The geometry of arbitrage and the existence of competitive equilibrium

Abstract

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 (1974) and Werner (1987) are equivalent and imply the compactness of the set of utility possibilities. Moreover, we show that if agents' sets of useless net trades are linerly independent, then the Hart-Werner conditions are equivalent to the stronger conditon of no-unbounded-arbitrage due to Page (1987) - 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 preferneces 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