Abstract
This paper investigates the dynamics of a nonlinear Cournot-type duopoly game with differentiated goods, linear demand and cost functions for two bounded rational players that have different objective functions. Specifically, the first player is a public company and cares about the social welfare and the second player is a private company which cares only about its own profit maximization. The game is modeled with a system of two difference equations. The stability analysis of the fixed points are analyzed and complex dynamic features including period doubling bifurcations of the unique Nash equilibrium is also investigated. Numerical simulations are carried out to show the complex behavior of the models’ parameters. We show that a higher (lower) degree of the speed of adjustment and a lower (higher) degree of the parameter of product differentiation destabilize (stabilize) the economy. The chaotic features are justified numerically via computing Lyapunov numbers, sensitive dependence on initial conditions, bifurcation diagrams and strange attractors.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G. Sarafopoulos, K. Papadopoulos, On a Cournot duopoly game with differentiated goods, heterogeneous expectations and a cost function including emission costs. Scientif. Bullet. Economic Sci. 1(1), 11–22 (2017)
G. Sarafopoulos, K. Papadopoulos, Chaos in oligopoly models. Int. J. Product. Manage. Assessment Technologies (IJPMAT) 7(1), Article 4 (2018)
G. Sarafopoulos, K. Papadopoulos, On the stability of a Cournot dynamic game under the influence of information, Knowledge E Publishing, ISSN 2518-668X, 131–144 (2018)
G. Sarafopoulos, On a duopoly game with homogeneous players and a quadratic demand function. East-West J. Econom. Business XX(2), 39–48 (2017)
G. Sarafopoulos, K. Papadopoulos, Complexity in a Bertrand Duopoly Game with heterogeneous players and differentiated goods. Springer Proc. Business Econ. 15–26 (2019)
H.N. Agiza, On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons Fractals 10, 1909–1916 (1999)
H.N. Agiza, On the stability, bifurcations, chaos and chaos control of Kopel map. Chaos Solitons Fract. 11, 1909–1916 (2004)
H.N. Agiza, A.S. Hegazi, A.A. Elsadany, Complex dynamics and synchronization of duopoly game with bounded rationality. Math. Comput. Simulat. 58, 133–146 (2002)
A. Agliari, L. Gardini, T. Puu, Some global bifurcations related to the appearance of closed invariant curves. Math Comput Simulation 68, 201–219 (2005)
A. Agliari, L. Gardini, T. Puu, Global bifurcations in duopoly when the Cournot point is destabilized via a subcritical Neimark bifurcation. International Game Theory Review 8, 1–20 (2006)
G.I. Bischi, M. Kopel, Equilibrium selection in a nonlinear duopoly game with adaptive expectations. J. Econ. Behav. Organ. 46, 73–100 (2001)
M. Kopel, Simple and complex adjustment dynamics in Cournot duopoly models. Chaos Solitons Fract. 12, 2031–2048 (1996)
T. Puu, The chaotic duopolists revisited. J Econ. Behav. Org. 37, 385–394 (1998)
G. Sarafopoulos, Complexity in a duopoly game with homogeneous players, convex, log linear demand and quadratic cost functions. Proc. Econ. Finance 33, 358–366 (2015)
H.N. Agiza, A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput. 149, 843–860 (2004)
H.N. Agiza, A.A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Phys. A 320, 512–524 (2003)
W.J. Den Haan, The importance of the number of different agents in a heterogeneous asset—pricing model. J. Econ. Dynam. Control 25, 721–746 (2001)
L. Fanti, L. Gori, The dynamics of a differentiated duopoly with quantity competition. Econ. Model. 29, 421–427 (2012)
F. Tramontana, Heterogeneous duopoly with isoelastic demand function. Econ. Model. 27, 350–357 (2010)
J. Zhang, Q. Da, Y. Wang, The dynamics of Bertrand model with bounded rationality. Chaos, Solitons Fractals 39, 2048–2055 (2009)
A.K. Naimzada, G. Ricchiuti, Complex dynamics in a monopoly with a rule of thumb. Appl. Math. Comput. 203, 921–925 (2008)
S.S. Askar, Complex dynamic properties of Cournot duopoly games with convex and log-concave demand function. Operations Research Letters 42, 85–90 (2014)
A. Naimzada, L. Sbragia, Oligopoly games with nonlinear demand and cost functions: two boundedly rational adjustment processes. Chaos, Solitons Fractals 29, 707–722 (2006)
Y. Tanaka, Irrelevance of the choice of strategic variables in duopoly under relative profit maximization. Econ. Business Lett. 2(2), 75–83 (2013)
A.A. Elsadany, Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization. Appl. Math. Comput. 294, 253–263 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Sarafopoulos, G., Papadopoulos, K. (2020). Dynamics of a Cournot Duopoly Game with Differentiated Goods Between Public and Private Firms. In: Skiadas, C., Dimotikalis, Y. (eds) 12th Chaotic Modeling and Simulation International Conference. CHAOS 2019. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-39515-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-39515-5_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-39514-8
Online ISBN: 978-3-030-39515-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)