資源描述：
《recursive macroeconomic theory dynamic programming》由會(huì)員上傳分享，免費(fèi)在線閱讀，更多相關(guān)內(nèi)容在工程資料-天天文庫(kù)。
1、Chapter3DynamicProgrammingThischapterintroducesbasicideasandmethodsofdynamicprogramming.1Itsetsoutthebasicelementsofarecursiveoptimizationproblem,describesakeyfunctionalequationcalledtheBellmanequation,presentsthreemethodsforsolvingtheBellmanequation,andgivestheBenvenis
2、te-Scheinkmanformulaforthederivativeoftheoptimalvaluefunction.Let’sdivein.3.1.SequentialproblemsLetβ∈(0,1)beadiscountfactor.Wewanttochooseanin?nitesequenceof“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.To?ndthepolicyfunctionhweneedtoknowanotherfunctionV(x)thatexpressestheoptimalvalueoftheoriginalproblem,startingfromanarbitraryinitialconditionx∈X.Thisiscalledthevalue
5、function.Inparticular,de?ne∞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,wehaveexchangedtheoriginalproblemof?ndinganin?nitesequenceo
8、fcontrolsthatmaximizesexpression(3.1.1)fortheprob-lemof?ndingtheoptimalvaluefunctionV(x)andafunctionhthatsolve
