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
3.7Limitedarbitrage
l
i=1De nition3.8εsatis eslimitedarbitrageif,(LA)Di=φ
Thismeansthatthereexistsoneprice,thesameforalltraders,atwhichthetradestheycana ordonlyincreasetheirutilitiesbylimited,orbounded,amounts.Whenglobalconesareindependentoftheinitialendowments,thiscondition(LA)issatis edsimultaneouslyateverysetofendowments.Theconceptoflimitedarbitragecanbeinterpretedintermsofgainsfromtrade:
m gainsfromtrade=G(ε)=sup(ui(xi) ui(ei)),
i=1
where(x1,···,xm)∈A.
Condition(C)Letz=(x1,···,xm)∈A.Ifasequence(xn)∈Asatis esm nnn j(xj)forn>N. x →∞andxi∈Pi(xi),then Ns.t.ei xi∈/j=iP
i=1
Itisusefultoshowtheconnectionbetweenlimitedarbitrageandthenotionof’no-arbitrage’usedin nance.Theconceptsaregenerallydi erent,butincertaincasestheycoincide.In nancialmarkets,anarbitrageopportunityexistswhengainscanbemadeatnocostor,equivalently,bytakingnorisks.Thesimplestillustrationofthelinkbetweenlimitedarbitrageandno-arbitrageisaneconomyεwherethetraders’initialendowmentsarezero,ei=0,andthenormalizedgradientofaclosedsetofindi erencevectorsde neaclosedset.Hereno-arbitrageattheinitialendowmentsmeansthattherearenotradeswhichcouldincreasethetrader’sutilitiesatzerocost:gainsfromtrademustbezero.Bycontrast,limitedarbitragemeansthatnotradercanincreaseutilitybeyondagivenboundatzerocost:gainsfromtrademustbebounded.
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
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