Relative profit maximization in duopoly: difference or ratio

We compare formulations of relative profit maximization in duopoly with differentiated goods, 1) (Difference case) maximization of the difference between the profit of one firm and that of the other firm, 2) (Ratio case) maximization of the ratio of the profit of one firm to the total profit. We show that in asymmetric duopoly the equilibrium output of the more efficient (lower cost) firm in the ratio case is larger than that in the difference case and the price of the good of the more efficient firm in the ratio case is lower than that in the difference case. For the less efficient firm (higher cost firm) we obtain the converse results