Stability, Bifurcation, and Chaos in N-Firm Nonlinear Cournot Games

Abstract

An 𝑁-firm production game known as oligopoly will be examined with isoelastic price function and linear cost under al Cournot competition. After the best responses of the firms are determined, a dynamic system with adaptive expectations is introduced. It is first shown that the local asymptotic behavior of the system is identical with that of the adaptive adjustment process in which the firms cautiously determine their outputs. Dynamic analysis is confined to two special cases, one in which 𝑁 is divided into two groups and the other in which 𝑁 is divided into three groups. Then stability conditions will be derived and the global behavior of the equilibria will be illustrated including chaos control. Lastly the two- and three-group models are compared with two-firm (duopoly) and three-firm (triopoly) models to shed light on roles of the number of the firms.

1. Introduction

The state sequence of discrete dynamic systems will be considered as time series, with a deterministic rule to obtain the consecutive state variables. Among the large variety of dynamic economic systems the oligopoly models have a very special place, since the long-term behavior of the state trajectories has many different possibilities including global asymptotic stability, limit cycles with increasing number of periods and even chaos. Following Cournot [1] many researchers worked on developing more realistic models and on examining their properties. The existence and uniqueness of the equilibrium was the main focus in earlier studies, and later the researchers turned their focus to the dynamic extensions of these models. A comprehensive summary of earlier results is given in Okuguchi [2], and their multiproduct extensions with different model variants are discussed in Okuguchi and Szidarovszky [3]. Most studies considered concave oligopolies with monotonic best responses. Puu [4] replaces a linear demand function (the most popular form) with an isoelastic demand function (the second popular form), which can be derived by assuming a Cobb-Douglas type utility function of the market. In this case the best responses are unimodal, making equilibrium and stability analyses more complicated and more different from those under monotonic best responses. In particular, chaotic dynamics emerges through a Neimark-Sacker bifurcation when inherited nonlinearities becomes stronger. Agiza and Elsadany [5] have investigated discrete-time Cournot duopolies with heterogeneous players. Richter and Stolk [6] have introduced a new method of controlling coexisting chaotic attractors in Cournot triopolies by means of steering the systems dynamics from one attractor to another. See Puu and Sushko [7], Puu [8] and Bischi et al. [9] for comprehensive summary of recent developments in the theory of nonlinear oligopolies.

This work continues an earlier paper of Matsumoto [10], where chaos control for nonlinear duopolies was examined. The number of the firms is generalized to 𝑁≥2. After determining best responses of the firms and giving detailed equilibrium analysis, it constructs an adjustment process with adaptive expectations in which the expectations are adaptively updated. To simplify dynamic analysis, attentions are confined to two special cases, one in which the 𝑁 firms are divided into two groups (i.e., two-group model) and the other in which the 𝑁 firms are divided into three groups (i.e., three-group model). It is shown that these models generate complex dynamics involving chaos. It is further shown that complex dynamics could be stabilized by proper selection of the speeds of adjustment.

The paper develops as follows. In Section 2 the general model will be introduced, the best responses of the firms and the Cournot equilibrium will be determined. In Section 3 we will show that the local stability properties of the adjustment process with adaptive expectations are identical with those of the adaptive adjustment process in which the firms cautiously adjust their outputs in the next period. In Section 4 two special cases, two- and three-group models, will be analyzed both theoretically and numerically in which the dynamics is two- and three-dimensional, respectively. The stability regions, where chaos is controlled, will be shown and their dependence on the number of firms will be illustrated. Section 5 will conclude the paper.

2. Nonlinear Oligopoly Models

It is assumed that a homogeneous market is supplied by 𝑁 firms. For the sake of mathematical simplicity only one product is considered. Let 𝑥_𝑖 denote the production output of firm 𝑖, then 𝑆_𝑖=∑_𝑗≠𝑖𝑥_𝑗 is the output of the rest of the industry and 𝑆=∑^𝑁_𝑖=1𝑥_𝑖 is the total output of the industry. We assume isoelastic price function 𝑝=1/𝑆 and linear cost functions 𝐶_𝑖(𝑥_𝑖)=𝑐_𝑖𝑥_𝑖 as in the duopoly model of Puu [8]. Since the firms make decisions about their production levels simultaneously, the firms do not know the outputs of the rivals when their decisions are made. Each firm 𝑖 can have only an expectation (prediction) of the output of the rest of the industry, 𝑆^𝑒_𝑖. So the expected profit of firm 𝑖 can be given as Π^𝑒_𝑖=𝑥_𝑖𝑥_𝑖+𝑆^𝑒_𝑖−𝑐_𝑖𝑥_𝑖(𝑖=1,2,…,𝑁). (2.1) Notice that this function is strictly concave in 𝑥_𝑖.

The strict concavity of Π^𝑒_𝑖 implies that with any given value of 𝑆^𝑒_𝑖 the profit maximizing output level of firm 𝑖 can be computed as𝑥_𝑖=𝑓_𝑖𝑆^𝑒_𝑖=𝑆^𝑒_𝑖𝑐_𝑖−𝑆^𝑒_𝑖if0<𝑆^𝑒_𝑖<1𝑐_𝑖,0if1𝑐_𝑖≤𝑆^𝑒_𝑖. (2.2) This function is continuous, piecewise differentiable, and in interval [0,1/𝑐_𝑖] it is strictly concave in 𝑆^𝑒_𝑖. Then the best response dynamic process is 𝑥_𝑖(𝑡+1)=𝑓_𝑖𝑆^𝑒_𝑖(𝑡+1)for𝑖=1,2,…,𝑁. (2.3) Dynamic characteristics are sensitive to the expectation formation. In this study we first consider naive expectation in which the firms assume that the output of the rest of the industry remains the same as in the previous period: 𝑆^𝑒_𝑖(𝑡+1)=_𝑗≠𝑖𝑥_𝑗(𝑡), (2.4) and call it a naive system. It is well known that the naive system is a special case of the best reply dynamics with adaptive expectations. In the appendix, local stability conditions for dynamic systems with adaptive expectations are derived. It is, as will be seen, useful to determine the stability of not only naive systems but also that of the controlled systems.

