On a Bertrand Duopoly Game With Heterogeneous Expectations and Emissions Costs
This study was based on the dynamics of a nonlinear Bertrand-type duopoly game with differentiated goods, linear demand and a cost function that included emissions costs. This duopoly game was modeled with a system of two difference equations. Existence and stability of equilibria of this system were studied. It was shown that the model gave more complex chaotic and unpredictable trajectories as a consequence of changes in the speed of adjustment parameter and horizontal product differentiation parameter. Numerical simulations showed that a higher value of the speed of adjustment and a higher or lower (negative) value of product differentiation (weaker or fiercer competition) can destabilize the economy. The chaotic features were justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. Also, it was shown that in this case of a duopoly game, there were stable trajectories, and a higher (lower) degree of product differentiation did not tend to destabilize the economy.
Keywords: Bertrand duopoly game, discrete dynamical system, heterogeneous expectations, stability, chaotic behavior
 Agiza HN. On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons & Fractals. 1999;10(11):1909-1916.
 Agiza HN, Elsadany AA. Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Applied Mathematics and Computation. 2004;149:843-860.
 Agiza HN, Elsadany AA. Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Physica A. 2003;320:512-524.
 Agiza HN, Hegazi AS, Elsadany AA. Complex dynamics and synchronization of duopoly game with bounded rationality. Mathematics and Computers in Simulation. 2002;58:133-146.
 Agliari A, Gardini L, Puu T. Some global bifurcations related to the appearance of closed invariant curves. Mathematics and Computers in Simulation. 2005;68:201-219.
 Agliari A, Gardini L, Puu T. Global bifurcations in duopoly when the Cournot point is destabilized via a subcritical Neimark bifurcation. International Game Theory Review. 2006;8:1-20.
 Askar SS. On complex dynamics of monopoly market. Economic Modelling. 2013;31:586-589.
 Askar SS. Complex dynamic properties of Cournot duopoly games with convex and log-concave demand function. Operations Research Letters. 2014;42:85-90.
 Baumol WJ, Quandt RE. Rules of thumb and optimally imperfect decisions. American Economic Review. 1964;54(2):23-46.
 Bischi GI, Kopel M. Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior and Organization. 2001;46:73-100.
 Den Haan WJ. The importance of the number of different agents in a heterogeneous asset-pricing model. Journal of Economic Dynamics and Control. 2001;25:721-746.
 Elsadany A.A., Awad A.M. Nonlinear dynamics of Cournot duopoly game with social welfare. Electronic Journal of Mathematical Analysis and Applications. 2016;4:173-191.
 Elaydi S. An introduction to difference equations. 3rd ed. New York: Springer Verlag; 2005.
 Fanti L, Gori L. The dynamics of a differentiated duopoly with quantity competition. Economic Modelling. 2012;29:421-427.
 Gandolfo G. Economic dynamics. Berlin: Springer; 1997.
 Gao Y. Complex dynamics in a two dimensional noninvertible map. Chaos Solitons and Fractals. 2009;39:1798-1810.
 Hommes CH. Handbook of computational economics, agent-based computational economics 2. Tesfatsion L, Judd KL, editors. Elsevier Science; 2006. Heterogeneous agent models in economics and finance; p.1109-1186.
 Kopel M. Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals. 1996;12:2031-2048.
 Kulenovic M, Merino O. Discreete dynamical systems and difference equations with mathematica. Chapman & Hall/Crc; 2002.
 Naimzada AK, Ricchiuti G. Complex dynamics in a monopoly with a rule of thumb. Applied Mathematics and Computation. 2008;203:921-925.
 Naimzada AK, Sbragia L. Oligopoly games with nonlinear demand and cost functions: Two bounded rational adjustment processes. Chaos, Solitons & Fractals. 2006;29:707-722.
 Puu T. The chaotic monopolist. Chaos, Solitons and Fractals. 1995;5(1):35–44.
 Puu T. The chaotic duopolists revisited. Journal of Economic Behavior & Organization. 1998;37:385–94.
 Puu T. Chaos in duopoly pricing. Chaos Solitons and Fractals. 1991;1:573–81.
 Puu T. Nonlinear dynamical systems in economics. Lines M, editor. Springer Wien New York: CISM; 2005. Complex oligopoly dynamics; p. 165–86.
 Sarafopoulos G. Οn the dynamics of a duopoly game with differentiated goods. Procedia Economics and Finance. 2015;19:146-153.
 Sarafopoulos G. Complexity in a duopoly game with homogeneous players, convex, log linear demand and quadratic cost functions. Procedia Economics and Finance. 2015;33:358-366.
 Sarafopoulos G, Papadopoulos K. On a Cournot duopoly game with differentiated goods, heterogeneous expectations and a cost function including emission costs. Scientific Bulletin - Economic Sciences. 2017;16(1):11-22.
 Sarafopoulos G, Papadopoulos K. Economic and financial Challenges for Balkan and eastern European Countries. Springer Proceedings in Business and Economics. Sykianakis N., Polychronidou P., Karasavvoglou A., editors. Springer International Publishing; 2019: Complexity in a Bertrand duopoly game with heterogeneous players and differentiated goods; p. 15-26.
 Sarafopoulos G, Papadopoulos K. Economic and financial Challenges for Balkan and eastern European Countries. Springer Proceedings in Business and Economics. Janowicz-Lomott M., Lyskawa K., Polychronidou P., Karasavvoglou A., editors. Springer Cham; 2020 On a Bertrand dynamic game with differentiated goods, heterogeneous expectations and asymmetric cost functions; p. 223-241.
 Singh N, Vives X. Price and quantity competition in a differentiated duopoly. The RAND Journal of Economics. 1984;15:546–554.
 Tramontana F. Heterogeneous duopoly with isoelastic demand function. Economic Modelling. 2010;27:350-357.
 Westerhoff F. Nonlinear expectation formation, endogenous business cycles and stylized facts. Studies in Nonlinear Dynamics and Econometrics. 2006;10(4):Article 4.
 Wu W, Chen Z, Ip WH. Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model. Nonlinear Analysis: Real World Applications. 2010;11:4363-4377.
 Zhang J, Da Q, Wang Y. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling. 2007;24:138-148.
 Zhang J, Da Q, Wang Y. The dynamics of Bertrand model with bounded rationality. Chaos, Solitons & Fractals. 2009;39:2048-2055.