Discrete Dynamics in Nature and Society

Volume 2018, Article ID 7630395, 7 pages

https://doi.org/10.1155/2018/7630395

## Exploring General Equilibrium Points for Cournot Model

^{1}School of Economics and Management, Xidian University, Xi’an 710126, China^{2}Department of Economics and Management, Yuncheng University, Yuncheng 044000, China

Correspondence should be addressed to Bingyuan Gao; moc.361@gnauyb

Received 2 May 2018; Accepted 30 July 2018; Published 16 August 2018

Academic Editor: Guang Zhang

Copyright © 2018 Bingyuan Gao and Yueping Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In Cournot model, when there are many competitions, the competitive equilibrium becomes chaotic. It is extremely difficult to derive the general equilibrium points. There is no previous research to explore a further problem with the general equilibrium points of n-contenders in Cournot model. In this paper, a general equilibrium Cournot game is proposed based on an inverse demand function. A market spatial structure model is built. Intermediate value theorem, as a realistic method, is introduced to handle a general competitive equilibrium in Cournot model. The number and stability in general equilibrium points are detected by means of celestial motion theory and spatial agglomeration competition model. The existence of general equilibrium points and the stability of Cournot equilibrium points, which are new and future complement of previously known results. Numerical simulations are given to support the research results.

#### 1. Introduction

There is no previous research using mean value theorem approach to handle a further problem with the general equilibrium points of n-contenders in the Cournot model. The case of economists generally divides market structure into four types: monopolies, monopolistic competition, oligopolies, and complete competition. This article studies incomplete market according to oligopoly market. Auguste Cournot (1838) [1] introduced a model to analyze the market structure of oligopoly, which locates between monopoly and perfect competition, and now it has become a very important concept in microeconomics. Later, many scholars study Cournot model from many different aspects. There are two different topics involved: (a) the Couront equilibrium must exist with the competitive equilibrium when the number of firms is finite, and, (b) with the increasing number of firms, the competing equilibrium state must be stable. By now, the second question raised grave doubts. In this paper, we only consider the equilibrium point of oligopoly market structure. When there are more firms in the Cournot model, how to find the general stable competitive equilibrium point and how many general competitive equilibrium points there are, which is stable or unstable point. In [2], the authors consider that there exists a stable equilibrium point in the Cournot duopoly model. Is this the general equilibrium point? That is not defined. They analyze the existence of solutions of the model in the form of nonlinear second-order difference equation. R. D. Theocharis (1960) [3] research shows when there are two sellers in the Cournot model, the Cournot solution is always dynamically stable; if the number is three, the result shows we shall get finite oscillations about the equilibrium district. If the number is greater than three, the state will always be instability. In CLSC competitive system is composed of one manufacturer and one retailer, which will be easier to enter into chaos [4]. In [5], a dynamic duopoly Cournot game is studied from different adjustment mechanisms and expectations. Anming Zhang and Yimin Zhang (1996) [6] derived stability conditions for a multiproduct duopoly. The work [7] analyzes oligopolistic markets, and it proves existence of nontrivial fulfilled-expectations equilibrium. Tönu Puu (1996) [8] simulated the dynamic process with three oligopolists. Though it is unstable, the process found the Cournot equilibrium point of three oligopolists. If the Cournot equilibrium exists, it is stable; Tönu Puu (2008) [9] derived the equilibrium district. H. N. Agiza (1998) [10] research showed that there exist stable fixed points in the model of three competitors. A dynamic repeated game system composed of two manufacturers and one retailer is investigated [11]. In a dynamic Cournot model, the limiting production level of the game is a random variable [12]. When competitor is four, the state of equilibrium is still stable. A dynamic four-oligopolist game with different expectations is modeled by four-dimensional nonlinear difference equations. The result shows that four-oligopolist game model behaves chaotically with the variation of the parameters (Xiaosong Pu and Junhai Ma (2013) [13]. E. Ahmed and H. N. Agiza (1998) [14] tried to find the fixed point of n-competitors by the model. The dynamical system of n-competitors in a Cournot game is derived, and the stability of its fixed point is studied. The fixed point is relative to c (cost) and n. When the number of competing firms becomes arbitrarily large, prices converge to marginal costs [15]. But, so far, we are unable to derive a general result relating to stability and the number n of competitors. Many researchers have also paid a great attention to the controlling dynamics of Cournot games, such as Agiza and Elsadany [16], Chen and Chen [17], Holyst and Urbanowicz [18], and Elabbasy et al. [19]. This paper presents a new Cournot duopoly game. The main advantage of this game is that the outputs are nonnegative for all times. The chaotic behavior of the Cournot duopoly game has been stabilized on the Nash equilibrium point by using delay feedback control method (H. N. Agiza, A. A. Elsadany, and M. M. El-Dessoky (2013) [20]. Junhai Ma and Fang Wu (2014) [21] explored a discrete triopoly dynamical game from three aspects, and the parameters of bounded rationality can evoke chaos in identical market.

