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1、Chapter1BasicOverlappingGenerationsModel§1.1SpecializedUtilityandProductionFunctions1.ThemodelAclosedeconomyisdefinedtobecomposedofnumerousindividualsandfirms,andagovernment.Thegenerationbornatthebeginningofperiodtiscalledgenerationt.Thepopulationgrowsatrate,whereNtisthepopulationofgenera
2、tiont.1.1IndividualsIndividualsareidenticalandlivefortwoperiods:workingperiodandretirementperiod.Intheworkingperiod,eachindividualprovidesinelasticallyoneunitoflabortoearnwage.Heorsheconsumespartofhisorherincomesandsavestherest.Intheretirementperiod,heorsheconsumesthesavingswithaccruedint
3、erest.Eachindividualderivesutilityfromhisorherworking-periodconsumptionandtheretirement-periodconsumption.Theutilityisdescribedbyanadditivelyseparablelogarithmicfunction.Thuseachindividualsolvesthefollowingmaximizationproblem:,(1)s.t.,(2),(3)whereisthediscountfactor,thewage,thesavings,and
4、theinterestrate.Thefirst-orderconditionfortheutilitymaximizationis.(4)Thisexpressionimpliesthattheutilitylossfromreducingoneunitofworking-periodconsumptionisequaltotheutilitygainfromincreasingunitsofretirement-periodconsumptiondiscountedbyθ.1.2FirmsFirmsproduceahomogenouscommodityincompet
5、itivemarkets.TheproductionisdescribedbyCobb-Douglasfunctionor,whereistheoutputinperiodt,thecapitalstock,thecapital-laborratio,andtheoutput-laborratio.Firmsactcompetitively,rentingcapitaltothepointwherethemarginalproductofcapitalisequaltoitsrentalrate,andhiringlabortothepointwherethemargin
6、alproductoflaborisequaltothewage:,(5)。(6)1.3GoodsmarketThesavingsoftheyounginperiodtgeneratethecapitalstockinperiodt+1,.(7)2.DynamicEquilibriumAcompetitiveequilibriumforthismarketeconomyisasequenceasthatsatisfiesequations(1)-(7)forallt,giventheinitialcondition,.Substitutingequations(2),(3
7、)and(5)-(7)into(4)givesadynamicsystem:.(8)Assumethatthereexistsunique,stableandnonoscillatorysteadystateequilibrium(k).Inordertofindthestabilitycondition,welinearizeDynamicSystem(8)aroundthesteadystate.Somemanipulationgives,(9)where,.Theassumptionthattheequilibriumi