Without losing generality we may assume that at the equilibrium all firms have positive outputs, otherwise we can ignore the firms with zero equilibrium output values and decrease the value of 𝑁. Assuming a positive equilibrium, then, from the definitions of naive expectations and the reaction function of firm 𝑖, we get 𝑆^𝑒_𝑖=_𝑗≠𝑖𝑥_𝑗,𝑥_𝑖=𝑆^𝑒_𝑖𝑐_𝑖−𝑆^𝑒_𝑖. (2.5) Since the total industry output is 𝑆=𝑥_𝑖+𝑆^𝑒_𝑖,from the second equation we have 𝑆^𝑒_𝑖=𝑐_𝑖𝑆², (2.6) that is, 𝑥_𝑖=𝑆−𝑐_𝑖𝑆². (2.7) Adding this equation for all values of 𝑖 and denoting the sum of the marginal costs by 𝐶=∑^𝑁_𝑖=1𝑐_𝑖 gives 𝑆=𝑁𝑆−𝐶𝑆². (2.8) Therefore there is a trivial equilibrium with 𝑆=𝑥₁=⋯=𝑥_𝑁=0, and a nontrivial positive equilibrium with 𝑆^𝑐=𝑁−1𝐶, (2.9) where the Cournot output of firm 𝑖 becomes𝑥^𝑐_𝑖=(𝑁−1)𝐶−𝑐_𝑖(𝑁−1)𝐶². (2.10) The superscript “𝑐” is attached to variables to indicate that they are computed at the Cournot equilibrium. Our concern is on the nontrivial point, (𝑥^𝑐_𝑖)_{𝑖=1,2,…,𝑁}, and thus no further considerations will be given to the trivial point. For a positive Cournot output, the following inequality has to be satisfied:𝐶−𝑐_𝑖(𝑁−1)>0or𝑐_𝑖<𝐶.𝑁−1 (2.11) This always holds for 𝑁=2, and necessarily holds for 𝑁>2 if the marginal costs, 𝑐_𝑖, are sufficiently close to each other. In the rest of this paper, we assume that this condition is satisfied. We also need to guarantee that in (2.2) the first case applies. Substituting 𝑐_𝑖𝑆² for 𝑆^𝑒_𝑖 and arranging terms yield 𝐶>𝑐_𝑖(𝑁−1). Thus the condition 𝑆^𝑒_𝑖<1/𝑐_𝑖 can be reduced to the nonnegativity condition (2.11).

Adaptive expectations are generalizations of naive expectation where the expected output of the rest of the industry is computed as 𝑆^𝑒_𝑖(𝑡+1)=1−𝛼_𝑖𝑆^𝑒_𝑖(𝑡)+𝛼_{𝑖𝑗≠𝑖}𝑥_𝑗(𝑡). (2.12) Notice that in the case of 𝛼_𝑖=0, the expectation 𝑆^𝑒_𝑖(𝑡) of firm 𝑖 remains constant and therefore the same best response is chosen at all time periods 𝑡, which is not equilibrium strategy in general. If 𝛼_𝑖=1, then this formula reduces to naive expectation. The best response dynamic process with adaptive expectations is a 2𝑁-dimensional system with state variables 𝑥_𝑖 and 𝑆^𝑒_𝑖(𝑖=1,2,…,𝑁). It is easy to see that the steady state 𝑥_𝑖, 𝑆_𝑖 of the system and the Cournot equilibrium of the static 𝑁-firm oligopoly coincide: 𝑥_𝑖=𝑥^𝑐_𝑖 and 𝑆_𝑖=𝑆^𝑐_𝑖. In the appendix the local stability conditions for system with adaptive expectations are derived. Combining inequalities (2.11) and (A.15)from the appendix with 𝛼_𝑖=1gives 𝐶4(𝑁−1)<𝑐_𝑖<𝐶𝑁−1for𝑖=1,2,…,𝑁, (2.13) and the eigenvalue equation (A.16) of the appendix reduces to the following:^𝑁_𝑖=1𝛾_𝑖𝛾_𝑖+𝜆=1,where𝛾_𝑖≡𝜕𝑓_𝑖𝑆^𝑒_𝑖𝜕𝑆^𝑒_𝑖attheequilibrium. (2.14) It can be written as the quadratic and cubic equations, 𝜆²−𝛾₁𝛾₂=0for𝑁=2,−𝜆³+𝛾₁𝛾₂+𝛾₁𝛾₃+𝛾₂𝛾₃𝜆+2𝛾₁𝛾₂𝛾₃=0for𝑁=3. (2.15)

3. General Stability Conditions

In this section, we first consider the dynamic process in which firms cautiously adjust their outputs, that is, the output in the next period is a weighted average of the current output and the best reply with naive expectations: 𝑥_𝑖(𝑡+1)=𝛼_𝑖𝑓_𝑖𝑆^𝑒_𝑖(𝑡+1)+1−𝛼_𝑖𝑥_𝑖(𝑡)for𝑖=1,2,…,𝑁, (3.1) or, equivalently,𝑥_𝑖(𝑡+1)=𝛼_𝑖∑_𝑗≠𝑖𝑥_𝑗(𝑡)𝑐_𝑖−_𝑗≠𝑖𝑥_𝑗(𝑡)+1−𝛼_𝑖𝑥_𝑖(𝑡). (3.2) This is commonly known as the adaptive adjustment process. Here we assume that firm 𝑖 moves into the direction toward its profit maximizing output, and reaches it only for 𝛼_𝑖=1. Since this adjustment process describes the best reply dynamics with inertia, we call it the inertia control system. Here 𝛼_𝑖 is the inertia or control parameter of firm 𝑖 and assumed to be positive and not greater than unity. It is easy to see that the fixed point of the inertia control system is the same as that of the naive system. The Jacobian of the system has the form 𝐇^𝑐=1−𝛼₁𝛾₁𝛼₁⋅𝛾₁𝛼₁𝛾₂𝛼₂1−𝛼₂⋅𝛾₂𝛼₂𝛾⋅⋅⋅⋅_𝑁𝛼_𝑁𝛾_𝑁𝛼_𝑁⋅1−𝛼_𝑁. (3.3)

In the appendix, we show that the nonzero eigenvalues of the Jacobian of the system with adaptive expectations are the eigenvalues of matrix 𝐇=1−𝛼₁𝛾₂𝛼₁⋅𝛾_𝑁𝛼₁𝛾₁𝛼₂1−𝛼₂⋅𝛾_𝑁𝛼₂𝛾⋅⋅⋅⋅₁𝛼_𝑁𝛾₂𝛼_𝑁⋅1−𝛼_𝑁. (3.4) It is easy to see that the characteristic polynomials of matrices 𝐇 and 𝐇^𝑐 are identical since 𝐑⁻¹𝐇^𝑐𝐑=𝐇 with the diagonal matrix 𝐑=diag𝛾₁,𝛾₂,…,𝛾_𝑁. (3.5) If any or more 𝛾_𝑖=0, then the continuity of the characteristic polynomial coefficients in the matrix elements implies the result. Hence the local stability conditions of dynamics with adaptive expectations and with inertia control are identical. So in the rest of this paper we will consider only inertia control which also contains models with best response dynamics as special case with 𝛼_𝑖=1.

