recursive macroeconomic theory dynamic programming

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1、Chapter3DynamicProgrammingThischapterintroducesbasicideasandmethodsofdynamicprogramming.1Itsetsoutthebasicelementsofarecursiveoptimizationproblem,describesakeyfunctionalequationcalledtheBellmanequation,presentsthreemethodsforsolvingtheBellmanequation,andgivestheBenvenis

2、te-Scheinkmanformulaforthederivativeoftheoptimalvaluefunction.Let’sdivein.3.1.SequentialproblemsLetβ∈(0,1)beadiscountfactor.Wewanttochooseaninfinitesequenceof“controls”{u}∞tomaximizett=0∞βtr(x,u),(3.1.1)ttt=0nsubjecttoxt+1=g(xt,ut),withx0∈IRgiven.Weassumethatr(xt,ut)kis

3、aconcavefunctionandthattheset{(xt+1,xt):xt+1≤g(xt,ut),ut∈IR}isconvexandcompact.Dynamicprogrammingseeksatime-invariantpolicyfunctionhmappingthestatextintothecontrolut,suchthatthesequence{u}∞generatedbyiteratingthetwofunctionsss=0ut=h(xt)(3.1.2)xt+1=g(xt,ut),startingfromi

4、nitialconditionx0att=0,solvestheoriginalproblem.Asolutionintheformofequations(3.1.2)issaidtoberecursive.TofindthepolicyfunctionhweneedtoknowanotherfunctionV(x)thatexpressestheoptimalvalueoftheoriginalproblem,startingfromanarbitraryinitialconditionx∈X.Thisiscalledthevalue

5、function.Inparticular,define∞V(x)=maxβtr(x,u),(3.1.3)0tt{us}∞s=0t=01Thischapteraimstothereadertostartusingthemethodsquickly.Wehopetopromotedemandforfurtherandmorerigorousstudyofthesubject.InparticularseeBertsekas(1976),BertsekasandShreve(1978),StokeyandLucas(withPrescot

6、t)(1989),Bellman(1957),andChow(1981).ThischaptercoversmuchofthesamematerialasSargent(1987b,chapter1).–103–104DynamicProgrammingwhereagainthemaximizationissubjecttoxt+1=g(xt,ut),withx0given.Ofcourse,wecannotpossiblyexpecttoknowV(x0)untilafterwehavesolvedtheproblem,butlet

7、’sproceedonfaith.IfweknewV(x0),thenthepolicyfunctionhcouldbecomputedbysolvingforeachx∈Xtheproblemmax{r(x,u)+βV(˜x)},(3.1.4)uwherethemaximizationissubjectto˜x=g(x,u)withxgiven,and˜xdenotesthestatenextperiod.Thus,wehaveexchangedtheoriginalproblemoffindinganinfinitesequenceo

8、fcontrolsthatmaximizesexpression(3.1.1)fortheprob-lemoffindingtheoptimalvaluefunctionV(x)andafunctionhthatsolve