消费者最优选择(3)

ChapterFiveChoice消费者最优选择WhereAreWeDoinginThisChapter?Aftermodelingaconsumer’schoicesetandhispreference(representedbyutilityfunctions),wenowputthemtogetherandmodelhowhe/shemakesoptimalchoice.Inmathematicalterms,thisisaconstrainedmaximizationproblem;Ineconomics,thisisarationalchoiceproblem.RationalConstrainedChoiceAffordablebundlesx1x2MorepreferredbundlesRationalConstrainedChoiceThemostpreferredaffordablebundleiscalledtheconsumer’sORDINARYDEMANDatthegivenpricesandbudget.Ordinarydemandswillbedenotedbyx1*(p1,p2,m)andx2*(p1,p2,m).RationalConstrainedChoiceWhenx1*0andx2*0thedemandedbundleisINTERIOR.Ifbuying(x1*,x2*)costs$mthenthebudgetisexhausted.RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.(a)(x1*,x2*)exhauststhebudget;p1x1*+p2x2*=m.RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.(b)Theslopeoftheindiff.curveat(x1*,x2*)equalstheslopeofthebudgetconstraint.RationalConstrainedChoice(x1*,x2*)satisfiestwoconditions:(a)thebudgetisexhausted;p1x1*+p2x2*=m(b)theslopeofthebudgetconstraint,-p1/p2,andtheslopeoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).ComputingOrdinaryDemands-aCobb-DouglasExample.SupposethattheconsumerhasCobb-Douglaspreferences.Uxxxxab(,)1212ComputingOrdinaryDemands-aCobb-DouglasExample.SupposethattheconsumerhasCobb-Douglaspreferences.ThenUxxxxab(,)1212MUUxaxxab11112MUUxbxxab22121ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSisMRSdxdxUxUxaxxbxxaxbxabab211211212121//.ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSisAt(x1*,x2*),MRS=-p1/p2soMRSdxdxUxUxaxxbxxaxbxabab211211212121//.*22*11xpxpba(A)ComputingOrdinaryDemands-aCobb-DouglasExample.(x1*,x2*)alsoexhauststhebudgetsopxpxm1122**.(B)ComputingOrdinaryDemands-aCobb-DouglasExample.SowehavediscoveredthatthemostpreferredaffordablebundleforaconsumerwithCobb-DouglaspreferencesUxxxxab(,)1212is(,)(),().**()xxamabpbmabp1212ComputingOrdinaryDemands-aCobb-DouglasExample.x1x2xamabp11*()xbmabp22*()Uxxxxab(,)1212RationalConstrainedChoice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*)isatacornersolutiontotheproblemofmaximizingutilitysubjecttoabudgetconstraint.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1p2.ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2xyp11*x20*MRS=-1Slope=-p1/p2withp1p2.(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=ax1ThemostpreferredaffordablebundleExamplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)p1x1*+p2x2*=m(b)x2*=ax1*Summary:ThreeStepstoFindtheOptimalChoiceoftheConsumerStep1:Drawthebudgetset;Step2:Drawtheindifferencecurves;Step3:Locatethepointofoptimalchoiceandcalculatethesolution.