Multimarket oligopolies with restricted market access

We study the existence of Cournot equilibria in multimarket oligopolies under the additional restriction that every firm may sell its product only to a limited number of markets simultaneously. This situation naturally arises if market entry is subject to a valid license and each firm holds a fixed number of licenses only, or, equivalently, if the firms’ short-term assets only suffice to serve up to a certain number of markets. We allow for firm-specific market reaction functions modeling heterogeneity among products. As our main result, we show the existence of a Cournot equilibrium under the following assumptions stated informally below: (i) cost functions are convex; (ii) the marginal return functions strictly decrease for strictly increased own quantities and non-decreased aggregated quantities; (iii) for every firm, the firm-specific price functions across markets are identical up to market-specific shifts. While assumptions (i) and (ii) are frequently imposed in the literature on single market oligopolies, only assumption (iii) seems limiting. We show, however, that if it is violated, there are games without a Cournot equilibrium.