Abstract: This article surveys some recent progress on arbitrage and equilibrium in asset exchange economies. Using the basic geometry of arbitrage, the relationships between various no-arbitrage conditions appeared in the literature are presented. The rel
andthecompactnessofthesetofutilitypossibilitiesareequivalent.Thus,whenallequilibriaareParetooptimal forexample,whenlocalnonsatiationholds inconsequentialarbitrageisnecessaryandsu cientfortheexistenceofanequilib-rium.Byfurtherstrengtheningthisnonsatiationcondition,Pageetal.[48]obtainasecondwelfaretheoremforexchangeeconomiesallowingshortsales.Inaddition,underweakuniformityonlythattheconditionsofHartandWernerconditionsimplyinconsequentialarbitrage.Undertheassumptionofnohalf-linesinindi erencesur-faces,theconditionsofHartandWernerconditionsandinconsequentialarbitrageareequivalent.
Danaetal.[16]introducetheconceptofstrongunboundedarbitrageandshowthattheabsenceofstrongunboundedarbitragedirectlyimpliesthecompactnessoftheindividuallyrationalutilityset.Thisresultseemstobethe rstwhichinfersthecompactnessofUfromano-arbitragecondition.Undertheassumptionoflocalnonsatiationatrationalallocations,Danaetal.[16]showthatcompactnessofutilitypossibilitiesissu cientfortheexistenceofanequilibrium.
Allouch[4]alsointroducesthecompactnesswithpartialpreordercondition(anewcondition,boundedarbitrageintroducedinAllouch[2]),whicheliminatestheproblemofunboundednessbyrequiringeverysequenceofattainableandindividu-allyrationalallocationstobedominatedbyanincreasingpreferencesubsequenceconvergingtoanattainableallocation,andtherefore,impliestheexistenceofacom-petitiveequilibrium.Allouch[2]alsoshowsthatiflocalsatiationisruledout,thenhisconditionofboundedarbitrageisequivalenttothecompactnessofutilitypossi-bilities.Allouch’sresultisimpliedbyHart[27]andPage[41],butisequivalenttoDanaetal.[16]inthecaseofutility-representablepreferences.ThecompactnesswithpartialpreorderconditionisweakerthantheclassicalcompactnessofAthesetofindividuallyrationalandattainableallocations.
Underadi erentsetofassumptionsontheeconomicmodel,Chichilniskyde nedarbitrageasanopportunityforanagenttoincreasehisutilitycostlesslybeyondthelevelassociatedwithanygivenvectorinhisconsumptionset.Chichilniskyintro-ducesanewcondition,calledlimitedarbitragewhichrulesoutsucharbitrage,andassertsthatwithinthecontextofhermodel,limitedarbitrageisnecessaryandsu -cientfortheexistenceofanequilibrium.Chichilniskyalsoclaimsthatherconditionisnecessaryandsu cientforboundednessofgainsfromtrade.Becauseofsomeambiguousnesses,thede nitiongivenbyChichilniskymaybe awed(seeMonterioetal.[35]).ThisambiguousnessesdisappearinChichilnisky[12].ChichilniskyandHeal[14]presentlimitedarbitrageisnecessaryandsu cientfortheexistenceofanequilibriumandthecorein niteorin niteeconomies.
ThestrongerconditionsofHammond[26]andPage[41]implytheexistenceofanequilibrium,withoutuniformityconditions,byguaranteeingthecompactnessofthesetofrationalallocation,whiletheweakerconditionsofHart[27]andWerner
[55]requireweakuniformityofpreferencestoguaranteethecompactnessofutilitypossibilities,andthereforetoguaranteetheexistenceviatheDanaetal.[16]result.
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