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MACROECONOMICFinalExam,May8thYouhave115minutesforthiam.Neitherbooks,norclassnotesarepermitted.Please,readthroughthewholeexambearting.Notethatquestionsarenotequallyweighted,andpointswithinaquestionmayhavedi?erentweight.The umscoreis100.Ifyouthinkyouneedtomakeadditionalassumptionstoanswersomequestions,statethemclearlyinyourQuestion1:Aggregation?(30Consideraneconomyinhabitedby2typesofinfiniylivedagents,indexedbyi=1,2.Denotebyμithemeasureofagentsiandnormalizethetotalmeasureofagentstoone,i.e.μμi
=1.Preferencesaretimeseparable,definedoverstreamsofconsumption,givenP U t=0βtu(cit),wheretheperiodutilityfunctionisu(cit)= .Theaggregate technologyisYt=f(Kt),andKt+1=(1?δ)Kt+Yt?Ct.Therearecompetitivespotmarketsforthefinalgood,pricedatpt,andcompletefinancialmarkets.Consideranrrow-aebreuequilibriumforthiseconomy,wheretheinitialendowmentsofthetwotypesarea0,a0 (20pts)Doesthiseconomyadmitarepresentativeagentrepresentation?Proveyouranswerformally.(10pts)Stateasetofconditions,asgeneralaspossible,underwhichaneconomywithheterogeneousconsumersadmitsasingle-agentrepresentation.Question2:Economywithtaxevasion(40Considerastationaryeconomypopulatedbyacontinuumofmeasureoneofinfini ylived,ex-anteequalagentswithpreferencesoversequencesofconsumptionandleisuregivenby βtu(cit)+eψitv(1?hit)Agentsfaceindividualshockstotheirpreferenceforleisureψit,whichfollowthestochasticprocessψit=ρψi,t?1+ηit,withηit~N(0,ση).Agentscansavebutcannotborrow.ProductiontakescethroughtheaggregatetechnologyCt+Kt+1?(1?δ)Kt= whereCt,KtandHtare,respectively,aggregateconsumption,aggregatecapital,andaggregatehoursattimet.Laborandassetmarketsarecompetitiveandclear,everyperiod,withpriceswtandrt,respectively.Theernmenttaxescapital eatafixedflatrateτ.Taxrevenuesarereturnedtoagentsastax-exemptlump-sumtransfersb.Householdscanevadetaxesbydecidingeveryperiodtthefractionofcapital eφittodeclareintheirtaxreturn,i.e.,thefractionof eonwhichtheypaytaxes.Letxitbethetotalundeclaredtaxesattimet.TURNTheernment,knowingthatagentsmayhaveevadedtaxesattimet?1,attimetcanmonitorandperfectlyverifythepastperiodindividualtaxreturns.Letπbetheprobabilitythat,attimet,thetimet?1taxreturnedofahouseholdissubjecttomonitoring.Thehouseholdfindsoutwhetherhert?1periodtaxreturnismonitoredatthebeginningofperiodt,i.e.,beforeconsumptiondecisionsaretaken.Intheeventthehouseholdiscaught,attimetthetaxagencygivesherafineequaltoz(xi,t?1),wherexi,t?1isthetaxamountduefromthepastperiod,withz(0)=0andz0(xi,t?1)>1.(10pts)Writedowntheproblemofthehouseholdinrecursiveform,makingexplicittheindividualandtheaggregatestatevariables.(10pts)Writedowntheindividualfirst-ordernecessaryconditionthatcharacterizestheoptimaltaxevasionchoice.(10pts)DefineastationaryrecursivecompetitiveequilibriumforthisSupposethat,inordertoverifyπindividuals’taxreturns,theernmentfacesanadmin-istrativemonitoringexpenditurem(π),withm(0)=0,m0(π)>0,m00(π)>0.(10pts)Writedowntheproblemofabenevolenternmentthatchoosesthefractionπofagentstomonitorinorderto theernmentfacesinsettingπ?Question3:Stochasticgrowthmodel pletemarkets(30Consideraneconomypopulatedbyacontinuumofmeasureoneofinfiniylived,ex-anteequalagentswithpreferencesoversequencesofconsumptiongivenby βtu(cit)Agentsfaceindividualuninsurableendowmentshocksεittolaborproductivity,whereεit{ε1,...,εN}. ocess ernedbytheMarkovchainπε(ε0,ε)1Pr{εi,t+1=ε0|εit=ε}Production cethroughtheaggregatetechnologyYt=ztKαH whereKtand are,respectively,aggregatecapitalandaggregatee?ciencyunitsoflaborattimet.Capitaldepreciatesatrateδ.Aggregatetotalfactorproductivityzt∈{zb,zg}followsaMarkovprocessπz(z0,z)=Pr{zt+1=z0|zt=z}.Agentscanholdtwoassets:capitalaitandarisk-bondbit.Letqtbethepriceofbondattimet:buyingbunitsoftherisk-bondatpriceqtatt?imetg¢aranteesbunitsfinalgoodnextperiod.Assumethatagentsfaceborrowinglimits?aˉ,?ˉbonthetwoLaborandassetmarketsarecompetitiveandclear,everyperiod,withpriceswt(perunit)andrt,(10pts)Writedowntheproblemofthehouseholdinrecursiveform,makingexplicittheindividualandtheaggregatestatevariables.(10pts)Definearecursivecompetitiveequilibriumforthis(10pts)Isthisaneasyorhardproblemtosolve?Exinyouranswerandoutlineanumericalalgorithmthatsolvesfortheequilibriumallocationsandprices.ENDOF宏觀經(jīng)濟理論期末考試,2006年5月8i1,2μii磷μi=1i=1P∞itUt=0βtu(cit),其中周期效用函數(shù)為u(citc11σσYt=f(Kt),Kt+1=(1.δ)Kt+YtCT。最終商品有競爭性的現(xiàn)貨市場(定價為pt)和完整的金融市場??紤]該經(jīng)濟體的-魯均衡,其中兩種類型的初始稟賦為。1,a20a。XhiE0βtu(cit)+eψitv(1.hit)給出。時間=0N(0,σηCt+Kt+1(1.δ)KttCt、KtHtt期都是競爭性的和清晰的,價格分別為wt和rt。xitt1πt,t.11申報表在t期開始時(即做出消費決策之前)進(jìn)行監(jiān)控。如果該家庭被抓,在時間t,稅務(wù)機構(gòu)給她一個等于z(xi,t.1)的.ne,其中xi,t.1是過去一段時間應(yīng)繳的稅額,z(0)=0且z0(xi,t.1)>1。10(10題。政府在設(shè)定π時面臨哪些權(quán)衡?
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