Arbitrage and Equilibrium in Asset Exchange Economies A Surv(16)

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

shortsales,theseaboveno-arbitrageconditionsguaranteecompactnessofthesetofrationalutilitypossibilities.

[A.5] i,ei∈intXiand xi∈Ai,Pi(xi)isrelativelyopeninXi.

Assumption[A.5]allowsustoconcludethataquasi-equilibriumfortheeconomyisinfactanequilibriumfortheeconomy.Thefollowingtheoremstatescompactnessofrationalutilitypossibilitiesissu cientfortheexistenceofaquasi-equilibriumduetoDanaetal.[16].

Theorem4.1(compactnessoftheutilitysetissu cientforexistence)

Letεbeaneconomysatisfyingassumption[A.1],[A,2],and[A,4].IfthesetofrationalutilitypossibilitiesUiscompact,thenεhasaquasi-equilibrium.Moreover,if[A.5]holds,thenεhasanequilibrium.

PuttingtogetherTheorem4.1andTheorem3.1,3.2,3.2,3.4,3.5and3.6,wecansummarizetherelationshipbetweentheno-arbitrageconditionswehavediscussedandexistenceofequilibriumasfollows:

Theorem4.2(No-arbitrageconditionsimplyingexistence)

Letεbeaneconomysatisfying[A.1],[A.2]and[A.4].Thefollowingstatementsaretrue:

1.IfICholds,thenεhasaquasi-equilibrium.

2.Ifinadditiontheeconomysatis es[A.3],(weakuniformity),then

(a)ifWNMAholds,thenεhasaquasi-equilibrium,

(b)ifNAPSholds,thenεhasaquasi-equilibrium.

3.IfNUBAholds,thenεhasaquasi-equilibrium.

4.IfNSUBAholds,thenεhasaquasi-equilibrium.

5.Ifboundedarbitrageholds,thenεhasaquasi-equilibrium.

Remark4.1part2(b)oftheaboveTheoremimprovesupontheexistenceresultofWerner’sinthefollowingsense.Allouchetal.[3]showthatanextendedversionofWerner’sno-arbitragepriceconditionissu cientforexistenceunderweakuni-formity[A.3].Wernerinhisproofofexistencerequiresthestrongerconditionofuniformity[A’.3].However,Wernermakesadi erentassumptionconcerningnon-satiation.Inparticular,Wernerassumes[WNS]ratherthanlocalnonsatiationasAllouchetal.[5]do.InAllouchetal.[3],theyinvestigatetherelationshipbetweenexistenceandnonsatiationusingourextendedversionofWerner’sno-arbitrage-pricesystemcondition.Part2(a)improvesupontheexistenceresultofHart.Inpartic-ular,Allouchetal.[5]extendHart’sconditiontoanabstractgeneralequilibriummodelandshowthatHart’sconditionissu cientforexistenceunderweakunifor-mity[A.3].LikeWerner,Hartinhisproofofexistencerequiresthatthestrongerconditionofuniformity[A’.3]hold.

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