Arbitrage, equilibrium, and nonsatiation - Nizar Allouch; Cuong Le Van; Frank H. Page (Queen Mary University of London, Department of Economics, 2004)
In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (Econometrica, 1987) proved the existence of a competitive equilibrium, under a price no-arbitrage condition, without assuming either local or global nonsatiation. Werner's existence result contrasts sharply with classical existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. We make two main contributions to the theory of arbitrage and competitive equilibrium. First, we show that, in general, in unbounded exchange economies (for example, asset exchange economies allowing short sales), even if some agents' preferences are satiated, the absence of arbitrage is sufficient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades - that is, as long as agents' preferences satisfy weak nonsatiation. Second, we provide a new approach to proving existence in unbounded exchange economies. The key step in our new approach is to transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy are equilibria of the original economy. What our approach makes clear is that it is precisely the condition of weak nonsatiation - a condition considerably weaker than local or global nonsatiation - that makes possible this transformation. Moreover, as we show via examples, without weak nonsatiation, existence fails.
Keywords: C62; D50; 330; Arbitrage; Asset market equilibrium; Nonsatiation; Recession cones; Arbitragegesch\~AƒÂft; Vollkommener Wettbewerb; Theorie