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
De nition3.5Theeconomyεsatis estheNAPSconditionif
m
i=1Si=φ.
Remark3.1Notethatiftheeconomyεsatis esWerner’snonsatiationcondition,i.e.,Ri\Li=φ, i,thentheNAPSconditiongiveninDe nition3.5abovereducestowerner’soriginalconditiongiveninDe nition3.3.
Lemma3.1Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsaretrue:
1.Foranyi,suchthatRi\Li=φ,wehave:
⊥Si={p∈L⊥i|p·y>0, y∈(Ri∩Li)\{0}}.
002. i=1,···,m,Si= ri(Ri)where(Ri)isthepolarconeofRi.
Pageetal.[48]showthatunder[A.1]-[A.2],[A’.3]andWNS],WNMAholdsifm andonlyifSiW=φ(i.e.,Hart’sconditionholdsifandonlyifWerner’sconditionholds).Allouch[5]extendthisresultbyproving,under[A.1]-[A.2]only,thatWNMAm holdsifandonlyifSi=φ.
i=1i=1
Theorem3.3Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsareequivalent:
1.εsatis esWNMA.
2.εsatis esNAPS.
PageandWooders[45]statethatifLi={0}, i,thenNUBAholdsifandonlym ifSiW=φ.Infact,thisresultisaconsequenceofasharperresult:
i=1
Corollary3.1Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsareequivalent:
1.εsatis esNUBA.
m 2.Si=φ.andthelinearityspacesarelinearlyindependent.
i=1
Remark3.2BytheCorollary3.1thereisanabsenceofarbitrageopportunitiesifandonlyifthereexistsapricesystemlimitingarbitrageopportunitiescontainedintheL⊥ispacesandtherearenoarbitrageopportunitiesinthelinearityspaces.Thus,whenthelinearityspacesareequaltozero,nonemptinessofthesetofno-arbitragem prices(i.e.,Si=φ.)isnecessaryandsu cienttoruleoutarbitrageopportunitiesintheeconomy.
186i=1