Markets with Production: A Polynomial Time Algorithm and a Reduction to Pure Exchange

Abstract

The classic Arrow-Debreu market model captures both production and consumption, two equally important blocks of an economy, however most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. In this paper we show two new results on markets with production. Our first result gives a polynomial time algorithm for Arrow-Debreu markets under piecewise linear concave (PLC) utilities and polyhedral production sets provided the number of goods is constant. This is the first polynomial time result for the most general case of Arrow-Debreu markets. Our second result gives a novel reduction from an Arrow-Debreu market M (with production firms) to an equivalent exchange market M' such that the equilibria of M are in one-to-one correspondence with the equilibria of M'. Unlike the previous reduction by Rader where M' is artificially constructed, our reduction gives an explicit market M' and we also get: (i) when M has concave utilities and convex production sets (standard assumption in Arrow-Debreu markets), then M' has concave utilities, (ii) when M has PLC utilities and polyhedral production sets, then M' has PLC utilities, and (iii) when M has nested CES-Leontief utilities and nested CES-Leontief production, then M' has nested CES-Leontief utilities.