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
subspacesofusefulnettradesiscompact.Ifinaddition,weaklyuniformissatis ed,thenHart’sconditionalsoimpliesthecompactnessofthesetofrationalutilitypossibilities.
Werner[55]introducestheno-arbitragepricesystemconditiononpriceswhichrequiresthattherebeanonemptysetofpricessuchthateachpricecontainedinthisnon-emptysubsetassignsapositivevaluetoanyvectorofusefulnettradesbelongingtoanyagent.Wernerthenassumesthatforeachagentthesetofusefulnettradesatendowmentsisnon-empty.Werner’s[55]conditionofno-arbitragepricesystemimpliesdirectlythecompactnessofthesetofutilitypossibilitiesbutallowthesetofrationalallocationstobeunbounded.AnespeciallyintriguingaspectofWerner’sexistenceresultisthatitdoesnotrequirelocalorglobalnonsatiation(seeWerner
[55],Theorems1).Thiscontrastssharplywithclassicalexistenceresultsforboundedexchangeeconomieswhichrequire,atminimum,thatagents’preferencesbegloballynonsatiatedatrationalallocations(e.g.,seeDebreu[17],GaleandMas-Colell[19],andBergstrom[7]).
Allouchetal.[3]extendWerner’spriceno-arbitrageconditiontoallowforweaknonsatiation-andinparticular,toallowforthepossibilitythatsomeagentshaveemptysetsofusefulnettradesatsomerationalallocations.Allouchetal.[3]showthatthisextendedpriceno-arbitrageconditionisequivalenttoHart’s[27]weakno-market-arbitragecondition.
Page[41]introducestheconditionofno-unbounded-arbitrageconditiononnettradesstrongerthanHart’condition,whichrequiresthatallmutuallycompatiblearbitrageopportunitiesbetrivial.Page’s[41]conditionisequivalenttothecom-pactnessofthesetofrationalallocations,andthereforeimpliesthecompactnessofthesetofrationalutilitypossibilities.Moreover,weshowthatifagentsutility-constantsubspaces(atendowments)arelinearlyindependent,thenHart[27]weakno-market-arbitragecondition,Werner[55]no-arbitragepriceconditionandPage
[41]no-unbounded-arbitrageconditionareequivalent,andinturn,allareequiv-alenttothecompactnessofthesetofrationalallocations,andthereforeimpliesthecompactnessofthesetofrationalutilitypossibilities.Becausetheno-arbitrageconditionofHammond[26] overlappingexpectations isstatedintermsofprop-ertiesofthesubjectiveprobabilitydistributionsofassetreturns,itisdi culttomakecomparisonsinanabstractgeneralequilibriumsettingbetweenHammond’sconditionandotherno-arbitrageconditions.Page[41]showsthatunderverymildconditionsonutilityfunctionsandassetreturndistributions,Hammond’sconditionofoverlappingexpectationsisequivalenttono-unbounded-arbitrage.
Pageetal.[48]introducetheconceptofinconsequentialarbitrageand,inthecontextofamodelallowingshort-salesandhalf-linesinindi erencesurfaces,provethatinconsequentialarbitrageissu cientfortheexistenceofanequilibrium.More-over,withaslightlystrongerconditionofnonsatiationthanthatrequiredfortheexistenceofanequilibriumandwithamilduniformityconditiononarbitrageop-portunities,inconsequentialarbitrage,theexistenceofaParetooptimalallocation,
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