Relative profit maximization and Bertrand equilibrium with quadratic cost functions

We study the Bertrand equilibrium in duopoly in which two firms produce a homogeneous good under quadratic cost functions, and they seek to maximize the weighted sum of their absolute and relative profits. We show that there exists a range of the equilibrium price in duopolistic equilibria. This range of equilibrium price is narrower and lower than the range of the equilibrium price in duopolistic equilibria under pure absolute profit maximization, and the larger the weight on the relative profit, the narrower and lower the range of the equilibrium price. In this sense relative profit maximization by the firms is more aggressive than absolute profit maximization.