Free entry oligopoly, Cournot, Bertrand and relative profit maximization

We study a symmetric free entry oligopoly in which firms produce differentiated goods so as to maximize their relative profits. The relative profit of each firm is the difference between its profit and the average of the profits of other firms. We show that whether firms determine their outputs or prices, the equilibrium price when firms maximize their relative profits is lower than the equilibrium price when firms maximize their absolute profits, but the equilibrium number of firms under relative profit maximization is smaller than the equilibrium number of firms under absolute profit maximization. This is because each firm is more aggressive and produces larger output under relative profit maximization than under absolute profit maximization.