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1、JournalofMathematicalEconomics2(1975)9-15.@North-HollandPublishingCompanyANEQUILJBRIUMEXISTENCETHEOREMFORAGENERALMODELWITHOUTORDEREDPREFERENCES*D.GALEandA.MAS-COLELLUniversityofCalifornia,Berkeley,Calif.94720,U.S.A.ReceivedSeptember1974,revisedversionreceivedOctober19741.IntroductionIn
2、arecentpaperthesecondauthorhasshownthatsomeoftheusualhypothesesonconsumerspreferencesarenotneededfortheproofofexistenceofaWalrasianGeneralEquilibrium[Mas-Cole11(1974)].Specifically,itisnotnecessarythatpreferencescomefromapreferenceordering.Theonlyorderpropertyrequiredisirreflexivity(me
3、aningthatagivencommoditybundleisnotpreferredtoitself).Thepropertiesofnon-satiation,continuityandconvexityofpreferredsetsturnouttobesufficienttoobtaintheexistenceresult.ThemainpurposeofthepresentnoteistogiveasecondproofofthisfactwhichseemssimplerthanthatofMas-Cole11(1974),andnomorelengt
4、hyorcomplicatedthantheknownequilibriumexistenceproofswhichuseorderedpreferences.Intwoadditionalrespectsthemodelstudiedheregeneralizestheusualequilibriummodel.ThestandardWalras,Arrow-Debreutheorytreatswhatmightbecalledthelaissez-fairemodelinwhicheachagentsincomeiswhateverhegetsfromselli
5、nggoodsplushisshareoftheprofitsofanyfirminwhichhemayownstock.Inthepresentmodeltheincomeofaconsumermaybeanycontinuousfunctionoftheprices,sothelaissez-faireincomefunctionisincluded,butsoalsowouldanyruleforassigningincometoconsumers(e.g.,accordingtohisabilityorhisneedorthecolorofhiseyes).
6、Anotherwayofsayingthisisthatthemodelincludesthepossibilityofarbitrarylumpsumtransfersofin-comeamongconsumers,asmightbeachieved,forexample,byaprogramofincometaxesandsubsidies.Thissubstantialeconomicgeneralizationrequiresnochangewhateverinthemathematicalargument.Thesecondgeneralizationco
7、ncernsproduction.Theonlyrequirementonourproductionset,besidestheusualclosure,convexity,andfreedisposal,isthatitin-tersectthepositiveorthantinaboundedset.Thismeansthattheusualassumption*ResearchforthispaperhasbeensupportedbyNSFGrantGP-30961X,andOfficeofNavalResearchContractNOO014-69-A
