Arbitrage and Equilibrium in Asset Exchange Economies A Surv(14)

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

3.5Strongunboundedarbitrage

Danaetal.[16]re ned“nounboundedarbitrage”conditionofPage[41]andpro-videdanewconceptofno-arbitrage,called“nostrongunboundedarbitrage”(NSUBA).De nition3.7A“strongunboundedarbitrage”isanunboundedarbitrageywiththeproperty(P):Thereexistsequencesλn∈R+andyn∈(Rl)msuchthat:(i)λn→+∞andyn→y;

(ii)e+λnyn∈A, n;

n(iii) x∈A, isuchthatui(ei+λnyi)>ui(xi).

ThefollowingtheoremshowsthatnostrongunboundedarbitragedirectlyimpliesthecompactnessofU.Thisresultseemstobethe rstwhichinfersthecompactnessofUfromano-arbitragecondition.

Theorem3.5Letεbeaneconomysatisfying[A.1]-[A.2].Ifthereisnostrongunboundedarbitrage(NSUBA),thenUiscompact.

3.6Boundedarbitrage

Allouch[2]introducedanewcondition,boundedarbitrage(Thecompactnesswithpartialpreorder(CPP)conditioncalledinAllouch[4]),de nedasfollows:

Theeconomysatis esboundedarbitrage(CPP)ifforallsequencesofrationalallocations{xn}n∈Athereexists

asubsequence{xnk)}k∈A;

arationalallocationz∈Aandn kasequencez Xiconvergingtozsuchthat

i=1

kkzj∈Pj(xn

j).

Here,followingGaleandMas-Colell(1975),theaugmentedpreferencecorrespon- dencexj→Pj(xj)isgivenby

j∈Xj:x j=(1 λ)xj+λxjfor0≤λ≤1,xj∈Pj(xj)}.Pj(xj):={x

Thus,byboundedarbitrage,foreverysequenceofrationalallocationsthereisasubsequencethatisaugmentedpreference-dominatedbyasequenceconvergingtoarationalallocation.Notethatboundedarbitrageimpliesglobalnonsatiationatrationalallocations.

Theorem3.6Letεbeaneconomysatisfying[A.1]-[A.2].Thenboundedarbitragedirectlyimpliescompactnessofthesetofutilitypossibilities.Inaddition,ifεsatis eslocalnonsatiation,thenboundedarbitrageisequivalenttocompactnessofthesetofutilitypossibilities.

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