Continuous-time dynamic games for the Cournot adjustment process for competing oligopolists

doi:10.1016/j.amc.2012.12.078

Applied Mathematics and Computation

Volume 219, Issue 12, 15 February 2013, Pages 6400-6409

https://doi.org/10.1016/j.amc.2012.12.078 Get rights and content

Abstract

Under the assumption of an iso-elastic demand function, we consider a continuous-time dynamic Cournot adjustment game for n oligopolists. We first show the existence and uniqueness of a positive Cournot equilibrium and, for any n, we show that positive response functions are bounded in time. For the n = 3 case, we analytically show that the positive equilibrium is locally asymptotically stable, while, for n > 3, we are able to give some asymptotic results. As an application, we then introduce a proportional tax or subsidy to the model, and discuss the influence of such policies on the positive equilibrium. Furthermore, we briefly discuss a generalization of the results to non-iso-elastic demand functions. Comments on possible areas of future work are given.

Introduction

The study of complex dynamics in duopolistic games can be traced back to Rand [16], who considered complex dynamics in duopoly models. Continuous oligopoly dynamics were then considered under a more general framework by Chiarella and Khomin [5], where it was demonstrated that for continuous oligopoly dynamics, the most complex dynamics are limit cycles. For an excellent review of the literature on complex oligopoly, see Rosser [18].

Puu [11] showed that both periodicity and chaotic dynamics can be found in a discrete-time two-player game modeling Cournot duopoly by considering an iso-elastic market demand curve with price scaling as the reciprocal of the total output of both firms. We should note that a similar model was considered by Furth [7], who took cost functions which were cubic functions of a firms’ own output (in contrast to the linear cost functions of Puu [11]). In both cases, the reaction functions are unimodal. Kopel [8] later considered unimodal reaction functions of a logistic form, and observed periodic and complex behaviors in the discrete-time dynamics. Recently, Matsumoto [9] has discussed the control of chaotic dynamics emergent in Puu’s discrete-time duopoly model via feedback control.

Later, Puu [12], [13] extended this model to a discrete-time game for three oligopolists. Later, variations on this theme (and that of the study of chaos for the standard two player duopoly model) have appeared in the literature; such studies include Agiza et al. [2], Agliari et al. [3], Elabbasy et al. [6], Puu and Panchuk [15], Richter and Stolk [17], Tramontana et al. [19], Wu et al. [20]. Recently, Puu [14] has discussed the fact that increasing the number of players can lead to destabilization of equilibrium values.

Taking the continuous-time limit of the discrete n-player Cournot adjustment game, we obtain an n-dimensional nonlinear dynamical system governing each firm’s best-response quantity outputs. We outline the discrete model and mathematical preliminaries in Section 2. Then, in Section 3, we formulate the continuum model, and discuss the solution to the equilibrium equations. We find that there exists a unique positive solution, which represents the Cournot equilibrium. The corresponding long-run profits may then be found for each player, in closed form. In Section 4, we show that the mathematical solutions to the resulting dynamical system (the best-response quantities governed by the Cournot adjustment game) are always bounded in time. From here, in Section 5 we restrict out attention to the three-player Cournot adjustment game, and study the stability of the Cournot equilibrium. We prove that whenever the Cournot equilibirum is positive, it must be locally asymptotically stable. However, our proof does not extend to higher dimensions, and hence it is possible that for four or more players, the equilibrium could become unstable. This indeed was the trend noticed the in literature on the discrete-time Game: adding players would, in some cases, destabilize the equilibrium. In Section 6, we make some comments on the stability properties of the general n-player game. As an application of the model, in Section 7 we introduce a proportional tax or subsidy to the model, and discuss the influence of such policies on the equilibrium. In Section 8, we make some comments on the extension of the present results to different demand curves. Finally, in Section 9, we make some concluding remarks and discuss some prospects for future work.

Section snippets

Mathematical preliminaries

Under the assumption of an iso-elastic demand function, price p is reciprocal to output quantity Q, i.e. pQ = 1. We shall assume that supply always equals demand, and that the output quantity is the sum of the output of n oligopolists (henceforth “players”), i.e. Q = q₁ + … + q_n. Furthermore, we assume that marginal costs $c_{k}$ are constant for each firm (we discuss relaxing this assumption in Section 8). Under such assumptions, the vector of profit functions for each of the n-players reads $v (q_{1}, \dots q_{n}) = (\frac{q_{1}}{Q} - c_{1})$

Continuous-time dynamic game

Taking the continuum limit of the discrete-time model, we arrive at ${\dot{q}}_{k} = \sqrt{\frac{Q - q_{k}}{c_{k}}} - Q, for all k = 1, 2, \dots, n,$ where ${\dot{q}}_{k}$ denotes the time derivative of the kth player’s output quantity. The right hand side of each equation is slightly different from that of the corresponding equation in the discrete model, as we replace q_k(t + 1) − q_k(t) with ${\dot{q}}_{k}$ in the continuous-time model. As we shall see, the additional −q_k term due to the continuous-time assumption leads to greater stability in the solutions. These

