微觀經(jīng)濟學(xué)（第八版-英文）課件：Individual and Market Demand

4.1

IndividualDemand4.2 IncomeandSubstitutionEffects4.3

MarketDemand4.4

ConsumerSurplus4.5

NetworkExternalities4.6

EmpiricalEstimationofDemand Appendix:DemandTheory—AMathematicalTreatment

IndividualandMarketDemandCHAPTEROUTLINEOuranalysisofdemandproceedsinsixsteps:

Webeginbyderivingthedemandcurveforanindividualconsumer.

Withthisfoundation,wewillexaminetheeffectofapricechangeinmore

detail.

Next,wewillseehowindividualdemandcurvescanbeaggregatedto

determinethemarketdemandcurve.

Wewillgoontoshowhowmarketdemandcurvescanbeusedtomeasure

thebenefitsthatpeoplereceivewhentheyconsumeproducts,aboveand

beyondtheexpenditurestheymake.

Wethendescribetheeffectsofnetworkexternalities—i.e.,whathappens

whenaperson’sdemandforagoodalsodependsonthedemandsof

other

people.6. Finally,wewillbrieflydescribesomeofthemethodsthateconomistsusetoobtainempiricalinformationaboutdemand.IndividualDemand4.1PriceChangesAreductioninthepriceoffood,withincomeandthepriceofclothingfixed,causestheconsumertochooseadifferentmarketbasket.Inpanel(a),thebasketsthatmaximizeutilityforvariouspricesoffood(pointA,$2;B,$1;D,$0.50)traceouttheprice-consumptioncurve.Part(b)givesthedemandcurve,whichrelatesthepriceoffoodtothequantitydemanded.(PointsE,G,andHcorrespondtopointsA,B,andD,respectively).EFFECTOFPRICECHANGESFIGURE4.1●

price-consumptioncurveCurvetracingtheutility-maximizingcombinationsoftwogoodsasthepriceofonechanges.●

individualdemandcurveCurverelatingthequantityofagoodthatasingleconsumerwillbuytoitsprice.TheIndividualDemandCurveTheindividualdemandcurvehastwoimportantproperties:1. Thelevelofutilitythatcanbeattainedchangesaswemovealongthe

curve.2.

Ateverypointonthedemandcurve,theconsumerismaximizingutilitybysatisfyingtheconditionthatthemarginalrateofsubstitution(MRS)offoodforclothingequalstheratioofthepricesoffoodandclothing.IncomeChanges●

income-consumptioncurveCurvetracingtheutility-maximizingcombinationsoftwogoodsasaconsumer’sincomechanges.EFFECTOFINCOMECHANGESFIGURE4.2Anincreaseinincome,withthepricesofallgoodsfixed,causesconsumerstoaltertheirchoiceofmarketbaskets.Inpart(a),thebasketsthatmaximizeconsumersatisfactionforvariousincomes(pointA,$10;B,$20;D,$30)traceouttheincome-consumptioncurve.Theshifttotherightofthedemandcurveinresponsetotheincreasesinincomeisshowninpart(b).(PointsE,G,andHcorrespondtopointsA,B,andD,respectively.)NormalversusInferiorGoodsANINFERIORGOODFIGURE4.3Anincreaseinaperson’sincomecanleadtolessconsumptionofoneofthetwogoodsbeingpurchased.Here,hamburger,thoughanormalgoodbetweenAandB,becomesaninferiorgoodwhentheincome-consumptioncurvebendsbackwardbetweenBandC.EngelCurvesENGLECURVESFIGURE4.4Engelcurvesrelatethequantityofagoodconsumedtoincome.In(a),foodisanormalgoodandtheEngelcurveisupwardsloping.In(b),however,hamburgerisanormalgoodforincomelessthan$20permonthandaninferiorgoodforincomegreaterthan$20permonth.●

EngelcurveCurverelatingthequantityofagoodconsumedtoincome.WecanderiveEngelcurvesforgroupsofconsumers.Thisinformationisparticularlyusefulifwewanttoseehowconsumerspendingvariesamongdifferentincomegroups.EXAMPLE4.1CONSUMEREXPENDITURESINTHEUNITEDSTATESTABLE4.1ANNUALU.S.HOUSEHOLDCONSUMEREXPENDITURESINCOMEGROUP(2009$)EXPENDITURES($)ON:LESSTHAN$10,00010,000–19,99920,000–29,99930,000–39,99940,000–49,99950,000–69,99970,000AND

