Arbitrage and Equilibrium in Asset Exchange Economies A Surv(5)

Whenthetraders’utilitiesarelinearfunctions,thetwoconceptscoincide.Lemma3.2Theeconomyεsatis eslimitedarbitrageifandonlyifithasboundedgainsfromtradewhichareattainable,i.e., x ∈Asuchthat:

m ui(x G(ε)=(i) ui(ei))<∞.

i=1

4Su cientconditionsforexistenceofequilibriumUndertheassumptionoflocalnonsatiationatrationalallocations,Danaetal.[16]showthatcompactnessofutilitypossibilitiesissu cientfortheexistenceofanequi-librium.Intheabovesection,ineconomicmodelsofexchangeeconomiesallowing

189

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.

190

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

5Necessaryandsu cientconditionsforexistence

ofequilibrium

Inthislastsection,iftheeconomysatis estheadditionalconditionofweakno-half-lines,thentheconclusionsofTheorem4.2canbegreatlystrengthened.Inparticular,underweakno-half-linestheconditionsofHartandWernerandinconsequentialarbitrage,areequivalent,andallareequivalenttothecompactnessofthesetofthesetofrationalutilitypossibilitiesandtheexistenceofequilibrium.

i(xi),ify∈Rl,satis esui(xi+λy)=[A.6][WeakNo-half-lines] xi∈P

Ui(xi), λ≥0,theny∈Li.

Iftheeconomysatis esuniformity[A’.3]aswellasweakno-half-lines,thenanypotentialarbitrage(i.e.,anynettradevectorcontainedinanyagent’sarbitragecone)iseitheradirectioninwhichtheagent’sutilityiseventuallyincreasingoradirectioninwhichtheagent’sutilityiseventuallyincreasingoradirectionoradirectioninwhichtheagent’sutilityisconstant.

Anagent’sutilityiseventuallyincreasingatxiindirectionyiifgivenanyλ≥0, thereexistsaλ>λsuchthatui(xi+λyi)>ui(xi+λyi).

Lemma5.1Letεbeaneconomysatisfyingassumption[A.1],[A,2],[A’.3],[A,4],and[A.6].Thenanyequilibriumpriceisano-arbitrageprice.

Theorem5.1Letεbeaneconomysatisfyingassumption[A.1],[A,2],[A’.3],[A,4],

[A.5],and[A.6].Thenthefollowingstatementsareequivalent:

1.εsatis estheno-arbitragepricesystemcondition(Werner[55]).

2.εsatis estheweak-no-arbitragecondition(Hart[27]).

3.εsatis esinconsequentialarbitrage(Pageetal.[48]).

4.εsatis esboundedarbitrage(CPP)(Allouch[2,4]).

5.Thesetofrationalutilitypossibilities,U,iscompact.

6.εhasanequilibrium.

Ifwestrengthentheweakno-half-linesconditionthenno-unbounded-arbitrage(NUBA)andcompactnessofthesetrationalallocationscanbeaddedtoourlistofequivalence.

i(xi),ify∈Rl,satis esui(xi+λy)=Ui(xi), λ≥[A’.6][No-half-lines] xi∈P

0,theny=0.

Corollary5.1Letεbeaneconomysatisfyingassumption[A.1],[A,2],[A’.3],

[A,4],[A.5],and[A’.6].Thenthefollowingstatementsareequivalent:

1.εsatis estheno-arbitragepricesystemcondition(Werner[55]).

2.εsatis estheweak-no-arbitragecondition(Hart[27]).

191

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

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8.εsatis estheno-unbounded-arbitragecondition(Page[41]).εsatis esboundedarbitrage(CPP)(Allouch[2,4]).Thesetofrationalallocations,A,iscompact.εsatis esinconsequentialarbitrage(Pageetal.[48]).Thesetofrationalutilitypossibiliti …… 此处隐藏：5609字，全部文档内容请下载后查看。喜欢就下载吧 ……