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
Thesetofindividuallyrationalutilitypossibilitiesisgivenby
A={(xi)∈m
i=1Xi| m i=1xi=m i=1ei i(ei), i}andxi∈P
WeshalldenotenyAitheprojectionofAontoXi.
Thesetofindividuallyrationalutilitypossibilitiesisgivenby
U(ε)={(vi)∈Rm| x∈A,suchthatui(ei)≤vi≤ui(xi), i}
TheParetofrontierP(ε)isthesetofundominatedvectorsinU:
P(ε)={U∈U(ε):～ V∈U(ε)withV>U}.
De nition2.1(a)Arationalallocationx ∈Atogetherwithanonzerovectorofpricesp ∈Rlisanequilibriumfortheeconomyε
(i)ifforeachagentiandx∈Xi,ui(x)>ui(x i)impliesp·x>p·ei,and
(ii)ifforeachagenti,p ·x=p ·ei.
(b)Arationalallocationx ∈Atogetherwithanonzerovectorofpricesp ∈Rlisaquasi-equilibriumfortheeconomyε
(i)ifforeachagentiandx∈Xi,ui(x)>ui(x i)impliesp·x≥p·ei,and
(ii)ifforeachagenti,p ·x=p ·ei.
Given(x ,p )aquasi-equilibrium,itiswell-knownthatifforeachagenti,(i)
p ·x<p ·eiforsomex∈Xiand(ii)Pi(x i)isrelativelyopeninXi,then(x,p)
isanequilibrium.Conditions(i)and(ii)willbesatis edif,forexample,foreachagenti,ei∈intXi,http://www.77cn.com.cningirreducibilityassumptions,onecanalsoshowthataquasi-equilibriumisanequilibrium.
Wenowintroduceour rsttwoassumptionsforagentsi=1,2,···,m,
[A.1]Xiisclosedandconvexwithei∈Xi,
[A.2]uiisuppersemicontinuousandquasi-concave.
i(xi)isconvexandclosedUnderthesetwoassumptions,theweakpreferredsetP
forxi∈Xi.
2.1
2.1.1Arbitrage,Uniformity,andNonsatiationArbitrage
De nition2.2Theithagent’sarbitrageconeatxi∈Xiastheclosedconvexconecontainingtheorigingivenby
i(xi)andλ≥0,x
i+λyi∈P i(xi)}. i(xi)={yi∈Rl| x
i∈PO+P
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