Dynamic bertrand oligopoly

Andrew Ledvina, Ronnie Sircar

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


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 ofsmall 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.

Original languageEnglish (US)
Pages (from-to)11-44
Number of pages34
JournalApplied Mathematics and Optimization
Issue number1
StatePublished - Feb 2011

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics


  • Asymptotic expansions
  • Bertrand oligopoly
  • Nash equilibrium
  • Stochastic differential games
  • Substitutable goods
  • Systems of HJB equations


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