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On the (in)stability of nonlinear feedback solutions in a dynamic duopoly with renewable resource exploitation

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  • Lambertini, Luca
  • Mantovani, Andrea

Abstract

We revisit Fujiwara’s (2008) linear–quadratic differential duopoly game to show that the degenerate nonlinear feedback identified by the tangency point with the stationary state line is indeed unstable, given the dynamics of the natural resource exploited by firms. To do so, we fully characterise the continuum of nonlinear feedback solution via Rowat’s (2007) method, thereby identifying the infinitely many stable nonlinear feedback equilibria. This entails that Rowat’s method can be used in games where each player’s instantaneous payoff depends quadratically on all players’ controls.

Suggested Citation

  • Lambertini, Luca & Mantovani, Andrea, 2016. "On the (in)stability of nonlinear feedback solutions in a dynamic duopoly with renewable resource exploitation," Economics Letters, Elsevier, vol. 143(C), pages 9-12.
    • Handle: RePEc:eee:ecolet:v:143:y:2016:i:c:p:9-12
    DOI: 10.1016/j.econlet.2016.03.015
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    1. Lambertini, Luca & Mantovani, Andrea, 2014. "Feedback equilibria in a dynamic renewable resource oligopoly: Pre-emption, voracity and exhaustion," Journal of Economic Dynamics and Control, Elsevier, vol. 47(C), pages 115-122.
    2. Dockner, Engelbert J. & Sorger, Gerhard, 1996. "Existence and Properties of Equilibria for a Dynamic Game on Productive Assets," Journal of Economic Theory, Elsevier, vol. 71(1), pages 209-227, October.
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    4. Itaya, Jun-ichi & Shimomura, Koji, 2001. "A dynamic conjectural variations model in the private provision of public goods: a differential game approach," Journal of Public Economics, Elsevier, vol. 81(1), pages 153-172, July.
    5. Rowat, Colin, 2007. "Non-linear strategies in a linear quadratic differential game," Journal of Economic Dynamics and Control, Elsevier, vol. 31(10), pages 3179-3202, October.
    6. Benchekroun, Hassan, 2008. "Comparative dynamics in a productive asset oligopoly," Journal of Economic Theory, Elsevier, vol. 138(1), pages 237-261, January.
    7. Rubio, Santiago J. & Casino, Begona, 2002. "A note on cooperative versus non-cooperative strategies in international pollution control," Resource and Energy Economics, Elsevier, vol. 24(3), pages 251-261, June.
    8. Benchekroun, Hassan, 2003. "Unilateral production restrictions in a dynamic duopoly," Journal of Economic Theory, Elsevier, vol. 111(2), pages 214-239, August.
    9. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329.
    10. Fujiwara, Kenji, 2008. "Duopoly can be more anti-competitive than monopoly," Economics Letters, Elsevier, vol. 101(3), pages 217-219, December.
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    Citations

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    Cited by:

    1. Benchekroun, Hassan & Ray Chaudhuri, Amrita & Tasneem, Dina, 2020. "On the impact of trade in a common property renewable resource oligopoly," Journal of Environmental Economics and Management, Elsevier, vol. 101(C).
    2. L. Lambertini & G. Leitmann, 2017. "On the attainment of the maximum sustainable yield in the Verhulst-Lotka-Volterra model," Working Papers wp1112, Dipartimento Scienze Economiche, Universita' di Bologna.
    3. Luca Lambertini & Arsen Palestini, 2018. "Voluntary Export Restraints in a Trade Model with Sticky Price: Linear and Nonlinear Feedback Solutions," Dynamic Games and Applications, Springer, vol. 8(3), pages 507-518, September.
    4. N. Baris Vardar & Georges Zaccour, 2020. "Exploitation of a Productive Asset in the Presence of Strategic Behavior and Pollution Externalities," Mathematics, MDPI, vol. 8(10), pages 1-28, October.
    5. Michele Bisceglia, 2020. "Optimal taxation in a common resource oligopoly game," Journal of Economics, Springer, vol. 129(1), pages 1-31, January.
    6. Yuankan Huang & Takehiro Inohara, 2023. "Stable Markov perfect equilibria in the asymmetric differential-game duopoly with a renewable resource," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 11(1), pages 45-63, April.
    7. Lambertini, Luca, 2021. "Regulating the tragedy of commons: Nonlinear feedback solutions of a differential game with a dual interpretation," Energy Economics, Elsevier, vol. 100(C).
    8. Feichtinger, Gustav & Lambertini, Luca & Leitmann, George & Wrzaczek, Stefan, 2022. "Managing the tragedy of commons and polluting emissions: A unified view," European Journal of Operational Research, Elsevier, vol. 303(1), pages 487-499.

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    More about this item

    Keywords

    Differential games; Renewable resources; Feedback strategies;
    All these keywords.

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets
    • Q2 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Renewable Resources and Conservation

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