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On Cournot–Bertrand competition with differentiated products

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  • S. Askar

Abstract

In this paper, we present a game theoretic framework for Cournot–Bertrand competition based on a nonlinear price function. The competition is between two firms and is assumed to take place in terms of pricing decision and quantity produced. However, the proposed objective function has not been used in literature before, yet the throughput obtained in this paper generalizes some of the existing results in literature. The competitive interaction between firms is described and analyzed using best-reply reaction, proposed adaptive adjustment and bounded rationality approach. The condition of stability of Nash equilibrium (NE) is induced by these approaches. Interestingly, we prove that there exists exactly a unique NE. Furthermore, it is noticed that when firms adopt best-reply and the proposed adaptive adjustment, the firms’ strategic variables become asymptotically stable. On the contrary, when the bounded rationality is used both quantity and price behave chaotically due to bifurcation occurred. Copyright Springer Science+Business Media New York 2014

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  • S. Askar, 2014. "On Cournot–Bertrand competition with differentiated products," Annals of Operations Research, Springer, vol. 223(1), pages 81-93, December.
    • Handle: RePEc:spr:annopr:v:223:y:2014:i:1:p:81-93:10.1007/s10479-014-1612-8
    DOI: 10.1007/s10479-014-1612-8
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    References listed on IDEAS

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    Cited by:

    1. Askar, S.S., 2021. "On complex dynamics of Cournot-Bertrand game with asymmetric market information," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    2. Jian Ni & Lap Keung Chu & Shoude Li, 2018. "Financial hedging and competitive strategy for value-maximizing firms under quantity competition," Annals of Operations Research, Springer, vol. 264(1), pages 391-407, May.
    3. 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.
    4. Askar, S.S., 2022. "On the dynamics of Cournot duopoly game with private firms: Investigations and analysis," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    5. Ren, Jing & Sun, Hao & Xu, Genjiu & Hou, Dongshuang, 2023. "Convergence of output dynamics in duopoly co-opetition model with incomplete information," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 209-225.
    6. Sung Hee Lee & Kun Soo Park & Yong Won Seo, 2016. "Multinational Firm’s Production Decisions under Overlapping Free Trade Agreements: Rule of Origin Requirements and Environmental Regulation," Sustainability, MDPI, vol. 9(1), pages 1-16, December.
    7. Duarte Brito & Margarida Catalão-Lopes, 2023. "Cournot–Bertrand endogenous behavior in a differentiated oligopoly with entry deterrence," Theory and Decision, Springer, vol. 95(1), pages 55-78, July.
    8. Anne-Christine Barthel & Eric Hoffmann, 2020. "On the existence and stability of equilibria in N-firm Cournot–Bertrand oligopolies," Theory and Decision, Springer, vol. 88(4), pages 471-491, May.
    9. S. S. Askar & A. Al-khedhairi, 2019. "Cournot Duopoly Games: Models and Investigations," Mathematics, MDPI, vol. 7(11), pages 1-15, November.
    10. Duarte Brito & Margarida Catalão-Lopes, 2023. "Profit raising entry under mixed behavior," Journal of Economics, Springer, vol. 138(1), pages 51-72, January.
    11. Zhang, Ming & Wang, Guanghui & Xu, Jin & Qu, Cunquan, 2020. "Dynamic contest model with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    12. S. S. Askar & Mona F. EL-Wakeel & M. A. Alrodaini, 2018. "Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products," Complexity, Hindawi, vol. 2018, pages 1-13, February.
    13. Sameh S Askar & Abdulrahman Al-Khedhairi, 2020. "Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game," Mathematics, MDPI, vol. 8(9), pages 1-16, August.
    14. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2015. "Dynamic Cournot duopoly games with nonlinear demand function," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 427-437.
    15. Bian, Junsong & Lai, Kin Keung & Hua, Zhongsheng & Zhao, Xuan & Zhou, Guanghui, 2018. "Bertrand vs. Cournot competition in distribution channels with upstream collusion," International Journal of Production Economics, Elsevier, vol. 204(C), pages 278-289.
    16. 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.
    17. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2016. "The arising of cooperation in Cournot duopoly games," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 535-542.
    18. Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
    19. Kittaka, Yuta & Matsushima, Noriaki & Saruta, Fuyuki, 2022. "Negative effect of price-matching policy on traditional retailers in a dual-channel supply chain with different content formats," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 161(C).

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