Inconsequential arbitrage

Abstract

We introduce the concept of inconsequential arbitrage and, in the context of a model allowing short-sales and half-lines in indifference surfaces, we prove that inconsequential arbitrage is sufficient for existence of equilibrium. With a slightly stronger condition of local nonsatiation than required for existence of equilibrium and with a mild uniformity condition on arbitrage opportunities, we show that the existence of a pareto-optimal allocation implies inconsequential arbitrage, implying that inconsequential arbitrage is necessary and sufficient for existence of an equilibrium. By further strengthening our nonsatiation condition, we obtain a second welfare theorem for exchange economies allowing short sales. To further understand inconsequential arbitrage, we introduce the notion of exhaustible arbitrage and we show that any inconsequential arbitrage is exhaustible. We also compare inconsequential arbitrage to the conditions limiting arbitrage of hart (1974) and Werner (1987), as well as to the conditions recently introduced by Dan, Le Van, and Magnien (1999) and Allouch (1999). For example, we show that the condition of Hart (translated to a general equilibrium setting) and the condition of Werner are equivalent. We then show that the Hart/Werner conditions imply inconsequential arbitrage. To highlight the extent to which we extend Hart and Werner, we construct an example of an exchange economy in which inconsequential arbitrage holds (and is necessary and sufficient for existence), while the Hart/Werner conditions do not hold. Finally, under additional conditions on the model, we show that if agents' indifference surfaces contain no half lines, then inconsequential arbitrage, the Hart/Werner conditions, the Dana, Le Van, and Magnien condition, and Allouch's conditions are all equivalent - and in turn, equivalent to the existence of equilibrium