Stability of the Cournot-Nash Equilibrium in Standard Oligopoly

2009

The theory of oligopolies is a particularly active area of research using applied mathematics to answer questions that arise in microeconomics. It basically studies the occurrence of equilibria and their stability in market models involving few firms and has a history that goes back to the work of Cournot in the 19th century. More recently, interest in this approach has been revived, owing to important advances in analogous studies of Nash equilibria in game theory. In this paper, we first attempt to highlight the basic ingredients of this theory for a concrete model involving two firms. Then, after reviewing earlier work on this model, we describe our modifications and improvements, presenting results that demonstrate the robustness of the approach of nonlinear dynamics in studying equilibria and their stability properties. On the other hand, plotting the profit functions resulting from our modified model we show that its behavior is more realistic than that of other models reporte...

Dynamic Games and Applications

We consider a model of evolutionary competition between adjustment processes in the Cournot oligopoly model and investigate the effect of increasing the number of firms. Our focus is on Nash play versus a general short-memory adaptive adjustment process. We find that, although Nash play has a stabilizing influence, a sufficient increase in the number of firms in the market tends to make the Cournot-Nash equilibrium unstable. This shows that the famous result by Theocharis (Rev Econ Stud 1960), that Cournot oligopoly markets are unstable for more than three firms, is robust, although the instability threshold increases in the presence of Nash firms. We establish that both the existence and the level of this threshold depend on the information costs associated with Nash play. Moreover, the interaction between adjustment processes naturally leads to the emergence of complicated endogenous fluctuations as the number of firms increases, even when demand and costs are linear. Keywords Stability of Cournot-Nash equilibrium • n-Player Cournot games • Evolutionary competition • Endogenous fluctuations JEL Classification C72 • C73 • D43 An earlier version of this paper circulated under the title "On the stability of the Cournot equilibrium: An evolutionary approach".

2011

We construct an evolutionary version of Theocharis (1960)'s seminal work on the stability of equilibrium in multi-player quantity-setting oligopolies. Two sets of behavioral heuristics are investigated under …xed and endogenously evolving fractions: (myopic) Cournot …rms vs. Nash …rms and Cournot …rms vs. rational …rms. The analysis with evolutionary competition between these heuristics nests the famous Theocharis instability threshold, n = 3, as a special case and shows that Theocharis'result is quite robust.

2011

In a simple agent-based model of a small oligopoly nonrenewable natural resource model, the agents, communicating solely through the market price, sometimes exhibit collusion-like behavior, sometimes Cournot-like behavior. The collusion-like behavior is shown to arise when dierences between the agents are small. Conversely, the Cournot-like behavior is shown to result from dierences in production decisions based on dierences in the agents. Close examination of the Cournot-like behavior indicates that the outcome results from a misinterpretation of market price response. This motivates investigation into how additional qualitative information about the market leads to quantitative improvements in the estimated marginal price.

SSRN Electronic Journal, 2000

Studies in Nonlinear Dynamics & Econometrics, 2000

Within a classical discrete-time Cournot oligopoly model with linear demand and quadratic cost functions, minimum and maximum production constraints are imposed in order to explore their effects on the dynamic of the system. Due to the presence of such constraints, the dynamic model assumes the form of a continuous piecewise linear map of the plane. The study of Nash equilibria of the oligopoly game, together with an analytical and numerical investigation of the different kinds of attractors of the dynamical system, shows how the presence of production constraints generates so called border collision bifurcations, a kind of global bifurcations recently introduced in the literature on non-smooth dynamical systems, which gives rise to a quite rich spectrum of dynamic scenarios, characterized by drastic changes in the qualitative dynamic properties of the system. * Acknowledgments. We thank Carl Chiarella, Michael Kopel and Ferenc Szidarovszky for their illuminating discussions on oligopoly models, Laura Gardini and Iryna Sushko for helpful comments and suggestions about border collision bifurcations.

Nonlinear Dynamics, 2015

In this paper we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark-Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark-Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium.

Communications in Nonlinear Science and Numerical Simulation, 2015

We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a "Local Monopolistic Approximation" (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditions on reactivity and marginal costs under which the solution converges to the Cournot-Nash equilibrium. Moreover, we compare the stability regions of the proposed oligopoly to a similar one, in which the LMA firm is replaced by a best response firm, which is more rational than the LMA firm. We show that, depending on costs ratio, the equilibrium can lose its stability in two different ways, through both a flip and a Neimark-Sacker bifurcation. We show that the nonlinear, noninvertible map describing the model can give rise to several coexisting stable attractors (multistability). We analytically investigate the shape of the basins of attractions, in particular proving the existence of regions known in the literature as lobes.

Applied Mathematics and Computation, 2008

In this paper, we analyze the equilibrium effects of a monopolist with bounded rationality. Assuming that the entire (monotonic) demand function is unknown, she employs a rule of thumb to produce a quantity that guarantees the largest profits. The steady state of the map is equal to the level of price that maximizes profits, as can be seen in the classical microeconomic theories. However, complex dynamics can arise, especially when the reaction coefficient to variation in profits is high.

Journal of Computational and Applied Mathematics, 2012

Guirao and Rubio [6] introduces an economic model, which generalizes the classical duopoly of Cournot type, where the competitors are located around a circle or a line and each rm competes à la Cournot with its right and left neighboring. For the case of having three and four players we describe completely the bifurcations of equilibria in terms of the production costs of each rm and we study the stability of them. Moreover, for the case of four players we provide some information on twoperiodic orbits.