In the case of concave oligopolies (see, e.g., Szidarovszky and Chiarella [11], Bischi et al. [9]) it is proved that −1<𝛾_𝑖≤0,all eigenvalues are real and they are inside the unit circle if and only if for all 𝑖, 𝛼_𝑖<21+𝛾_{𝑖𝑁𝑖=1},𝛼_𝑖𝛾_𝑖2−𝛼_𝑖1+𝛾_𝑖>−1. (3.6) These conditions are clearly satisfied if the speeds 𝛼_𝑖 of adjustments are sufficiently small. However in the case of isoelastic inverse demand functions there is the possibility of complex eigenvalues, so no such simple general stability conditions can be derived. In the next section stability in two special cases will be examined both theoretically and numerically.

4. Stability Conditions in Special Models

In this section we will focus on two cases: one case where the industry consists of two groups of identical firms and the other case with three groups of identical firms. Our aim is to see whether the unstable naive system can be stabilized by the inertia control method.

4.1. Two Groups of Firms

Assume there are two groups of firms in a sense that firms of the same group produce the same output with the same marginal cost and have the same speed of adjustment. So the 𝑁 firms are divided into two groups. Without loss of generality, we can assume that the first 𝑁_𝑎 firms are in the first group and the remaining 𝑁_𝑏 firms in the second group, where 𝑁_𝑎+𝑁_𝑏=𝑁 and 𝑁_𝑎=(1/𝑤)𝑁 with 𝑤>1. By assuming 𝑁/𝑁−1<𝑤<𝑁,we will not consider the extreme division in which 𝑁 is divided into 1 and 𝑁−1. We denote the outputs produced by the firms of the two groups by 𝑥 and 𝑦, so 𝑥₁=⋅⋅⋅=𝑥_𝑁𝑎=𝑥,𝑥_𝑁𝑎+1=⋅⋅⋅=𝑥_𝑁=𝑦, (4.1) the two kinds of marginal costs are denoted by 𝑎 and 𝑏, so 𝑐₁=⋅⋅⋅=𝑐_𝑁𝑎=𝑎,𝑐_𝑁𝑎+1=⋅⋅⋅=𝑐_𝑁=𝑏. (4.2) It is also assumed that the firms use adaptive output adjustments (3.2) with naive expectation with the two values of speeds of adjustment 𝛼₁=⋅⋅⋅=𝛼_𝑁𝑎=𝛼,𝛼_𝑁𝑎+1=⋅⋅⋅=𝛼_𝑁=𝛽. (4.3) We further assume, without any losses of generality, that the firms in the first group are more efficient than the ones in the second in a sense that 𝑎<𝑏. (4.4) Accordingly, the marginal cost ratio of 𝑏 over 𝑎, which we denote by ℎ, is greater than unity, 𝑏ℎ=𝑎>1. (4.5) The sum of the marginal costs and the derivatives of the reaction functions evaluated at the Cournot equilibrium are, respectively, 𝐶=𝑁_𝑎𝑎+𝑁_𝑏𝛾𝑏,_𝑎=𝐶2𝑎(𝑁−1)−1,𝛾_𝑏=𝐶2𝑏(𝑁−1)−1. (4.6)

From (2.10), the equilibrium outputs of the firms are 𝑥^𝑐=(𝑁−1)(𝐶−𝑎(𝑁−1))𝐶²,𝑦^𝑐=(𝑁−1)(𝐶−𝑏(𝑁−1))𝐶², (4.7) where 𝐶−𝑎(𝑁−1)=𝑎𝑁_𝑏()()𝑁ℎ−1+1>0,𝐶−𝑏𝑁−1=𝑎_𝑎𝑁−1.𝑁−𝑤−ℎ (4.8) The first inequality is always true because ℎ>1, so 𝑥^𝑐 is always positive. The second equation indicates that 𝑦^𝑐 is positive if 𝑁_𝑎>1 and the marginal cost ratio is bounded from above:ℎ<ℎ_𝑁≡𝑁=𝑁𝑁−𝑤𝑁^𝑎_𝑎.−1 (4.9) Notice that 𝑁_𝑎>1 implies that the denominator of the marginal cost ratio, ℎ_𝑁, is positive. It can be seen that ℎ_𝑁 decreases in 𝑁 and is approaching unity as 𝑁 converges to infinity with fixed values of 𝑤 or 𝑁_𝑎 converges to infinity regardless of the value of 𝑤. Since the ratio ℎ is assumed to be greater than unity, this implies that it becomes more difficult to have a positive Cournot equilibrium as the number of the firms in the industry increases. Similarly, condition (A.15) for each group can be written as 4()ℎ<𝑁−1−𝛼𝑁_𝑎𝛼𝑁_𝑏forthefirstgroup,ℎ>𝛽𝑁_𝑎4(𝑁−1)−𝛽𝑁_𝑏forthesecondgroup. (4.10) However neither inequality is effective, since for 𝑁≥3 and 𝛼,𝛽∈(0,1],𝛽𝑁_𝑎4(𝑁−1)−𝛽𝑁_𝑏𝑁<1≤<4()𝑁−𝑤𝑁−1−𝛼𝑁_𝑎𝛼𝑁_𝑏, (4.11) which indicates that if ℎ fulfills (4.9), then it also satisfies (A.15).

In this case, (A.16) assumes the form 1+𝑁_𝑎𝛼𝛾_𝑎1−𝛼1+𝛾_𝑎−𝜆+𝑁_𝑏𝛽𝛾_𝑏1−𝛽1+𝛾_𝑏−𝜆=0, (4.12) since 𝑁_𝑎 and 𝑁_𝑏 firms have identical parameters, in which case 𝛼_𝑖=𝛼, 𝛾_𝑖=𝛾_𝑎(1≤𝑖≤𝑁_𝑎) and 𝛼_𝑖=𝛽, 𝛾_𝑖=𝛾_𝑏(𝑁_𝑎+1≤𝑖≤𝑁_𝑎+𝑁_𝑏). Notice that with notation 𝑅_𝑎=1+𝛾_𝑎 and 𝑅_𝑏=1+𝛾_𝑏, the above equation reads 1−𝛼𝑅_𝑎−𝜆1−𝛽𝑅_𝑏−𝜆+𝑁_𝑎𝛼𝛾_𝑎1−𝛽𝑅_𝑏−𝜆+𝑁_𝑏𝛽𝛾_𝑏1−𝛼𝑅_𝑎−𝜆=0. (4.13) This is a quadratic equation in 𝜆,𝜆²+𝑝𝜆+𝑞=0 (4.14) with coefficients 𝑝=𝛼𝑅_𝑎1−𝑁_𝑎+𝑁_𝑎+𝛽𝑅_𝑏1−𝑁_𝑏+𝑁_𝑏𝑅−2,𝑞=−𝛼_𝑎1−𝑁_𝑎+𝑁_𝑎−𝛽𝑅_𝑏1−𝑁_𝑏+𝑁_𝑏+𝛼𝛽𝑅_𝑎𝑅_𝑏(1−𝑁)+𝑅_𝑎𝑁_𝑏+𝑅_𝑏𝑁_𝑎+1. (4.15)

It is well-known that the roots of the quadratic equation (4.14) are inside the unit circle if and only if 𝑞−1<0,𝑞+𝑝+1>0,𝑞−𝑝+1>0. (4.16) The left hand side of the second stability condition can be rewritten as 𝑁𝑞+𝑝+1=^𝑎+ℎ𝑁_𝑏²𝛼𝛽(),4ℎ𝑁−1 (4.17) which is always positive. This indicates that the characteristic equation (4.14) does not have a root equal to unity. The other two conditions are hard to be simplified and explained in general. Therefore we make a specializing assumption that the speeds of adjustment are the same for the two groups.

