《微观经济学》清华大学-Tsinghua Micro Ch14 021025

ChapterFourteenConsumer’sSurplusMonetaryMeasuresofGains-to-TradeYoucanbuyasmuchriceasyouwishatRMB1perkilogramonceyouenterthegasolinemarket.Q:Whatisthemostyouwouldpaytoenterthemarket?A:Youwouldpayuptothedollarvalueofthegains-to-tradeyouwouldenjoyonceinthemarket.Howcansuchgains-to-tradebemeasured?MonetaryMeasuresofGains-to-TradeThreesuchmeasuresare:Consumer’sSurplusEquivalentVariation,andCompensatingVariation.Onlyinonespecialcircumstancedothesethreemeasurescoincide.MonetaryMeasuresofGains-to-TradeSupposericecanbeboughtonlyinlumpsofonekilogram.User1todenotethemostasingleconsumerwouldpayfora1stkilogram--callthisherreservationpriceforthe1stkilogram.r1isthedollarequivalentofthemarginalutilityofthe1stkilogram.$EquivalentUtilityGainsNowthatshehasonekilogram,user2todenotethemostshewouldpayfora2ndkilogram--thisisherreservationpriceforthe2ndkilogram.r2isthedollarequivalentofthemarginalutilityofthe2ndgallon.$EquivalentUtilityGainsGenerally,ifshealreadyhasn-1kilogramsofricethenrndenotesthemostshewillpayforannthkilogram.rnisthedollarequivalentofthemarginalutilityofthenthkilogram.$EquivalentUtilityGainsr1+…+rnwillthereforebethedollarequivalentofthetotalchangetoutilityfromacquiringnkilogramsofriceatapriceof$0.Sor1+…+rn-pGnwillbethedollarequivalentofthetotalchangetoutilityfromacquiringnkilogramsofriceatapriceof$pGeach.$EquivalentUtilityGainsAplotofr1,r2,…,rn,…againstnisareservation-pricecurve.Thisisnotquitethesameastheconsumer’sdemandcurveforrice.$EquivalentUtilityGains$EquivalentUtilityGainsReservationPriceCurveforRice0246810Rice(kilograms)($)Res.Values123456r1r2r3r4r5r6Whatisthemonetaryvalueofourconsumer’sgain-to-tradinginthericemarketatapriceof$pG?$EquivalentUtilityGainsThedollarequivalentnetutilitygainforthe1stkilogramis$(r1-pG)andis$(r2-pG)forthe2ndkilogram,andsoon,sothedollarvalueofthegain-to-tradeis$(r1-pG)+$(r2-pG)+…foraslongasrn-pG0.$EquivalentUtilityGains$EquivalentUtilityGainsReservationPriceCurveforGasoline0246810Gasoline(gallons)($)Res.Values123456r1r2r3r4r5r6pG$EquivalentUtilityGainsReservationPriceCurveforGasoline0246810Gasoline(gallons)($)Res.Values123456r1r2r3r4r5r6pG$valueofnetutilitygains-to-tradeSupposericecanbepurchasedinanycontinuousquantitythen...$EquivalentUtilityGains$EquivalentUtilityGainsRiceRes.PricespGReservationPriceCurveforRice$valueofnetutilitygains-to-tradeUnfortunately,estimatingaconsumer’sreservation-pricecurveisdifficult,so,asanapproximation,thereservation-pricecurveisreplacedwiththeconsumer’sordinarydemandcurve.$EquivalentUtilityGainsAconsumer’sreservation-pricecurveisnotquitethesameasherordinarydemandcurve.Whynot?Areservation-pricecurvedescribessequentiallythevaluesofsuccessivesingleunitsofacommodity.Anordinarydemandcurvedescribesthemostthatwouldbepaidforqunitsofacommoditypurchasedsimultaneously.Consumer’sSurplusApproximatingthenetutilitygainareaunderthereservation-pricecurvebythecorrespondingareaundertheordinarydemandcurvegivestheConsumer’sSurplusmeasureofnetutilitygain.Consumer’sSurplusConsumer’sSurplusGasoline($)ReservationpricecurveforgasolineOrdinarydemandcurveforgasolineConsumer’sSurplusriceReservationpricecurveforriceOrdinarydemandcurveforricepGThedifferencebetweentheconsumer’sreservation-priceandordinarydemandcurvesisduetoincomeeffects.But,iftheconsumer’sutilityfunctionisquasilinearinincomethentherearenoincomeeffectsandConsumer’sSurplusisanexactmeasureofgains-to-trade.Consumer’sSurplusConsumer’sSurplusUxxvxx(,)()1212Theconsumer’sutilityfunctionisquasilinearinx2.Takep2=1.Thentheconsumer’schoiceproblemistomaximizeUxxvxx(,)()1212subjecttopxxm112.Consumer’sSurplusThatis,choosex1tomaximizevxmpx().111Thefirst-orderconditionisvxp'()110Thatis,pvx11'().Thisistheequationoftheconsumer’sordinarydemandforcommodity1.Consumer’sSurplusOrdinarydemandcurve,p1pvx11'()x1*x1'p1'CSisexactlytheconsumer’sutilitygainfromconsumingx1’unitsofcommodity1.CSvxdxpxx'()'''111101vxvpx()()'''1110Consumer’sSurplusisanexactdollarmeasureofutilitygainedfromconsumingcommodity1whentheconsumer’sutilityfunctionisquasilinearincommodity2.OtherwiseConsumer’sSurplusisanapproximation.Consumer’sSurplusThechangetoaconsumer’stotalutilityduetoachangetop1isapproximatelythechangeinherConsumer’sSurplus.Consumer’sSurplusConsumer’sSurplusp1x1*x1'p1'p1(x1),theinverseordinarydemandcurveforcommodity1Consumer’sSurplusp1x1*x1'CSbeforep1(x1)p1'Consumer’sSurplusp1x1*x1'CSafterp1x1p1(x1)p1'Consumer’sSurplusp1x1*x1'x1LostCSp1(x1),inverseordinarydemandcurveforcommodity1.p1p1'Consumer’sSurplusp1x1*x1'x1LostCS1'1pp11*1dp)p(xCSx1*(p1),theconsumer’sordinarydemandcurveforcommodity1.p1p1'measuresthelossinConsumer’sSurplus.TwoadditionaldollarmeasuresofthetotalutilitychangecausedbyapricechangeareCompensatingVariationandEquivalentVariation.CompensatingVariationandEquivalentVariationp1rises.Q:Whatistheleastextraincomethat,atthenewprices,justrestorestheconsumer’soriginalutilitylevel?A:TheCompensatingVariation.CompensatingVariationCompensatingVariationx2x1x1'u1x2'p1=p1’p2isfixed.mpxpx11122'''Com