Quantity Precommitment and Bertrand Competition: A Dynamic Games Approach

Abstract

In this paper we look at a new way to combine both quantity precommitment and price competition in a dynamic games framework. In each period players choose to invest in production capacities for the next period and also engage in a price competition. Production is free up to capacity and has steeply increasing costs if exceeded. The demand is allocated according to a quadratic utility function, which allows for product differentiation. This way we avoid rationing rules, that are frequently used in these kind of models. We use real algebraic geometry to explicitly solve the subgames played each period. This information allows us to guess that the value function is piecewise quadratic. Our results are that if costs for exceeding production capacity go to infinity, then the equilibrium converges to an extreme case, where production is free up to capacity and excess has infinite marginal cost. This limiting approach allows us to compare our model to the ones with homogeneous goods and rationing. We also construct a similar Cournot like model as a benchmark. Computational experiments show that, in the limit, the outcome of the price competition with quantity precommitment equals the one in the Cournot case. This extends a similar result for static two stage games. However if we move away from the limit case, i.e. to a case where marginal costs are finite, the two models no longer agree.