Assumption (𝛼=𝛽). The feasible region in terms of the adjustment speeds and the marginal cost ratios for which the Cournot equilibrium is positive is 𝑃_𝑁=(𝛼,ℎ)∣0<𝛼≤1,1<ℎ<ℎ_𝑁, (4.18) where ℎ_𝑁 is the upper bound of the marginal cost ratio defined in (4.9). Since it is determined by 𝑁 and 𝑤, the feasible region is presented by a rectangular, 𝑃_𝑁=(0,1]×(1,ℎ_𝑁) which decreases in𝑁 (i.e., 𝑃_𝑁+1⊂𝑃_𝑁).
The locus of 𝑞−1=0 is the Neimark-Sacker boundary on which the Cournot equilibrium changes stability through a pair of complex conjugates and a closed invariant curve is born. We first examine the possibility of the Neimark-Sacker bifurcation. Under 𝛼=𝛽 and ℎ=ℎ_𝑁, the left hand side of the first stability condition, 𝑞−1, has the form 𝛼()𝑞−1=𝑁−1𝑁4ℎ𝑁−𝑤²2()𝛼−1+𝑁)(𝑁−𝑤𝑁().𝑁−1 (4.19) It is natural to confine our analysis to the case where 𝑁,𝑁_𝑎 and 𝑁_𝑏 are integers. In consequence, the relation 𝑁≥2𝑤 must hold to have that 𝑁_𝑎=𝑁/𝑤 is an integer, since 𝑁>𝑤 has already been assumed. Then substituting 𝑁/2 for 𝑤 yields 2(1+𝑁)(𝑁−𝑤)𝑁()≥𝑁−1𝑁+1𝑁−1>1. (4.20) It then follows from the assumption 𝛼≤1 that 𝑞−1<0. In other words, the 𝑞−1=0 locus does not cross the ℎ=ℎ_𝑁 locus. We may go from this to the conclusion that the first stability condition 𝑞−1<0is always satisfied for any (𝛼,ℎ)∈𝑃_𝑁, given 𝑤 and 𝑁. Our first result on Cournot dynamics is summarized as follows.

Theorem 4.2. If 𝛼=𝛽, 𝑁/(𝑁−1)<𝑤<𝑁and all of 𝑁, 𝑁_𝑎 and 𝑁_𝑏 are integers, then the Neimark-Sacker bifurcation does not occur in the feasible region 𝑃_𝑁.

We then examine the possibility of flip bifurcation. The locus of 1−𝑝+𝑞=0 is the flip boundary on which at least one of the eigenvalues is equal to −1. Crossing this boundary the equilibrium point goes through a period-doubling cascade to chaos. Solving 1−𝑝+𝑞=0 for α under 𝛼=𝛽 and ℎ=ℎ_𝑁 yields2𝛼=𝑁()()−𝑁−1𝑁−𝑤)(𝑁+1(𝑁−𝑤)²(𝑁+1)²−4𝑁(𝑁−𝑤)(𝑁+1). (4.21) Given 𝑤 and 𝑁, the above equation determines a threshold value of α at which the flip boundary crosses the ℎ=ℎ_𝑁 locus. It, however, still has a complicated expression. So, instead of analytic study, we will consider an important special case and perform numerical simulations. For further simplification, we set 𝑤=2(i.e., the industry consists of two groups with equal size) and increase the number of the firms by two from 𝑁=4 to 𝑁=10to find the threshold value of α when the number of the firms in the industry increases: 𝛼₄=43when𝛼𝑁=4,₆=47−√1915≃0.704when𝛼𝑁=6,₈=27−√39314≃0.513when𝛼𝑁=8,₁₀=811−2√1945≃0.406when𝑁=10. (4.22) For 𝑁=4, the threshold value 𝛼₄ is greater than unity. This implies that the flip boundary and the ℎ=ℎ_𝑁 locus do not intersect. Thus we see that the Cournot equilibrium is stable in the feasible region 𝑃_𝑁 when 𝑁=4.On the other hand, the threshold value 𝛼_𝑖 for 𝑖=6,8,10 is less than unity. This implies that the flip boundary with 𝑁≥6 intersects the ℎ=ℎ_𝑁 locus and then divides the corresponding feasible region 𝑃_𝑁into two parts, stability region and instability region. Figure 1 illustrates three intersections (denoted by three dots) of the flip boundaries and the ℎ=ℎ_𝑖 locus for 𝑖=6,8,10 and the separation of the feasible regions. When 𝑁=6, the corresponding feasible region 𝑃₆ is a rectangular having the ℎ=ℎ₆ locus as its upper boundary. The locus denoted by 𝐹₆ is the flip boundary with 𝑁=6 and divides the feasible region 𝑃₆into two regions: the stability region is the largest rectangular with light-gray in the left of the flip boundary and the instability region is the white region in the right of the flip boundary and below the ℎ=ℎ₆ locus. In the same way, it is shown in Figure 1 that the flip boundaries with 𝑁=8 and 𝑁=10 denoted by 𝐹₈ and 𝐹₁₀divide the corresponding feasible regions into two parts. The smallest rectangular with dark-gray is the stability region with 𝑁=10 and the remaining middle rectangular is the one with 𝑁=8. Since the ℎ=ℎ_𝑁 locus and the flip boundary shift downward and leftward, respectively, the feasible and stable region becomes smaller as the number of firms 𝑁 increases.

Figure 1 

Stability and instability regions for 𝑁=6,8,10.

