第十四章

ChapterFourteen,ConsumersSurplus消费者剩余,Structure,MoneyequivalentofutilitygainstotradeConsumerssurplusChangesinconsumerssurplusCompensatingandequivalentvariationsProducerssurplus,MonetaryMeasuresofGains-to-Trade,Youcanbuyasmuchgasolineasyouwishat$1pergallononceyouenterthegasolinemarket.Q:Whatisthemostyouwouldpaytoenterthemarket?A:Youwouldpayuptothedollarvalueofthegains-to-tradeyouwouldenjoyonceinthemarket.,Howcansuchgains-to-tradebemeasured?Threesuchmeasuresare:ConsumersSurplusEquivalentVariation（等价变换）,andCompensatingVariation（补偿变换）.Onlyinonespecialcircumstancedothesethreemeasurescoincide.,MonetaryMeasuresofGains-to-Trade,User1todenotethemostaconsumerwouldpayfora1stgallon-reservationprice（保留价格）forthe1stgallon.r1isthedollarequivalentofthemarginalutilityofthe1stgallon.user2todenotethemostshewouldpayfora2ndgallon-thisisherreservationpriceforthe2ndgallon.r2isthedollarequivalentofthemarginalutilityofthe2ndgallon.,$EquivalentUtilityGains,Generally,ifshealreadyhasn-1gallonsofgasolinethenrndenotesthemostshewillpayforannthgallon.rnisthedollarequivalentofthemarginalutilityofthenthgallon.,$EquivalentUtilityGains,r1+rnwillbethedollarequivalentofthetotalchangetoutilityfromconsumingngallonsofgasolineatapriceof$0.Sor1+rn-pGnwillbethedollarequivalentofthetotalchangetoutilityfromconsumingngallonsofgasolineatapriceof$pGeach.,$EquivalentUtilityGains,$EquivalentUtilityGains,1,2,3,4,5,6,r1,r2,r3,r4,r5,r6,Whatisthemonetaryvalueofourconsumersgain-to-tradinginthegasolinemarketatapriceof$pG?,$EquivalentUtilityGains,Thedollarequivalentnetutilitygainforthe1stgallonis$(r1-pG)andis$(r2-pG)forthe2ndgallon,andsoon,sothedollarvalueofthegain-to-tradeis$(r1-pG)+$(r2-pG)+foraslongasrn-pG0.,$EquivalentUtilityGains,$EquivalentUtilityGains,1,2,3,4,5,6,r1,r2,r3,r4,r5,r6,pG,$EquivalentUtilityGains,1,2,3,4,5,6,r1,r2,r3,r4,r5,r6,pG,$EquivalentUtilityGains,1,2,3,4,5,6,r1,r2,r3,r4,r5,r6,pG,$valueofnetutilitygains-to-trade,Ifgasolinecanbepurchasedinanyquantitythen.,$EquivalentUtilityGains,$EquivalentUtilityGains,Gasoline,($)Res.Prices,ReservationPriceCurveforGasoline,$EquivalentUtilityGains,Gasoline,($)Res.Prices,pG,ReservationPriceCurveforGasoline,$EquivalentUtilityGains,Gasoline,($)Res.Prices,pG,ReservationPriceCurveforGasoline,$valueofnetutilitygains-to-trade,Quasi-LinearUtilityExample,U(x,y)=v(x)+yReservationpricesforthefirst3units:r1=v(1)-v(0)r2=v(2)-v(1)r3=v(3)-v(2)r1+r2+r3=v(3)-v(0)=v(3)Thisisthegrossbenefit（毛收益）ofconsuming3unitsofgoodx.,Quasi-LinearUtility,1,2,3,4,5,6,r1,r2,r3,r4,r5,r6,Quasi-LinearUtility,Whenx=3,y=m-3p,U(3,m-3p)=v(3)+m-3pIfconsume0goodx,theny=mU(0,m)=mThenetbenefitofconsumingx(gains-to-trade)U(3,m-3p)U(0,m)=v(3)-3pIfnunitsofxthenutilitygainisv(n)pnThisisthenetbenefit（净收益）.,Quasi-LinearUtility,1,2,3,4,5,6,r1,r2,r3,r4,r5,r6,p,Unfortunately,estimatingaconsumersreservation-pricecurveisdifficult,so,asanapproximation,weusetheconsumersordinarydemandcurve.ThisapproximationgivestheConsumersSurplusmeasureofnetutilitygain.,$EquivalentUtilityGains,Aconsumersreservation-pricecurveisnotquitethesameasherordinarydemandcurve.Whynot?Areservation-pricecurvedescribessequentiallythevaluesofsuccessivesingleunitsofacommodity.Anordinarydemandcurvedescribesthemostthatwouldbepaidforqunitsofacommoditypurchasedsimultaneously.,ConsumersSurplus,But,iftheconsumersutilityfunctionisquasilinearinincomethenConsumersSurplusisanexact$measureofgains-to-trade.Ifincomeeffectsaresmall,thentheapproximationisgood.,ConsumersSurplus,ConsumersSurplus,Theconsumersutilityfunctionisquasilinearinx2.,Takep2=1.Thentheconsumerschoiceproblemistomaximize,subjectto,ConsumersSurplus,Theconsumersutilityfunctionisquasilinearinx2.,Takep2=1.Thentheconsumerschoiceproblemistomaximize,subjectto,ConsumersSurplus,Thatis,choosex1tomaximize,Thefirst-orderconditionis,Thatis,Thisistheequationoftheconsumersordinarydemandforcommodity1.,ConsumersSurplus,Ordinarydemandcurve,p1,CS,ConsumersSurplus,Ordinarydemandcurve,p1,CS,ConsumersSurplus,Ordinarydemandcurve,p1,CS,ConsumersSurplus,Ordinarydemandcurve,p1,CS,isexactlytheconsumersutilitygainfromconsumingx1unitsofcommodity1.,InterpretingConsumersSurplus,Netbenefitofconsumingnunitsofthegood:UtilityminusexpenditureCompensationneededtogiveupconsumingtheproduct.,Thechangetoaconsumerstotalutilityduetoachangetop1isapproximatelythechangeinherConsumersSurplus.,ChangeinConsumersSurplus,ConsumersSurplus,p1,p1(x1),theinverseordinarydemandcurveforcommodity1,ConsumersSurplus,p1,CSbefore,p1(x1),ConsumersSurplus,p1,CSafter,p1(x1),ConsumersSurplus,p1,LostCS,p1(x1),inverseordinarydemandcurveforcommodity1.,ConsumersSurplus,p1,LostCS,x1*(p1),theconsumersordinarydemandcurveforcommodity1.,measuresthelossinConsumersSurplus.,TwoadditionaldollarmeasuresofthetotalutilitychangecausedbyapricechangeareCompensatingVariationandEquivalentVariation.,CompensatingVariationandEquivalentVariation,p1rises.Q:Whatistheleastextraincomethat,atthenewprices,justrestorestheconsumersoriginalutilitylevel?,CompensatingVariation,p1rises.Q:Whatistheleastextraincomethat,atthenewprices,justrestorestheconsumersoriginalutilitylevel?A:TheCompensatingVariation.,CompensatingVariation,CompensatingVariation,x2,x1,u1,p1=p1,p2isfixed.,CompensatingVariation,x2,x1,u1,u2,p1=p1p1=p1”,p2isfixed.,CompensatingVariation,x2,x1,u1,u2,p1=p1p1=p1”,p2isfixed.,CompensatingVariation,x2,x1,u1,u2,p1=p1p1=p1”,p2isfixed.,CV=m2-m1.,p1rises.Q:Whatistheleastextraincomethat,attheoriginalprices,justrestorestheconsumersoriginalutilitylevel?A:TheEquivalentVariation.,EquivalentVariation,EquivalentVariation,x2,x1,u1,p1=p1,p2isfixed.,EquivalentVariation,x2,x1,u1,u2,p1=p1p1=p1”,p2isfixed.,EquivalentVariation,x2,x1,u1,u2,p1=p1p1=p1”,p2isfixed.,EquivalentVariation,x2,x1,u1,u2,p1=p1p1=p1”,p2isfixed.,EV=m1-m2.,ConsumersSurplus,CompensatingVariationandEquivalentVariation,Sowhen