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Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational

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  • Joaquín Andaluz
  • Gloria Jarne

Abstract

This paper compares the dynamic of Cournot and Bertrand duopolies with vertical product differentiation and under bounded rationality. We find that an increase in the product differentiation degree destabilizes the Nash equilibrium under quantity competition, while the Bertrand–Nash equilibrium becomes more stable. From a global dynamic analysis, we show that an increase in the firms’ adjustment speed constitutes a source of complexity in both models. There is a cascade of flip bifurcations leading to increasingly complex attractors, and there are global bifurcations generating complex basins of attraction. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Joaquín Andaluz & Gloria Jarne, 2016. "Stability of vertically differentiated Cournot and Bertrand-type models when firms are boundedly rational," Annals of Operations Research, Springer, vol. 238(1), pages 1-25, March.
    • Handle: RePEc:spr:annopr:v:238:y:2016:i:1:p:1-25:10.1007/s10479-015-2057-4
    DOI: 10.1007/s10479-015-2057-4
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    Cited by:

    1. Gori, Luca & Sodini, Mauro, 2017. "Price competition in a nonlinear differentiated duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 557-567.
    2. Wentao Yi & Chunqiao Tan, 2019. "Bertrand Game with Nash Bargaining Fairness Concern," Complexity, Hindawi, vol. 2019, pages 1-22, August.
    3. Yu, Yu & Yu, Weisheng, 2021. "The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    4. Mai, Fengxia & Zhang, Jianxiong & Sun, Xiaojie, 2021. "Dynamic analysis of pricing model in a book supply chain," International Journal of Production Economics, Elsevier, vol. 233(C).
    5. Andaluz, J. & Elsadany, A.A. & Jarne, G., 2023. "Dynamic behavior in a Cournot duopoly with social responsibility," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    6. Joaquín Andaluz & Gloria Jarne, 2019. "On price stability and the nature of product differentiation," Journal of Evolutionary Economics, Springer, vol. 29(2), pages 741-762, April.

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    More about this item

    Keywords

    Vertical product differentiation; Bertrand competition; Cournot competition; Bounded rationality; Dynamic stability; Bifurcation; C62; D43; L13;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets

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