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
n[A.4][localNonsatiation] xi∈Ai, {yi}n Xiwith
n i(xi)forallUi(yi)>ui(xi), n,thatis,forallagentsi,Pi(xi)= andclPi(xi)=P
xi∈Ai.
[A’.4][globalNonsatiation]thattheeconomyεsatis esnonsatiationatra-tionalallocationsifforanyrationalallocation(x1,···,xn)∈A,Pi(xi)= ,forallagentsi.
Wernerthenassumesthatforeachagentthesetofusefulnettradesatendow-mentsisnonempty,ratherthanassumeglobalorlocalnosatiation.Inparticular,Wernerassumesthat
[WNS][Wernernonsatiation]Ri\Li=φ, i.
Thisassumptionisweakerthantheclassicalassumptions.Allouchetal.[3]weakenWerner’snonsatiationassumptionasfollows:
[Weaknonsatiation]forallagentsi, xi∈Ai,ifPi(xi)= ,thenn→+∞nlimyi=xiand
i(xi)\Li(xi)= .O+P
2.1.4Thegeometryofarbitrage
LetL⊥i(xi)denotethespaceorthogonalagenti’ssubspaceLi(xi)ofuselessnettradesatxi.ThevectorspaceRlcanbedecomposedintothedirectsumofthelinearityspaceLi(xi)anditsorthogonalcomplement,L⊥i(xi).Thus,givenxi∈Xi,wehave
Rl=L⊥i(xi) Li(xi),
andthus,eachvectorx∈Rlhasauniquerepresentationatthesumoftwovectors,
lonefromLi(xi)andonefromL⊥i(xi).inparticular,foreachx∈R,thereexist
uniquelytwovectors,y∈L⊥i(xi)andz∈Li(xi),suchthatx=y+z.
Lemma2.1Letε=(Xi,ui,ei)mi=1beaneconomysatisfying[A.1]-[A.2].Thefol-lowingstatementsaretrue:
1. i, xi∈Xi,
i(xi)=(P i(xi)∩L⊥(xi))⊕Li(xi),(a)Pi
+ + (b)OPi(xi)=(OPi(xi)∩L⊥i(xi))⊕Li(xi).
2.Ifinaddition[A.3]holds(i.e.,ifweakuniformityholds),then
i(xi)and yi∈Li.ui(xi+yi)=ui(xi), xi∈P
⊥⊥3.LetA⊥betheprojectionofAontom
i=1Li.ThenAisclosedandconvex.
4.LetO+(A),O+(A⊥)denotetherecessionconesofAandA⊥respectively.Thenmmm O+(A⊥)={(yi)∈(Ri∩L⊥yi∈Li}.i)|
i=1i=1i=1
183