Cournot Equilibrium Research Articles

The Output Competition Game between FDI Port Enterprises and Incumbent Port Enterprises

With the opening to the outside world, port market entry and exit become relatively free in China. In this context, there will be a game of market entry and anti-entry between foreign direct investment (FDI) port enterprises and incumbent ones. Cournot and Dixit Model were both used to study the output competition game between above two kinds of enterprises. Cournot equilibrium shows that FDI port enterprises' output falls as the its own marginal cost increases but increases as the competitors' marginal cost increases. In Dixit Model, by researching the reaction function based on the production scale, marginal cost and scale cost, the selection conditions of FDI port enterprises' market entry strategy were studied. Results show that enterprises entry strategy selection is relative to the limitation and equilibrium output.

Price Increasing Timing of Competitive Perishable Products

AbstractDynamic pricing has been proven to be an effective tool to increase revenue in many industries. We discuss the optimal time point to increase price of perishable products under duopoly competition in revenue management. We propose a game-theoretic model to describe the price increasing timing problem of competitive perishable products in the same market. By solving this problem, we show the existence of Stackelberg equilibrium point and Cournot equilibrium point when choosing the optimal price switching time. To illustrate our results, we also present a numerical example. Our results are applicable for decision makers to determine the optimal time to stop discounting.

A Simple Cournotian Model of Production and Trade

We provide a two good model of oligopolistic production and trade with one good being commodity money. There is the usual demand function of the consumers for the produced good that producer-sellers face. Each seller is a budget constrained preference maximizer and derives utility (or satisfaction) from consuming bundles comprising commodity money and the produced good. We define a competitive equilibrium strategy profile and a Cournotian equilibrium and show that under our assumptions both exist. We further show that at a competitive equilibrium strategy profile, each seller maximizes profits given his own consumption of the produced good and the price of the produced good, the latter being determined by the inverse demand function. Similarly we show that at a Cournotian the sellers are at a Cournot equilibrium given their own consumption of the produced good. Assuming sufficient differentiability of the cost functions we show that at a competitive equilibrium each seller either sets price equal to marginal cost or exhausts his capacity of production; at a Cournotian equilibrium each seller either sets marginal revenue equal to marginal cost or exhausts his capacity of production. We also study the evolution of Cournotian strategies as the sellers and buyers are replicated. As the number of buyers and sellers go to infinity any sequence of interior symmetric Cournotian equilibrium strategies admits a convergent subsequence, which converges to an interior symmetric competitive equilibrium strategy. In a final section we discuss the Bertrand Edgeworth price setting game and show that a Bertrand Edgeworth equilibrium must be a derived from a competitive equilibrium price. Here we show that if at a symmetric competitive equilibrium, the sellers consume positive quantities of the produced good then the competitive equilibrium cannot be a Bertrand Edgeworth equilibrium. Thus, if at all symmetric competitive equilibria the sellers consume positive amounts of the produced good, then a Bertrand Edgeworth equilibrium simply does not exist.

The Analysis of Pricing Power of Preponderant Metal Mineral Resources under the Perspective of Intergenerational Equity and Social Preferences: An Analytical Framework Based on Cournot Equilibrium Model

This paper combines intergenerational equity equilibrium and social preferences equilibrium with Cournot equilibrium solving the technological problem of intergenerational equity and strategic value compensation confirmation, achieving the effective combination between sustainable development concept and value evaluation, thinking and expanding the theoretical framework for the lack of pricing power of mineral resources. The conclusion of the theoretical model and the numerical simulation shows that intergenerational equity equilibrium and social preferences equilibrium enhance international trade market power of preponderant metal mineral resources owing to the production of intergenerational equity compensation value and strategic value. However, the impact exerted on Cournot market power by social preferences is inconsistent: that is, changes of altruistic Cournot equilibrium and reciprocal inequity Cournot equilibrium are consistent, while inequity aversion Cournot equilibrium has the characteristic of loss aversion, namely, under the consideration of inequity aversion Cournot competition, Counot-Nash equilibrium transforms monotonically with sympathy and jealousy of inequity aversion.

Aggregation with Cournot Competition: the Le Chatelier Samuelson Principle

This paper studies the aggregate substitution and expansion effects triggered by changes in input prices in a context where firms supply a homogeneous commodity and compete in quantities a la Cournot. We derive a sufficient condition for the existence of a Cournot equilibrium and show that this condition also ensures that the Le Chatelier-Samuelson principle is satisfied in the aggregate at the Cournot equilibrium, although it may not be satisfied at the firm level.

