practical dynamic programming

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1、Chapter4PracticalDynamicProgramming4.1.ThecurseofdimensionalityWeoftenencounterproblemswhereitisimpossibletoattainclosedformsforiteratingontheBellmanequation.Thenwehavetoadoptnumericalapproxi-mations.Thischapterdescribestwopopularmethodsforobtainingnumeri

2、calapproximations.Thefirstmethodreplacestheoriginalproblemwithanotherproblemthatforcesthestatevectortoliveonafiniteanddiscretegridofpoints,thenappliesdiscrete-statedynamicprogrammingtothisproblem.The“curseofdimensionality”impelsustokeepthenumberofpointsinth

3、ediscretestatespacesmall.Thesecondapproachusespolynomialstoapproximatethevaluefunction.Judd(1998)isacomprehensivereferenceaboutnumericalanalysisofdynamiceconomicmodelsandcontainsmanyinsightsaboutwaystocomputedynamicmodels.4.2.Discrete-statedynamicprogramm

4、ingWeintroducethemethodofdiscretizationofthestatespaceinthecontextofaparticulardiscrete-stateversionofanoptimalsavingsproblem.Aninfinitelylivedhouseholdlikestoconsumeonegoodthatitcanacquirebyspendinglaborincomeoraccumulatedsavings.Thehouseholdhasanendowmen

5、toflaborattimet,st,thatevolvesaccordingtoanm-stateMarkovchainwithtransitionmatrixPandstatespace[¯s1,s¯2,...,s¯m].Iftherealizationoftheprocessattis¯si,thenattimetthehouseholdreceiveslaborincomeofamountws¯i.Thewagewisfixedovertime.Weshallsometimesassumethatm

6、is2,andthatsttakesonvalue0inanunemployedstateand1inanemployedstate.Inthiscase,whastheinterpretationofbeingthewageofemployedworkers.Thehouseholdcanchoosetoholdasingleassetindiscreteamountsat∈AwhereAisagrid[a1

7、PracticalDynamicProgrammingendpointsofthegridAisimportant,aswedescribeindetailinchapter18onincompletemarketmodels.Theassetbearsagrossrateofreturnrthatisfixedovertime.Thehousehold’smaximumproblem,forgivenvaluesof(w,r)andgiveninitialvalues(a,s),istochooseapo

8、licyfor{a}∞tomaximize00t+1t=0∞Eβtu(c),(4.2.1)tt=0subjecttoct+at+1=(r+1)at+wstct≥0(4.2.2)at+1∈Awhereβ∈(0,1)isadiscountfactorandrisfixedrateofreturnontheassets.Weassumethatβ(1+r)<1.Hereu(c)isastrictlyincreasing,concav