Arbitrage and Equilibrium in Asset Exchange Economies A Surv(13)

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

3.4Inconsequentialarbitrage

Pageetal.[48]extendedtheHart[27]modeltoanabstractgeneralequilibriumsettingwithoutuniformityconditionsandintroduceaconditionlimitingarbitrage,calledinconsequentialarbitrage(IC).Theirconditionisweakerthattheweakno-market-arbitrageconditionandimpliescompactnessoftheutilitysetU.Asetoftradesy=(y1,···,ym)∈Rlmisanarbitrageintheeconomyεifyisthelimitofsomesequence{λnxn}nwhereλn↓0and{xn}n Aisasequenceofrationalallocations.Theydenotethesetosallarbitragesby

arb(ε)={y∈Rlm| {xn}n Aandλn↓0suchthaty=limλnxn}n→+∞

andtheydenoteby

arbseq(y)={{xn}n A| λn↓0suchthaty=limλnxn}n→+∞

thesetofallarbitragesequencescorrespondingtoy∈arb(ε).

y∈arb(ε)isintheback-upset,denotedbybus(ε),ifforally∈arb(ε)and{xn}n∈arbseq(y),thereexistsan >0suchthatforallnsu cientlylarge

nnxni yi∈Xiandui(xi yi)≥ui(xi), i.

De nition3.6Theeconomyεsatis esthe(IC)conditionif

arb(ε) bus(ε).

Inwords,anarbitragey∈arb(ε)isinconsequential(i.e.iscontainedinthebackupsetatendowmentsbus(ε)))ifforsu cientlylargeallocationsx∈Ainthey=(y1,···,yn)‘directions’fromtheendowmentω,eachagentjcanreducehisconsumptionbyasmallamountinthe yjdirectionwithoutreducinghisutility.Theorem3.4Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsaretrue:

1.NUBAholds ICholds.

2.If,inaddition,[A.3]holds,thenWNMAholds ICholds.

3.ICholds Uiscompact.

Theabovetheoremshowsthatunderweakuniformcondition,theHart/Wernerconditionsimplyinconsequentialarbitrage.Ingeneral,nounboundedarbitrageimpliesinconsequentialarbitrage.Whiletheconditionofnounboundedarbitragefocusesonexpandingutilitynondecreasingorincreasingtrades,inconsequentialar-bitragefocusesoncontractingnettradeswithoutdecreasingutility.Meanwhile,inconsequentialarbitrageinconsequentialarbitragedirectlyimpliescompactnessofthesetofutilitypossibilities.

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