On the polynomial time computation of equilibria for certain exchange economies

The problem of computing equilibria for exchange economies has recently started to receive a great deal of attention in the theoretical computer science community. It has been shown that equilibria can be computed in polynomial time in various special cases, the most important of which are when traders have linear, Cobb-Douglas, or a range of CES utility functions. These important special cases are instances when the market satisfies a property called weak gross substitutability. Classical results in economics, which theoretical computer scientists (including us) appear to have been hitherto unaware of, show that the equilibrium prices in such markets are characterized by an infinite number of linear inequalities and therefore form a convex set. In this paper, we show that under fairly general assumptions, there are polynomial-time algorithms to compute equilibria in such markets. To the best of our knowledge, these are the first polynomial-time algorithms for exchange markets under the general setting of weak gross substitutability. To show this result, we need to build on the proofs that characterize the equilibria as a convex set.As a consequence, we obtain alternative polynomial-time algorithms for computing equilibria with linear, Cobb-Douglas, a range of CES, as well as certain other non-homogeneous utility functions that satisfy weak gross substitutability. Unlike previous polynomial-time algorithms, our approach does not make use of the specific form of these utility functions and is in this sense more general. We expect our framework to work or be readily adaptable to handle other exchange markets, provided that the utility functions satisfy weak gross substitutability.