A Rational Convex Program for Linear Arrow-Debreu Markets

TL;DR

This paper introduces a new flow-type convex program for linear Arrow-Debreu markets that simplifies the understanding of equilibrium solutions, proving existence, rationality, and convexity of equilibrium prices.

Contribution

It presents a novel convex program with new features, providing simple proofs for existence, rationality, and the convex polyhedral structure of equilibrium prices.

Findings

Provides a necessary and sufficient condition for equilibria.

Proves the rationality of equilibrium solutions.

Shows equilibrium prices form a convex polyhedral set.

Abstract

We give a new, flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known [Nenakov, Primak 83; Jain 07; Cornet '89], our program exhibits several new features. It gives a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani. As a consequence we also obtain a simple new proof of Mertens's result that the equilibrium prices form a convex polyhedral set.