y'

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'

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.

Benefit-Cost AnalysisyPrice The free-market equilibrium

Supply

Demand

QD, QSq0

Benefit-Cost AnalysisyPrice The free-market equilibrium

and the gains from tradeSupply

and the gains from tradegenerated by it.

CSCSp0

Demand

QD, QSq0

Benefit-Cost AnalysisyPrice The gain from freely

trading the q th unitSupply

Consumer’s

trading the q1th unit.

Consumer’sgain

PS Producer’sgain

Demand

QD, QSq0q1

Benefit-Cost AnalysisyPrice The gains from freely

trading the units fromSupply

trading the units fromq1 to q0.

Consumer’s

Consumer’sgains

PS Producer’sgains

Demand

QD, QSq0q1

Benefit-Cost AnalysisyPrice The gains from freely

trading the units fromSupply

trading the units fromq1 to q0.

Consumer’s

Consumer’sgains

Demand

QD, QSq0q1

Benefit-Cost AnalysisyPrice

Consumer’sAny regulation that

Consumer’sgains

causes the unitsfrom q1 to q0 to be

not traded destroysthese gains. This

loss is the net costof the regulation.

QD QSqq

QD, QSq0q1

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

TaxRevenuet

QD, QSq0q1

Deadweight So does a floor

gLoss price set at pf

QD, QSq0q1

Deadweight So does a floorgLoss price set at pf,

a ceiling price setgat pcCS

QD, QSq0q1

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, QSq0q1Revenue received by holders of ration coupons.

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