Computation of Fisher-Gale Equilibrium by Auction

We study the Fisher model of a competitive market from the algorithmic perspective. For that, the related convex optimization problem due to Gale and Eisen-berg (Ann Math Stat 30(1):165-168, 1959) 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. Our goal is to examine 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 consumption bundle to a common numeraire. 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 tatonnement 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 is justified by a global rate of convergence. In the worst case, the number of price updates needed to achieve an epsilon-tolerance is proportional to 1/epsilon(2).