Endogenous timing in a mixed duopoly

This paper addresses the issue of endogenizing the equilibrium solution when a private - domestic or foreign - firm competes in the quantities with a public, welfare maximizing firm. Theoretical literature on mixed oligopolies, in fact, provides results and policy implications that crucially rely on the notion of equilibrium assumed, either sequential or simultaneous. In the framework of the endogenous timing model of Hamilton and Slutsky (1990), we show that simultaneous play never emerges as the equilibrium of mixed duopoly games. We provide sufficient conditions for the emergence of public and/or private leadership equilibria. These results are in sharp contrast with those obtained in private duopoly games in which simultaneous play is the general result. We show that the key difference lies in the fact that the objective of a welfare maximizing firm is generally increasing in the rival's output, while the contrary holds for private firms. We develop a comprehensive analysis of a mixed duopoly considering both the cases of domestic and international competition, and the possible strategic complementarity and substitutability. From a methodological viewpoint we make large use of the basic results of the theory of supermodular games in order to avoid extraneous assumptions such as concavity, existence and uniqueness of the equilibria.