清华大学中级微观经济学讲义清华李稻葵14

1、http:/ fourteenconsumers surpluswhat do we do in this chapter?u after studying consumers choices, we now examine consumers welfareuthe key question is: how much has a consumer gained in welfare from purchasing certain units of a good?uthree such measures are:lconsumers surpluslequivalent variation,

2、andlcompensating variation.uonly in one special circumstance do these three measures coincide.monetary measures of gains-to-tradeusuppose rice can be bought only in lumps of one kilogram.uuse r1 to denote the most a single consumer would pay for a 1st kilogram - call this her reservation price for t

3、he 1st kilogram.ur1 is the dollar equivalent of the marginal utility of the 1st kilogram.monetary equivalent utility gainsunow 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.ur2 is the dollar equivalent of th

4、e marginal utility of the 2nd gallon.monetary equivalent utility gainsugenerally, if she already has n-1 kilograms of rice then rn denotes the most she will pay for an nth kilogram.urn is the dollar equivalent of the marginal utility of the nth kilogram.monetary equivalent utility gainsur1 + + rn wi

5、ll therefore be the dollar equivalent of the total change to utility from acquiring n kilograms of rice at a price of $0.uso r1 + + rn - pgn will be the dollar equivalent of the total change to utility from acquiring n kilograms of rice at a price of $pg each.monetary equivalent utility gainsua plot

6、 of r1, r2, , rn, against n is a reservation-price curve. this is not quite the same as the consumers demand curve for rice.monetary equivalent utility gainsmonetary equivalent utility gainsreservation price curve for rice0246810rice (kilograms)($) res.values123456r1r2r3r4r5r6uwhat is the monetary v

7、alue of our consumers gain-to-trading in the rice market at a price of $pg?monetary equivalent utility gainsuthe dollar equivalent net utility gain for the 1st kilogram is $(r1 - pg)uand is $(r2 - pg) for the 2nd kilogram, uand so on, so the dollar value of the gain-to-trade is$(r1 - pg) + $(r2 - pg

8、) + for as long as rn - pg 0.monetary equivalent utility gainsmonetary equivalent utility gainsreservation price curve for gasoline0246810gasoline (gallons)($) res.values123456r1r2r3r4r5r6pgmonetary equivalent utility gainsreservation price curve for gasoline0246810gasoline (gallons)($) res.values12

9、3456r1r2r3r4r5r6pg$ value of net utility gains-to-tradesuppose rice can be purchased in any continuous quantity then .monetary equivalent utility gainsmonetary equivalent utility gainsriceres.pricespgreservation price curve for rice$ value of net utility gains-to-tradeuunfortunately, estimating a co

10、nsumers reservation-price curve is difficult,uso, as an approximation, the reservation-price curve is replaced with the consumers ordinary demand curve.$ equivalent utility gainsua reservation-price curve describes sequentially the values of successive single units of a commodity.uan ordinary demand

11、 curve describes the most that would be paid for q units of a commodity purchased simultaneously.the reservation price curve is different from the ordinary demand curve !uapproximating the net utility gain area under the reservation-price curve by the corresponding area under the ordinary demand cur

12、ve gives the consumers surplus measure of net utility gain.consumers surplusconsumers surplusgasoline($) reservation price curve for gasolineordinary demand curve for gasolineconsumers surplusricereservation price curve for riceordinary demand curve for ricepguthe difference between the consumers re

13、servation-price and ordinary demand curves is due to income effects.ubut, if the consumers utility function is quasilinear in income then there are no income effects and consumers surplus is an exact measure of gains-to-trade. consumers surplusconsumers surplusu xxv xx(,)()1212 the consumers utility

14、 function isquasilinear in x2.take p2 = 1. then the consumerschoice problem is to maximizeu xxv xx(,)()1212 subject top xxm1 12 .consumers surplusthat is, choose x1 to maximizev xmp x().11 1 the first-order condition is v xp()110 that is,pvx11 ().this is the equation of the consumersordinary demand

