On convergence of the sequential joint maximizatio

WorkingPaperONCONVERGENCEOFTHESEQUENTIALJOINTMAXIMIZATIONMETHODFORAPPLIEDEQUILIBRIUMPROBLEMSYuriERMOLIEVGuntherFISCHERVladimirNORKINWP-96-118October1996IIASAInternationalInstituteforAppliedSystemsAnalysisA-2361LaxenburgAustriaTelephone:432236807Fax:[phone]E-Mail:[e-mail]ENTIALJOINTMAXIMIZATIONMETHODFORAPPLIEDEQUILIBRIUMPROBLEMSYuriERMOLIEVGuntherFISCHERVladimirNORKINWP-96-118October1996WorkingPapersareinterimreportsonworkoftheInternationalInstituteforAppliedSystemsAnalysisandhavereceivedonlylimitedreview.ViewsoropinionsexpressedhereindonotnecessarilyrepresentthoseoftheInstitute,itsNationalMemberOrganizations,orotherorganizationssupportingthework.IIASAInternationalInstituteforAppliedSystemsAnalysisA-2361LaxenburgAustriaTelephone:432236807Fax:[phone]E-Mail:[e-mail]ofthesequentialjointmaximizationmethod(Rutherford[10])forsearch-ingeconomicequilibriaisstudiedinthecaseofCobb-Douglasutilityfunctions.ItisshownthatconvergenceiscloselyrelatedtothebehaviorofcertaininhomogeneousMarkovchains.Inparticular,convergencetakesplaceifeachgoodiseitherproducedoravailableintheeconomy.Keywords:Appliedequilibriumproblem,jointmaximizationmethod,Cobb-Douglasutility.iiiivContents1Introduction12Generalequilibriumproblem13Cobb-Douglasutilities74Thelackofgrosssubstitutability95Sequentialjointmaximizationmethod116Concludingremarks14vviONCONVERGENCEOFTHESEQUENTIALJOINTMAXIMIZATIONMETHODFORAPPLIEDEQUILIBRIUMPROBLEMSYuriERMOLIEVGuntherFISCHERVladimirNORKIN1IntroductionThesequentialjointmaximizationmethodwasproposedbyRutherford[10]asaheuristicprocedureforappliedequilibriumproblems.Itturnedouttobeeectiveinapplicationstorathercomplexintertemporalequilibriummodelsforintegratedassessmentofinterna-tionalenvironmentalpolicies(seeManne[6],ManneandRutherford[7]).Inthepresentpaperweanalyzesomeconvergencepropertiesofthemethod.WeconsiderthecaseofCobb-Douglasutilityfunctionswhichallowtoillustratethemainfeaturesoftheprocedureinthemostsimplemanner.Forexample,itisshownthatconvergenceofthejointmax-imizationmethodisrelatedtonewproblemsforinhomogeneousMarkovprocesses.Wealsoillustratetheconvergenceofthemethodwithoutrequiringthegrosssubstitutabilityassumptions.2GeneralequilibriumproblemLetusintroducesomenecessarynotations.Consideraneconomyconsistingofmcon-sumersandlproducers.EachconsumerkischaracterizedbyautilityfunctionU(xk),consumptionvectorxk2QkRn,initialendowmentwk2Rn+andshareskiinprotsofproduceri,Pmk=1ki=1.Produceriischaracterizedbythesetoffeasibleactivityvectorsyi2YiRnandaproductionvector-functiongi(yi)=(gi1(yi);:::;gin(yi)).Letp2Rn+denoteapricevectorofgoodsintheeconomy,x=(x1;:::;xm),y=(y1;:::;yl),Q=Q1:::Qm,Y=Y1:::Yl.1Demandforgoodsintheeconomyisgeneratedaccordingtotheprincipleofutilitymaximization:itisassumedthateachconsumerkchoosesavectory+kofgoodsthatmaximizeshis/herutilitysubjecttoabudgetconstraint(2)andothers,forexample,environmentalconstraints(3):Uk(xk)!maxxk;(1)pxkIk(y;p);(2)xk2Qk2Rn;(3)whereincomefunctionIk(y;p)hastheform:Ik(y;p)=pwk+lXi=1kipgi(yi);mXk=1ki=1;(4)wherepgi(yi)denotesaninnerproductofvectorspandgi(yi).Thisapproachallowstogenerateanarbitrarynumberofdemandfunctionsxk(Ik;p)bychoosingappropriateutilityfunctionsUk(xk).Producerichoosestheproductionlevelsyifromtheprotmaximization:pgi(yi)!maxyi;(5)yi2YiRn:(6)Wealsoconsideramarketplayer(seeZangwillandGarcia[12]):p(mXk=1xkmXk=1wklXi=1gi(yi))!maxp;(7)p0;nXj=1pj=1:(8)Vectorsx,yandpconstituteageneralequilibriumifvectorsxkaresolutionsof(1)-(3)forxedp=p,y=y,k=1;:::;m;yiisasolutionof(5)-(6)forxedp=p,i=1;:::;l,andp=pisasolutionof(7)-(8)forxedx=x,y=y,i.e.thefollowingmaterialandnancialbalancesarefullled:mXk=1xkW+G(y);(9)pmXk=1xk=p(W+G(y));(10)whereW=Pmk=1wk,G(y)=Pli=1gi(yi)(component-wisesummation).Thusageneralequilibriumx;y;pisinfactaNashequilibriumoftheappropriategamewith(m+l+1)players.2Weusesomecommonassumptions:(i)utilityfunctionsUk(xk)areconcaveandcontinuousonQk;(ii)setsQkareclosedandconvex,02QkRn+;(iii)productionfunctionsgij(yi)areconcave,i=1;:::;m;j=1;:::;n;(iv)setsYi,i=1;:::;l;areconvexcompacts,YiRn+;(v)foranyproductj=1;:::;nthereexistactivityvectorsyi2YisuchthatWj+Gj(y)0.Letusnotethatthecaseofnonlinearfunctionsgi(yi)(insteadoftraditionalgi(yi)=yi)isimportantwhendecompositionschemesareused(see,forexample,[3]).IfutilitiesUk(),k=1;:::;m;arepositivelyhomogeneousandincomefunctionsIk(y;p):=tk,k=1;:::;m,areconstant,thenthegeneralequilibriumproblemisre-ducedtoanoptimizationproblem(seeEisenbergandGale[2],Gale[4],Eisenberg[1],Polterovich[8],[9]).Denition2.1FunctionU(x);x2Q;iscalledpositivelyhomogeneouswithdegreeonaconeQ2Rnifforanyx2Qandr0U(rx)=rU(x):Thefollowingpositivelyhomogeneousutilityfunctionsareoftenused:U(x)=x11:::xnn,Pnj=1j=1,0j1(Cobb-Douglasfunction);U(x)=min1infx1=a1;:::;xn=ang,aj0(Leontieffunction);U(x)=Pni=1cixi,ci0(linearfunction).Theorem2.1Assumeinadditionto(i)-(v)that(vi)functionUkispositivelyhomogeneouswithdegreekandnonnegativeonQk,setQkisaconewiththe