ABOVEEntertainment1,0411,0251,5041,9702,0082,6114,733OwnedDwelling1,8802,0833,1174,0384,8476,47312,306RentedDwelling3,1723,3593,2283,2963,2952,9772,098HealthCare1,2221,9172,5362,6842,9373,4544,393Food3,4293,5294,4154,7375,3846,4209,761Clothing7999271,0801,2251,3361,6082,850EXAMPLE4.1CONSUMEREXPENDITURESINTHEUNITEDSTATESENGELCURVESFORU.S.CONSUMERSAverageper-householdexpendituresonrenteddwellings,healthcare,andentertainmentareplottedasfunctionsofannualincome.Healthcareandentertainmentarenormalgoods,asexpendituresincreasewithincome.Rentalhousing,however,isaninferiorgoodforincomesabove$30,000.FIGURE4.5SubstitutesandComplementsTwogoodsaresubstitutesifanincreaseinthepriceofoneleadstoanincreaseinthequantitydemandedoftheother.Twogoodsarecomplementsifanincreaseinthepriceofonegoodleadstoadecreaseinthequantitydemandedoftheother.Twogoodsareindependentifachangeinthepriceofonegoodhasnoeffectonthequantitydemandedoftheother.Thefactthatgoodscanbecomplementsorsubstitutessuggeststhatwhenstudyingtheeffectsofpricechangesinonemarket,itmaybeimportanttolookattheconsequencesinrelatedmarkets.Afallinthepriceofagoodhastwoeffects:

Consumerswilltendtobuymoreofthegoodthathasbecome

cheaperandlessofthosegoodsthatarenowrelativelymore

expensive.Thisresponsetoachangeintherelativepricesofgoods

iscalledthesubstitutioneffect.

Becauseoneofthegoodsisnowcheaper,consumersenjoyan

increaseinrealpurchasingpower.Thechangeindemandresulting

fromthischangeinrealpurchasingpoweriscalledtheincomeeffect.IncomeandSubstitutionEffects4.2SubstitutionEffectsubstitutioneffectChangeinconsumptionofagoodassociatedwithachangeinitsprice,withthelevelofutilityheldconstant.IncomeEffectincomeeffectChangeinconsumptionofagoodresultingfromanincreaseinpurchasingpower,withrelativepricesheldconstant.TotalEffect(F1F2)=SubstitutionEffect(F1E)+IncomeEffect(EF2)InFigure4.6,thetotaleffectofachangeinpriceisgiventheoreticallybythesumofthesubstitutioneffectandtheincomeeffect:INCOMEANDSUBSTITUTIONEFFECTS:NORMALGOODAdecreaseinthepriceoffoodhasbothanincomeeffectandasubstitutioneffect.TheconsumerisinitiallyatA,onbudgetlineRS.Whenthepriceoffoodfalls,consumptionincreasesbyF1F2astheconsumermovestoB.ThesubstitutioneffectF1E(associatedwithamovefromAtoD)changestherelativepricesoffoodandclothingbutkeepsrealincome(satisfaction)constant.TheincomeeffectEF2(associatedwithamovefromDtoB)keepsrelativepricesconstantbutincreasespurchasingpower.FoodisanormalgoodbecausetheincomeeffectEF2ispositive.FIGURE4.6INCOMEANDSUBSTITUTIONEFFECTS:INFERIORGOODFIGURE4.7TheconsumerisinitiallyatAonbudgetlineRS.Withadecreaseinthepriceoffood,theconsumermovestoB.Theresultingchangeinfoodpurchasedcanbebrokendownintoasubstitutioneffect,F1E(associatedwithamovefromAtoD),andanincomeeffect,EF2(associatedwithamovefromDtoB).Inthiscase,foodisaninferiorgoodbecausetheincomeeffectisnegative.However,becausethesubstitutioneffectexceedstheincomeeffect,thedecreaseinthepriceoffoodleadstoanincreaseinthequantityoffooddemanded.UPWARD-SLOPINGDEMANDCURVE:THEGIFFENGOODFIGURE4.8Whenfoodisaninferiorgood,andwhentheincomeeffectislargeenoughtodominatethesubstitutioneffect,thedemandcurvewillbeupward-sloping.TheconsumerisinitiallyatpointA,but,afterthepriceoffoodfalls,movestoBandconsumeslessfood.BecausetheincomeeffectF2F1islargerthanthesubstitutioneffectEF2,thedecreaseinthepriceoffoodleadstoalowerquantityoffooddemanded.ASpecialCase:TheGiffenGoodGiffengoodGoodwhosedemandcurveslopesupwardbecause

