# American Institute of Mathematical Sciences

May  2022, 27(5): 2817-2831. doi: 10.3934/dcdsb.2021161

## Monopoly conditions in a Cournot-Theocharis oligopoly model under adaptive expectations

 Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, C/ Doctor Fleming sn, 30202, Cartagena, Spain

* Corresponding author: J. S. Cánovas

Received  October 2020 Revised  March 2021 Published  May 2022 Early access  June 2021

Fund Project: The authors have been partially supported by the Grant MTM2017-84079-P from Agencia Estatal de Investigación (AEI) y Fondo Europeo de Desarrollo Regional (FEDER)

We consider the Cournot-Theocharis oligopoly model, where firms make their choices under adaptive expectations. Following [2], we assume that quantities cannot be negative, which implies that the model is nonlinear. The stability of the equilibrium point in the general case is analyzed. We focus on the conditions for which the number of competitors is reduced to a monopoly. In particular, we find necessary and sufficient conditions giving an analytic proof of the convergence to oligopoly to monopoly.

Citation: Jose S. Cánovas, María Muñoz-Guillermo. Monopoly conditions in a Cournot-Theocharis oligopoly model under adaptive expectations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2817-2831. doi: 10.3934/dcdsb.2021161
##### References:
 [1] J. S. Cánovas, Reducing competitors in a Cournot-Theocharis oligopoly model, Journal of Difference Equations and Applications, 15 (2009), 153-165.  doi: 10.1080/10236190802006415. [2] J. S. Cánovas, T. Puu and M. Ruíz, The Cournot-Theocharis problem reconsidered, Chaos, Solitons & Fractals, 37 (2008), 1025-1039.  doi: 10.1016/j.chaos.2006.09.081. [3] N. Chrysanthopoulos and G. P. Papavassilopoulos, Adaptive rules for discrete-time Cournot games of high competition level markets, Operational Research, (2019). doi: 10.1007/s12351-019-00522-z. [4] G. W. Evans and S. Honkapohja, Expectations, economics of, in International Encyclopedia of the Social and Behavioral Sciences, Elsevier, (2001), 5060–5067 doi: 10.1016/B0-08-043076-7/02245-2. [5] G. W. Evans, Expectations in macroeconomics adaptive versus eductive learning, Presses de Sciences Po (PFNSP), 52 (2001). doi: 10.3917/reco.523.0573. [6] C. H. Hommes, M. I. Ochea and J. Tuinstra, Evolutionary competition between adjustment processes in Cournot oligopoly: Instability and complex dynamics, Dyn. Games Appl., 8 (2018), 822-843.  doi: 10.1007/s13235-018-0238-x. [7] S. Keppler, Firm survival and the evolution of oligopoly, The RAND Journal of Economics, 33 (2002), 37-61. [8] W. R. Mann, Mean value methods in iteration, Proceedings American Mathematical Society, 4 (1953), 506-510.  doi: 10.1090/S0002-9939-1953-0054846-3. [9] A. Matsumoto and F. Szidarovszky, Theocharis Problem Reconsidered in Differentiated Oligopoly, Economics Research International, (2014), Article ID 630351, 12 pages. doi: 10.1155/2014/630351. [10] E. S. Mills, The use of adaptive expectations in stability analysis: A comment, The Quaterly Journal of Economics, 75 (1961), 330-335.  doi: 10.2307/1884208. [11] M. Nerlove, Adaptive expectations and cobweb phenomena, The Quaterly Journal of Economics, 72 (1958), 227-240.  doi: 10.2307/1880597. [12] T. Palander, Konkurrens och marknadsjmvikt vid duopol och oligopol, Ekon Tidskr, 41 (1939), 123-145.  doi: 10.2307/3437997. [13] R. D. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134.  doi: 10.2307/2296135.

