Arbitrage and Equilibrium in Asset Exchange Economies A Surv(4)

[WNS][Wernernonsatiation]Ri\Li=φ, i.

De nition3.3

is edif,Theeconomyεsatis esthe[WNS],theNAPSconditionissat-m

i=1SiW=φ,

where

SiW={p∈Rl|p·y>0, y∈Ri\Li}

isWerner’sconeofno-arbitrageprices.

Allouchetal.[5]extendedWerner’sconditiontoallowforthepossibilitythatforsomeagentthesetofusefulnettradesisempty,thatis,toallowforthepossibilitythatforsomeagent,Ri\Li=φ.Moreimportantly,Allouchetal.[5]proved,underverymildconditions,thattheirextendedversionofWerner’sconditionisequivalenttoHart’scondition.ThisresultextendsanearlierresultbyPageetal.[48]ontheequivalenceofHartandWernerconditions.

De nition3.4Foreachagenti,de ne

WSiifRi\Li=φ,Si=ifRi\Li=φ.L⊥i

185

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

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

3.4Inconsequentialarbitrage

Pageetal.[48]extendedtheHart[27]modeltoanabstractgeneralequilibriumsettingwithoutuniformityconditionsandintroduceaconditionlimitingarbitrage,calledinconsequentialarbitrage(IC).Theirconditionisweakerthattheweakno-market-arbitrageconditionandimpliescompactnessoftheutilitysetU.Asetoftradesy=(y1,···,ym)∈Rlmisanarbitrageintheeconomyεifyisthelimitofsomesequence{λnxn}nwhereλn↓0and{xn}n Aisasequenceofrationalallocations.Theydenotethesetosallarbitragesby

arb(ε)={y∈Rlm| {xn}n Aandλn↓0suchthaty=limλnxn}n→+∞

andtheydenoteby

arbseq(y)={{xn}n A| λn↓0suchthaty=limλnxn}n→+∞

thesetofallarbitragesequencescorrespondingtoy∈arb(ε).

y∈arb(ε)isintheback-upset,denotedbybus(ε),ifforally∈arb(ε)and{xn}n∈arbseq(y),thereexistsan >0suchthatforallnsu cientlylarge

nnxni yi∈Xiandui(xi yi)≥ui(xi), i.

De nition3.6Theeconomyεsatis esthe(IC)conditionif

arb(ε) bus(ε).

Inwords,anarbitragey∈arb(ε)isinconsequential(i.e.iscontainedinthebackupsetatendowmentsbus(ε)))ifforsu cientlylargeallocationsx∈Ainthey=(y1,···,yn)‘directions’fromtheendowmentω,eachagentjcanreducehisconsumptionbyasmallamountinthe yjdirectionwithoutreducinghisutility.Theorem3.4Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsaretrue:

1.NUBAholds ICholds.

2.If,inaddition,[A.3]holds,thenWNMAholds ICholds.

3.ICholds Uiscompact.

Theabovetheoremshowsthatunderweakuniformcondition,theHart/Wernerconditionsimplyinconsequentialarbitrage.Ingeneral,nounboundedarbitrageimpliesinconsequentialarbitrage.Whiletheconditionofnounboundedarbitragefocusesonexpandingutilitynondecreasingorincreasingtrades,inconsequentialar-bitragefocusesoncontractingnettradeswithoutdecreasingutility.Meanwhile,inconsequentialarbitrageinconsequentialarbitragedirectlyimpliescompactnessofthesetofutilitypossibilities.

187

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

3.5Strongunboundedarbitrage

Danaetal.[16]re ned“nounboundedarbitrage”conditionofPage[41]andpro-videdanewconceptofno-arbitrage,called“nostrongunboundedarbitrage”(NSUBA).De nition3.7A“strongunboundedarbitrage”isanunboundedarbitrageywiththeproperty(P):Thereexistsequencesλn∈R+andyn∈(Rl)msuchthat:(i)λn→+∞andyn→y;

(ii)e+λnyn∈A, n;

n(iii) x∈A, isuchthatui(ei+λnyi)>ui(xi).

ThefollowingtheoremshowsthatnostrongunboundedarbitragedirectlyimpliesthecompactnessofU.Thisresultseemstobethe rstwhichinfersthecompactnessofUfromano-arbitragecondition.

Theorem3.5Letεbeaneconomysatisfying[A.1]-[A.2].Ifthereisnostrongunboundedarbitrage(NSUBA),thenUiscompact.

3.6Boundedarbitrage

Allouch[2]introducedanewcondition,boundedarbitrage(Thecompactnesswithpartialpreorder(CPP)conditioncalledinAllouch[4]),de nedasfollows:…… 此处隐藏：5896字，全部文档内容请下载后查看。喜欢就下载吧 ……