微观经济学第七版(平狄克版)课后习题答案 第四章.pdf

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1、Chapter4:AppendixCHAPTER4APPENDIXDEMANDTHEORY–AMATHEMATICALTREATMENTEXERCISES1.Whichofthefollowingutilityfunctionsareconsistentwithconvexindifferencecurvesandwhicharenot?a.U(X,Y)=2X+5Yb.U(X,Y)=(XY)0.5c.U(X,Y)=Min(X,Y),whereMinistheminimumofthetwovaluesofXandY.Indifferencemapsforthet

2、hreeutilityfunctionsarepresentedinFigures4A.1(a),4A.1(b),and4A.1(c).Thefirstisaseriesofstraightlines,thesecondisaseriesofhyperbolasandthethirdisaseriesofL-shapedcurves.Onlythesecondutilityfunctionhasstrictlyconvexindifferencecurves.Tographtheindifferencecurveswhichrepresenttheprefer

3、encesgivenbyU(X,Y)=2X+5Y,setutilityequaltosomelevel,U0,andsolveforYtogetU02Y=−X.55Sincethisistheequationforastraightline,theindifferencecurvesarelinearwithU02interceptandslope−.Thegraphshowsthreeindifferencecurvesforthree55differentvaluesofU,whereU0

4、igure4A.1(a)69Copyright©2009PearsonEducation,Inc.PublishingasPrenticeHall.Chapter4:AppendixTographtheindifferencecurvesthatrepresentthepreferencesgivenby0.5U(X,Y)=(XY),setutilityequalagivenlevelU0andsolveforYtoget2U0Y=.XByplugginginafewvaluesforXandsolvingforY,youwillbeabletographth

5、eindifferencecurveforutilityvalueU0,whichisillustratedinFigure4A.1(b),alongwiththeindifferencecurveforalargerutilityvalue,U1.YU1U0XFigure4A.1(b)TographtheindifferencecurveswhichrepresentthepreferencesgivenbyU(X,Y)=Min(X,Y),firstnotethatutilityfunctionsofthisformresultinindifferencec

6、urvesthatareL-shapedandrepresentacomplementaryrelationshipbetweenXandY.Inthiscase,foranygivenlevelofutilityU0,theminimumvalueofXandYwillalsobeequaltoU0.IfXincreasesbutYdoesnot,utilitywillnotchange.IfbothXandYchange,thenutilitywillchange,andwewillmovetoadifferentindifferencecurve.See

7、thefollowingtablewhichillustrateshowtheutilityvaluedependsontheamountsofXandYintheconsumptionbundle.XYU101010101210121212121111811889870Copyright©2009PearsonEducation,Inc.PublishingasPrenticeHall.Chapter4:AppendixYU1U1U0UoU0U1XFigure4A.1(c)2.Showthatthetwoutilityfunctionsgivenbelowg

8、eneratetheidentical