We consider the problem of computing market equilibria and show three results. (i) For exchange economies satisfying weak gross substitutability we analyze a simple discrete version of tâtonnement, and prove that it converges to an approximate equilibrium in polynomial time. This is the first polynomial-time approximation scheme based on a simple tâtonnement process. It was only recently shown, using vastly more sophisticated techniques, that an approximate equilibrium for this class of economies is computable in polynomial time. (ii) For Fisher's model, we extend the frontier of tractability by developing a polynomial-time algorithm that applies well beyond the homothetic case and the gross substitutes case. (iii) For production economies, we obtain the first polynomial-time algorithms for computing an approximate equilibrium when the consumers' side of the economy satisfies weak gross substitutability and the producers' side is restricted to positive production.

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Over the last few years, the problem of computing market equilibrium prices for exchange economies has received much attention in the theoretical computer science community. Such activity led to a flurry of polynomial time algorithms for various restricted, yet significant, settings. The most important restrictions arise either when the traders' utility functions satisfy a property known as gross substitutability or when the initial endowments are proportional (the Fisher model). In this paper, we experimentally compare the performance of some of these recent algorithms against that of the most used software packages. In particular, we evaluate the following approaches: (1) the solver PATH, available under GAMS/MPSGE, a popular tool for computing market equilibrium prices; (2) a discrete version of a simple iterative price update scheme called tâtonnement; (3) a discrete version of the welfare adjustment process; (4) convex feasibility programs that characterize the equilibrium in some special cases. We analyze the performance of these approaches on models of exchange economies where the consumers are equipped with utility functions, which are widely used in real world applications. The outcomes of our experiments consistently show that many market settings allow for an efficient computation of the equilibrium, well beyond the restrictions under which the theory provides polynomial time guarantees. For some of the approaches, we also identify models where they are are prone to failure.A preliminary version of this paper appears in [Codenotti et al. 2005a]. This paper also includes additional results from [Codenotti et al. 2005c].

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