**1. Introduction**

Examining oligopoly models is a very frequently studied research area in mathematical economics. Based on the pioneering work of Cournot [1], many researchers were devoted to this interesting and challenging model and its variants and extensions. One frequently studied extension is obtained by considering the dynamic behavior of the firms. These models can be divided into several categories including linear and nonlinear models, discrete and continuous time scales, best response, and gradient adjustments. For discrete time scales Theocharis [2] showed that the equilibrium of *n*-firm linear oligopolies without product differentiation is asymptotically stable if *n* = 2, marginally stable if *n* = 3 and unstable if *n* > 3. For continuous time scales, McManus and Quandt [3] showed that the equilibrium is always asymptotically stable in the linear case regardless of the values of the positive speeds of adjustments. These classical results already indicated that the dynamic properties of the equilibrium strongly depends on the selection of time scales. Several generalizations and extensions were then introduced and studied in the literature. The early results up to the mid-70s are summarized in Okuguchi [4] and their multiproduct generalizations are presented in Okuguchi and Szidarovszky [5]. Different aspects of the classical Theocharis model were then examined by several authors including Canovas et al. [6], Hommes et al. [7], Lampart [8], Puu [9,10], Matsumoto and Szidarovszky [11] among others. Nonlinear models are discussed in Bischi et al. [12] and their extensions including delays are examined in Matsumoto and Szidarovszky [13].

In this paper, we reconsider the classical Theocharis model by examining the dynamic behavior of linear *n*-firm oligopolies without product differentiation and with the additional assumption that the firms face both implementation and information delays. As it is well known that in the linear case best response and gradient adjustment processes are equivalent with different speeds of adjustments, we deal only with best response dynamics. It is assumed that the firms face equal delays in both types. If the implementation and information delays are equal, then the model is equivalent with a single delay

case mathematically. In this case, we show that the equilibrium is oscillatory asymptotically stable if the common delay is sufficiently small, at the threshold Hopf bifurcation occurs with cyclic and for larger delays expanding cyclic trajectories. If the delays are different, then a two-delay model is obtained. The stability switching curves are first constructed and then the directions of stability switches are determined. Growth rate dynamics result in nonlinear systems, their local linearizations around the equilibrium result in linear dynamics, that is equivalent to the best response case. So the local dynamics of the two systems are equivalent. Simulation studies verify and illustrate the theoretical findings of the paper. Even in the very special case of linear models, our analysis discovered several aspects of the dynamics which were not studied in the literature before. The importance of examining linear models is verified in addition to the fact that linearized nonlinear models have the same mathematical structures.

This paper develops as follows. Section 3 introduces the best response dynamics. First stability switching curves are constructed and then the case of equal delays is discussed in detail. Growth rate dynamics are introduced in Section 4. First, the stability switching curves are shown and then the directions of stability switches are determined. In both sections, numerical results and simulation studies verify and illustrate the theoretical results. Section 5 offers conclusions and outlines further research directions.

#### **2. Model**

The classical oligopoly model is presented reconsidering the classical results of Theocharis [2] and McManus and Quandt [3]. In the model, *n* firms are producing a homogeneous output. The price function is assumed to be linear,

$$p = a - b \sum\_{j=1}^{n} x\_j$$

where *a* > 0 is the maximum price, *b* > 0 is the slope of the price function and *xj* is firm *j*'s output. The production cost is also assumed to be linear with no fixed cost. The marginal cost of firm *j* is denoted by *cj*, being positive. The profit function of firm *i* is defined by

$$
\pi\_i = \left( a - b \sum\_{j=1}^n x\_j \right) x\_i - c\_i x\_i.
$$

Under the Cournot competition, the firms decide how much to produce. As we focus only on interior solutions (If the optimal output level of a firm is zero, then the firm leaves the industry, so we can igonore such firms), the first-order condition of firm *i* for profit maximization is

$$\frac{\partial \pi\_i}{\partial \mathbf{x}\_i} = a - 2b\mathbf{x}\_i - b\sum\_{j\neq i}^n \mathbf{x}\_j - c\_i = 0$$

and the second-order condition is satisfied,

$$\frac{\partial^2 \pi\_i}{\partial x\_i^2} = -2b < 0.$$

The best reply function is obtained through the first-order condition and depends on the choices of other firms,

$$
\mathfrak{x}\_i^\* = \frac{a - c\_i - b\sum\_{j\neq i}^n \mathfrak{x}\_j}{2b}.
$$

Let us introduce a new notation,

$$\alpha\_i = \frac{a - c\_i}{2b}, \beta = \frac{1}{2} \text{ and } Q = \sum\_{j=1}^n x\_j.$$

*Mathematics* **2020**, *8*, 1615

and make the conventional assumption:

**Assumption 1.** *ci* = *c for all i and a* > *c.*

Assumption 1 implies *α<sup>i</sup>* = *α* > 0 for all *i*. As each firm makes an optimal choice at the Cournot equilibrium, its best reply function is written as

$$
\alpha\_i^\* = \frac{a - \beta Q^\*}{1 - \beta}.
$$

The aggregate output of all firms is obtained by adding the individual outputs,

$$Q^\* = \sum\_{i=1}^n \mathbf{x}\_i^\* = n \frac{\boldsymbol{\alpha} - \beta Q^\*}{1 - \beta}$$

that is solved for *Q*∗ to have

$$Q^\* = \frac{na}{1 + (n-1)\,\beta}.$$

Substituting *Q*∗ into the best reply gives the individual output values at the Cournot equilibrium,

$$\mathbf{x}\_i^c = \frac{\alpha}{1 + (n - 1)\beta} \text{ for } i = 1, 2, \dots, n.$$