Our extension from the two-firm (i.e., duopoly) model to the two-group model reveals interesting features of the nonlinear oligopoly. See Puu [8] for a theoretical and numerical analysis of the two-firm (duopoly) and three-firm (triopoly) models. To see them, we perform numerical simulations in two models and compare the results. To this end, in the two-group model, we take the set of parameters 𝑁=6, 𝑤=2 and choose the two bifurcation parameters: one is the common adjustment coefficient α and the other is the cost ratio ℎ=𝑏/𝑎 between the two groups where 𝑎=1 is taken for simplicity. The dynamic system (3.2) with the two groups has now the special form: 𝑥(𝑡+1)=(1−𝛼)𝑥(𝑡)+𝛼2𝑥(𝑡)+3𝑦(𝑡)𝑎−(𝑡)(𝑡),𝑦()=()𝑦(𝑡)2𝑥+3𝑦𝑡+11−𝛼+𝛼3𝑥(𝑡)+2𝑦(𝑡)𝑏−3𝑥(𝑡)+2𝑦(𝑡). (4.23) On the other hand, in the two-firm model, replacing 𝑁=6 with 𝑁=2 reduces the above dynamic system to 𝑥(𝑡+1)=(1−𝛼)𝑥(𝑡)+𝛼𝑦(𝑡)𝑎(𝑡),𝑦()=()𝑦(𝑡)−𝑦𝑡+11−𝛼+𝛼𝑥(𝑡)𝑏−𝑥(𝑡). (4.24)

Changing α from 0.6 to 1.0 and ℎ (actually 𝑏) from 1 to 1.5(=ℎ₆) generates the two-parameter bifurcation diagram shown as the left part of Figure 2 while changing α from 0.9 to 1.0 and ℎ from 3+2√2 to 6.5 yields the two-parameter bifurcation diagram, the right part of Figure 2. Different colors in the (𝛼,ℎ) plane indicate different periods of cycles up to 16. Periodic points with a period larger than 16 and aperiodic points are colored in gray. The solution becomes infeasible if the parameter values are selected from the white region. The upward sloping black curve in the left diagram and the downward sloping black curve in the right diagram are the flip boundary of the two-group model and the Neimark-Sacker boundary of the two-firm (i.e., duopoly) model, respectively. As the destabilizing scenario is concerned, the comparison shows two issues. First, the stationary state is destabilized through a flip bifurcation in the two-group model and by a Neimark-Sacker bifurcation in the two-firm model. Even though both models can generate complex dynamics involving chaos, the rout to chaos is different. In particular, the Cournot equilibrium goes through a period-doubling cascade to chaos in the two-group model while the Cournot equilibrium bifurcates to periodic cycles, quasiperiodic cycle and then to chaotic fluctuation in the two-firm model. Second, the loss of stability occurs for a relatively low production cost ratio in the two-group model and for a high ratio in the two-firm model. Furthermore, the left diagram indicates that the value of the adjustment speed seems to be a main source of flip bifurcation since the almost similar bifurcation takes place as α increases regardless of the value of ℎ. This implies that the adjustment speed can be an effective control variable in the two-group model. For any value of 𝑤, we have qualitatively the same result. Our second result on Cournot dynamics in the two-group model is summarized as follows.

Figure 2 

Bifurcation diagrams in the (𝛼,ℎ) plane for the two-group model (a) and for the two-firm model (b).

Theorem 4.3. If 𝛼=𝛽, 𝑤=2 and 𝑁=6, then the flip boundary divides the feasible region into two parts, stability region and instability region in which chaos emerges via the period-doubling bifurcation.

4.2. Three Groups of Firms

Assume that the industry consists of three groups with 𝑁_𝑎, 𝑁_𝑏 and 𝑁_𝑐 firms where 𝑁_𝑎+𝑁_𝑏+𝑁_𝑐=𝑁 and 𝑁_𝑎=(1/𝑤_𝑎)𝑁 and 𝑁_𝑏=(1/𝑤_𝑏)𝑁 with 𝑤_𝑎> and 𝑤_𝑏>1. The common outputs of the three groups of firms are denoted by 𝑥,𝑦 and 𝑧,𝑥₁=⋅⋅⋅=𝑥_𝑁𝑎=𝑥,𝑥_𝑁𝑎+1=⋅⋅⋅=𝑥_{𝑁𝑎+𝑁𝑏}=𝑦,𝑥_{𝑁𝑎+𝑁𝑏+1}=⋅⋅⋅=𝑥_𝑁=𝑧, (4.25) the marginal costs are 𝑎, 𝑏 and 𝑐 with 𝑐₁=⋅⋅⋅=𝑐_𝑁𝑎=𝑎,𝑐_𝑁𝑎+1=⋅⋅⋅=𝑐_{𝑁𝑎+𝑁𝑏}=𝑏,𝑐_{𝑁𝑎+𝑁𝑏+1}=⋅⋅⋅=𝑐_𝑁=𝑐, (4.26) and the speeds of adjustment are denoted by α, β and 𝛾,𝛼₁=⋅⋅⋅=𝛼_𝑁𝑎=𝛼,𝛼_𝑁𝑎+1=⋅⋅⋅=𝛼_{𝑁𝑎+𝑁𝑏}=𝛽,𝛼_{𝑁𝑎+𝑁𝑏+1}=⋅⋅⋅=𝛼_𝑁=𝛾. (4.27) As in the previous section, we assume that the firms use adaptive output adjustments with naive expectations, and the firms in the first group are the most efficient in a sense that their marginal cost is the smallest, 𝑎<min{𝑏,𝑐}. (4.28) The marginal cost ratios are denoted by ℎ and 𝑘, 𝑏ℎ=𝑎𝑐>1,𝑘=𝑎>1. (4.29)

The Cournot outputs are obtained from (2.10), 𝑥^𝑐=𝑁−1𝐶²((,𝑦𝐶−𝑎𝑁−1))_𝑐=𝑁−1𝐶²((,𝑧𝐶−𝑏𝑁−1))_𝑐=𝑁−1𝐶²(𝐶−𝑐(𝑁−1)) (4.30) with 𝐶=𝑁_𝑎𝑎+𝑁_𝑏𝑏+𝑁_𝑐𝑐. Then it can be seen that left hand sides of the positivity conditions (2.11) have the forms 𝐶−𝑎(𝑁−1)=𝑎𝑁_𝑏(ℎ−1)+𝑁_𝑐()()𝑁𝑘−1+1>0,𝐶−𝑏𝑁−1=𝑎_𝑐𝑘−𝑁_𝑎+𝑁_𝑐−1ℎ+𝑁_𝑎,𝐶−𝑐(𝑁−1)=𝑎−𝑁_𝑎+𝑁_𝑏−1𝑘+𝑁_𝑏ℎ+𝑁_𝑎. (4.31) The first inequality is always true. In order to have 𝑦^𝑐,𝑧^𝑐>0, we have to assume that 𝑁𝑘>^𝑐+𝑁_𝑎−1𝑁_𝑐𝑁ℎ−^𝑎𝑁_𝑐,𝑁𝑘<^𝑏𝑁_𝑎+𝑁_𝑏𝑁−1ℎ+^𝑎𝑁_𝑎+𝑁_𝑏.−1 (4.32) Let the right hand sides of the first and the second inequalities be denoted by 𝑓_𝑏(ℎ) and 𝑓_𝑐(ℎ). There is no guarantee that 𝑓_𝑏(ℎ)>1, but 𝑓_𝑐(ℎ)>1 always for ℎ>1. Then the set of the marginal cost ratios that generate positive Cournot equilibrium can be defined by 𝑁_𝑁=(ℎ,𝑘)∣max1,𝑓_𝑏(ℎ)<𝑘<𝑓_𝑐(ℎ),ℎ>1. (4.33)

