中级微观经济学习题

1CentralUniversityofFinanceandEconomicsSchoolofEconomicsIntermediateMicroeconomics,Fall2017HW1（Date:Friday,November17,2017）Question1.True/False/Uncertain(6points)Takethenon-italicizedstatementsasgiven.IndicatewhethertheitalicizedstatementsareTRUE,FALSE,orUNCERTAINandexplainwhy.a)Ifutilityisgivenby12Uxx,theconsumermaystopconsumingonegoodaltogetherinresponsetoaslightchangeinprice.b)Supposedemandisgivenby10QP.Supposethereisonlyonefirmservingthemarket.Itsrevenuesaregivenby*()PQP.Increasingpriceincreasesrevenue.Question2.(12points)Supposeaconsumer’spreferencesarerepresentedbytheutilityfunction11(,)Uxyxy.Thepriceforgoodxisxp,thepriceforgoodyisyp,andtheconsumerhasmtospend.a)Derivetheconsumer’sdemandforgoodxasafunctionofxp,yp,andm.(Note:Thesolutionisinterior.)b)Calculatethepriceelasticityofdemandforgoodx.c)Calculatetheincomeelasticityforgoodx.d)Isthegoodxanormalgoodoraninferiorgood?Question3.(12points)Supposeaconsumerhasademandfunctionoftheform:(,,,)xyzxpppm2111()420400xzxmmppp.Supposethatwhilethepriceofgoodxdecreasesfrom4to2,incomeandotherpricesremainconstantat100,3ympand2zp.Findthechangeinquantitydemanded.FindthemagnitudesoftheSlutskysubstitutioneffectandincomeeffect.Question4.(15points)Daveisdecidinghowmuchtoworkinthecomingyear.Hederivesutilityfromconsumption,C,buthealsoreallylikestakingleisuretimeL.Hemustdividehis2availablehoursbetweenworkandleisureforeveryhourofleisurehetakeshemustworkonefewerhours.Thefunctionthatdescribeshispreferencesisgivenby:3144(,)UCLCLHecanearnawageofw,andsupposethepriceofconsumptionisgivenbyp=1.Finally,Davereceivessomenon-wageincome,m,whichdoesnotdependonthenumberofhoursheworks.a)WhatisDave’sfullincomeifhecanwork2000hoursinayear?Writedownhisbudgetconstraint.b)SolveDave’sutilitymaximizationproblemandwritedownhisoptimalconsumptionofCandL.HowdoesDave’sconsumptiondependonwandm?Isleisureanormalgood?c)SupposeDave’swagesareinitiallygivenbyw=10andhisnon-laborincomebym=100.Findhisoptimallevelsofleisureandconsumption.IfDave’swageincreasestow=20,whatarehisnewlevelsofleisureandconsumption?d)Graphicallydepicttheincomeandsubstitutioneffectsonleisurehoursassociatedwiththechangeinwagefrom10to20.Dotheincomeandsubstitutioneffectshavethesamesign?Whatistheintuitionforthis?Question5.(30points)Peterspendshisincomeonbooks(1x)andCD(2x).a)Supposethepriceofabookis110p,thepriceofaCDis25p,andPeter’sdailybudgetis40m.ShowgraphicallyPeter’sbudgetconstraint,markinghisrealincomesintermsofbooksandCDs.Onthesamegraph,showhowhisbudgetsetisaffectedbyagiftof2CDs(assumethathecanalwaysdisposethegift).b)Peter’spreferencesaregivenbythefollowingutilityfunction1212(,)2lnUxxxx-FindPeter’smarginalrateofsubstitution(MRS)foranybundle12(,)xx(givetheformulaforMRS).-WhatisthevalueofMRSatconsumptionbundle(5,8)(giveanumber)?-SupposePeter“consumes”5booksand8CDsandonetakesaway0.0001ofabook.WhatiscompensationintermsofCDsissufficienttomakePeterindifferent?-Depicttheindifferencecurvemapinacommodityspace.Marktheslopeoftheindifferencecurveatbundle(5,8).c)Given1212(,)2lnUxxxx,writedownthenecessaryconditionsfortheoptimalchoice(assumeinteriorsolution).Provideeconomicintuitionbehindthetwoconditions.3d)Usingthetwoconditionsfromc)findtheoptimalconsumptionlevelsofbothtypesofcommodities12(,)xxfor:-1210,5,ppand40m(givetwonumbers)Andafterthepriceofabookincreases:-1230,5,ppand40m(givetwonumbers)Iseachofthesolutionsinterior?Illustratethechangeonthegraph.e)Ind),isthemarginalutilityofthelastdollarinvestedinbooksandCDsequal?(Findtwonumbersfortheparametersbeforeandafterthechange.)Incasetheyarenot,explainwhynotequalizingthemarginalutilityofthelastdollarisconsistentwithoptimum.Question6.(10points)Consideramodelwithtwoperiods,eachwithonecompositeconsumptiongood.Pricesarethestableandnormalizedto1.Supposeaconsumerisendowedwithincome20inthefirstperiodand60inthesecondperiod.Theconsumerreceivesa10%interestrateonsavingsandissubjecttoa20%interestrateonborrowing.Thebanksprohibittheconsumerfromborrowingmorethan20.a)Findthisconsumer’sintertemporalbudgetconstraint.b)Carefullygraphthebudgetset.Labelallintercepts.Question7.(15points)TheStone-Gearyutilityfunction:12121122(,)()()UxxxxItisanimportantgeneralizationoftheCobb-DouglasUtilityFunction.Considerthespecialcasewhere11,21,10and20.Thepriceforgood1xis1p,thepriceforgood2xis2p,andtheconsumerhasmtospend.a)Graphtheindifferencecurvesassociatedwiththispreference.b)Derivetheconsumer’sdemandforgood1xasafunctionof1p,2p,andm.c)Aregood1xand2xgrosssubstitutes,grosscomplements,orunrelatedgoods?