Dynamic Economics-2

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1、ITheoryThispageintentionallyleftblank2TheoryofDynamicProgramming2.1OverviewThemathematicaltheoryofdynamicprogrammingasameansofsolvingdynamicoptimizationproblemsdatestotheearlycontribu-tionsofBellman(1957)andBertsekas(1976).Foreconomists,thecontributionsofSargent(1

2、987)andStokeyandLucas(1989)provideavaluablebridgetothisliterature.2.2IndirectUtilityIntuitively,theapproachofdynamicprogrammingcanbeunder-stoodbyrecallingthethemeofindirectutilityfrombasicstaticcon-sumertheoryorareducedformprofitfunctiongeneratedbytheoptimizationof

3、afirm.Thesereducedformrepresentationsofpay-offssummarizeinformationabouttheoptimizedvalueofthechoiceproblemsfacedbyhouseholdsandfirms.Aswewillsee,thetheoryofdynamicprogrammingtakesthisinsighttoadynamiccontext.2.2.1ConsumersConsumerchoicetheoryfocusesonhouseholdsthat

4、solveVðI;pÞ¼maxuðcÞcsubjecttopc¼I;wherecisavectorofconsumptiongoods,pisavectorofpricesandIisincome.1Thefirst-orderconditionisgivenby1.AssumethatthereareJcommoditiesinthiseconomy.Thispresentationassumesthatyouunderstandtheconditionsunderwhichthisoptimizationproblemh

5、asasolutionandwhenthatsolutioncanbecharacterizedbyfirst-orderconditions.8Chapter2ujðcÞ¼lforj¼1;2;...;J;pjwherelisthemultiplieronthebudgetconstraintandujðcÞisthemarginalutilityfromgoodj.HereVðI;pÞisanindirectutilityfunction.Itisthemaximizedlevelofutilityfromthecurre

6、ntstateðI;pÞ.Someoneinthisstatecanbepredictedtoattainthislevelofutility.Onedoesnotneedtoknowwhatthatpersonwilldowithhisincome;itisenoughtoknowthathewillactoptimally.Thisisverypowerfullogicandunderliestheideabehindthedynamicprogrammingmodelsstudiedbelow.Toillustrat

7、e,whathappensifwegivetheconsumerabitmoreincome?WelfaregoesupbyVIðI;pÞ>0.Cantheresearcherpredictwhatwillhappenwithalittlemoreincome?Notreallysincetheoptimizingconsumerisindifferentwithrespecttohowthisisspent:ujðcÞ¼VIðI;pÞforallj:pjItisinthissensethattheindirectutil

8、ityfunctionsummarizesthevalueofthehouseholdsoptimizationproblemandallowsustodeterminethemarginalvalueofincomewithoutknowingmoreaboutconsumptionfunctions