Determination of Transmission Capacity for a Two-node Market

We examine a two-node competitive market, taking into account the losses under transmission and the costs of transmission line construction. We state a welfare maximization problem and show how to solve it. Moreover, we analyze how the flow of good between markets affects the benefit of transmission system, consumer surplus and producers’ profit. We also consider a two-node market under imperfect competition and study the Cournot equilibrium depending on the transmission capacity.

Strategy Equilibrium in Stackelberg Model with Transmission Congestion in Electricity Market

Nash Cournot Equilibrium (NCE) has been widely used in a competitive electricity market to analyze generation firms’ strategic production quantities. Congestion on a transmission network may lead to a mixed strategy NCE. Mixed strategy is complicated to understand, difficult to compute, and hard to implement in practical market. However, Stackelberg model based equilibrium does not have any mixed strategy, even under congestion in a transmission line. A guide to understanding mixed strategy equilibrium is given by analyzing a cycling phenomenon in the players’ best choices. This paper connects the concept of leader-follower in Stackelberg model with relations between generation firms on both sides of the congested line. From the viewpoint of social welfare, the surplus analysis is presented for comparison between the NCE and the Stackelberg equilibrium (SE).

Endogenous timing in a mixed duopoly

This paper applies the framework of endogenous timing in games to mixed quantity duopoly, wherein a private—domestic or foreign—firm competes with a public, welfare-maximizing firm. A central goal of the paper is to present a unified and general treatment of the basic question of what constitutes the appropriate solution concept—Cournot or Stackelberg—in such duopolies. We show that simultaneous play never emerges as a subgame-perfect equilibrium of the extended game, in sharp contrast to private duopoly games. We demonstrate that this result is due to the objective function of the public firm being increasing in the rival’s output (instead of decreasing for a private firm). We provide sufficient conditions for the emergence of public and/or private leadership equilibrium. In all cases, private profits and social welfare are higher than under the corresponding Cournot equilibrium. We make extensive use of the basic results from the theory of supermodular games in order to avoid common extraneous assumptions such as concavity, existence and uniqueness of the different equilibria, whenever possible. Some policy implications are drawn, in particular those relating to the merits of privatization.

Endougenous Timing in a Mixed Duopoly

This paper applies the framework of endogenous timing in games to mixed quantity duopoly, wherein a private – domestic or foreign – firm competes with a public, welfare maximizing firm. We show that simultaneous play never emerges as a subgame-perfect equilibrium of the extended game, in sharp contrast to private duopoly games. We provide sufficient conditions for the emergence of public and/or private leadership equilibrium. In all cases, private profits and social welfare are higher than under the corresponding Cournot equilibrium. From a methodological viewpoint we make extensive use of the basic results from the theory of supermodular games in order to avoid common extraneous assumptions such as concavity, existence and uniqueness of the different equilibria, whenever possible. Some policy implications are drawn, in particular those relating to the merits of privatization.

On the convergence to the Cournot equilibrium in a productive asset oligopoly

We derive a feedback equilibrium of a dynamic Cournot game where production requires exploitation of a renewable asset. As in the classical Cournot model, quantity-setting firms compete in the same market for a given homogeneous good. We show that, when the asset stock grows sufficiently fast, the unique globally asymptotically stable steady state of the dynamic Cournot game corresponds to the static Cournot solution. Initial differences between firms’ production rates due to asymmetric allocations of asset stocks tend to disappear over time. When instead the asset stock grows slowly, the system does not converge to any stationary point. We also show that, within the class of linear feedback equilibrium strategies, besides the couple of strategies that stabilizes the states for every possible initial conditions, there exists another couple which is more efficient, in that it leads to higher stationary equilibrium profits for both firms, closer to the collusive outcome. Finally, we show that, as the discount rate approaches zero, there exist multiple linear feedback equilibrium strategies that induce a price trajectory that converges asymptotically to a price which is above the static Cournot equilibrium price.

Intra-Industry Trade with Identical Products in Cournot and Stackelberg Models of Duopoly

This paper analyzes and compares Cournot and Stackelberg duopoly equilibria focusing on identical products under multilateral free trade. The author...

