A continuous-time heterogeneous duopoly model with delays

Abstract

In this paper, the continuous-time version with delays of the model of Tramontana (Econ Model 27(1):350–357, 2010) is presented. The stability of the equilibrium point is discussed by analyzing the distribution of roots of associated characteristic equation. It is found that combining marginal costs with time delays gives rise to different economic scenarios, where stability switches may appear and Hopf bifurcations occur within certain range of parameters.

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References

  1. Bischi, G.I., Naimzada, A.K., Sbragia, L.: Oligopoly games with local monopolistic approximation. J. Econ. Behav. Organ. 62(3), 371–388 (2007)

    Article  Google Scholar 

  2. Bischi, G.I., Lamantia, F., Radi, D.: An evolutionary cournot model with limited market knowledge. J. Econ. Behav. Organ. 116, 219–238 (2015)

    Article  Google Scholar 

  3. Dieci, R., Westerhoff, F.: Stability analysis of a cobweb model with market interactions. Appl. Math. Comput. 215(6), 2011–2023 (2009)

    Google Scholar 

  4. Ezekiel, M.: The cobweb theorem. The Quarterly Journal of Economics, pages 255–280 (1938)

    Article  Google Scholar 

  5. Gori, L., Guerrini, L., Sodini, M.: A continuous time cournot duopoly with delays. Chaos Solitons Fract. 79, 166–177 (2015a)

    Article  Google Scholar 

  6. Gori, L., Guerrini, L., Sodini, M.: Hopf bifurcation and stability crossing curves in a cobweb model with heterogeneous producers and time delays. Nonlinear Anal. Hybrid Syst. 18, 117–133 (2015b)

    Article  Google Scholar 

  7. Gori, L., Guerrini, L., Sodini, M.: Different modelling approaches for time lags in a monopoly. Essays in Economic Dynamics, pp. 81–98. Springer (2016)

  8. Gori, L., Guerrini, L., Sodini, M.: A characterisation of duopoly dynamics with frictions in production adjustments. J. Evolut. Econ. 27(5), 963–988 (2017)

    Article  Google Scholar 

  9. Hommes, C.H.: Adaptive learning and roads to chaos: the case of the cobweb. Econ. Lett. 36(2), 127–132 (1991)

    Article  Google Scholar 

  10. Hommes, C.H.: Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand. J. Econ. Behav. Organ. 24(3), 315–335 (1994)

    Article  Google Scholar 

  11. Matsumoto, A., Nakayama, K.: Nonlinear cobweb model with time delays.International Conference on Modeling, Simulation and AppliedMathematics, pages 402–406. (2015)

  12. Matsumoto, A., Szidarovszky, F.: Nonlinear delay monopoly with bounded rationality. Chaos Solitons Fract. 45(4), 507–519 (2012)

    Article  Google Scholar 

  13. Matsumoto, A., Szidarovszky, F.: Discrete and continuous dynamics in nonlinear monopolies. Appl. Math. Comput. 232, 632–642 (2014)

    Google Scholar 

  14. Matsumoto, A., Szidarovszky, F.: The asymptotic behavior in a nonlinear cobweb model with time delays. Discrete Dynamics in Nature and Society, 2015 (2015)

    Article  Google Scholar 

  15. Pecora, N., Sodini, M.: A heterogenous cournot duopoly with delay dynamics: Hopf bifurcations and stability switching curves. Commun. Nonlinear Sci. Numer. Simul. 58, 36–46 (2018)

    Article  Google Scholar 

  16. Puu, T.: The chaotic monopolist. Chaos Solitons Fract. 5(1), 35–44 (1995)

    Article  Google Scholar 

  17. Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Continuous Discrete Impuls. Syst. Ser. A 10, 863–874 (2003)

    Google Scholar 

  18. Tramontana, F.: Heterogeneous duopoly with isoelastic demand function. Econ. Model. 27(1), 350–357 (2010)

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge two anonymous reviewers for comments and suggestions. The usual disclaimer applies.

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Correspondence to Giovanni Campisi.

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Brianzoni, S., Campisi, G. & Guerrini, L. A continuous-time heterogeneous duopoly model with delays. Decisions Econ Finan 41, 259–275 (2018). https://doi.org/10.1007/s10203-018-0227-y

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Keywords

  • Heterogeneous players
  • Duopoly model
  • Time delay