内容讲义教程final

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MACROECONOMICTHEORYII

FinalExam

Youhave2hoursforthiam.Neitherbooks,norclassnotesarepermitted.Please,readthroughthewholeexambearting.Notethatquestionsarenotequallyweighted,andpointswithinaquestionmayhavedifferentweight.Theumscoreis

100.Ifyouthinkyouneedtomakeadditionalassumptionstoanswersomequestions,statethemclearlyinyouranswer.

Question1:Life-cycleInequality(20pts)

Consideracohortofindividualsbornatage=0wholiveuntilageEachindividual

discountsthefutureatrate0andhasperiodutilitydefinedoverconsumption

andhoursworked:

1− 1+

()= −

1− 1+

Let,thedistasteforwork,beheterogeneousacrossindividualsbutfixedoverthelife-cycle.Individualsareendowedwithheterogeneouslevelsofinitialwealth0and,throughouttheirlife,theyaresubjecttoidiosyncraticproductivityshocksdrawnfromaMarkovchainΓ(+1)with∈{¯1¯}forall().

(10pts)Assumecompletemarkets,andwritedownnt’sproblem.Arguethat,withthismarketstructure,consumptioninequalityacrossindividualsofthiscohortwillnotgrowoverthelife-cycle.

(10pts)Supposethatisstochastic.Can,inthiscase,thecomplete-marketsallocationbeconsistentwithinequalitygrowinerthelife-cycle?And,whatistheimplicationforthegrowthofinequalityinhoursworkedoverthelife-cycle?

Question2:UnemploymentInsurance(40pts)

Consideraneconomywherehouseholdshaveperiodutility(1−)whereisconsumptionandishoursworkedwith000and0Householdscanbeineitheroneoftwostates∈{}denotingemployment()andunemployment().Theemployment-statusshockfollowsaMarkovchainΓ(0).Employedworkers

supplyhourstothemarketandreceivewagesperhour.Unemploymentmeansbeingoutofwork(=0)withnoearnings.Theleft-wingernmentinpowerrunsanunemploymentinsurance(UI)systemfortheunemployed.Everyperiodofunemployment,anindividualreceives·˜,where˜areearnings(hourlywagetimeshours)inherlastjoband0istherecementrateofUI.

Thereisnoaggregateuncertainty.Householdscansavethroughrisk-claimstophysicalcapital,andborrowuptothenaturallimit(whichiswhat?).Theaggregate

productiontechnologyisastandardCRSproductionfunctionincapitalandtotalhours.Laborandcapitalmarketsarecompetitive.

TURNOVER

TheernmentfinancesUItransfersbytaxingemployedworkersproportionaytotheirlaborearningsatrate,butonlyuptoaumlevelofearnings¯i.e.,theportionofearningsabove¯iemptfromtheUItax.Theernmentissuesanamountofone-periodrisk-debt0everyperiod.

(10pts)Statethehouseholdproblemindynamicprogrammingformulation.De-finearecursivestationaryequilibriumforthiseconomy.Stateconditionsfortheexistenceofanequilibrium.

(20pts)Supposethattheright-wingpartyisthinkingaboutrunningforelectionsandasksyou(theeconomist)tocomputethewelfarechangeofcompleyeliminatingUIbenefits(andassociatedtaxes).Howwouldyoumakethiscomputation?Whatwouldyouexpecttohappentoequilibriumpricesandallocationsduringthetransition?

(10pts)Supposenowthattheleft-wingparty,worriedaboutbeingtakenoutofoffice,isthinkingofloweringtaxesbyestablishingatime-limittoUIbenefits(i.e.,after

periodsofunemployment,UIbenefitpireandarerecedbyafixedsubsistencetransfer).Howwouldyouchangethehouseholdproblemstatedinpoint1)incasethisisthewayUIgetsimplemented?

Question3:Demographicshocks(40pts)

Consideraneconomywhere,atanydatehouseholdssurviveuntil+1withfixedprobability∈(01)Everyperiod,anewcohortofhouseholdsofsizeenterstheeconomy.followsafirst-orderMarkovchainΓ(0)Theassetsofthedeceased

householdsareseizedbytheernmentwhodistributesthemequallyamongallthenewborn.Householdutilityis(1−)whereisconsumptionandhoursworked,and∈(01)isthediscountfactor.Householdsaresubjecttoidiosyncraticproductivityshockswhichfollowthefirst-orderMarkovchainΓ(0)Householdscansavethrough

risk-claimstophysicalcapital,butcannotborrow.Theaggregateproductiontech-nologyisastandardCRSproductionfunctionofcapitalandaggregateefficiency-weightedhours.