Abstract
Since Kopel’s duopoly model was proposed about 3 decades ago, there are almost no analytical results on the equilibria and their stability in the asymmetric case. The first objective of our study is to fill this gap. This paper analyzes the asymmetric duopoly model of Kopel analytically by using several tools based on symbolic computations. We discuss the possibility of the existence of multiple positive equilibria and establish conditions for a given number of positive equilibria to exist. The possible positions of the equilibria in Kopel’s model are also explored. Furthermore, in the asymmetric model of Kopel, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish conditions for the local stability of equilibria for the first time. The occurrence of chaos in Kopel’s model seems to be supported by observations through numerical simulations, which, however, is challenging to prove rigorously. The second objective of this paper is to prove the existence of snapback repellers in Kopel’s map, which implies the existence of chaos in the sense of Li–Yorke according to Marotto’s theorem.
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References
Agiza, H. N. (1999). On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons & Fractals, 10(11), 1909–1916.
Agiza, H. N., Bischi, G. I., & Kopel, M. (1999). Multistability in a dynamic Cournot game with three oligopolists. Mathematics and Computers in Simulation, 51(1–2), 63–90.
Anderson, D. R., Myran, N. G., & White, D. L. (2005). Basins of attraction in a Cournot duopoly model of Kopel. Journal of Difference Equations and Applications, 11(10), 879–887.
Bischi, G. I., & Kopel, M. (2001). Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior & Organization, 46(1), 73–100.
Bischi, G. I., Mammana, C., & Gardini, L. (2000). Multistability and cyclic attractors in duopoly games. Chaos, Solitons & Fractals, 11(4), 543–564.
Bistritz, Y. (1984). Zero location with respect to the unit circle of discrete-time linear system polynomials. Proceedings of the IEEE, 72(9), 1131–1142.
Buchberger, B. (1985). Gröbner bases: An algorithmic method in polynomial ideal theory. In N. K. Bose (Ed.), Multidimensional systems theory (pp. 184–232). Dordrecht: Reidel.
Cánovas, J. S., & Muñoz-Guillermo, M. (2018). On the dynamics of Kopel’s Cournot duopoly model. Applied Mathematics and Computation, 330, 292–306.
Cavalli, F., & Naimzada, A. (2015). Nonlinear dynamics and convergence speed of heterogeneous Cournot duopolies involving best response mechanisms with different degrees of rationality. Nonlinear Dynamics, 81(1), 967–979.
Collins, G. E., & Hong, H. (1991). Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3), 299–328.
Collins, G. E., & Loos, R. (1983). Real zeros of polynomials. In B. Buchberger, G. Collins, & R. Loos (Eds.), Computer algebra: Symbolic and algebraic computation (pp. 83–94). New York: Springer.
Cooper, R. (1994). Equilibrium selection in imperfectly competitive economies with multiple equilibria. The Economic Journal, 104(426), 1106.
Echenique, F., & Komunjer, I. (2009). Testing models with multiple equilibria by quantile methods. Econometrica, 77(4), 1281–1297.
Elsadany, A. A., & Awad, A. M. (2016). Dynamical analysis and chaos control in a heterogeneous Kopel duopoly game. Indian Journal of Pure and Applied Mathematics, 47(4), 617–639.
Eskandari, Z., Alidousti, J., & Ghaziani, R. K. (2021). Codimension-one and-two bifurcations of a three-dimensional discrete game model. International Journal of Bifurcation and Chaos, 31(02), 2150023.
Gardini, L., Sushko, I., Avrutin, V., & Schanz, M. (2011). Critical homoclinic orbits lead to snap-back repellers. Chaos, Solitons & Fractals, 44(6), 433–449.
Govaerts, W., & Ghaziani, R. K. (2008). Stable cycles in a Cournot duopoly model of Kopel. Journal of Computational and Applied Mathematics, 218(2), 247–258.
Huang, B., & Niu, W. (2019). Analysis of snapback repellers using methods of symbolic computation. International Journal of Bifurcation and Chaos, 29(04), 1950054.
Jin, M., Li, X., & Wang, D. (2013). A new algorithmic scheme for computing characteristic sets. Journal of Symbolic Computation, 50, 431–449.
Jing, Z., Chang, Y., & Guo, B. (2004). Bifurcation and chaos in discrete FitzHugh–Nagumo system. Chaos, Solitons & Fractals, 21(3), 701–720.
Jury, E., Stark, L., & Krishnan, V. (1976). Inners and stability of dynamic systems. IEEE Transactions on Systems, Man, and Cybernetics, 10, 724–725.
Kopel, M. (1996). Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals, 7(12), 2031–2048.
Kubler, F., & Schmedders, K. (2010). Tackling multiplicity of equilibria with Gröbner bases. Operations Research, 58(4–Part–2), 1037–1050.
Lazard, D., & Rouillier, F. (2007). Solving parametric polynomial systems. Journal of Symbolic Computation, 42(6), 636–667.
Li, B., Liang, H., Shi, L., & He, Q. (2022). Complex dynamics of Kopel model with nonsymmetric response between oligopolists. Chaos, Solitons, & Fractals, 156, 111860-1–14.
Li, B., Zhang, Y., Li, X., Eskandari, Z., & He, Q. (2023). Bifurcation analysis and complex dynamics of a Kopel triopoly model. Journal of Computational and Applied Mathematics, 426, 115089.
Li, C., & Chen, G. (2003). An improved version of the Marotto Theorem. Chaos, Solitons & Fractals, 18(1), 69–77.
Li, T.-Y., & Yorke, J. A. (1975). Period three implies chaos. The American Mathematical Monthly, 82(10), 985–992.
