人大中级微观经济学讲义IM Course 3

ChoiceandDemandRecap•Consumer’soptimalchoices:–(x1*,x2*)satisfiestwoconditions:–(a)thebudgetisexhausted;p1x1*+p2x2*=m–(b)tangency:theslopeofthebudgetconstraint,-p1/p2,andtheslopeoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).x1x1x2MRS=-2,-P1/P2=-1MeaningoftheTangencyCondition•Consumer’smarginalwillingnesstopayequalsthemarketexchangerate.•Supposeataconsumptionbundle(x1,x2),MRS=-2,-P1/P2=-1–Theconsumeriswillingtogiveup2unitofx2toexchangeforanadditionalunitofx1–Themarketallowshertogiveuponly1unitofx2toobtainanadditionalx1•(x1,x2)isnotoptimalchoice•Shecanbebetteroffincreasingherconsumptionofx1.SolvingUtilityMaximization•Solvefor2simultaneousequations.–Tangency–Budgetconstraint•Transformingthebudgetconstraintandsubstitutingintoutilityfunction,wecanobtainanunconstrainedoptimizationproblem.•TheLagrangianmultipliermethod,i.e.,constrainedoptimizationincalculus.MarshallianDemandfunction•Demandfunctiongivestheoptimalamountofeachgoodasafunctionofthepricesandincomefacedbytheconsumer.Wewritetheordinarydemandfunctionas(,,)(,,)xyxyxxppmyyppmQuantityFunctionComputingOrdinaryDemands-aCobb-DouglasExample.SowehavediscoveredthatthemostpreferredaffordablebundleforaconsumerwithCobb-Douglaspreferences1212(,)abUxxxxis**1212(,),.()()()ambmxxabpabpRationalConstrainedChoice:Summary•Whenx1*0andx2*0and(x1*,x2*)exhauststhebudget,andindifferencecurveshaveno‘kinks’,theordinarydemandsareobtainedbysolving:•(a)p1x1*+p2x2*=y•(b)theslopesofthebudgetconstraint,-p1/p2,andoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).RationalConstrainedChoice•Butwhatifx1*=0?•Orifx2*=0?•Ifeitherx1*=0orx2*=0thentheordinarydemand(x1*,x2*)isatacornersolution(角点解)totheproblemofmaximizingutilitysubjecttoabudgetconstraint.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Budgetline:Slope=-p1/p2withp1p2.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1p2.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2xyp22*x10*MRS=-1Slope=-p1/p2withp1p2.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2xyp11*x20*MRS=-1Slope=-p1/p2withp1p2.ExamplesofCornerSolutions--thePerfectSubstitutesCaseSowhenU(x1,x2)=x1+x2,themostpreferredaffordablebundleis(x1*,x2*)where0,py)x,x(1*2*1and2*2*1py,0)x,x(ifp1p2ifp1p2.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1=p2.yp1yp2ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2Allthebundlesintheconstraintareequallythemostpreferredaffordablewhenp1=p2.yp2yp1ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Whichisthemostpreferredaffordablebundle?ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2ThemostpreferredaffordablebundleExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2ThemostpreferredaffordablebundleNoticethatthe“tangencysolution”isnotthemostpreferredaffordablebundle.Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=0U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=-MRS=0U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=-MRS=0MRSisundefinedU(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Whichisthemostpreferredaffordablebundle?Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1ThemostpreferredaffordablebundleExamplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)p1x1*+p2x2*=mExamplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)p1x1*+p2x2*=m(b)x2*=ax1*Examplesof‘Kinky’Solutions--thePerfectComplementsCase(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitutionfrom(b)forx2*in(a)givesp1x1*+p2ax1*=mwhichgives.appamx;appmx21*221*1Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1xmpap112*xampap212*ChoosingTaxes:VariousTaxes•Quantitytax:onx:(p+t)x•Valuetax:onpx:(1+t)px–Alsocalledadvaloremtax•Lumpsumtax:T•Incometax:–CanbeproportionalorlumpsumIncomeTaxvs.QuantityTax•Originalbudget:p1x1+p2x2=m•Afterquantitytax:(p1+t)x1+p2x2=m•Atoptimalchoice(x1*,x2*)–(p1+t)x1*+p2x2*=m(5.2)–Taxrevenue:R*=tx1*•Withanincometax,budgetis:p1x1+p2x2=m-tx1*Incomevs.QuantityTax•Proposition:(x1*,x2*)isaffordableunderincometax•Equivalentto:provethat(x1*,x2*)satisfiesbudgetconstraintunderincometax.•Or,budgetconstraintholdsatpoint(x1*,x2*).p1x1*+p2x2*=m-tx1*•Whichistrueaccordingto(5.2).•Itisnotanoptimalchoicebecausepricesaredifferent.•Conclusion:Theoptimalchoicemustbemorepreferredto(x1*,x2*)PropertiesofDemandF