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1、AdvancedMicroeconomicsTopic3:ConsumerDemandPrimaryReadings:DL–Chapter5;JR-Chapter3;Varian,Chapters7-9.3.1MarshallianDemandFunctionsLetXbetheconsumer'sconsumptionsetandassumethattheX=Rm+.Foragivenpricevectorpofcommoditiesandthelevelofincomey,theconsumertriestos
2、olvethefollowingproblem:maxu(x)subjecttop×x=yxÎX·Thefunctionx(p,y)thatsolvestheaboveproblemiscalledtheconsumer'sdemandfunction.·ItisalsoreferredastheMarshalliandemandfunction.OthercommonlyknownnamesincludeWalrasiandemandcorrespondence/function,ordinarydemandfu
3、nctions,marketdemandfunctions,andmoneyincomedemands.·Thebindingpropertyofthebudgetconstraintattheoptimalsolution,i.e.,p×x=y,istheWalras’Law.·Itiseasytoseethatx(p,y)ishomogeneousofdegree0inpandy.Examples:(1)Cobb-DouglasUtilityFunction:Fromtheexampleinthelastlec
4、ture,theMarshalliandemandfunctionsare:where.(2)CESUtilityFunctions:ThentheMarshalliandemandsare:wherer=r/(r-1).AndthecorrespondingindirectutilityfunctionisgivenbyLetusderivetheseresults.Notethattheindirectutilityfunctionistheresultoftheutilitymaximizationprobl
5、em:DefinetheLagrangianfunction:11TheFOCsare:Eliminatingl,wegetSotheMarshalliandemandfunctionsare:withr=r/(r-1).Sothecorrespondingindirectutilityfunctionisgivenby:3.2OptimalityConditionsforConsumer’sProblemFirst-OrderConditionsTheLagrangianfortheutilitymaximiza
6、tionproblemcanbewrittenasL=u(x)-l(p×x-y).Thenthefirst-orderconditionsforaninteriorsolutionare:(1)11Rewritingthefirstsetofconditionsin(1)leadstowhichisadirectgeneralizationofthetangencyconditionfortwo-commoditycase.x2u(x1,x2=uslope=-MRS21slope=-p1/p2x1Sufficien
7、cyofFirst-OrderConditionsProposition:Supposethatu(x)iscontinuousandquasiconcaveonRm+,andthat(p,y)>0.Ifuifdifferentiableatx*,and(x*,l*)>0solves(1),thenx*solvetheconsumer'sutilitymaximizationproblematpricespandincomey.Proof.Wewillusethefollowingfactwithoutaproof
8、:·Forallx,x'³0suchthatu(x')³u(x),ifuisquasiconcaveanddifferentiableatx,thenÑu(x)(x'-x)³0.NowsupposethatÑu(x*)existsand(x*,l*)>0solves(1).Then,Ñu(x*)=l*p,p×x*=y.Ifx*isnotutility-max
