Periodicity Induced by Production Constraints in Cournot Duopoly Models with Unimodal Reaction Curves
In the Cournot duopoly game with unimodal piecewise-linear reaction functions (tent maps) proposed by Rand (J Math Econ, 5:173–184, 1978) to show the occurrence of robust chaotic dynamics, a maximum production constraint is imposed in order to explore its effects on the long run dynamics. The presence of such constraint causes the replacement of chaotic dynamics with asymptotic periodic behaviour, characterized by fast convergence to superstable cycles. The creation of new periodic patters, as well as the possible coexistence of several stable cycles, each with its own basin of attraction, are described in terms of border collision bifurcations, a kind of global bifurcation recently introduced in the literature on non-smooth dynamical systems. These bifurcations, caused by the presence of maximum production constraint, give rise to quite particular bifurcation structures. Hence the duopoly model with constraints proposed in this paper can be seen as a simple exemplary case for the exploration of the properties of piecewise smooth dynamical systems.
KeywordsOligopoly games Dynamical systems Constraints Border collision bifurcations
JEL classificationC61 C73 L13
This work is developed in the framework of the research project on “Dynamic Models for behavioural economics” financed by DESP-University of Urbino.
- Agliari, A., Bischi, G.I., & Gardini L. (2002). Some methods for the global analysis of dynamic games represented by noninvertible maps. In T. Puu & I. Sushko (Eds.), Oligopoly dynamics: models and tools. Springer Verlag.Google Scholar
- Banerjee, S., Karthik, M. S., Yuan, G., & Yorke, J. A. (2000a). Bifurcations in one-dimensional piecewise smooth maps—theory and applications in switching circuits. IEEE Transactions on Circuits and System I: Fundamental Theory and Applications, 47(3), 389–394.Google Scholar
- Banerjee, S., Ranjan, P., & Grebogi, C. (2000b). Bifurcations in two-dimensional piecewise smooth maps—theory and applications in switching circuits. IEEE Transactions on Circuits and System I: Fundamental Theory and Applications, 47(5), 633–643.Google Scholar
- Bischi, G. I., Chiarella, C., Kopel, M., & Szidarovszky, F. (2010). Nonlinear oligopolies: Stability and bifurcations. Springer-Verlag.Google Scholar
- Cournot, A. (1838). Recherches sur les principes matematiques de la theorie de la richesse. Paris: Hachette.Google Scholar
- Di Bernardo, M., Budd, C. J., Champneys, A. R., & Kowalczyk, P. (2008). Piecewise-smooth dynamical systems. London: Springer Verlag.Google Scholar
- Hahn, F. (1962). The stability of the Cournot solution. Journal of Economic Studies, 29, 329–331.Google Scholar
- Leonov, N. N. (1959). Map of the line onto itself. Radiofisica, 3(3), 942–956.Google Scholar
- Leonov, N. N. (1962). Discontinuous map of the straight line. Dokl. Acad. Nauk. SSSR., 143(5), 1038–1041.Google Scholar
- Mira, C. (1978). Sur les structure des bifurcations des diffeomorphisme du cercle. C.R.Acad. Sc. Paris 287 Series A, 883–886.Google Scholar
- Mosekilde, E., Zhusubaliyev, Z. T. (2003). Bifurcations and chaos in piecewise-smooth dynamical systems. World Scientific.Google Scholar
- Okuguchi, K. (1964). The stability of the Cournot oligopoly solution: A further generalization. 287. Journal of Economic Studies, 31, 143–146.Google Scholar
- Sushko, I., Avrutin, V., & Gardini, L. (2015). Bifurcation structure in the skew tent map and its application as a border collision normal form. Journal of Difference Equations and Applications. doi: 10.1080/10236198.2015.1113273.
- Sushko, I., Gardini, L., & Avrutin, V. (2016). Nonsmooth One-dimensional maps: Some basic concepts and definitions. Journal of Difference Equations and Applications, 1–56. doi: 10.1080/10236198.2016.1248426.
- Szidarovszky, F. (1999). Adaptive expectations in discrete dynamic oligopolies with production adjustment costs. Pure Mathmatics and Application, 10(2), 133–139.Google Scholar