the(negative)incomeeffectislargerthanthesubstitutioneffect.THEEFFECTSOFAGASOLINETAXEFFECTOFAGASOLINETAXWITHAREBATEFIGURE4.9Agasolinetaxisimposedwhentheconsumerisinitiallybuying1200gallonsofgasolineatpointC.Afterthetaxtakeseffect,thebudgetlineshiftsfromABtoADandtheconsumermaximizeshispreferencesbychoosingE,withagasolineconsumptionof900gallons.However,whentheproceedsofthetaxarerebatedtotheconsumer,hisconsumptionincreasessomewhat,to913.5gallonsatH.Despitetherebateprogram,theconsumer’sgasolineconsumptionhasfallen,ashashislevelofsatisfaction.EXAMPLE4.2●

marketdemandcurve Curverelatingthequantityofagood

thatallconsumersinamarketwillbuytoitsprice.MarketDemand4.3TABLE4.2DETERMININGTHEMARKETDEMANDCURVE(1)

PRICE($)(2)

INDIVIDUALA(UNITS)(3)

INDIVIDUALB(UNITS)(4)

INDIVIDUALC(UNITS)(5)

MARKETUNITS161016322481325326101840471150246FromIndividualtoMarketDemandSUMMINGTOOBTAINAMARKETDEMANDCURVEFIGURE4.10Themarketdemandcurveisobtainedbysummingourthreeconsumers’demandcurvesDA,DB,andDC.Ateachprice,thequantityofcoffeedemandedbythemarketisthesumofthequantitiesdemandedbyeachconsumer.Atapriceof$4,forexample,thequantitydemandedbythemarket(11units)isthesumofthequantitydemandedbyA(nounits),B(4units),andC(7units).Theaggregationofindividualdemandsintomarketbecomesimportantinpracticewhenmarketdemandsarebuiltupfromthedemandsofdifferentdemographicgroupsorfromconsumerslocatedindifferentareas.Twopointsshouldbenoted:

Themarketdemandcurvewillshifttotherightasmoreconsumersenter

themarket.

Factorsthatinfluencethedemandsofmanyconsumerswillalsoaffect

marketdemand.ElasticityofDemandDenotingthequantityofagoodbyQanditspricebyP,thepriceelasticityofdemandis(4.1)INELASTICDEMANDWhendemandisinelastic,thequantitydemandedisrelativelyunresponsivetochangesinprice.Asaresult,totalexpenditureontheproductincreaseswhenthepriceincreases.ELASTICDEMANDWhendemandiselastic,totalexpenditureontheproductdecreasesasthepricegoesup.

UNIT-ELASTICDEMANDCURVEFIGURE4.11ISOELASTICDEMANDWhenthepriceelasticityofdemandis?1.0ateveryprice,thetotalexpenditureisconstantalongthedemandcurveD.isoelasticdemandcurveDemandcurvewithaconstantpriceelasticity.TABLE4.3PRICEELASTICITYANDCONSUMEREXPENDITURESDEMANDIFPRICEINCREASES,EXPENDITURESIFPRICEDECREASES,EXPENDITURESInelasticIncreaseDecreaseUnitelasticAreunchangedAreunchangedElasticDecreaseIncreaseSpeculativeDemand●

