Figure 1.
Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue
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November
2021, 26(11): 5905-5923.
doi: 10.3934/dcdsb.2021069
Stability switching and its directions in cournot duopoly game with three delays
| 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 |
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.
Mathematics Subject Classification:
34A34, 34C23, 34D20, 34K18.
Citation:
Akio Matsumoto, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete and Continuous Dynamical Systems - B,
2021, 26
(11)
: 5905-5923.
doi: 10.3934/dcdsb.2021069
![]()
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. |
show all references
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. |
Figure 2.
Stability switching curves
Figure 6.
Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue
Figure 7.
Stability switching curves
Figure 8.
Nonlinear adjustment coefficient functions
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