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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