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##### References:
 [1] J. S. Cánovas, Reducing competitors in a Cournot-Theocharis oligopoly model, Journal of Difference Equations and Applications, 15 (2009), 153-165.  doi: 10.1080/10236190802006415. [2] J. S. Cánovas, T. Puu and M. Ruíz, The Cournot-Theocharis problem reconsidered, Chaos, Solitons & Fractals, 37 (2008), 1025-1039.  doi: 10.1016/j.chaos.2006.09.081. [3] N. Chrysanthopoulos and G. P. Papavassilopoulos, Adaptive rules for discrete-time Cournot games of high competition level markets, Operational Research, (2019). doi: 10.1007/s12351-019-00522-z. [4] G. W. Evans and S. Honkapohja, Expectations, economics of, in International Encyclopedia of the Social and Behavioral Sciences, Elsevier, (2001), 5060–5067 doi: 10.1016/B0-08-043076-7/02245-2. [5] G. W. Evans, Expectations in macroeconomics adaptive versus eductive learning, Presses de Sciences Po (PFNSP), 52 (2001). doi: 10.3917/reco.523.0573. [6] C. H. Hommes, M. I. Ochea and J. Tuinstra, Evolutionary competition between adjustment processes in Cournot oligopoly: Instability and complex dynamics, Dyn. Games Appl., 8 (2018), 822-843.  doi: 10.1007/s13235-018-0238-x. [7] S. Keppler, Firm survival and the evolution of oligopoly, The RAND Journal of Economics, 33 (2002), 37-61. [8] W. R. Mann, Mean value methods in iteration, Proceedings American Mathematical Society, 4 (1953), 506-510.  doi: 10.1090/S0002-9939-1953-0054846-3. [9] A. Matsumoto and F. Szidarovszky, Theocharis Problem Reconsidered in Differentiated Oligopoly, Economics Research International, (2014), Article ID 630351, 12 pages. doi: 10.1155/2014/630351. [10] E. S. Mills, The use of adaptive expectations in stability analysis: A comment, The Quaterly Journal of Economics, 75 (1961), 330-335.  doi: 10.2307/1884208. [11] M. Nerlove, Adaptive expectations and cobweb phenomena, The Quaterly Journal of Economics, 72 (1958), 227-240.  doi: 10.2307/1880597. [12] T. Palander, Konkurrens och marknadsjmvikt vid duopol och oligopol, Ekon Tidskr, 41 (1939), 123-145.  doi: 10.2307/3437997. [13] R. D. Theocharis, On the stability of the Cournot solution on the oligopoly problem, Review of Economic Studies, 27 (1960), 133-134.  doi: 10.2307/2296135.
For $n = 3$, $a = b = 1$, $c_1 = c_2 = 0.8$ and $c_3 = 0.2$, the joint graphs of $h_1$ (dashed), $h_2$ (dot-dashed) and the identity map (thin) are depicted for values of $\lambda$ equal to $0.475$ (a), $0.6$ (b) and $0.975$ (c)
(a) For $a = b = 1$, $c_1 = 0.6$, $c_2 = 0.2$ and $\lambda = 0.5$, we depict the time series of orbits of the production of both firms with random initial conditions. The production of the first firm (dots) does not vanish, although it converges to 0. The production of second firm (squares) converges to the monopoly equilibrium. (b) With the same parameters but changing $c_1 = 0.7$ the orbit is firm 1 is constant to 0 after some time
For $n = 7$, $a = b = 1$, $c_i = 0.75$, $i = 1,...,6$, $c_7 = 0.4$, and $\lambda = 0.9>\lambda _1 = 0.8888888888888891$, we depict the time series of the orbits of the firms 1 (dots) and 7 (squares) with random initial conditions for all the firms on $[0,005]$ (a) and $[0,0.05]$ (b). For (a) we find that the orbits converges to a periodic orbit (note that dots and squares overlap at 0) while for (b) the firms 1 to 6 will disappear from the market
For $n = 10$, $a = b = 1$, $c_1 = 0.9$, $c_i = c_{i-1}-0.01$, for $i = 2,...,8$, $c_9 = 0.74$, $c_{10} = 0.4$ and $\lambda = 0.94$, we find that $q_n = 0.0037440758293838957$. We depict the time series of the orbits of firms 1 (dots) and 10 (squares) with initial conditions $(0,0,...,q_n)$, which is a periodic point of period two. The point $(0,0,...,0)$ is also periodic with the same period 2
For $n = 10$, $a = b = 1$, $c_1 = 0.9$, $c_i = c_{i-1}-0.01$, for $i = 2,...,8$, $c_9 = 0.74$ and $c_{10} = 0.4$ the map $q_n(\lambda)$ is plotted between $[0,1]$ and $[0.9,1]$, respectively. After the asymptote, obtain the positive values of $q_n$ mentioned in Remark 4
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