Computation of Market Equilibria by Convex Programming

The market equilibrium problem consists of finding a set of prices and allocations of goods to economic agents such that each agent maximizes her utility, subject to her budget constraints, and the market clears. Since the nineteenth century, economists have introduced models that capture the notion of market equilibrium. In 1874, Walras published the "Elements of Pure Economics," in which he describes a model for the state of an economic system in terms of demand and supply, and expresses the supply equal demand equilibrium conditions (Walras, 1954). In 1936, Wald gave the first proof of the existence of an equilibrium for the Walrasian system, albeit under severe restrictions (Wald, 1951). In 1954, Nobel laureates Arrow and Debreu proved the existence of an equilibrium under much milder assumptions (Arrow and Debreu, 1954). The market equilibrium problem can be stated as a fixed point problem, and indeed the proofs of existence of a market equilibrium are based on either Brouwer's or Kakutani's fixed point theorem, depending on the setting (see, e. g., the beautiful monograph (Border, 1985) for a friendly exposition of the main results in this vein).