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1、Chapter8:OptimizationoverTimeSHI,HaoAntaiCollege,ShanghaiJiaoTongUniversityNovember3,20101IntroductionOtherthanmakingdecisionsunderuncertainty,agentsinoureconomyalsooftenmakedecisionsovertime.Asinthecaseofuncertainty,thestudyofoptimizationovertimerequiresnonewgeneralprinciples.
2、Thevariablestobechosenwillpertaintodi¤erentdates,butwecanalwaysstackthemtogetherinonelargevectorx,andthegeneralproblemremainsoneofmaximizingthevalueofafunctionf(x)subjecttoavectorequalityconstraintg(x)andavectorinequalityconstrainth(x).Atthetimewhenthedecisionistaken,theknowled
3、geoffuturetastesandtechnologymaybeveryimperfect.Butthissimplyrequiresustocapturetheuncertaintiesandattitudestoriskinthefunctionf,g,andh.Oncealltheconsiderationsareincorporatedintheobjectivefunctionandtheconstraints,theformaltheoryofthepreviouschapterscontinuestoapply.Thoughstud
4、yingoptimizationinvolvingtimedoesnotrequireanybasicallynewtheory,suchproblemsoftenhaveaspecialstructure.So,itisstillnecessarytotreatsuchproblemsasaspecialtopic.Themostimportantaspectofthisspecialstructureistheexistenceofstock-‡owrelationshipsamongthevariablesatsuccessivepointsi
5、ntime.Someofthevariables,labeledasywiththeap-propriatetime-subscriptorargument,havethedimensionsofastock.Others,labeledasz,havethedimensionsofa‡ow.Oftentimes,thestocksarecalledstatevariables,andthe‡owscontrolvariables.Example1Suppose,inaclosedeconomywithonlyoneagent,ktisthecapi
6、talthathehasattimet.Hedecidestoputlaborltinproduction,hencehasoutputf(kt;lt;t)wheretheargumenttdenotestheproductiontechnologytheagenthasattimet.Withoutputf(kt;lt;t),hedecidestoconsumectandsavetherestitfortimet+1.it;hence,isalsocalledtheinvestmentattimet.Thestatevariableisk,andt
7、hecontrolvariablesarelandc.1Supposethedepreciationtttrateofcapitalis.Capitalattimet+1isdeterminedbykt+1=it+(1 )kt=f(kt;lt;t) ct+(1 )kt.k0isgiven.1Strictlyspeaking,itisalsoacontrolvariable.Butyoumayrealizethataslongasltandctarealreadydetermined,it=F(kt;lt;t) ctisuniquelydeter
8、mined.Thatis,thevalueofitisnotfreelysethere.1Inadditio