Characteristic equation (A.16) now can be written in the form 1+𝑁_𝑎𝛼𝛾_𝑎1−𝛼1+𝛾_𝑎−𝜆+𝑁_𝑏𝛽𝛾_𝑏1−𝛽1+𝛾_𝑏−𝜆+𝑁_𝑐𝛾𝛾_𝑐1−𝛾1+𝛾_𝑐−𝜆=0. (4.34) Introducing notation 𝑅_𝑎=1+𝛾_𝑎, 𝑅_𝑏=1+𝛾_𝑏 and 𝑅_𝑐=1+𝛾_𝑐, it is reduced to a cubic equation, −𝜆³+𝑝𝜆²+𝑞𝜆+𝑟=0, (4.35) where the coefficients are 𝑝=−𝑁_𝑎+𝑁_𝑎−1𝑅_𝑎𝛼+−𝑁_𝑏+𝑁_𝑏−1𝑅_𝑏𝛽+−𝑁_𝑐+𝑁_𝑐−1𝑅_𝑐𝛾+3,𝑞=−2−𝑁_𝑎+𝑁_𝑎−1𝑅_𝑎𝛼+−𝑁_𝑏+𝑁_𝑏−1𝑅_𝑏𝛽+−𝑁_𝑐+𝑁_𝑐−1𝑅_𝑐𝛾+−𝑁_𝑏𝑅_𝑎−𝑁_𝑎𝑅_𝑏+𝑁_𝑎+𝑁_𝑏−1𝑅_𝑎𝑅_𝑏+𝛼𝛽−𝑁_𝑐𝑅_𝑎−𝑁_𝑎𝑅_𝑐+𝑁_𝑎+𝑁_𝑐−1𝑅_𝑎𝑅_𝑐+𝛼𝛾−𝑁_𝑐𝑅_𝑏−𝑁_𝑏𝑅_𝑐+𝑁_𝑏+𝑁_𝑐−1𝑅_𝑏𝑅_𝑐𝛽𝛾−3,𝑟=−𝑁_𝑎+𝑁_𝑎−1𝑅_𝑎𝛼+−𝑁_𝑏+𝑁_𝑏−1𝑅_𝑏𝛽+−𝑁_𝑐+𝑁_𝑐−1𝑅_𝑐𝛾−−𝑁_𝑏𝑅_𝑎−𝑁_𝑎𝑅_𝑏+𝑁_𝑎+𝑁_𝑏−1𝑅_𝑎𝑅_𝑏−𝛼𝛽−𝑁_𝑐𝑅_𝑎−𝑁_𝑎𝑅_𝑐+𝑁_𝑎+𝑁_𝑐−1𝑅_𝑎𝑅_𝑐−𝛼𝛾−𝑁_𝑐𝑅_𝑏−𝑁_𝑏𝑅_𝑐+𝑁_𝑏+𝑁_𝑐−1𝑅_𝑏𝑅_𝑐+𝛽𝛾−𝑁_𝑐𝑅_𝑎𝑅_𝑏−𝑁_𝑏𝑅_𝑎𝑅_𝑐−𝑁_𝑎𝑅_𝑏𝑅_𝑐−𝑁_𝑎+𝑁_𝑏+𝑁_𝑐−1𝛼𝛽𝛾. (4.36) The Cournot equilibrium is locally asymptotically stable if all eigenvalues are less than unity in absolute value. The most simplified form of the necessary and sufficient conditions for the cubic equation to have roots only inside the unit circle is 1−𝑝+−𝑞−𝑟>0,1+𝑝+−𝑞+𝑟>0,1+𝑞+𝑝𝑟−𝑟²>0 (4.37) as it has been proved in Farebrother [12] and Okuguchi and Irie [13]. It is easy to show that ()𝑁1−𝑝+−𝑞−𝑟=𝑁−1_𝑎+ℎ𝑁_𝑏+𝑘𝑁_𝑐³𝛼𝛽𝛾8ℎ𝑘(𝑁−1)³>0, (4.38) which implies that unity is not a root of the cubic equation.

It seems tedious to examine the remaining two conditions in general, instead we numerically confirm the stability region in the special case with 𝑁=6 and 𝑤₁=𝑤₂=3. The qualitatively same results can be obtained for any 𝑁(>4), 𝑤₁ and 𝑤₂.In this case we have three groups with two firms in each of them. The region for positive Cournot equilibrium is surrounded by the 𝑘=𝑓_𝑎(ℎ) and 𝑘=𝑓_𝑏(ℎ) loci. Substituting 𝑁_𝑎=𝑁_𝑏=𝑁_𝑐=2 into these functions determines the nonnegativity region 𝑁₆=)∣2(ℎ,𝑘32ℎ+33>𝑘>2ℎ−1,ℎ>1,𝑘>1. (4.39)

The positivity and stability regions under naive expectation (with 𝛼=1) and those under adaptive adjustment (with 𝛼=0.7) are given in Figures 3(a) and 3(b). In each figure, the outer curve is the Neimark-Sacker boundary, the inner curve is the flip boundary and the two straight lines are the 𝑘=𝑓_𝑏(ℎ) and the 𝑘=𝑓_𝑐(ℎ) loci where the former is steeper than the latter. The light gray area bounded by the two straight lines is the positivity region and the dark gray area illustrates the stability region. Their intersection is the feasible and stability region. It is seen in Figure 3(a) that the positivity region is completely outside the stability region when the expectations are naively formed. This implies that the positive Cournot equilibrium is always unstable under naive expectation. This result reminds us the classical result of Theocharis [14] that the stability of a nondifferentiated Cournot equilibrium can be confirmed only in the duopoly framework if the expectations are naively formed and the price and cost functions are linear. It is, in turn, seen in Figure 3(b) that about half of the positivity region is included in the stability region for adaptive adjustment, which implies that an unstable Cournot equilibrium under naive expectations could be stabilized by the adaptive adjustment system.

(a) 𝛼=1

(b) 𝛼=0.7

Figure 3 

Stability and feasible regions of the naive and adaptive systems.

