Cournot and Bertrand competition with asymmetric costs in a mixed duopoly revisited

We investigate a differentiated mixed duopoly in which private and public firms can choose to strategically set prices or quantities when the public firm is less efficient than the private firm. Thus, if the Singh and Vives assumption of positive primary outputs holds, (i) Bertrand competition or quantity-price competition can occur depending on the degree of public firm's inefficiency when the goods are substitutes. (ii) regardless of its inefficiency, there can be always sustained Bertrand competition when the goods are complements. (iii) the ranking of a private firm's profit is not reversed. However if we relax the parameter restriction imposed implicitly by Singh and Vives (i.e., we adopt Zanchettin (2006) assumption) to allow for a wider range of cost asymmetry, there can be always sustained multiple subgame Nash perfect equilibria in the contract stage by each critical value of the public firm's inefficiency. In particular, Cournot and Bertrand competition coexist if its inefficiency is sufficiently small or large.