The purpose of this article is to investigate how the introduction of the shadow cost of public funds in the utilitarian measure of the economywide welfare affects the behavior of a welfare maximizer public firm in a mixed duopoly. We prove that when firms play simultaneously, the mixed-Nash equilibrium can dominate any Cournot equilibria implemented after a privatization, with or without efficiency gains. This can be true both in terms of welfare and of public firm's profit. When we consider endogenous timing, we show that either mixed- Nash, private leadership or both Stackelberg equilibria can result as subgameperfect Nash equilibria (SPNE). As a consequence, the sustainability of sequential equilibria enlarges the subspace of parameters such that the market performance with an inefficient public firm is better than the one implemented after a full-efficient privatization. Absent efficiency gains, privatization always lowers welfare