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A Note on Bertrand Competition under Quadratic Cost Functions


  • Jacek Prokop
  • Michał Ramsza
  • Bartłomiej Wiśnicki


The authors focus on a model of competition known in economics as Bertrand competition. They investigate how the outcome of Bertrand competition changes when linear cost functions are replaced by quadratic functions in the model. Named after French mathematician Joseph Louis Francois Bertrand (1822-1900), Bertrand competition describes interactions among firms and a market situation in which firms make their output and pricing decisions based on the assumption that their competitors will not change their own prices. Prokop, Ramsza and Wiśnicki show that the introduction of quadratic cost functions in the model leads to a qualitative change in competition between firms. In this case, the standard assumption that a firm with a lower price is interested in taking over the entire market is not only unrealistic (even in the absence of capacity constraints), but also irrational from the profit-maximization viewpoint, the authors say. Therefore the results of previous studies are misleading, according to Prokop, Ramsza and Wiśnicki. They argue that the so-called Bertrand duopoly model should be adjusted to better capture the behavior of firms under quadratic cost functions. The authors relax the assumption that the output of firms is determined by existing demand. They offer a modified Bertrand duopoly model in which firms competing in prices are free to choose their level of production depending on the market demand for their products at specific prices. “Under the modified assumptions, the static price competition game of identical duopolists has no symmetric equilibrium in pure stra- tegies,” the authors conclude. Their research shows that, under quadratic cost functions, it is possible to expect price fluctuations on oligopolistic markets rather than a stable equilibrium situation described by the standard model of price competition.

Suggested Citation

  • Jacek Prokop & Michał Ramsza & Bartłomiej Wiśnicki, 2015. "A Note on Bertrand Competition under Quadratic Cost Functions," Gospodarka Narodowa, Warsaw School of Economics, issue 2, pages 5-14.
  • Handle: RePEc:sgh:gosnar:y:2015:i:2:p:5-14

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    References listed on IDEAS

    1. Jean Tirole, 1988. "The Theory of Industrial Organization," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262200716, January.
    2. Atsuhiro Satoh & Yasuhito Tanaka, 2013. "Relative profit maximization and Bertrand equilibrium with quadratic cost functions," Economics and Business Letters, Oviedo University Press, vol. 2(3), pages 134-139.
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    More about this item


    Bertrand competition; quadratic cost functions; static price competition;

    JEL classification:

    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets
    • L41 - Industrial Organization - - Antitrust Issues and Policies - - - Monopolization; Horizontal Anticompetitive Practices
    • O31 - Economic Development, Innovation, Technological Change, and Growth - - Innovation; Research and Development; Technological Change; Intellectual Property Rights - - - Innovation and Invention: Processes and Incentives


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