哈佛Ariel Pakes实证产业组织dl3.pdf

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1、IOClassNotes:SingleAgentDynamics,Example3,S-sPolicies.ArielPakes.November4,2009ThisexampleistakenfromDixit(1989).Theanalysisinthatarticleisincontinuoustime.Thesenotesdevelopthediscretetimeanalogueofthatarticleandshowhowonecanapplythetechniqueswealreadyintroducedtoanalyzethisproblem.Dixitmotivate

2、shisexamplesbytheproblemofarmde-cidingwhethertoenter(ifitiscurrentlyout)orexit(ifitiscurrentlyin)amarketwhentherm'sentry-exitdecisiondoesnotaectthepriceprocessandtherearesunkcostsofentry(andpossiblyofexit).Asimilartypeofanalysiscouldbeap-pliedtothemanyphenomenathatinvolveswitchingbetweendier

3、entregimeswhentherearecoststoswitching(menucostsforprices,producingdierentgoods,switchingbetweendier-enttypesoffuels,orpreparinglandfordierentcrops,arebutafewoftheexamples).Theseproblemsareallasubclassofasetofproblemssometimesreferredtoas(s;S)problems.Theoriginalproofoftheoptimalityof(s;S)pol

4、iciesisduetoScarf(
).Alsowewilluseadirectproofofthecontractionpropertyhere,ratherthanBlackwell'stheorem,sothisshouldprovideanexampleofhowtousethealternativeproofstrategy.1Thebasicframeworkforthesingleagentanalysisinvolveseitherlookingattwovaluefunctions,oneforbeingineachstate,orlookingatonevalue

5、functiondenedontwostatevariables,i:e.onX=(0;1)R+.ItisstraightforwardtogeneralizetoKdiscretestates.LetS=ff=(f1;f2)2B(X)B(X)g.Superscriptswilldenoteelements(couples)inS,andsubscriptswilldenotetheelementofthecouple.Denethemetric1212'(f;f)=maxmaxxjfi(x)fi(x)jiThenwecanshowthatthecouple(S;')isab

6、anachspace.WewilllookforapairoffunctionsinS,onethatgivesthevalueofbeingactivewhentheprice"=x,andonethatgivesthevalueofbeingapotentialentrantwhentheprice=x.Theeconomicsoftheproblemhasthiscouple,say(A(x);O(x))satisfyingthefollowingfunctionalequationonS,Z000A(x)=xw+maxfA(x),O(x)lgP(dxjx)Z00

7、0O(x)=+maxfA(x)k,O(x)gP(dxjx)where2(0;1).Let01R(Tf)=xw+maxff(x0);f(x0)lgP(dx0jx)@1(x)A12[Tf](x)=(Tf)=+Rmaxff(x0)k;f(x0)gP(dx0jx)2(x)12whereP=fP(
jx);x2Xgisafamilywhichisstochasticallyincreasinginx.Youshouldbeabletopro