Arbitrage and Equilibrium in Asset Exchange Economies A Surv(8)

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

2.1.2Uniformity

Asetcloselyrelatedtotheithagent’sarbitrageconeisthelinearityspaceLi(xi)of i(xi)givenbyO+P

i(xi)andλ∈R,x +λyi∈P i(xi)}.Li(xi)={yi∈Rl| xi∈Pi

ThesetLi(xi)consistsofthezerovectorandallthenonzerovectorsyisuchthatfor i(xi)),anyvectorzionthelinethroughx

ieachxiweaklypreferredtoxi(i.e.xi∈P inthedirectionyi,zi=xi+λyi,isalsoweaklypreferredtoxi.ThesetLi(xi)isa i(xi).subspaceofRl,andisthelargestsubspacecontainedinthearbitrageconeO+P

Moreover,sinceRlis nite-dimensional,Li(xi)isaclosedsubspaceofRl.Asetofnettradesy=(y1,···,yl)isuselessforconsumeriifui(x+y)=ui(x)=ui(x y)forallx∈Xi;Asetofnettradesy=(y1,···,yl)isusefulforconsumeriifui(x+y)≥ui(x)forallx∈Xi,andyisnotuseless[Werner(1987)].

i(ei), i.[A.3][WeakUniformity]Li(xi)=Li:=L(ei), xi∈P

i(ei), iandyi∈Li(xi),Underweakuniformity,forallxi∈P

ui(xi+yi)=ui(xi).

FollowingtheterminologyofWerner(1987),werefertoarbitrageopportunities i(xi)suchthatyi∈O+P

ui(xi+λyi)=ui(xi)forallλ∈( ∞,∞)

asuselessatxi.Thus,under[A.3],theuselesssetisthelinearityspaceLi(xi)of i(xi);andtheusefulsetisO+P i(xi)\Li(xi).O+P

Werner[55]makesauniformityassumptionstrongerthanuniformityofuselessnettrades(i.e.,strongerthanweakuniformity,[A,3]).Werner[55]assumesthateachagent’sarbitrageconeisinvariantwithrespecttothestartingpointofthetrading(i.e.,xi),aslongasthestartingpointisweaklypreferredtotheagent’s i(ei)).Thatis,Wernerassumes:Inparticular,endowment(i.e.,aslongas,xi∈P

Wernerassumesthatallarbitrageopportunitiesareuniform.Statedformally,

i(ei):=Ri, xi∈P i(ei), i. i(xi)=O+P[A’.3][WeakUniformity]O+P

Notethatifuniformity[A’.3]holds,thenweakuniformity[A.3]holdsautomati- i(ei), i.cally.Thatis[A’.3]impliesthatLi(xi)=Li, xi∈P

2.1.3Nonsatiation

Classicalexistenceresultsforboundedexchangeeconomieswhichrequire,atmini-mum,globalnonsatiationatrationalallocations.

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