On the Non-existence of Equilibrium: From Thornton to Arrow

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Among the many contributions to economic science made by Arrow, one of the most important is certainly the proof of the existence of an equilibrium for a competitive economy.1 The case where no equilibrium exists even though indifference curves, production functions, and so on, are fairly well behaved is a useful one to show the necessity of proving the existence of equilibrium. Mill (1869) indicates that one of the first examples of the non-existence of equilibrium consists of the counter examples to equilibrium theory given in W. T. Thornton’s On Labour (1869) though Thornton himself was concerned not so much with the non-existence of equilibrium as the possibility of trade at disequilibrium prices. The Thornton-Mill examples of the non-existence of equilibrium are remarkable because they are due to the discontinuity of demand curves; other unsuccessful attempts to show the non-existence of equilibrium failed because of their assumption of continuity. The Thornton-Mill examples, as well as the example of Wald (1951), are, however, not so serious to equilibrium theory, if we consider, not the Walrasian tâtonnement with recontract, but the non-tâtonnement without recontract. From such non-tâtonnement point of view, a truly important example of the non-existence of equilibrium is the one given by Arrow, that is, the case where a Pareto optimal allocation cannot be viewed as a competitive equilibrium.2

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