Another related body of literature deals with Cournot competition in spatial models. Static spatial Cournot-Nash oligopoly models were formulated on optimization and variational inequality theories (Harker, 1986) [22]. Anderson and Neven (1991) and Hamilton, Thisse, and Weskamp (1989) [23] study spatial discrimination under Cournot oligopoly where locations are strategic choice variables. These papers establish that Cournot competition yields spatial agglomeration. In S. P. Anderson and D. J. Neven work (1991) [24], the research of their article shows the competition between Cournot oligopolists that discriminate over space leads to spatial agglomeration. At the same time, the authors implicate that firms do not necessarily earn supernormal profit by the free-entry equilibrium state. Byung-Wook Wie and Roger L. Tobin (1997) [25] developed a methodology for the study of dynamic oligopolistic competition in serving spatially separated markets. In [26], in spatial markets, locations and quantities are selected sequentially. Many firms are agglomerate equidistantly from each other in a circular city. Numerous firms agglomerate at the center in a linear city. Corrado Benassi (2014) [27] explored the spatial Cournot competition with two firms and shows the existence of dispersion equilibrium, the existing state of which is not unimodal and that asymmetry is not too stronger than normal distribution. The behavior of the tow- and three-group is compared with that of tow-firm and three-firm to shed light on roles of the number of the firms. The local asymptotic behavior of the system is identical with that of the adaptive adjustment in the Cournot model (Akio Matsumoto and Ferenc Szidarovszky (2011) [28]. The work [29] shows symmetric two-player oligopoly game over continuous action spaces can be extended to a population game in which each player interacts with a subset of the entire population.

However, so far, we are not in a position to derive a general result relating stability and instability to the number n of competition in the Cournot model. The aim of our study is how to find the general stable and unstable equilibrium points with a method in the Cournot model. This article studies further Cournot model between the district of the two points of monopoly and complete competition. The district that we define is chaos. Whether or not the chaotic district is stable, we will do our utmost to find at least a fixed equilibrium point. How many points are in the chaotic district: that is the interesting point of this article. Our result shows that we find the general equilibrium points in that chaotic district. Although the research seems slight, it may have remarkable effects on the stability of the system.

#### 2. Model Formulation

In oligopoly competition, we may find inner optional equilibrium production of two firms in the Cournot model. Now, we assume the total production of two firms: . They produce homogeneous product. On the similar level of product, we define market price: , which is the inverse demand function. The cost function of any firm is assumed , i=1, 2 (we omit the factor prices in the cost function because we will assume that they are fixed). Then, the most profit of firm i may be denoted The condition of inner most optimal profit per firm is satisfied with Cournot model: If there are n firms in the market, formula (2) is also denoted: where

and is market share of i firm’s production in whole output, so where

where is the derivative of elasticity of market demand.

Proposition 1. *When , **In this situation, the profit of market does not exist. Every firm in the market is an infinitesimal share of the market, which becomes a completely competitive market. The Cournot equilibrium is close to a completely competitive equilibrium.*

Proposition 2. *When , we have exactly the monopoly market condition. There is only one company on the market. The question is raised. When the number of firms is little, we might find the equilibrium production of competition in the Cournot model. In reality, some firms locate between monopoly and complete competition structure. Whether or not we can find the optimal quantity between them is what we will study.*

*Definition 3. *Any firm i has common cost

*Definition 4. *The inverse demand function is expressed as and* parameter *isAccording to (8) and (9), the profit of firm j is (11) is in accordance with Lagrange’s mean value theorem.

Lemma 5. *When , it is only a firm in the market. It belongs to a monopoly firm. Now, we will find the most value by taking partial derivative of : *

Lemma 6. *, and there are more firms in the market, which is completely competitive market.**Formula (11) is taken partial derivative: *

*Definition 7. * is average production. Equation (14) also is denoted:

Proposition 8. *When *

#### 3. A General Equilibrium Point

Based on Lagrange’s mean value theorem, consider the following.

*Assumption 9. * is a function defined on a closed interval . Then, there exists z in the open interval . The derivative of at z equals the difference quotient

More explicitly,In this article, we construct a market space structure model. The point m is defined as a monopoly market structure. The point n is on behalf of a perfect competitive market structure. Cournot competition locates between m and n. Some studies show there are many equilibrium points in the district. In this model, at least a general equilibrium point is derived. The model can provide acquiring optimal production for firms in incomplete competition market, such as Figure 1.