Stability switching and its directions in cournot duopoly game with three delays

Akio Matsumoto; Ferenc Szidarovszky

doi:10.3934/dcdsb.2021069

Article Contents

2021, Volume 26, Issue 11: 5905-5923. Doi: 10.3934/dcdsb.2021069

This issue Previous Article Public debt dynamics under ambiguity by means of iterated function systems on density functions Next Article On dynamics in a medium-term Keynesian model

Stability switching and its directions in cournot duopoly game with three delays

Akio Matsumoto^1, , and
Ferenc Szidarovszky^2,

1.
Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan
2.
Department of Mathematics, Corvinus University, Budapest, Fövám tér 8, 1093, Hungary

Received: August 31, 2020

Revised: October 31, 2020

Early access: March 2021

Published: November 2021

The authors appreciate the constructive suggestions and comments by anonymous referees which have greatly helped to improve the paper. The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566). The usual disclaimers apply

Abstract Full Text(HTML) Figure(8) Related Papers Cited by

Abstract

A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.

Keywords:

Mathematics Subject Classification: 34A34, 34C23, 34D20, 34K18.

Citation:

Full Text(HTML)

Figure 1. Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue

Download: Full-size image PowerPoint slide

Figure 2. Stability switching curves

Download: Full-size image PowerPoint slide

Figure 3. Dynamics at point $ a $ for various values of $ \tau _{1} $ and $ \tau _{2} $

Download: Full-size image PowerPoint slide

Figure 4. Dynamics at point $ b $ for various values of $ \tau _{1} $ and $ \tau _{2} $

Download: Full-size image PowerPoint slide

Figure 5. Dynamics at point $ c $ for various values of $ \tau _{2} $ and $ \tau _{3} $

Download: Full-size image PowerPoint slide

Figure 6. Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue

Download: Full-size image PowerPoint slide

Figure 7. Stability switching curves

Download: Full-size image PowerPoint slide

Figure 8. Nonlinear adjustment coefficient functions

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	R. Bellman and K. L. Cooke, Difference-Differential Equations, Academic Press, New York, 1963.
[2]	G. I. Bischi, C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-02106-0.
[3]	K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis and Applications, 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8.
[4]	A. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richessess, Researches into the Mathematical Principles of Theory of Wealth, Hachette, Paris, 1833, Kelley, New York, 1960.
[5]	J. W. Friedman, Game Theory with Applications to Economics, Oxford University Press, New York, Oxford, 1986.
[6]	L. Gori, L. Guerrini and M. Sodini, A continuous time Cournot duopoly with delays, Chaos, Solitons and Fractals, 79 (2015), 166-177. doi: 10.1016/j.chaos.2015.01.020.
[7]	K. Gu, S. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, Journal of Mathematical Analysis and Applications, 311 (2005), 231-252. doi: 10.1016/j.jmaa.2005.02.034.
[8]	L. Guerrini, A. Matsumoto and F. Szidarovszky, Delay Cournot duopoly models revisited, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 093113, 10 pp. doi: 10.1063/1.5020903.
[9]	F. Hahn, The stability of the Cournot oligopoly solution, Review of Economic Studies, 29 (1962), 329-331.
[10]	T. D. Howroyd and A. M. Russel, Cournot oligopoly models with time delays, Journal of Mathematical Economics, 13 (1984), 97-103. doi: 10.1016/0304-4068(84)90009-0.
[11]	X. Lin and H. Wang, Stability analysis of delay differential equations with two discrete delays, Canadian Applied Mathematics Quarterly, 20 (2012), 519-533.
[12]	A. Matsumoto and F. Szidarovszky, Dynamic Oligopolies with Time Delays, Springer Nature, Singapore, 2018. doi: 10.1007/978-981-13-1786-6.
[13]	A. Matsumoto and F. Szidarovszky, Nonlinear Cournot duopoly with implementation delays, Chaos, Solitons and Fractals, 79 (2015), 157-165. doi: 10.1016/j.chaos.2015.05.010.
[14]	A. Matsumoto, F. Szidarovszky and H. Yosida, Dynamics in linear Cournot duopolies with two time delays, Computational Economics, 38 (2011), 311-327.
[15]	M. McManus and R. Quandt, Comments on the stability of the Cournot oligopoly model, Review of Economic Studies, 28 (1964), 136-139.
[16]	H. Nikaidô and K. Isoda, Note on noncooperative convex games, Pacific Journal of Mathematics, 5 (1955), 807-815. doi: 10.2140/pjm.1955.5.807.
[17]	K. Okuguchi, Expectations and Stability in Oligopoly Models, Springer, Berlin, 1976.
[18]	K. Okuguchi and F. Szidarovszky, The Theory of Oligopoly with Multi-product Firms, 2nd. ed, Springer, Berlin, 1999. doi: 10.1007/978-3-662-02622-9.
[19]	J. B. Rosen, Existence and uniqueness of equilibrium points for concave $n$-person games, Econometrica, 33 (1965), 520-534. doi: 10.2307/1911749.
[20]	Y. Song, M. Han and Y. Peng, Stability and Hopf bifurcation in a competitive Lotka-Volterra system with two delays, Chaos, Solitions and Fractals, 22 (2004), 1139-1148. doi: 10.1016/j.chaos.2004.03.026.
[21]	F. Szidarovszky, On the oligopoly game, DM-70-1, Karl Marx University of Economics, Budapest, Hungary, 1970.
[22]	R. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134.
[23]	J. Zhang, Z. Jin, J. Yan and G. Sun, Stability and Hopf bifurcation in a delayed competition system, Nonlinear Analysis, 70 (2009), 658-670. doi: 10.1016/j.na.2008.01.002.