Equilibrium co-existence of public and private firms and the plausibility of price competition

We consider a differentiated product duopoly where a regulated firm competes with a private firm. The instrument of regulation is the level of privatization. First, the regulator determines the level of privatization to maximize social welfare. Then both firms endogenously choose the mode of competition (that is, whether to compete in price or quantity). Finally, the two firms compete in the market. Under a very general demand specification, we show that when the products are imperfect substitutes (complements), there is co-existence of private and public (strictly partially privatized) firms. Moreover, in the second stage, the firms compete in prices