speculativedemand Demanddrivennotbythedirectbenefitsoneobtainsfromowningorconsumingagoodbutinsteadbyanexpectationthatthepriceofthegoodwillincrease.EXAMPLE4.3THEAGGREGATEDEMANDFORWHEATDomesticdemandforwheatisgivenbytheequationQDD=1430?55PwhereQDDisthenumberofbushels(inmillions)demandeddomestically,andPisthepriceindollarsperbushel.ExportdemandisgivenbyQDE=1470?70PwhereQDEisthenumberofbushels(inmillions)demandedfromabroad.Toobtaintheworlddemandforwheat,wesettheleftsideofeachdemandequationequaltothequantityofwheat.Wethenaddtherightsideoftheequations,obtainingQDD+QDE=(1430?55P)+(1470?70P)=2900?125PEXAMPLE4.3THEAGGREGATEDEMANDFORWHEATTHEAGGREGATEDEMANDFORWHEATFIGURE4.12ThetotalworlddemandforwheatisthehorizontalsumofthedomesticdemandABandtheexportdemandCD.Eventhougheachindividualdemandcurveislinear,themarketdemandcurveiskinked,reflectingthefactthatthereisnoexportdemandwhenthepriceofwheatisgreaterthanabout$21perbushel.EXAMPLE4.4THEDEMANDFORHOUSINGTherearesignificantdifferencesinpriceandincomeelasticitiesofhousingdemandamongsubgroupsofthepopulation.TABLE4.4PRICEANDINCOMEELASTICITIESOFTHEDEMANDFORROOMSGROUPPRICEELASTICITYINCOMEELASTICITYSingleindividuals–0.100.21Married,headofhouseholdagelessthan30,1child–0.250.06Married,headage30–39,2ormorechildren–0.150.12Married,headage50orolder,1child–0.080.19Inrecentyears,thedemandforhousinghasbeenpartlydrivenbyspeculativedemand.Speculativedemandisdrivennotbythedirectbenefitsoneobtainsfromowningahomebutinsteadbyanexpectationthatthepricewillincrease.EXAMPLE4.5THELONG-RUNDEMANDFORGASOLINEWouldhighergasolinepricesreducegasolineconsumption?Figure4.13providesaclearanswer:Mostdefinitely.GASOLINEPRICESANDPERCAPITACONSUMPTIONIN10COUNTRIESThegraphplotspercapitaconsumptionofgasolineversusthepricepergallon(convertedtoU.S.dollars)for10countriesovertheperiod2008to2010.Eachcirclerepresentsthepopulationofthecorrespondingcountry.FIGURE4.13●

consumersurplus

Differencebetweenwhataconsumeriswillingtopayforagoodandtheamountactuallypaid.ConsumerSurplus4.4ConsumerSurplusandDemandConsumersurplusisthetotalbenefitfromtheconsumptionofaproduct,lessthetotalcostofpurchasingit.Here,theconsumersurplusassociatedwithsixconcerttickets(purchasedat$14perticket)isgivenbytheyellow-shadedarea:$6+$5+$4+$3+$2+$1=$21CONSUMERSURPLUSFIGURE4.14CONSUMERSURPLUSGENERALIZEDFIGURE4.15Forthemarketasawhole,consumersurplusismeasuredbytheareaunderthedemandcurveandabovethelinerepresentingthepurchasepriceofthegood.Here,theconsumersurplusisgivenbytheyellow-shadedtriangleandisequalto1/2

($20?$14)

6500=$19,500.APPLYINGCONSUMERSURPLUSConsumersurplushasimportantapplicationsineconomics.Whenaddedovermanyindividuals,itmeasurestheaggregatebenefitthatconsumersobtainfrombuyinggoodsinamarket.Whenwecombineconsumersurpluswiththeaggregateprofitsthatproducersobtain,wecanevaluateboththecostsandbenefitsnotonlyofalternativemarketstructures,butofpublicpoliciesthatalterthebehaviorofconsumersandfirmsinthosemarkets.THEVALUEOFCLEANAIRVALUINGCLEANERAIRFIGURE4.16EXAMPLE4.6Theyellow-shadedtrianglegivestheconsumersurplusgeneratedwhenairpollutionisreducedby5partsper100millionofnitrogenoxideatacostof$1000perpartreduced.Thesurplusiscreatedbecausemostconsumersarewillingtopaymorethan$1000foreachunitreductionofnitrogenoxide.Althoughthereisnoactualmarketforcleanair,peopledopaymoreforhouseswheretheairiscleanthanforcomparablehousesinareaswithdirtierair.●

networkexternality Wheneachindividual’sdemanddependsonthepurchasesofotherindividuals.NetworkExternalities4.5PositiveNetworkExternalitiesApositivenetworkexternalityexistsifthequantityofagooddemandedbyatypicalconsumerincreasesinresponsetothegrowthinpurchasesofotherconsumers.Ifthequantitydemandeddecreases,thereisanegativenetworkexternality.●

bandwagoneffect Positivenetworkexternalityinwhichaconsumerwishestopossessagoodinpartbecauseothersdo.POSITIVENETWORKEXTERNALITYFIGURE4.17Withapositivenetworkexternality,thequantityofagoodthatanindividualdemandsgrowsinresponsetothegrowthofpurchasesbyotherindividuals.Here,asthepriceoftheproductfallsfrom$30to$20,thebandwagoneffectcausesthedemandforthegoodtoshifttotheright,fromD40toD80.NEGATIVENETWORKEXTERNALITY:SNOBEFFECTFIGURE4.18NegativeNetworkExternalities●

