Relative profit maximization and irrelevance of leadership in Stackelberg model

We study the Stackelberg equilibrium in a symmetric duopoly with differentiated goods in which each firm maximizes its relative profit that is the difference between its profit and the profit of the rival firm. We show that the equilibrium output and price of the good of the leader and those of the follower are equal, that is, the role of leader or follower is irrelevant to the equilibrium, and the equilibrium outputs and prices do not change between the case where the firms are quantity setting firms and the case where the firms are price setting firms. We assume that demand functions are linear and symmetric, the marginal costs of the firms are common and constant, and the fixed costs are zero.