IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2020i1p32-d468104.html
   My bibliography  Save this article

Delay Cournot Duopoly Game with Gradient Adjustment: Berezowski Transition from a Discrete Model to a Continuous Model

Author

Listed:
  • Akio Matsumoto

    (Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan)

  • Ferenc Szidarovszky

    (Department of Mathematics, Corvinus University, Fovám tér 8, 1093 Budapest, Hungary)

  • Keiko Nakayama

    (Department of Economics, Chukyo University, 101-2, Yagoto Honmachi, Showa, Hagoya 466-8666, Japan)

Abstract

This paper investigates the asymptotical behavior of the equilibrium of linear classical duopolies by reconsidering the two-delay model with two different positive delays. In a two-dimensional analysis, the stability switching curves were first analytically determined. Numerical studies verified and illustrated the theoretical results. In the sensitivity analysis it was demonstrated that the inertia coefficient has a twofold effect: enlarges the stability region as well as simplifies the complicated dynamics with period-halving cascade. In contrary, the adjustment speed contracts the stability region and complicates simple dynamics with period-doubling bifurcation. In addition, for various values of τ 1 and τ 2 , a wide variety of dynamics appears ranging from simple cycle via a Hopf bifurcation to chaotic oscillations.

Suggested Citation

  • Akio Matsumoto & Ferenc Szidarovszky & Keiko Nakayama, 2020. "Delay Cournot Duopoly Game with Gradient Adjustment: Berezowski Transition from a Discrete Model to a Continuous Model," Mathematics, MDPI, vol. 9(1), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:32-:d:468104
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/1/32/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/1/32/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Gori, Luca & Guerrini, Luca & Sodini, Mauro, 2015. "A continuous time Cournot duopoly with delays," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 166-177.
    2. Matsumoto, Akio & Szidarovszky, Ferenc, 2014. "Discrete and continuous dynamics in nonlinear monopolies," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 632-642.
    3. Howroyd, T. D. & Russell, A. M., 1984. "Cournot oligopoly models with time delays," Journal of Mathematical Economics, Elsevier, vol. 13(2), pages 97-103, October.
    4. Akio Matsumoto & Ferenc Szidarovszky, 2018. "Dynamic Oligopolies with Time Delays," Springer Books, Springer, number 978-981-13-1786-6, December.
    5. Bischi, Gian-Italo & Stefanini, Luciano & Gardini, Laura, 1998. "Synchronization, intermittency and critical curves in a duopoly game," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 44(6), pages 559-585.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Luca Gori & Luca Guerrini & Mauro Sodini, 2017. "A characterisation of duopoly dynamics with frictions in production adjustments," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 963-988, November.
    2. Culda, Loredana Camelia & Kaslik, Eva & Neamţu, Mihaela, 2022. "Stability and bifurcations in a general Cournot duopoly model with distributed time delays," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    3. Gori, Luca & Guerrini, Luca & Sodini, Mauro, 2015. "A continuous time Cournot duopoly with delays," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 166-177.
    4. Matsumoto, Akio & Szidarovszky, Ferenc, 2015. "Nonlinear Cournot duopoly with implementation delays," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 157-165.
    5. Matsumoto, Akio & Szidarovszky, Ferenc, 2022. "The chaotic monopolist revisited with bounded rationality and delay dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    6. Serena Brianzoni & Giovanni Campisi & Luca Guerrini, 2018. "A continuous-time heterogeneous duopoly model with delays," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 259-275, November.
    7. Raul Villafuerte-Segura & Eduardo Alvarado-Santos & Benjamin A. Itza-Ortiz, 2019. "Conditions for stable equilibrium in Cournot duopoly models with tax evasion and time delay," Papers 1905.02817, arXiv.org, revised Oct 2019.
    8. Troy Tassier, 2013. "Handbook of Research on Complexity, by J. Barkley Rosser, Jr. and Edward Elgar," Eastern Economic Journal, Palgrave Macmillan;Eastern Economic Association, vol. 39(1), pages 132-133.
    9. Antonio Doria, Francisco, 2011. "J.B. Rosser Jr. , Handbook of Research on Complexity, Edward Elgar, Cheltenham, UK--Northampton, MA, USA (2009) 436 + viii pp., index, ISBN 978 1 84542 089 5 (cased)," Journal of Economic Behavior & Organization, Elsevier, vol. 78(1-2), pages 196-204, April.
    10. Yu Yu & Weisheng Yu, 2019. "The Complexion of Multi-period Stackelberg Triopoly Game with Bounded Rationality," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 457-478, January.
    11. Caravaggio, Andrea & Gori, Luca & Sodini, Mauro, 2022. "Endogenous preferences in a dynamic Cournot duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    12. Luís, Rafael & Mendonça, Sandra, 2024. "Local stability conditions for a n-dimensional periodic mapping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 15-30.
    13. Tamotsu Onozaki, 2018. "Nonlinearity, Bounded Rationality, and Heterogeneity," Springer Books, Springer, number 978-4-431-54971-0, December.
    14. Luciano Fanti & Luca Gori, 2013. "Stability Analysis in a Bertrand Duopoly with Different Product Quality and Heterogeneous Expectations," Journal of Industry, Competition and Trade, Springer, vol. 13(4), pages 481-501, December.
    15. S. Kabaivanov & V. Zhelinski & B. Zlatanov, 2021. "Coupled Fixed Points for Hardy-Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions," Papers 2110.01496, arXiv.org.
    16. Fanti, Luciano & Gori, Luca & Sodini, Mauro, 2012. "Nonlinear dynamics in a Cournot duopoly with relative profit delegation," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1469-1478.
    17. Fanti, Luciano & Gori, Luca, 2011. "The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited," MPRA Paper 33268, University Library of Munich, Germany.
    18. Fanti, Luciano & Gori, Luca & Sodini, Mauro, 2015. "Nonlinear dynamics in a Cournot duopoly with isoelastic demand," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 129-143.
    19. Matsumoto, Akio & Szidarovszky, Ferenc, 2012. "Nonlinear delay monopoly with bounded rationality," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 507-519.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:32-:d:468104. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.