Equivalance of Cournot and Bertrand equilibria in differentiated duopoly under relative profit maximization with linear demand

We investigate the relation between a Cournot equilibrium and a Bertrand equilibrium in a duopoly with differentiated substitutable 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 when firms maximize relative profits, a Cournot equilibrium and a Bertrand equilibrium coincide, and the equilibrium outputs under relative profit maximization are larger than the equilibrium outputs at the Cournot equilibrium and those at the Bertrand equilibrium under absolute profit maximization. We assume that demand functions for the goods of the firms are linear, the marginal costs of the firms are common and constant and the fixed costs are zero.