Computation of Fisher-Gale equilibrium by auction

Résumé

We study the Fisher model of a competitive market from the algorithmic perspective. For that, the related convex optimization problem due to Gale and Eisenberg [10] is used. The latter problem is known to yield a Fisher equilibrium under some structural assumptions on consumers’ utilities, e.g. homogeneity of degree 1, homotheticity etc. Our goal is to examine the applicability of the convex optimization framework by departing from these traditional assumptions. We just assume the concavity of consumers’ utility functions. For this case we suggest a novel concept of Fisher-Gale equilibrium by using consumers’ utility prices. The prices of utility transfer the utility of a consumption bundle to a common num´eraire. We develop a subgradient-type algorithm from Convex Analysis to compute a Fisher-Gale equilibrium via Gale’s approach. In order to decentralize prices, we additionally implement the auction design, i.e. consumers settle and update their individual prices and producers sell at the highest offer price. Our price adjustment is based on a tˆatonnement procedure, i.e. the prices change proportionally to consumers’ individual excess supplies. Historical averages of consumption are shown to clear the market of goods. Our algorithm enjoys a convergence rate. In worst case, the number of price updates needed to achieve the ε-tolerance is proportional to 1/ε² .