Price versus quantity in a mixed duopoly with foreign penetration

•We compare Bertrand and Cournot in a mixed duopoly.•We also endogenize the price-quantity choice.•We allow foreign competition in a domestic and in an integrated market.•Bertrand competition always yields higher welfare than Cournot.•Bertrand appears in a domestic market but not always in an integrated market.

We characterize the endogenous competition structure (in prices or quantities) in a differentiated duopoly between a public firm that maximizes domestic welfare and a private firm that can be owned by domestic or foreign investors. The market for which they compete can be domestic or integrated: in the first case Bertrand competition emerges endogenously and in the second case Cournot competition can emerge if the fraction of domestic consumers in the integrated market is low enough. We also determine the optimal degree of foreign penetration showing the optimality of a partial foreign ownership. Finally, we extend the model to increasing marginal cost confirming the robustness of the results.