Boundedness of solutions

In the present section, we show that the response functions are bounded for arbitrarily large n. Only non-negative reaction functions are of interest to us here, for economic reasons. So, note that admissible response functions with non-negative initial data must remain non-negative. In order for a reaction function with positive initial condition to go negative, it must first pass through zero. So, at zero quantity, there are three options: the function becomes positive again, the function

Dynamics of the three-player game

In the present section, we discuss the properties of the three-player game. When n = 3, the phase-space dimension is still small enough so that we may analyze the stability of the system. On the other hand, the three-player game has the minimal number of players needed in order to have the situation in which one player exits the market while the remaining players have positive equilibriums. The three player system becomes ${\dot{q}}_{1} = \sqrt{\frac{q_{2} + q_{3}}{c_{1}}} - (q_{1} + q_{2} + q_{3}), {\dot{q}}_{2} = \sqrt{\frac{q_{1} + q_{3}}{c_{2}}} - (q_{1} + q_{2} + q_{3}), {\dot{q}}_{3} = \sqrt{\frac{q_{1} + q_{2}}{c_{3}}} - (q_{1} + q_{2} + q_{3}) .$

Dynamics in the case of more than three players

While a stability result along the lines of Theorem 1 of Section 5 would be nice in the case where $n > 3$ , note that for such a case the analogous characteristic polynomial to (5.4) become quite complicated, and the method of proof employed for the three-player case is no longer useful. Indeed, for more than three players, the Cournot equilibrium may lose stability. As pointed out in Puu [14] for the discrete-time Cournot adjustment game, increasing the number of competitors in the Cournot

Example of equilibrium behavior: Introduction of a proportional tax or subsidy

We now introduce a proportional tax or subsidy to the continuous-time Cournot adjustment model, and discuss the influence of such policies on the equilibrium. In the case where a proportional tax or subsidy is added to the problem, the continuous time model becomes ${\dot{q}}_{1} = \sqrt{\frac{q_{2} + q_{3}}{c_{1} + τ_{1}}} - (q_{1} + q_{2} + q_{3}), {\dot{q}}_{2} = \sqrt{\frac{q_{1} + q_{3}}{c_{2} + τ_{2}}} - (q_{1} + q_{2} + q_{3}), {\dot{q}}_{3} = \sqrt{\frac{q_{1} + q_{2}}{c_{3} + τ_{3}}} - (q_{1} + q_{2} + q_{3})$ where $τ_{i} > 0$ denotes a tax and $τ_{i} < 0$ denotes a subsidy. Clearly, if $τ_{i} > 0$ for all $i = 1, 2, 3$ , there is no qualitative change in the behavior of the model solutions

Comments on the extension of these results to other demand curves

Note that the demand curve selected in Section 2, corresponding to $pQ = 1$ , suffers from one problem: when output tends to zero, price must tend to infinity to compensate, which is unreasonable. For this reason, authors sometimes introduce the parameter $p_{\max} > 0$ and employ the modified demand relation $p (Q + 1 / p_{\max}) = 1$ . At $Q = 0$ , $p = p_{\max}$ which is actually the maximum price. In this way, the hyperbolic curve defined by $pQ = 1$ has been shifted to the right by $p_{\max} > 0$ so that the demand curve is non-singular for

Concluding remarks

We have been able to show that for costs satisfying the restrictions that $C > (n - 1) c_{k}$ , for all $k = 1, 2, \dots, n$ there exists a unique positive equilibrium for a continuous-time dynamic game governing the actions of n oligopolists. For the three-player game, the positive equilibrium is always stable. Unlike in the discrete time model of Puu [12], other non-equilibrium behaviors, such as Hopf bifurcations, do not occur when a positive equilibrium exists, when n = 3. For all n, the dynamics are bounded.

Acknowledgment

The authors thank an anonymous reviewer for comments which have greatly improved the clarity of the paper. R.A.V. was supported by NSF Grant # 1144246.

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View full text

Continuous-time dynamic games for the Cournot adjustment process for competing oligopolists

Abstract

Introduction

Section snippets

Mathematical preliminaries

Continuous-time dynamic game

Boundedness of solutions

Dynamics of the three-player game

Dynamics in the case of more than three players

Example of equilibrium behavior: Introduction of a proportional tax or subsidy

Comments on the extension of these results to other demand curves

Concluding remarks

Acknowledgment

Chaos, Solitons and Fractals

Chaos, Solitons and Fractals

Journal of Economic Behavior and Organization

Chaos, Solitons and Fractals

Computers and Mathematics with Applications

Journal of Economic Theory

Chaos, Solitons and Fractals

Chaos, Solitons and Fractals

Chaos, Solitons and Fractals

Journal of Economic Behavior and Organization