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

In this note we investigate the relation between a Cournot equilibrium and a Bertrand equilibrium in a 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 will show that when firms maximize relative profits, a Cournot equilibrium and a Bertrand equilibrium coincide, and the equilibrium outputs under relative maximization is larger than both of the equilibrium outputs at the Cournot equilibrium and 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 constant and the fixed costs are zero.