Arbitrage and Equilibrium in Asset Exchange Economies A Surv(3)

Gi(xi)={y∈Rl: x∈Rl λx>0suchthatui(xi+λxy)>ui(x)}.

De nition2.4(Page[39])Theincreasingconecorrespondingtotheithagent’sutilityfunctionui(·)atconsumptionvectorxi∈Xiisgivenby,

Ii(xi)={y∈Rl:ui(xi+λy)>ui(x+µy)ifλ>µ≥0}.

InPageandWooders[45,46]thede nitionoftheincreasingconeisextendedtoaccommodatethickindi erencecurve:

i(xi)={y∈Rl: µ≥0, λ>µsuchthatui(xi+λy)>ui(x+µy)}.I

Chichinisky[12]modi esherarbitrageconditionbyusingtheincreasingcone i(xi),butalternativelystatedinherpaperas:I

G i(xi)={y∈Rl:¬ maxui(xi+λy)}.λ≥0

Themarketconeofconsumeriis

Di(xi)={z∈X: y∈G i(xi), z,y >0}

DiistheconvexconeofpricesassigningstrictlypositivevaluetoalldirectionsinGi(xi).

Let (ei)=I i;G (ei)=G ;Di(ei)=Di.Gi(ei)=Gi;Ii(ei)=Ii;Iii

i(xi)directiononNotethatiftheagent,startingatxi,tradesintheyi∈O+P

anyscaleλ≥0,thenhisutilitywillbenondecreasing.Inparticular,asetofnettradesy=(y1,···,ym)isanarbitrageopportunityatx=(x1,···,xm)if

m

i=1yi=0(i.e.,tradesaremutuallycompatible),

and

i(xi)foralli(i.e.,tradesstartingatxiareutilitynondecreasing).yi∈O+P

181

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.

182

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

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

5.LetB=O(A)++⊥m

i=1

+Li.Thenm

i=1O(A)={(yi)∈B|yi=0}.

3

3.1No-arbitrageconditionsandcompactnessWeaknomarketarbitrage

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