snobeffect Negativenetworkexternalityinwhichaconsumerwishestoownanexclusiveoruniquegood.Thesnobeffectisanegativenetworkexternalityinwhichthequantityofagoodthatanindividualdemandsfallsinresponsetothegrowthofpurchasesbyotherindividuals.Here,asthepricefallsfrom$30,000to$15,000andmorepeoplebuythegood,thesnobeffectcausesthedemandforthegoodtoshifttotheleft,fromD2toD6.FACEBOOKEXAMPLE4.7Byearly2011,withover600millionusers,Facebookbecametheworld’ssecondmostvisitedwebsite(afterGoogle).AstrongpositivenetworkexternalitywascentraltoFacebook’ssuccess.TABLE4.3FACEBOOKUSERSYEARFACEBOOKUSERS(MILLIONS)HOURSPERUSERPERMONTH2004120055.5200612<120075022008100320093505.520105007Networkexternalitieshavebeencrucialdriversformanymoderntechnologiesovermanyyears.EmpiricalEstimationofDemand4.6TABLE4.6DEMANDDATAYEARQUANTITY(Q)PRICE(P)INCOME(I)2004424102005720102006817102007131717200816102720091515272010191220201120920201222520(4.2)Usingthedatainthetableandtheleastsquaresmethod,thedemandrelationshipis:

TheStatisticalApproachtoDemandEstimationPriceandquantitydatacanbeusedtodeterminetheformofademandrelationship.ButthesamedatacoulddescribeasingledemandcurveDorthreedemandcurvesd1,d2,andd3thatshiftovertime.ESTIMATINGDEMANDFIGURE4.19TheFormoftheDemandRelationship

Weoftenfinditusefultoworkwiththeisoelasticdemandcurve,inwhichthepriceelasticityandtheincomeelasticityareconstant.Whenwritteninitslog-linearform,anisoelasticdemandcurveappearsasfollows:

SupposethatP2representsthepriceofasecondgood—onewhichisbelievedtoberelatedtotheproductwearestudying.Wecanthenwritethedemandfunctioninthefollowingform:

Whenb2,thecross-priceelasticity,ispositive,thetwogoodsaresubstitutes;whenb2isnegative,thetwogoodsarecomplements.(4.3)(4.4)EXAMPLE4.8THEDEMANDFORREADY-TO-EATCEREALTheacquisitionofShreddedWheatcerealsofNabisco

byPostCerealsraisedthequestionofwhetherPost

wouldraisethepriceofGrapeNuts,orthepriceof

Nabisco’sShreddedWheatSpoonSize.

Oneimportantissuewaswhetherthetwobrandswere

closesubstitutesforoneanother.Ifso,itwouldbemore

profitableforPosttoincreasethepriceofGrapeNuts

afterratherthanbeforetheacquisitionbecausethelost

salesfromconsumerswhoswitchedawayfromGrapeNutswouldberecoveredtotheextentthattheyswitchedtothesubstituteproduct.ThesubstitutabilityofGrapeNutsandShreddedWheatcanbemeasuredbythecross-priceelasticityofdemandforGrapeNutswithrespecttothepriceofShreddedWheat.Oneisoelasticdemandequationappearedinthefollowinglog-linearform:ThedemandforGrapeNutsiselastic,withapriceelasticityofabout?2.Incomeelasticityis0.62.thecross-priceelasticityis0.14.Thetwocerealsarenotveryclosesubstitutes.

InterviewandExperimentalApproachesto

DemandDeterminationAnotherwaytoobtaininformationaboutdemandisthroughinterviews.

