ChapterFourteenConsumer’sSurplus消費(fèi)者剩余StructureMoneyequivalentofutilitygainstotradeConsumer’ssurplusChangesinconsumer’ssurplusCompensatingandequivalentvariationsProducer’ssurplusMonetaryMeasuresofGains-to-TradeYoucanbuyasmuchgasolineasyouwishat$1pergallononceyouenterthegasolinemarket.Q:Whatisthemostyouwouldpaytoenterthemarket?A:Youwouldpayuptothedollarvalueofthegains-to-tradeyouwouldenjoyonceinthemarket.Howcansuchgains-to-tradebemeasured?Threesuchmeasuresare:Consumer’sSurplusEquivalentVariation（等價(jià)變換）,andCompensatingVariation（補(bǔ)償變換）.Onlyinonespecialcircumstancedothesethreemeasurescoincide.MonetaryMeasuresofGains-to-TradeReservationPriceMaximumwillingnesstopayforanadditionalunitofagoodTwogoods:good1(x1)andexpenditureonothers(x2);p2=1Bybudgetconstraint(p1x1+x2=m):Ifx1=0,thenx2=mIfx1=1,thenx2=m-p1Ifx1=2,thenx2=m-2p1Ifx1=3,thenx2=m-3p1ReservationPriceReservationpriceforthe1stunit(r1)u(0,m)=u(1,m-r1)r1isthedollarequivalentofthemarginalutilityofthe1stunit.Reservationpriceforthe2ndunit(r2)u(1,m-r2)=u(2,m-2r2)r2isthedollarequivalentofthemarginalutilityofthe2ndunit.Reservationpriceforthe3rdunit(r3)u(2,m-2r3)=u(2,m-3r3)Generally,ifshealreadyhasn-1gallonsofgasolinethenrndenotesthemostshewillpayforannthgallon.rnisthedollarequivalentofthemarginalutilityofthenthgallon.$EquivalentUtilityGainsr1+…+rnwillbethedollarequivalentofthetotalchangetoutilityfromconsumingngallonsofgasolineatapriceof$0.$EquivalentUtilityGains$EquivalentUtilityGains123456r1r2r3r4r5r6Whatisthemonetaryvalueofourconsumer’sgain-to-tradinginthegasolinemarketatapriceof$pG?$EquivalentUtilityGainsThedollarequivalentnetutilitygainforthe1stgallonis$(r1-pG)andis$(r2-pG)forthe2ndgallon,andsoon,sothedollarvalueofthegain-to-tradeis

$(r1-pG)+$(r2-pG)+…

foraslongasrn-pG>0.$EquivalentUtilityGainsSor1+…+rn-pGnwillbethedollarequivalentofthetotalchangetoutilityfromconsumingngallonsofgasolineatapriceof$pGeach.$EquivalentUtilityGains$EquivalentUtilityGains123456r1r2r3r4r5r6pG$EquivalentUtilityGains123456r1r2r3r4r5r6pG$EquivalentUtilityGains123456r1r2r3r4r5r6pG$valueofnetutilitygains-to-trade

Ifgasolinecanbepurchasedinanyquantitythen...$EquivalentUtilityGains$EquivalentUtilityGainsGasoline($)Res.

PricesReservationPriceCurveforGasoline$EquivalentUtilityGainsGasoline($)Res.

PricespGReservationPriceCurveforGasoline$EquivalentUtilityGainsGasoline($)Res.

PricespGReservationPriceCurveforGasoline$valueofnetutilitygains-to-tradeQuasi-LinearUtilityU(x,y)=v(x)+yReservationpricesforthefirst3units:v(0)+m=v(1)+m-r1

