Dynamic Bertrand Oligopoly

TL;DR

This paper analyzes dynamic Bertrand oligopolies with multiple firms competing through pricing, using static and dynamic models, to understand equilibrium strategies and market outcomes under uncertainty and heterogeneity.

Contribution

It introduces a stochastic differential game framework for dynamic Bertrand competition with heterogeneous firms and characterizes equilibrium features using analytical and numerical methods.

Findings

Consumers benefit from many similarly sized firms producing substitutable goods.

High substitutability does not always lead to significant price drops, especially with size disparities.

Asymptotic analysis provides insights into competitive behavior in the small-competition limit.

Abstract

We study continuous time Bertrand oligopolies in which a small number of firms producing similar goods compete with one another by setting prices. We first analyze a static version of this game in order to better understand the strategies played in the dynamic setting. Within the static game, we characterize the Nash equilibrium when there are $N$ players with heterogeneous costs. In the dynamic game with uncertain market demand, firms of different sizes have different lifetime capacities which deplete over time according to the market demand for their good. We setup the nonzero-sum stochastic differential game and its associated system of HJB partial differential equations in the case of linear demand functions. We characterize certain qualitative features of the game using an asymptotic approximation in the limit of small competition. The equilibrium of the game is further studied using numerical solutions. We find that consumers benefit the most when a market is structured with many firms of the same relative size producing highly substitutable goods. However, a large degree of substitutability does not always lead to large drops in price, for example when two firms have a large difference in their size.