Journal of Difference Equations

Volume 2015 (2015), Article ID 545364, 9 pages

http://dx.doi.org/10.1155/2015/545364

## A Note on a Modified Cournot-Puu Duopoly

Departamento de Matemática Aplicada y Estadstica, Universidad Politécnica de Cartagena, C/Doctor Fleming, s/n, 30202 Cartagena, Spain

Received 17 August 2014; Revised 10 December 2014; Accepted 10 December 2014

Academic Editor: Abdelalim A. Elsadany

Copyright © 2015 Jose S. Cánovas. 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

The aim of this paper is to analyze a classical duopoly model introduced by Puu in 1991 when lower bounds for productions are added to the model. In particular, we prove that the complexity of the modified model is smaller than or equal to the complexity of the seminal one by comparing their topological entropies. We also discuss whether the dynamical complexity of the new model is physically observable.

#### 1. Introduction

In this paper we study a model which is a modification of the well-known duopoly model introduced by Puu as follows (see [1]). Consider a market that consists of two firms which produce equivalent goods with isoelastic demand function: where , , are the outputs of each firm and is the price and , , are the constant marginal costs. Under these assumptions, we see that both firms maximize their profits, given by , , if The Cournot point, where both firms maximize their profits at the same time, is given by In addition, if , , are the production of both firms at time , then, under naive expectations on future, they plan the future production according to (2), and hence the dynamical model is given by The functions and are called reaction functions.

A detailed analysis of this model reveals that, when the firms are highly inhomogeneous, that is, when or are greater than 6.25, a paradoxical situation arises. We cite a sentence from [2]: “the disadvantages are that the model is no good for dealing with monopoly. As price and quantity are reciprocal, the revenue of a monopolistic firm would be constant, no matter how much the firm sells. On the other hand, any reasonable production cost function increases with output; so producing nothing is the best choice for lowering costs. With constant revenue, the obvious best choice is to actually produce nothing, so avoiding costs, and selling this nothing at an infinite price. The solution has no meaning in terms of substance.” Then, as is introduced in [2] “a more interesting model, satisfying the intuitive economic behavior, is that a state variable or can become very low, assuming a fixed low value, say , after which they can increase again.” Then, the reaction functions are as follows:

The new model is discontinuous, which makes its analysis more difficult than in the continuous case; for instance, the well-known period 3 implies that chaos is valid for continuous interval maps, but it is simple to show that it is not true in general for discontinuous maps. On the other hand, for some parameter values the production can be smaller than . For instance, we take and . Then, the square is invariant under the second iteration of the model and, inside this rectangle, the production of both firms can be as close to zero as we desire because the whole rectangle is an attractor for initial conditions inside it (see, e.g., [3]).

We propose a new model which keeps the idea of a minimal firm production as follows: where . Figure 1 shows the difference between the reactions functions for the discontinuous case and the model that we propose. Our aim is to analyze this new model and check what is the influence of the parameter on it.