Bowley's Analysis of Bilateral Monopoly and Sen's Liberal Paradox in Collective Choice Theory: A Note

Abstract

The purpose of this note is to show that A. L. Bowley's [1928] price leadership analysis of bilateral monopoly implicitly defines a social decision function that incorporates Sen's principle of minimal liberalism [Sen, 1970, Ch. 6]. This result establishes a direct connection between the failure of the price leadership solution to achieve maximum joint profits for both monopolists and Sen's liberal paradox, that any social decision function that is defined over a wide range of individual agent values and that incorporates the principle of minimal liberalism must violate Pareto optimality. Both liberalism, as formulated by Sen, and the price leadership analysis of bilateral monopoly introduced by Bowley share the property that each agent in the social decision process has a sphere of choice over which he is decisive. It is this property of limited-choice decisiveness that can lead, in specific instances of Sen's paradox and also in Bowley's analysis, to social decisions that are inconsistent with Pareto optimality. It is of interest to note that such limited-choice decisiveness may be found on liberal social ethics, as it is for Sen, or on descriptive features of specific roles in the institutional environment of a bargaining process, as it is for Bowley. Also the analysis of this paper can be extended to either Cournot type or von Stackelberg type duopoly theories to show that they define social decision functions that satisfy the minimal liberalism condition, thus giving rise, in specific circumstances, to the violation of Pareto optimality. These results are easily seen in a graphical analysis of price leadership models, as demonstrated in Section II. A rigorous formal proof is given in Section III.