Arbitrage and Equilibrium in Asset Exchange Economies A Surv(2)

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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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

Inconsequentialarbitrageandboundedarbitrageworkdi erently.Theyimplythecompactnessofthesetofutilitypossibilitieswithoutanytypeofuniformity,andtherefore,issu cientfortheexistencewithoutuniformity againviatheDanaetal.[16]result.

Usingthegeometryofarbitrage,Allouchetal.[5]sharpenandextendtheresultofPageetal.[48]showingtheequivalenceoftheconditionsofHart[27]andWerner

[55].Allouchetal.[5]establishthisequivalencewithoutanyassumptionsconcern-inguniformityornonsatiation.InPageetal.[48],theequivalenceofHartandWernerisobtainedassumingaveryweakformofnonsatiation(duetoWerner[55])andastrongformofuniformity(i.e.uniformityofarbitrageopportunities).Inaddi-tion,Allouchetal.[3]showunderweakuniformityonlythattheconditionsofHartandWernerimplytheconditionofinconsequentialarbitrage,introducedinPageetal.[48].Pageetal.[48]showthisaswell,butrequireWernernonsatiationandstronguniformity.Iftheeconomysatis esuniformityofarbitrageopportunities,lo-calnonsatiationatrationalallocationandweakno-half-lines,thentheHart-Wernerno-arbitrageconditionsandinconsequentialarbitrageareequivalent,andarenec-essaryandsu cientforthecompactnessofthesetofutilitypossibilitiesandtheexistenceofanequilibrium.Ifwestrengthentheweakno-half-linesconditiontoWerner’sconditionofno-half-lines,thentheHart-Wernerno-arbitrageconditionsandinconsequentialarbitrageareequivalenttono-unbounded-arbitrage,andallarenecessaryandsu cientforthecompactnessofthesetofrationalallocations,thecompactnessofthesetofutilitypossibilities,andtheexistenceofanequilibrium.Thepaperisorganizedasfollows.Basicmodelofanunboundedexchangeecon-omyandsomede nitionsarepresentedinSection2.Section3isdedicatedtopresenttherelationshipbetweenthevariousno-arbitrageconditionsfoundintheliteratureandthestrengthoftheboundednessimpliedbytheseconditions.InSection4,Suf- cientconditionsfortheexistenceofanequilibriumisaddressed.Finally,Section5showsthatundercertainconditions,thevariousno-arbitrageconditionsappearedintheliteratureareequivalentandnecessaryandsu cientfortheexistenceofanequilibrium.

2Themodel

Weconsideraneconomyε=(Xi,ui,ei)mi=1withmagentsandlgoods.Agenti hasconsumptionsetXi Rl,utilityfunctionui(.),andendowmentei,Agentispreferredsetatxi∈Xiis

Pi(xi)={x∈Xi|ui(x)>ui(xi)},

whiletheweakpreferredsetatxi∈Xiis

i(xi)={x∈Xi|ui(x)≥ui(xi)}.P

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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 condi …… 此处隐藏：5767字，全部文档内容请下载后查看。喜欢就下载吧 ……