Quantity competition vs. price competition under optimal subsidy in a mixed oligopoly

•We investigate the privatization neutrality theorem under optimal subsidy in mixed markets.•An optimal subsidy is provided by the government to a state-controlled firm and one or more private firms.•Both Cournot and Bertrand competition are considered.•The ownership of the state-controlled firm matters when firms compete sequentially.•Price (quantity) public leadership is equivalent to quantity (price) private leadership under duopoly.

This paper reconsiders the literature on the irrelevance of privatization in mixed markets within which both quantity and price competition are investigated under product differentiation. By allowing for partially privatization of a state-controlled firm, we explore competition under different timings of firms' moves and derive the conditions under which an optimal subsidy allows to achieve maximum efficiency. We show that, irrespective of the mode of competition, while the ownership of the controlled firm is irrelevant when firms play simultaneously, it matters when firms compete sequentially, requiring the leader to be publicly-owned for an optimal subsidy to restore the first-best. The paper also focuses on the extent to which a subsidy is needed to attain the social optimum in the considered scenarios, providing an ordering which highlights the subsidy equivalence between Cournot (Bertrand) private leadership and simultaneous Bertrand (Cournot) under duopoly, and the dominance of the former in oligopoly.