r1=v(1)-v(0)v(1)+m-r2=v(2)+m-2r2

r2=v(2)-v(1)v(2)+m-2r3=v(3)+m-3r3

r3=v(3)-v(2)r1+r2+r3=v(3)-v(0)=v(3)Thisisthegrossbenefit（毛收益）ofconsuming3unitsofgoodx.Quasi-LinearUtility123456r1r2r3r4r5r6Quasi-LinearUtilityWhenx=3,y=m-3p,U(3,m-3p)=v(3)+m-3pIfconsume0goodx,theny=mU(0,m)=mThenetbenefitofconsumingx(gains-to-trade)U(3,m-3p)–U(0,m)=v(3)-3pIfnunitsofxthenutilitygainisv(n)–pnThisisthenetbenefit（凈收益）.Quasi-LinearUtility123456r1r2r3r4r5r6pUnfortunately,estimatingaconsumer’sreservation-pricecurveisdifficult,so,asanapproximation,weusetheconsumer’sordinarydemandcurve.ThisapproximationgivestheConsumer’sSurplusmeasureofnetutilitygain.$EquivalentUtilityGainsAconsumer’sreservation-pricecurveisnotquitethesameasherordinarydemandcurve.Whynot?Areservation-pricecurvedescribessequentiallythevaluesofsuccessivesingleunitsofacommodity.Anordinarydemandcurvedescribesthemostthatwouldbepaidforqunitsofacommoditypurchasedsimultaneously.Consumer’sSurplusBut,iftheconsumer’sutilityfunctionisquasilinearinethenThereservationpricecurveisexactlythedemandcurveConsumer’sSurplusisanexact$measureofgains-to-trade.Ifeeffectsaresmall,thentheapproximationisgood.Consumer’sSurplusConsumer’sSurplusTheconsumer’sutilityfunctionis

quasilinearinx2.Takep2=1.Thentheconsumer’s

choiceproblemistomaximizesubjecttoConsumer’sSurplusTheconsumer’sutilityfunctionis

quasilinearinx2.Takep2=1.Thentheconsumer’s

choiceproblemistomaximizesubjecttoConsumer’sSurplusThatis,choosex1tomaximizeThefirst-orderconditionisThatis,

Thisistheequationoftheconsumer’s

ordinarydemandforcommodity1.Consumer’sSurplusOrdinarydemandcurve,p1CSisexactlytheconsumer’sutility

gainfromconsumingx1’

unitsofcommodity1.InterpretingConsumer’sSurplusNetbenefitofconsumingnunitsofthegood:

Compensationneededtogiveupconsumingtheproduct.Thechangetoaconsumer’stotalutilityduetoachangetop1isapproximatelythechangeinherConsumer’sSurplus.ChangeinConsumer’sSurplusConsumer’sSurplusp1p1(x1),theinverseordinarydemand

curveforcommodity1Consumer’sSurplusp1CSbeforep1(x1)Consumer’sSurplusp1CSafterp1(x1)Consumer’sSurplusp1LostCSp1(x1),inverseordinarydemand

curveforcommodity1.Consumer’sSurplusp1Lost

CSx1*(p1),theconsumer’sordinary

demandcurveforcommodity1.measuresthelossin

Consumer’sSurplus.TwoadditionaldollarmeasuresofthetotalutilitychangecausedbyapricechangeareCompensatingVariationandEquivalentVariation.CompensatingVariationandEquivalentVariationp1rises.Q:Whatistheleastextraethat,atthenewprices,justrestorestheconsumer’soriginalutilitylevel?CompensatingVariationA:TheCompensatingVariation.CompensatingVariationx2x1u1p1=p1’p2isfixed.CompensatingVariationx2x1u1u2p1=p1’

p1=p1”p2isfixed.CompensatingVariationx2x1u1u2p1=p1’

p1=p1”p2isfixed.CompensatingVariationx2x1u1u2p1=p1’

p1=p1”p2isfixed.CV=m2-m1.p1rises.Q:Whatistheleastextraethat,attheoriginalprices,justrestorestheconsumer’soriginalutilitylevel?EquivalentVariationA:TheEquivalentVariation.EquivalentVariationx2x1u1p1=p1’p2isfixed.EquivalentVariationx2x1u1u2p1=p1’

p1=p1”p2isfixed.EquivalentVariationx2x1u1u2p1=p1’

p1=p1”p2isfixed.EquivalentVariationx2x1u1u2p1=p1’

p1=p1”p2isfixed.EV=m1-m2.Consumer’sSurp