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Discrete and continuous dynamics in nonlinear monopolies

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  • Matsumoto, Akio
  • Szidarovszky, Ferenc

Abstract

Dynamic monopolies are investigated with discrete and continuous time scales by assuming general forms of the price and cost functions. The existence of the unique profit maximizing output level is proved. The discrete model is then constructed with gradient adjustment. It is shown that the steady state is locally asymptotically stable if the speed of adjustment is small enough and it goes to chaos through period-doubling cascade as the speed becomes larger. The non-negativity condition that prevents time trajectories from being negative is derived. The discrete model is converted into the continuous model augmented with time delay and inertia. It is then demonstrated that stability can be switched to instability and complex dynamics emerges as the length of the delay increases and that instability can be switched to stability as the inertia coefficient becomes larger. Therefore the delay has the destabilizing effect while the inertia has the stabilizing effect.

Suggested Citation

  • Matsumoto, Akio & Szidarovszky, Ferenc, 2014. "Discrete and continuous dynamics in nonlinear monopolies," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 632-642.
    • Handle: RePEc:eee:apmaco:v:232:y:2014:i:c:p:632-642
    DOI: 10.1016/j.amc.2014.01.101
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    1. Askar, S.S., 2013. "On complex dynamics of monopoly market," Economic Modelling, Elsevier, vol. 31(C), pages 586-589.
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    Cited by:

    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. 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.
    3. 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.
    4. Matsumoto, Akio & Szidarovszky, Ferenc, 2015. "Nonlinear Cournot duopoly with implementation delays," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 157-165.
    5. 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.
    6. Cavalli, Fausto & Naimzada, Ahmad, 2015. "Effect of price elasticity of demand in monopolies with gradient adjustment," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 47-55.
    7. Matsumoto, Akio & Szidarovszky, Ferenc, 2022. "The chaotic monopolist revisited with bounded rationality and delay dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    8. Al-Hdaibat, Bashir & Govaerts, Willy & Neirynck, Niels, 2015. "On periodic and chaotic behavior in a two-dimensional monopoly model," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 27-37.

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