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 nition2.3(Chichilnisky[10])Theglobalconecorrespondingtotheithagent’sutilityfunctionui(·)atconsumptionvectorxi∈Xiisgivenby,
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
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