# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021161
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## 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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021161
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##### References:
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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