Figure 4 is an enlargement of the south-west corner of Figure 3(b). It shows how the flip boundary shifts if the value of α changes. In the current example, solving 1+𝑝+(−𝑞)+𝑟=0 for α yields 0.8, given 𝑘=ℎ=2. This implies that the flip boundary with 𝛼=0.8 passes through the point (2,2), the vertex of the triangular part of the stable region. Thus if 𝛼≥0.8, then the positivity region is completely outside the stability region for adaptive adjustment, so no positive equilibrium becomes stable. On the other hand, solving 1+𝑝+(−𝑞)+𝑟=0 for α yields 2/3, given 𝑘=ℎ=1. This implies that the flip boundary with 𝛼=2/3 passes through the point (1,1). If 𝛼≤2/3, then the positivity region is entirely inside the stability region, in which case all nonnegative equilibria become stable. If 2/3<𝛼<0.8, then only a certain part of the nonnegativity region belongs to the stability region, and this part becomes larger if the value of α decreases. In particular, for 𝛼=0.7, the stability region is horizontally-striped and is the triangle with a base of the flip boundary with 𝛼=0.7, the most outer circle-wise curve. In the remaining light-gray area of the positivity region, the Cournot output is locally unstable. When the adaptive parameter α decreases, the flip boundary shifts inside accordingly. As a consequence, the stability region enlarges and the unstable region shrinks. Our third result on Cournot dynamics is numerically confirmed and then summarized as follows.

Figure 4 

Dependency of the stability region on α.

Theorem 4.4. In the special case of the three-group model where𝛼=𝛽=𝛾,𝑁=6 and 𝑤₁=𝑤₂=2, the Cournot equilibrium is locally unstable if 𝛼≥4/5, locally asymptotically stable if 𝛼≤2/3 while the feasible region is divided into the stability and instability regions by the flip boundary if 4/5>𝛼>2/3.

As the comparison of the three-group and the three-firm (triopoly) models, we, again, perform numerical simulations for these two models. The dynamic system in the three-group model becomes 𝑥(𝑡+1)=(1−𝛼)𝑥(𝑡)+𝛼𝑥(𝑡)+2𝑦(𝑡)+2𝑧(𝑡)𝑎−𝑥(𝑡)+2𝑦(𝑡)+2𝑧(𝑡),𝑦(𝑡+1)=1−𝛽𝑦(𝑡)+𝛽2𝑥(𝑡)+𝑦(𝑡)+2𝑧(𝑡)𝑏−2𝑥(𝑡)+𝑦(𝑡)+2𝑧(𝑡),𝑧(𝑡+1)=1−𝛾𝑧(𝑡)+𝛾2𝑥(𝑡)+2𝑦(𝑡)+𝑧(𝑡)𝑐−2𝑥(𝑡)+2𝑦(𝑡)+𝑧(𝑡), (4.40) where for simplicity, we assume that 𝛼=𝛽=𝛾, ℎ=𝑘, and 𝑎=1 implying 𝑏=𝑐. The dynamic system in the three-firm model (i.e., triopoly) can be constructed similarly. Selecting α and ℎ (in particular 𝑏) as the bifurcation parameters, Figure 5 illustrates the bifurcation diagrams in the (𝛼,ℎ) plane. The numerical investigations clearly reveal qualitatively the same issues as the ones we saw in Figure 2. Namely, first of all, the destabilizing process goes to chaos through a flip bifurcation with a lower production ratio in the three-group model and a Neimark-Sacker bifurcation with a higher ratio in the three-firm model. Second, the adjustment speed can be used to control unstable trajectories in the two- and three-group models. Comparing the bifurcation diagram of the two-group model and that of the three-group model shows the similarity of the destabilizing process in which period-doubling bifurcation takes place.

Figure 5 

Bifurcation diagrams in the (𝛼,ℎ) plane for the three-group model (a) and for the three-firm model (b).

In Figure 6, we present the bifurcation diagram in the (ℎ,𝑘) plane to draw attention to the feasibility of the solutions of the three-group model. The value 𝛼=0.8 is fixed. As already explored in Figure 4, ℎ and 𝑘 range from 1 to 2 and the stationary state is unstable for any combination of ℎ and 𝑘 from this region. Color has the same meanings as before. The feasible region of the three-group model is defined by two upward sloping curves, 𝑘=𝑓_𝑏(ℎ) and 𝑘=𝑓_𝑐(ℎ). Notice that although the area outside the feasible region is colored in the same way as the feasible region, the stationary point defined in that area becomes negative and thus economically meaningless. Bifurcation makes sense only in the feasible region from an economic point of view. Notice further that a trajectory that oscillates around the stationary point periodically or aperiodically may take negative values and thus become economically meaningless. One way is to confine the parameter choice in such a way that the resultant dynamics does not become infeasible. The other way is to reconstruct the dynamic system by taking into account the nonnegativity constraint explicitly. However, the former has the difficulty of deriving the confinement conditions in the currently considered model as many parameters are involved, and the latter makes the asymptotic behavior of the dynamic system significantly different and more difficult to analyze. Since our main concern is to control the unstable trajectories and our main conclusion is that the adjustment speed is an effective control parameter, which is supposed to hold in those models, we used the model without such modifications at the expense of some economic precision.

Figure 6 

2-parameter bifurcation diagram.

5. Conclusion

Discrete dynamic systems always generate time series. The asymptotic behavior of them is a fundamental research issue. In this paper a special class of economic models was examined. For the sake of mathematical simplicity we selected 𝑁-firm Cournot oligopolies without product differentiation, and with isoelastic price function. The reaction functions and the equilibrium were determined first, and then the asymptotic behavior of the equilibrium was illustrated in two special cases with two or three groups of identical firms. Stability conditions could be derived analytically in the first case, and the dependence of the asymptotic properties of the equilibrium on the number of firms was illustrated by computer simulation in the second case. The results of the nonlinear duopoly and triopoly models show that the Cournot equilibrium can be destabilized through a Neimark-Sacker bifurcation. We also found the following new dynamic phenomenon. In the multigroup models, the stationary state is destabilized through the Feigenbaum period doubling sequence, furthermore a Neimark-Sacker bifurcation can occur only in the infeasible regions in which the stationary state is negative. For 𝑁>4, the multigroup models are unstable if α is close to one and become stable if α is below a certain threshold, regardless of the production cost ratios. This implies that the main source of instability is the speeds of adjustment and thus the stationary state could be stabilized by selecting sufficiently small speeds of adjustment. That is, the multigroup models with 𝑁 firms are unstable under naive expectations but are controllable under adaptive adjustment process in which the speed of adjustment is the control parameter.

