The geometry of arbitrage and the existence of competitive equilibrium

We present the basic geometry of arbitrage, and use this basic geometry to shed new light on the relationships between various no-arbitrage conditions found in the literature. For example, under very mild conditions, we show that the no-arbitrage conditions of Hart [Journal of Economic Theory 9 (1974) 293] and Werner [Econometrica 55 (1987) 1403] are equivalent and imply the compactness of the set of utility possibilities. Moreover, we show that if agents' sets of useless net trades are linearly independent, then the Hart-Werner conditions are equivalent to the stronger condition of no-unbounded-arbitrage due to Page [Journal of Economic theory 41 (1987) 392]-and, in turn, all are equivalent to compactness of the set of rational allocations. We also consider the problem of existence of equilibrium. We show, for example, that under a uniformity condition on preferences weaker than Werner's uniformity condition, the Hart-Werner no-arbitrage conditions are sufficient for existence. With an additional condition of weak no-half-lines-a condition weaker than Werner's no-half-lines condition-we show that the Hart-Werner conditions are both necessary and sufficient for existence. (C) 2002 Published by Elsevier Science B.V.