A Dynamic Model of Leap-Frogging Investments and Bertrand Price Competition

We present a dynamic extension of the classic static model of Bertrand price competition that allows competing duopolists to undertake cost-reducing investments in an attempt to "leapfrog" their rival and attain, at least temporarily, low-cost leadership. The model resolves a paradox about investing in the presence of Bertrand price competition: if both firms simultaneously invest in the current state-of-the-art production technology and thereby attain the same marginal cost of production, the resulting price competition drives the price down to marginal cost and profits to zero. Thus, it would seem that neither firm can profit from undertaking the cost-reducing investment, so the firms should not have any incentive to undertake cost-reducing investments if they are Bertrand price competitors. We show this simple intuition is incorrect. We formulate a dynamic model of price and investment competition as a Markov-perfect equilibrium to a dynamic game. We show that even when firms start with the same marginal costs of production there are equilibria where one of the firms invests first, and leapfrogs its opponent. In fact, there are many equilibria, with some equilibria exhibiting asymmetries where there are extended periods of time where only one of the firms does most of the investing, and other equilibria where there are alternating investments by the two firms as they vie for temporary low cost leadership. Our model provides a new interpretation of the concept of a "price war". Instead of being a sign of a breakdown of tacit collusion, in our model price wars occur when one firm leapfrogs its opponent to become the new low cost leader.