Williamson-Notes on Macroeconomic Theory

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1、NotesonMacroeconomicTheorySteveWilliamsonDept.ofEconomicsWashingtonUniversityinSt.LouisSt.Louis,MO63130September2006Chapter1SimpleRepresentativeAgentModelsThischapterdealswiththesimplestkindofmacroeconomicmodel,whichabstractsfromallissuesofheterogeneityanddistributionamonge

2、conomicagents.Here,westudyaneconomyconsistingofarepresen-tativefirmandarepresentativeconsumer.Aswewillshow,thisisequivalent,undersomecircumstances,tostudyinganeconomywithmanyidenticalfirmsandmanyidenticalconsumers.Here,asinallthemodelswewillstudy,economicagentsoptimize,i.e.th

3、eymaximizesomeobjectivesubjecttotheconstraintstheyface.Thepreferencesofconsumers,thetechnologyavailabletofirms,andtheendowmentsofresourcesavailabletoconsumersandfirms,combinedwithoptimizingbehaviorandsomenotionofequilibrium,allowustousethemodeltomakepredictions.Here,theequili

4、briumconceptwewilluseiscompeti-tiveequilibrium,i.e.alleconomicagentsareassumedtobeprice-takers.1.1AStaticModel1.1.1Preferences,endowments,andtechnologyThereisoneperiodandNconsumers,whoeachhavepreferencesgivenbytheutilityfunctionu(c,c),wherecisconsumptionandcisleisure.Here,u

5、(·,·)isstrictlyincreasingineachargument,strictlyconcave,and12CHAPTER1.SIMPLEREPRESENTATIVEAGENTMODELStwicedifferentiable.Also,assumethatlimc→0u1(c,c)=∞,c>0,andlimc→0u2(c,c)=∞,c>0.Here,ui(c,c)isthepartialderivativewithrespecttoargumentiofu(c,c).Eachconsumerisendowedwithoneuni

6、toftime,whichcanbeallocatedbetweenworkandleisure.Eachconsumeralsoownsk0unitsofcapital,whichcanberentedtofirms.NThereareMfirms,whicheachhaveatechnologyforproducingconsumptiongoodsaccordingtoy=zf(k,n),whereyisoutput,kisthecapitalinput,nisthelaborinput,andzisaparameterrepresenti

7、ngtotalfactorproductivity.Here,thefunctionf(·,·)isstrictlyincreasinginbotharguments,strictlyquasiconcave,twicedifferentiable,andhomogeneousofdegreeone.Thatis,produc-tionisconstantreturnstoscale,sothatλy=zf(λk,λn),(1.1)forλ>0.Also,assumethatlimk→0f1(k,n)=∞,limk→∞f1(k,n)=0,lim

8、n→0f2(k,n)=∞,andlimn→∞f2(k,n)=0.1.1.2OptimizationInacompetitiveequilibrium,wecanat