Increasing cones, recession cones and global cones

Abstract

In this paper, we discuss and compare various cones used in the economics literature to analyze arbitrage in general equilibrium models with short selling. Our main result is that under certain conditions on an economic model, the closure of the increasing cone and the closure of the global cone are both equal to the arbitrage cone, the recession cone of the preferred set. It is known that in general, the Page-Wooders increasing cone may be strictly contained in the recession cone. We demonstrate that in general - for example, under the conditions of Werner (1987) or Page and Wooders (1996a, b) - the increasing cone is larger than the global cone; in fact the closure of the global cone may be strictly contained in the increasing cone. This shows that, except under special assumptions on the economic model, conditions based on the global cone are inadequate to ensure existence of equilibrium.