Stability of Monopoly

Abstract

THE STABILITY PROPERTIES of a competitive economy are well known-at least under the artificial tatonnement assumption which avoids the problem of who changes prices. This note demonstrates that similar, if not stronger, properties carry over to a monopolistic economywhere price formation is completely explained by the independent optimizing behavior of individual agents. The discussion may be interpreted as that of monopoly as the opposite polar case to that of competition or, equivalently, as that of the stability of the concept of equilibrium presented in an earlier paper [1]. Price adjustment is by individuals, following [3 and 4], through their perceived demand curves, following [2 and 5]. There are n commodities (the last being a numeraire) identified by ownership: agent i is endowed with one unit of commodity i and no other commodity. Each agent (i) chooses a price pi ) 0 for his commodity and a consumption x' e R% , so that a state of the economy is an array (p1. p, x . x), or (p, X). Given some existing state (p, X) each agent (i) first chooses pi to maximize his expected income pixi, where xi is his expected sales, given by his perceived linear demand curve (with negative slope one, say) xi(pi) = x-i + Pi - pi; here Xi = I 5i is the existing aggregate demand. Taking account of the constraint xi < 1 this gives