Continuity of Demand and the Direct Approach to Equilibrium Existence in Dual Banach Commodity Spaces - (Now published as 'Berge's Maximum Theorem with two topologies on the action set', in Economics Letters, 61 (1998), pp.285-291.)

Horsley, Anthony; Wrobel, Andrew J

Continuity of Demand and the Direct Approach to Equilibrium Existence in Dual Banach Commodity Spaces - (Now published as 'Berge's Maximum Theorem with two topologies on the action set', in Economics Letters, 61 (1998), pp.285-291.)

Published 1992

An extension of Berge's Maximum Theorem is given, with two different topologies on the choice set used for the two semicontinuity assumptions on preferences. It is used to establish the norm-to-weak* continuity of (truncated) excess-demand, announced without proof by Jones (1986). This significantly improves upon the continuity result of Florenzano (1983) and Van Zandt (1989) who use the finite topology on the price space. Though that result provides a sufficient basis for the direct, excess-demand method of proving equilibrium existence, its usefulness is limited by the extreme strength of the finite topology. With the norm topology used instead, demand continuity acquires an independent interest, particularly for practical implementations of the equilibrium solution. Other techniques needed to realize the full potential of the direct approach are also developed. These include a closedness result for the total production set and an additivity property of the asymptotic cone operation.