Arbitrage and Equilibrium in Asset Exchange Economies A Surv(10)

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

5.LetB=O(A)++⊥m

i=1

+Li.Thenm

i=1O(A)={(yi)∈B|yi=0}.

3

3.1No-arbitrageconditionsandcompactnessWeaknomarketarbitrage

Hart[27]introducedtheweakno-market-arbitragecondition(WNMA).Hart’scon-dition,aconditiononnettrades,requiresthatallmutuallycompatiblearbitrageopportunitiesbeuseless.

De nition3.1Theeconomyεsatis estheWNMAconditionif,

yi∈Riforalli,thenyi∈Liforalli.m i=1yi=0and

NextresulttellsusthatHart’sconditionisequivalenttotheconditionthatA⊥becompact.Moreimportantly,ittellusthatiftheeconomysatis esweakuniformity,thenHart’sconditionimpliesthatthesetofrationalutilitypossibilitiesiscompact.Theorem3.1Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsaretrue:

1.WNMAholdsifandonlyifA⊥iscompact.Inthiscase,

O+(A)={(yi)∈m

i=1Li|m i=1yi=0}.

2.Ifinaddition[A.3]holds,thenifεsatis esWNMA,thenthesetofrationalutilitypossibilities,Uiscompact.

3.2Nounboundedarbitrage

Page[41]introducedtheno-unbounded-arbitragecondition(NUBA).Page’scon-dition,aconditiononnettradesstrongerthanHart’s,requiresthatallmutuallycompatiblearbitrageopportunitiesbetrivial.

De nition3.2Theeconomyεsatis estheNUBAconditionif,m

i=1yi=0andyi∈Riforalli,thenyi=0foralli.yisanunboundedarbitrageifyi∈Riforallm i,yi=0forsomei,andyi=0.

i=1

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