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]).
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