Li, X. (2022). Analysis of the stability and the bifurcations of two heterogeneous triopoly games with an isoelastic demand. AIMS Mathematics, 7(10), 19388–19414.
Li, X., Chen, K., Niu, W., & Huang, B. (2023). Stability and chaos of the duopoly model of kopel: A study based on symbolic computations. arXiv:2304.02136.
Li, X., Li, B., & Liu, L. (2023). Stability and dynamic behaviors of a limited monopoly with a gradient adjustment mechanism. Chaos, Solitons & Fractals, 168, 113106-1–18.
Li, X., Mou, C., & Wang, D. (2010). Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case. Computers & Mathematics with Applications, 60(11), 2983–2997.
Li, X., & Wang, D. (2014). Computing equilibria of semi-algebraic economies using triangular decomposition and real solution classification. Journal of Mathematical Economics, 54, 48–58.
Liao, K.-L., & Shih, C.-W. (2012). Snapback repellers and homoclinic orbits for multi-dimensional maps. Journal of Mathematical Analysis and Applications, 386(1), 387–400.
Marotto, F. R. (1978). Snap-back repellers imply chaos in \({R}_n\). Journal of Mathematical Analysis and Applications, 63(1), 199–223.
Marotto, F. R. (2005). On redefining a snap-back repeller. Chaos, Solitons & Fractals, 25(1), 25–28.
Ming, H., Hu, H., & Zheng, J. (2021). Analysis of a new coupled hyperchaotic model and its topological types. Nonlinear Dynamics, 105(2), 1937–1952.
Mishra, B. (1993). Algorithmic algebra. New York: Springer.
Obstfeld, M. (1984). Multiple stable equilibria in an optimizing perfect-foresight model. Econometrica, 52(1), 223.
Peng, C.-C. (2007). Numerical computation of orbits and rigorous verification of existence of snapback repellers. Chaos: An Interdisciplinary Journal of Nonlinear Science, 17(1), 013107.
Puu, T. (1991). Chaos in duopoly pricing. Chaos, Solitons & Fractals, 1(6), 573–581.
Ren, J., Yu, L., & Siegmund, S. (2017). Bifurcations and chaos in a discrete predator-prey model with Crowley–Martin functional response. Nonlinear Dynamics, 90(1), 19–41.
Rionero, S., & Torcicollo, I. (2014). Stability of a continuous reaction–diffusion Cournot–Kopel duopoly game model. Acta Applicandae Mathematicae, 132(1), 505–513.
Schmitt-Grohé, S., & Uribe, M. (2021). Multiple equilibria in open economies with collateral constraints. The Review of Economic Studies, 88(2), 969–1001.
Torcicollo, I. (2013). On the dynamics of a non-linear duopoly game model. International Journal of Non-linear Mechanics, 57, 31–38.
Tramontana, F., & Elsadany, A. E. A. (2012). Heterogeneous triopoly game with isoelastic demand function. Nonlinear Dynamics, 68(1–2), 187–193.
Tramontana, F., Elsadany, A. A., Xin, B., & Agiza, H. N. (2015). Local stability of the Cournot solution with increasing heterogeneous competitors. Nonlinear Analysis: Real World Applications, 26, 150–160.
Wang, D. (2001). Elimination methods. Texts and Monographs in Symbolic ComputationNew York: Springer.
Wang, D., & Xia, B. (2005a). Algebraic analysis of stability for some biological systems. In Proceedings of the first international conference on algebraic biology (pp. 75–83). Tokyo: Universal Academy Press.
Wang, D., & Xia, B. (2005b). Stability analysis of biological systems with real solution classification. In Kauers, M. (Ed.), Proceedings of the 2005 international symposium on symbolic and algebraic computation (pp. 354–361). New York: ACM Press.
Wang, J., & Feng, G. (2010). Bifurcation and chaos in discrete-time BVP oscillator. International Journal of Non-linear Mechanics, 45(6), 608–620.
Wu, W., Chen, Z., & Ip, W. H. (2010). Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model. Nonlinear Analysis: Real World Applications, 11(5), 4363–4377.
Yang, L., Hou, X., & Xia, B. (2001). A complete algorithm for automated discovering of a class of inequality-type theorems. Science in China Series F, 44, 33–49.
Yang, L., & Xia, B. (2005). Real solution classifications of parametric semi-algebraic systems. In Algorithmic algebra and logic–proceedings of the A3L 2005 (pp. 281–289). Norderstedt: Herstellung und Verlag.
Yao, X., Li, X., Jiang, J., & Leung, A. Y. T. (2022). Codimension-one and -two bifurcation analysis of a two-dimensional coupled logistic map. Chaos, Solitons & Fractals, 164, 112651-1–9.
Zhang, Y., & Gao, X. (2019). Equilibrium selection of a homogenous duopoly with extrapolative foresight. Communications in Nonlinear Science and Numerical Simulation, 67, 366–374.
Zhao, M. (2021). Bifurcation and chaotic behavior in the discrete BVP oscillator. International Journal of Non-linear Mechanics, 131, 103687.
Acknowledgements
The authors are grateful to the anonymous referees for their helpful comments.
Funding
Wei Niu reports financial support was provided by Beijing Natural Science Foundation (No. 1212005). Kongyan Cheng reports financial support was provided by Social Development Science and Technology Project of Dongguan (No. 20211800900692).
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Li, X., Chen, K., Niu, W. et al. Stability and Chaos of the Duopoly Model of Kopel: A Study Based on Symbolic Computations. Comput Econ (2024). https://doi.org/10.1007/s10614-024-10608-2
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DOI: https://doi.org/10.1007/s10614-024-10608-2