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
NextresulttellsusthatPage’sconditionisequivalenttotheconditionthatAbecompact.Moreimportantly,ittellusthatifagents’linearityspacesarelinearlyindependent,thenHart’sconditionandPage’sconditionareequivalent.
Theorem3.2Letεbeaneconomysatisfying[A.1]-[A.2].Thefollowingstatementsareequivalent:
1.εsatis esNUBA.
2.Aiscompact.
3.A⊥iscompactandthelinearityspaces,Li,arelinearlyindependent.
4.εsatis esWNMAandthelinearityspaces,Li,arelinearlyindependent.
3.3Noarbitragepricesystem
Werner[55]introducedtheno-arbitragepricesystemcondition(NAPS).Werner’scondition,aconditiononprices,requiresthattherebeanonemptysetofpricessuchthateachpricecontainedinthisnonemptysubsetassignsapositivevaluetoanyvectorofusefulnettradesbelongingtoanyagent.Wernerthenassumedthatforeachagentthesetofusefulnettradesatendowmentsisnonempty.Inparticular,Wernerassumesthat
[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
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