Ex ante and ex post equilibrium supply curves

A number of writers have modelled imperfect markets using games in which the strategies are supply functions, that is, mappings from prices to quantities produced. Two representations of this problem have been analyzed, which may be referred to as ex ante and ex post, depending on whether strategies are chosen before or after demand shocks are observed. Both solutions give rise to an equilibrium supply curve, tracing out the locus of market equilibrium as demand varies in response to additive shocks. In this paper, we examine conditions under which equilibria in supply curves derived using the ex ante and ex post representations coincide. We show that under these conditions, the ex post solution is a dominant strategy equilibrium. Moreover, players are indifferent as to move order. For a familiar family of supply functions, we show that, for any given demand function, there exists a cost function such that the ex ante solution coincides with the ex post solution. These conditions generalize the case of linear demand and quadratic cost, analyzed by Klemperer and Meyer (Econometrica 57:1243–1277, 1989), and we demonstrate that all ex ante solutions derived in this way are unique. This result substantially extends the uniqueness result derived by Klemperer and Meyer (Econometrica 57:1243–1277, 1989) and suggests a path for further extensions. Further, we consider the relationship between the ex ante and ex post equilibrium concepts and the conjectural variations equilibrium concept of Bowley (The Mathematical Groundwork of Economics, 1924) and Bresnahan (Am Econ Rev 71:934–945, 1981). We show that, whenever ex ante and ex post equilibrium supply curves coincide, this equilibrium corresponds to a unique consistent conjectural variations equilibrium.