博弈论试题

Week7SeminarMaterial(Class5):Atwoperiodmodel/RBC

Questionsforreview

1.Howdoconsumerssaveinthetwo-periodmodel?

Inthetwo-periodmodel,consumerssavebybuyingbonds:inthecurrentperiodtheysaves,inthefutureperiodtheyreceive(1+r)s.2.Howdoestherepresentativeconsumersaveinthetwoperiodmodel?(notthesamequestion!)Therepresentativeconsumerinvestsinassets.Forexample,intheRBCmodel,sheinvestsinthefirm’scapital.3.Whatistheslopeoftheconsumersbudgetconstraint?Theslopeoftheconsumer’sbudgetconstraintis−(1+r).Problems

1.Aconsumersincomeinthecurrentperiodisy=100,andincomeinthefutureperiodisy=120.Heorshepayslump-sumtaxest=20inthecurrentperiodandt=10inthefutureperiod.Therealinterestrateis0.1,or10%perperiod.

(a)Determinetheconsumerslifetimewealth.

120−10110y󰀁−t󰀁

=100−20+=80+=180w=y−t+

1+r1+0.11.1

(b)Supposethatcurrentandfutureconsumptionareperfectcomplementsfortheconsumer

andthatheorshealwayswantstohaveequalconsumptioninthecurrentandfutureperiods.Drawtheconsumersindifferencecurves.

(c)Determinewhattheconsumersoptimalfirst-andsecond-periodconsumptionare,andwhat

optimalsavingis,andshowthisinadiagramwiththeconsumersbudgetconstraintandindifferencecurves.Istheconsumeralenderoraborrower?

Theconsumer’soptimalconsumptionbundlesimultaneouslysatisfies:

c=c󰀁

andthebudgetconstraint:

c+

1󰀁

c=w⇒(1+r)c+c󰀁=(1+r)w⇒c󰀁=(1+r)w−(1+r)c1+r1

c󰀁=1.1×180−1.1c⇒c󰀁=198−1.1c

Bysubstitutingforc=c󰀁intothebudgetconstraintweget:

c=198−1.1c⇒2.1c=198⇒c=c󰀁=94.2

Hencesavingisgivenby:

s=y−t−c=100−20−94.2=−14.2

i.e.theconsumerisaborrower.

Todrawthebudgetconstraint,notethatithasverticalintercept198andslope-1.1.TheequilibriumbundleisatpointA,whiletheendowmentpointisatE1.


110
94.2


E


1



80
94.2
(d)Assumecurrentincomeyrisesto110,withunchangedfutureincomewithnochangein

theinterestrate.Byhowmuchwillconsumptionincrease?

Nowtheconsumer’slifetimewealthis:

120−10110y󰀁−t󰀁

=110−20+=90+=190w=y−t+

1+r1+0.11.1

Hence,thebudgetconstraintbecomes:

c󰀁=1.1×190−1.1c⇒c󰀁=209−1.1c

Bysubstitutingforc=c󰀁intothebudgetconstraintweget:

c=209−1.1c⇒2.1c=209⇒c=c󰀁=99.5

Thus,consumptionwillincreaseby5.3.

(e)Nowassumeyalsorises,to130.Byhowmuchwillconsumptionincrease?

Inthiscase,theconsumer’slifetimewealthis:

y󰀁−t󰀁130−10120

w=y−t+=110−20+=90+=199.1

1+r1+0.11.1

Thebudgetconstraintbecomes:

c󰀁=1.1×199.1−1.1c⇒c󰀁=219.01−1.1c

Bysubstitutingforc=c󰀁intothebudgetconstraintweget:

c=219.01−1.1c⇒2.1c=219.01⇒c=c󰀁=104.2

Therefore,inthiscase,consumptionwillincreaseby10.

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2.Nowassumethatthesameconsumeristherepresentativeconsumerinarealbusinesscyclemodel,andthereforethattheirincomeisGDP.Assumethatlaboursupplyandthecurrentcapitalstockisfixed,andtheconsumertreatstheirincomeasexogenous,butthatitisinfactgeneratedbytheproductionfunctiony=zkαN.

(a)Supposethatatransitoryproductivityshockhitstheeconomy,raisingyto110.Bywhat

percentagehaszincreased?

zhasincreasedby10%.

(b)Whataffectwillthisshockhaveonthemarginalproductofcapital,andhenceonthe

returnearnedbytherepresentativeconsumer?

Themarginalproductofcapitalisgivenby:

MPK=zαkα−1N

HencealsoMPKwillincreaseby10%.

(c)Assumethatinthenextperiodzwillreverttoitsoriginalvalue.Giventhesameprefer-encesasabove,willtheconsumerincreaseconsumptioninthecurrentperiodbymoreorlessthantheincreaseinincome?(youdonotneedtogivepreciseanswers).

Therepresentativeconsumerwillincreaseconsumptioninthecurrentperiodbylessthantheincreaseincome.Theoptimalresponseistoconsumerelativelylittleoftheincreaseinincomeandspreadthebenefittofutureconsumptionbyinvestingincapital.

(d)Whatdoesthisimplyforthelevelofincomeinthenextperiod?(again,youdonotneed

togivepreciseanswers).

Thefactthattotalfactorproductivityhasreturnedtotheinitialleveldecreasesthemarginalprod-uctoflabour.Henceincomewillbesmallerbothbecauseofthereducedproductivityofcapitalandbecauseoflowerequilibriumemployment.

(e)Nowassumelaboursupplyisendogenous.Whatwouldyouexpecttohappentothereal

wageandthemarginalutilityofconsumptioninthetwoperiods?Hencewhatwouldyouexpecttohappentolaboursupply?

••••

Iftheproductivityshiftistemporary,thenthecurrentrealwageisgreaterthanthefutureone.Asconsumptiongoesupinbothperiods,marginalutilityofconsumptiondecreasesinbothperiods.

󰀁

Recalltheintratemporaloptimalitycondition:u󰀁l=wuC.

Assumeapositivelyslopedsupplycurve(substitutioneffect>incomeeffect).Thismeansthatas

󰀁

wageincreasesu󰀁l/uCmustincrease.

•Thismeansthatlaboursupplymustgoup.

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