Relative profit maximization in asymmetric oligopoly: Cournot and Bertrand equilibria

We analyze Bertrand and Cournot equilibria in an asymmetric oligopoly with more than two firms in which the firms produce differentiated substitutable goods and seek to maximize their relative profits instead of their absolute profits. Assuming linear demand functions and constant marginal costs we show the following results. If the marginal cost of a firm is lower (higher) than the average marginal cost over the industry, its output at the Bertrand equilibrium is larger (smaller) than that at the Cournot equilibrium, and the price of its good at the Bertrand equilibrium is lower (higher) than that at the Cournot equilibrium.