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Chaos, Solitons & Fractals

Volume 45, Issues 9–10, September–October 2012, Pages 1081-1085
Chaos, Solitons & Fractals

Stability of the Cournot equilibrium for a Cournot oligopoly model with n competitors

https://doi.org/10.1016/j.chaos.2012.05.007Get rights and content

Abstract

In this paper a Cournot-like model is constructed with an iso-elastic demand function for n competitors. The Cournot equilibrium is constructed for general constant unit costs. Finally, it is proved that for identical unit costs the Cournot point is a sink for two or three competitors and a saddle for more than four players.

Introduction

There are two opposing market forms in economics: competition and monopoly. In the case of competition, firms are numerous and hence small in comparison to the size of the total market, and they consider the market price to be exogenously determined. In a monopoly, only one firm supplies the market and supply influences the market price appreciably.

An oligopoly is a market form in which a market has a dominant influence on a small number of sellers (oligopolists). Since there are few sellers, each oligopolist is likely to be aware of the actions of the others. Each seller has an influence on, and is influenced by, the decisions of the other oligopolists. Hence, the planning of each oligopolist needs to take into account the responses of the other competitors. In an oligopoly, there are at least two firms controlling the market. If there are two sellers, it is called a duopoly; while if there are three competitors, it is known as a triopoly. The first treatment of oligopoly was proposed by A. Cournot, in 1838 [4], for a duopoly. Significant additions to the theory were made exactly one hundred years later by H. von Stackelberg [12].

A question in microeconomics is whether an increase in the number of competitors in a market defines a path to perfect competition. It was stated by Theocaris [13] (see also [8] page 237) that the oligopoly model produced under constant marginal costs with a linear demand function is neutrally stable for three competitors and unstable for more than three competitors. The argument for this fact can be found in [11]. As discussed in [11], linear demand functions are very easy to use, but they do not avoid negative supplies and prices, so it is possible to use them only for the study of local behavior. This problem can be solved by using nonlinear demand functions such as piecewise linear functions or other more complex functions, one of which was suggested by Puu [9] for a duopoly and later by Puu [10] for a triopoly using iso-elastic demand functions. These types of demand function were later studied by Agiza [1], [2] for a nonlinear (iso-elastic) demand function and constant marginal costs and it was concluded that this Cournot model for n competitors is neutrally stable if n=4 and is unstable if the number of competitors is greater than five (see also [11]).

The main aim of this paper is to consider Cournot points and discuss their stability while the number of players is increasing for the model with an iso-elastic demand function and under the assumption that the firms’ costs are identical. The terminology of dynamical systems is used, that is the Cournot point is identified as a fixed one.

A discrete dynamical system is an ordered pair (X,f) where X is standardly taken to be a compact metric space and f is a continuous map from X to, but not necessarily onto, X. So, X is invariant under f, that is f(X)X. A point xX is fixed if f(x)=x and a set of all fixed points of the map f is denoted by Fix(f). For tN, the t-th iterate of f is the t-fold composition ft=ff, where f0 is the identity map.

The paper is organized as follows. In the second section the Cournot iso-elastic model with n competitors is derived, and the model is constructed as a discrete dynamical system (Xn,Fn). In the third section it is proved that the Cournot point is a sink for n=2,3 and is a saddle for n>4. It is known that for n=4 the Cournot point is neutrally stable (see e.g. [1] or [11]). Finally, in the last section the analogous results are discussed that were given in the previous section on a model with the assumption that firms compete with their closest competitors in either direction.

Section snippets

The Cournot iso-elastic model for N competitors

The following construction is inspired by the work of Puu [9] who constructed the model for two competitors (later in [10] for three players). This model could be extended for n firms.

Assuming that the level of demand is reciprocal to price p, this represents an “iso-elastic” demand function reflecting a case where consumers always spend a constant sum on the commodity, regardless of price. Inverting the demand function givesp=1x1+x2++xn,where the total quantity in the denominator is the sum

Stability of the Cournot equilibrium of the model

A fixed point p for f:RnRn is called hyperbolic if Df(p) has no eigenvalues on the unite circle, where Df(p) is the Jacobian matrix of f at the point p. Such a hyperbolic point p is

  • 1.

    a sink fixed point if all eigenvalues of Df(p) are less than one in absolute value,

  • 2.

    a source fixed point if all eigenvalues of Df(p) are greater than one in absolute value,

  • 3.

    a saddle fixed point otherwise, i.e., if some eigenvalues of Df(p) are less and some larger than one in absolute value.

Proposition 1

[5]

Supposing that f:RnRn has

Conclusion

The Cournot point was constructed for (Xn,Fn) and for general unit costs in Theorem 3 and its stability discussed under the additional assumption that the unit costs are identical for all firms in Theorem 5. The Cournot point is a sink for two and three competitors according to Corollary 7 and is a saddle for more than four competitors by Corollary 8, under the assumption of constant marginal costs. It is known that for n=4 the Cournot point is neutrally stable (see e.g. [1] or [11]).

Acknowledgements

The author thanks those whose helpful comments has improved this paper.

This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).

The work was also supported by the Grant Agency of the Czech Republic, Grant No. P201/10/0887 and the Ministry of Education of the Czech Republic, project No. MSM6198910027.

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