Thisapproach,however,maynotsucceedwhenpeoplelackinformationorinterestorevenwanttomisleadtheinterviewer.Indirectmarketingexperiments,actualsalesoffersareposedtopotentialcustomers.Anairline,forexample,mightofferareducedpriceoncertainflightsforsixmonths,partlytolearnhowthepricechangeaffectsdemandforflightsandpartlytolearnhowcompetitorswillrespond.Alternatively,acerealcompanymighttestmarketanewbrand,withsomepotentialcustomersbeinggivencouponsranginginvaluefrom25centsto$1perbox.Theresponsetothecouponoffertellsthecompanytheshapeoftheunderlyingdemandcurve.Directexperimentsarereal,nothypothetical,butevenso,problemsremain.Thewrongexperimentcanbecostly,andthefirmcannotbeentirelysurethattheseincreasesresultedfromtheexperimentalchange;otherfactorsprobablychangedatthesametime.Moreover,theresponsetoexperiments—whichconsumersoftenrecognizeasshort-lived—maydifferfromtheresponsetopermanentchanges.Finally,afirmcanaffordtotryonlyalimitednumberofexperiments.AppendixtoChapter4DemandTheory—AMathematicalTreatmentSuppose,forexample,thatBob’sutilityfunctionisgivenbyU(X,Y)=logX+logY,whereXisusedtorepresentfoodandYrepresentsclothing.Inthatcase,themarginalutilityassociatedwiththeadditionalconsumptionofXisgivenbythepartialderivativeoftheutilityfunctionwithrespecttogoodX.Here,MUX,representingthemarginalutilityofgoodX,isgivenbyUtilityMaximization

Theconsumer’soptimizationproblemmaybewrittenas

(A4.1)subjecttotheconstraintthatallincomeisspentonthetwogoods:

(A4.2)StatingtheProblemFirst,wewritetheLagrangianfortheproblem.TheMethodofLagrangeMultipliers

Notethatwehavewrittenthebudgetconstraintas(A4.3)●

methodofLagrangemultipliers

Techniquetomaximizeorminimizeafunctionsubjecttooneormoreconstraints.

●

Lagrangian

Functiontobemaximizedorminimized,plusavariable(theLagrangemultiplier)multipliedbytheconstraint.DifferentiatingtheLagrangianWechoosevaluesofXandYthatsatisfythebudgetconstraint,thenthesecondterminequation(A4.3)willbezero.BydifferentiatingwithrespecttoX,Y,andlandthenequatingthederivativestozero,wecanobtainthenecessaryconditionsforamaximum.

(A4.4)SolvingtheResultingEquationsThethreeequationsin(A4.4)canberewrittenas

(A4.5)TheEqualMarginalPrincipleWecombinethefirsttwoconditionsabovetoobtaintheequalmarginalprinciple:

Tooptimize,theconsumermustgetthesameutilityfromthelastdollarspentbyconsumingeitherXorY.Tocharacterizetheindividual’soptimuminmoredetail,wecanrewritetheinformationin(A4.5)toobtain

(A4.6)(A4.7)MarginalRateofSubstitution(A4.8)IfU*isafixedutilitylevel,theindifferencecurvethatcorrespondstothatutilitylevelisgivenby

Rearranging,MarginalUtilityofIncomeTheLagrangemultiplierlrepresentstheextrautilitygeneratedwhen

thebudgetconstraintisrelaxed.Toshowhowtheprincipleworks,wedifferentiatetheutilityfunctionU(X,Y)totallywithrespecttoI:

(A4.9)Becauseanyincrementinincomemustbedividedbetweenthetwogoods,itfollowsthat

(A4.10)Substitutingfrom(A4.5)into(A4.9),weget

(A4.11)Substitutingfrom(A4.10)into(A4.11),weget

(A4.12)ThustheLagrangemultiplieristheextrautilitythatresultsfromanextradollarofincome.AnExample●

Cobb-Douglasutilityfunction UtilityfunctionU(X,Y

)=XaY1?a,

whereXandYaretwogoodsandaisaconstant.

TheCobb-Douglasutilityfunctioncanberepresentedintwoforms:andTofindthedemandfunctionsforXandY,giventheusualbudgetconstraint,wefirstwritetheLagrangian:

NowdifferentiatingwithrespecttoX,Y,andlandsettingthederivativesequaltozero,weobtain

Combiningtheseexpressionswiththelastcondition(thebudgetconstraint)givesus

Thefirsttwoconditionsimplythat

(A4.13)(A4.14)

DualityinConsumerTheory●

duality Alternativewayoflookingattheconsumer’sutility

maximizationdecision:Ratherthanchoosingthehighestindifferencecurve,givenabudgetconstraint,theconsumerchoosesthelowestbudgetlinethattouchesagivenindifferencecurve.Minimizingthecostofachievingaparticularlevelofutility:Minimize

subjecttotheconstraintthat

ThecorrespondingLagrangianisgivenby

Differentiating

withrespecttoX,Y,andμandsettingthederivativesequaltozero,wefindthefollowingnecessaryconditionsforexpenditureminimization:(A4.15)

and

Bysolvingthefirsttwoequations,