Appendix

In this appendix, we derive the stability conditions of the adaptive system in which the expectation is adaptively formulated, 𝑦^𝑒_𝑖(𝑡+1)=1−𝛼_𝑖𝑦^𝑒_𝑖(𝑡)+𝛼_{𝑖𝑗≠𝑖}𝑥_𝑗(𝑡). (A.1) Here 𝛼_𝑖∈(0,1] is the speed of adjustment of firm 𝑖 in assuming the expected aggregate output of the other firms. The stability conditions are also useful to determine the local dynamic behavior of the naive system as well as that of the inertia system, that is, the system where the firms with naive expectations adaptively adjust toward the best reply solution.

We consider the adjustment process with adaptive expectations first, since the one with naive expectations can be obtained by selecting the speeds of adjustment equal to unity (i.e., 𝛼_𝑖=1, 𝑖=1,2,…,𝑁). For 𝑖=1,2,…,𝑁, 𝑥_𝑖(𝑡+1)=𝛼_𝑖∑_𝑗≠𝑖𝑥_𝑗(𝑡)+1−𝛼_𝑖𝑦^𝑒_𝑖(𝑡)𝑐_𝑖−𝛼_{𝑖𝑗≠𝑖}𝑥_𝑗(𝑡)+1−𝛼_𝑖𝑦^𝑒_𝑖(𝑡),𝑦^𝑒_𝑖(𝑡+1)=𝛼_{𝑖𝑗≠𝑖}𝑥_𝑗(𝑡)+1−𝛼_𝑖𝑦^𝑒_𝑖(𝑡). (A.2) The Jacobian at the equilibrium has the form𝐉=0𝛾₁𝛼₁⋅𝛾₁𝛼₁𝛾₁1−𝛼₁𝛾0⋅0₂𝛼₂0⋅𝛾₂𝛼₂0𝛾₂1−𝛼₂𝛾⋅0⋅⋅⋅⋅⋅⋅⋅⋅_𝑁𝛼_𝑁⋅𝛾_𝑁𝛼_𝑁00⋅0𝛾_𝑁1−𝛼_𝑁0𝛼₁⋅𝛼₁1−𝛼₁𝛼0⋅0₂0⋅𝛼₂01−𝛼₂𝛼⋅0⋅⋅⋅⋅⋅⋅⋅⋅_𝑁⋅𝛼_𝑁00⋅01−𝛼_𝑁, (A.3) where 𝛾_𝑖≡𝜕𝑓_𝑖𝑦^𝑒_𝑖𝜕𝑦^𝑒_𝑖=𝐶2𝑐_𝑖(𝑁−1)−1. (A.4) Notice that condition (2.11) implies that 𝛾_𝑖>−1/2 for all 𝑖.

The eigenvalue equation has the form 𝐉𝐱=𝜆𝐱with𝐱=𝑢₁,…,𝑢_𝑁,𝑣₁,…,𝑣_𝑁^𝑇, (A.5) or equivalently,𝛾_𝑖𝛼_{𝑖𝑗≠𝑖}𝑢_𝑗+𝛾_𝑖1−𝛼_𝑖𝑣_𝑖=𝜆𝑢_𝑖𝛼,1≤𝑖≤𝑁,_{𝑖𝑗≠𝑖}𝑢_𝑗+1−𝛼_𝑖𝑣_𝑖=𝜆𝑣_𝑖,1≤𝑖≤𝑁. (A.6) Subtracting the 𝛾_𝑖-multiple of the second equation from the first one gives 𝜆𝑢_𝑖−𝛾_𝑖𝑣_𝑖=0. (A.7) The value 𝜆=0 cannot destroy stability, so we may assume 𝜆≠0. Then 𝑢_𝑖=𝛾_𝑖𝑣_𝑖, and by substituting it into the second equation, we have 𝛼_{𝑖𝑗≠𝑖}𝛾_𝑗𝑣_𝑗+1−𝛼_𝑖𝑣_𝑖=𝜆𝑣_𝑖,1≤𝑖≤𝑁. (A.8) This is the usual eigenvalue problem of the 𝑁×𝑁 matrix 𝐇=1−𝛼₁𝛾₂𝛼₁⋅𝛾_𝑁𝛼₁𝛾₁𝛼₂1−𝛼₂⋅𝛾_𝑁𝛼₂𝛾⋅⋅⋅⋅₁𝛼_𝑁𝛾₂𝛼_𝑁⋅1−𝛼_𝑁. (A.9) Notice that 𝐇=𝐃+𝐚𝐛^𝑇, (A.10) with 𝐚^𝑇=𝛼₁,𝛼₂,…,𝛼_𝑁,𝐛^𝑇=𝛾₁,𝛾₂,…,𝛾_𝑁,𝐃=diag1−𝛼₁1+𝛾₁,1−𝛼₂1+𝛾₂,…,1−𝛼_𝑁1+𝛾_𝑁. (A.11) The characteristic polynomial of 𝐇 can be decomposed by using the simple fact that if 𝐱,𝐲∈𝑅^𝑁, then det𝐈+𝐱𝐲^𝑇=1+𝐲^𝑇𝐱, (A.12) where 𝐈 is the 𝑁×𝑁 identity matrix. So we have det(𝐇−𝜆𝐈)=det𝐃+𝐚𝐛^𝑇()()−𝜆𝐈=det𝐃−𝜆𝐈det𝐈+𝐃−𝝀𝐈⁻¹𝐚𝐛^𝑇. (A.13) The roots of the first factor are 1−𝛼_𝑖(1+𝛾_𝑖) which are inside the unit circle if and only if −1<1−𝛼_𝑖1+𝛾_𝑖<1, (A.14) which occurs if and only if𝑐_𝑖>𝛼_𝑖𝐶4().𝑁−1 (A.15) The other eigenvalues are the roots of 1+^𝑁_𝑖=1𝛼_𝑖𝛾_𝑖1−𝛼_𝑖1+𝛾_𝑖−𝜆=0. (A.16) The Cournot equilibrium is locally asymptotically stable if all eigenvalues are less than unity in absolute value and is unstable if at least one eigenvalue is outside the unit circle. However, since parameters 𝛼₁,𝛼₂,…,𝑎_𝑁 can be selected arbitrarily in interval (0,1], there is a large flexibility in the location of the eigenvalues.

Acknowledgments

The authors are grateful for four anonymous referees and the editor, Weihon Huang, for their useful comments and critiques, which substantially improve the paper. They highly appreciate financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C), 21530172), Chuo University (Joint Research Grant 0981) and the Japan Economic Research Foundation. They also want to acknowledge the encouragement and support by Kei Matsumoto for the research leading to this paper. Part of this paper was done when the first author visited the Department of Systems and Industrial Engineering of the University of Arizona and the second author visited the Department of Economics of Chuo University. The usual disclaimer applies.