Cournot and Bertrand competition with asymmetric costs in a mixed duopoly

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, regardless of whether the goods are substitutes or complements, if the degree of public firm's inefficiency is sufficiently small, there exists a dominant strategy for both public and private firms that choose Bertrand competition, while there exists a dominant strategy only for the private firm that chooses Bertrand competition if the degree of inefficiency is sufficiently large. Consequently, we show that regardless of the nature of goods, (i) social welfare under Bertrand competition is determined in equilibrium, if the degree of public firm's inefficiency is sufficiently small; and (ii) if the degree of its inefficiency is sufficiently large, social welfare under which the private firm sets its price and the public firm sets its quantity is determined in equilibrium. Moreover, the ranking of a private firm's profit is not reversed.