Brief Announcement: Computation of Fisher-Gale Equilibrium by Auction

Abstract

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, [3], 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. We just assume the concavity of consumers’ utility functions. For this case we suggest a novel concept of Fisher-Gale equilibrium by introducing consumers’ utility prices. We develop a subgradient-type algorithm from Convex Analysis to compute a Fisher-Gale equilibrium by auction. In worst case, the number of price updates needed to achieve the $\varepsilon$-tolerance is proportional to $\frac{1}{\varepsilon ^2}$.