15、for commodity 1.consumers surplusordinary demand curve,p1pvx11 ()x1*x1p1csis exactly the consumers utility gain from consuming x1 units of commodity 1.csv xdxp xx ()111 101 v xvp x()( )11 10uconsumers surplus is an exact dollar measure of utility gained from consuming commodity 1 when the consumers

16、utility function is quasilinear in commodity 2. uotherwise consumers surplus is an approximation.consumers surplusuthe change to a consumers total utility due to a change to p1 is approximately the change in her consumers surplus. consumers surplusconsumers surplusp1x1*x1p1p1(x1), the inverse ordina

17、ry demand curve for commodity 1consumers surplusp1x1*x1cs beforep1(x1)p1consumers surplusp1x1*x1cs afterp1x1p1(x1)p1consumers surplusp1x1*x1x1lost csp1(x1), inverse ordinary demand curve for commodity 1.p1p1consumers surplusp1x1*x1x1lostcs 11pp11*1dp)p(xcsx1*(p1), the consumers ordinary demand curve

18、 for commodity 1.p1p1measures the loss in consumers surplus.utwo additional dollar measures of the total utility change caused by a price change are compensating variation and equivalent variation.compensating variation and equivalent variationup1 rises.uq: what is the least extra income that, at th

19、e new prices, just restores the consumers original utility level?ua: the compensating variation.compensating variationcompensating variationx2x1x1u1x2p1=p1p2 is fixed.mp xp x11 12 2 compensating variationx2x1x1x2x1x2u1u2p1=p1p1=p1”p2 is fixed.mp xp x11 12 2 p xp x1 12 2 compensating variationx2x1x1u

20、1u2x1x2x2x2x1p1=p1p1=p1”p2 is fixed.mp xp x11 12 2 p xp x1 12 2 22112xpxpm compensating variationx2x1x1u1u2x1x2x2x2x1p1=p1p1=p1”p2 is fixed.mp xp x11 12 2 p xp x1 12 2 22112xpxpm cv = m2 - m1.up1 rises.uq: what is the level of reduction in income such that the consumer is indifferent between the ori

21、ginal prices with the reduced income and the new prices with the original income? ua: the equivalent variation.equivalent variationup1 rises.uq: what is the level of income reduction such that, the consumer is indifferent between the new price with the old income and the old price with the reduced i

22、ncome?ua: the equivalent variation.equivalent variationequivalent variationx2x1x1u1x2p1=p1p2 is fixed.mp xp x11 12 2 equivalent variationx2x1x1x2x1x2u1u2p1=p1p1=p1”p2 is fixed.mp xp x11 12 2 p xp x1 12 2 equivalent variationx2x1x1u1u2x1x2x2x2x1p1=p1p1=p1”p2 is fixed.mp xp x11 12 2 p xp x1 12 2 mp xp

23、 x21 122 ev = m1 - m2.urelationship 1: when the consumers preferences are quasilinear, all three measures are the same.consumers surplus, compensating variation and equivalent variationuconsider first the change in consumers surplus when p1 rises from p1 to p1”.consumers surplus, compensating variat

24、ion and equivalent variationconsumers surplus, compensating variation and equivalent variationu xxv xx(,)()1212 ifthencs pv xvp x()()( )111 10 and so the change in cs when p1 risesfrom p1 to p1” is cscs pcs p ()()11 v xvp xv xvp x()( )()( ) 11 111 100 v xv xp xp x()()(). 111 111unow consider the cha

25、nge in cv when p1 rises from p1 to p1”.uthe consumers utility for given p1 isand cv is the extra income which, at the new prices, makes the consumers utility the same as at the old prices. that is, .consumers surplus, compensating variation and equivalent variationv xpmp xp()()*111 11 consumers surp

26、lus, compensating variation and equivalent variationv xmp x()11 1 v xmcvp x(). 111socvv xv xp xp x ()()() 111111 cs.unow consider the change in ev when p1 rises from p1 to p1”.uthe consumers utility for given p1 isand ev is the extra income which, at the old prices, makes the consumers utility the same as at the new prices. that is, .consumers surplus, compensating variation and equivalent variationv xpmp xp()()*111 11 consumers surplus, compensating variation and equivalent variationthat is,evv xv xp xp x ()()()