Cournot equilibrium model for electricity markets with interruptible load program and wind power

High wind power penetration presents many challenges to the flexibility and reliability of power system operation. In this environment, various demand response (DR) programs have received much attention. As an effective measure of DR programs, interruptible load (IL) programs have been widely used around the world. This paper addresses the concern of how the IL program impacts the equilibrium outcomes of electricity markets with wind power. First, a market demand model is developed to take consideration of the IL program. Next, a Cournot equilibrium model for electricity markets with an IL program and wind power is presented. The introduction of the IL program leads to a non-smooth equilibrium problem. To solve this equilibrium problem, a novel solution method is proposed. Finally, considering that wind power penetration will increase the risks faced by the conventional generators, the conditional value at risk is employed to measure the risk, so that the impact of the IL program on the generators’ risk can also be examined. Numerical examples are presented to verify the effectiveness of the method. It is shown that the IL program can lower market price and its volatility significantly. In addition, the IL program can help generators reduce their risks in the market, especially when the uncertainty in wind power output is relatively large.

COMPARING COURNOT DUOPOLY AND MONOPOLY WITH ASYMMETRIC COMPLEMENTARY GOODS

This paper concerns an industry with two complementary goods. It is demonstrated that if a good is active in Cournot equilibrium, then it is active in either one of the socially optimal solution and the monopoly solution; but, the converse is not true. When demands are linear, a good is active in the monopoly solution if and only if it is socially desirable. Therefore, the monopoly would be socially better than the Cournot duopoly in selecting products. We establish that each Cournot output falls below its corresponding monopoly output regardless of whether the solutions are interior or not. However, in the monopoly solution, a good should be cross-subsidized if and only if it is a supplemental good. This result has two implications. First, a small decrease in the output level of a supplemental good raises the welfare, implying that when there exists a supplement good, lower quantities are not necessarily worse in welfare terms. Second, if each good is a base good, then the Cournot equilibrium indeed is less welfare-efficient than the monopoly solution. In order to verify whether it is possible to have a welfare reversal, we consider the case where demands are linear. We demonstrate that prices are lower in the monopoly solution, and that lower prices are better in the welfare terms. These two results imply that welfare is higher in the monopoly solution. Again, these results hold regardless of whether the solutions are interior or not.

Profit loss in Cournot oligopolies

We compare the aggregate profit achieved at a Cournot equilibrium to the maximum possible, which would be obtained if the suppliers were to collude. We establish a lower bound on the profit of Cournot equilibria in terms of a scalar parameter that captures qualitative properties of the inverse demand function and the number of suppliers (or the maximum of the suppliers’ market shares). The lower bounds are tight when the inverse demand function is affine.

Can they beat the Cournot equilibrium? Learning with memory and convergence to equilibria in a Cournot oligopoly

This article analyses the ability of the learning firms in a Cournot oligopoly to discover market solutions more collusive that the Cournot equilibrium (CE). We start from the results of Vallee and Yildizoglu (J Econ Behav Organ 72:670---690, 2009) and of Alos-Ferrer (Int J Ind Organ 22:193---217, 2004), and qualify the role of random experimenting, social learning (imitation), and (updated) memory in helping firms to discover such solutions. Our new computational results show, in contradiction with Alos-Ferrer (2004), that memory and its continuous update can indeed allow firms to beat the CE, and benefit from significant periods with higher profits. We show that the results of the literature on evolutionary learning in oligopoly can analytically be characterized through the interaction of three forces, and indicate when these forces can yield more collusive outcomes. We confirm these results with complementary computational experiments that clearly show the role of long memory.

Spatial price discrimination in a symmetric barbell model: Bertrand vs. Cournot

AbstractThis paper investigates the theory of spatial discrimination for general demands and general transportation costs in a barbell model, where markets' locations are assumed to be at opposite endpoints of a line. Duopoly firms in the Bertrand model always differentiate maximally, whereas in the Cournot model the market demand structure is not so critical, but the functional form of the transportation cost plays a crucial role in determining equilibrium location. Non‐maximal distance appears in the Cournot equilibrium when a convex transportation cost is allowed, bringing about the findings that the equilibrium consumer surplus may be higher and the equilibrium profits may be lower under Cournot competition than under Bertrand competition.

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

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.

Consumer Rationing and Cournot Outcomes: Experimental Evidence

[Author Affiliation]Jason J Lepore, * Department of Economics, Orfalea Business College, California Polytechnic State University, San Luis Obispo, CA 93407, USA; E-mail: jlepore@calpoly.edu .Aric P Shafran, [dagger] Department of Economics, Orfalea Business College, California Polytechnic State University, San Luis Obispo, CA 93407, USA; E-mail: ashafran@calpoly.edu ; corresponding author.[Acknowledgment]The authors acknowledge the Orfalea College of Business for financial support. We thank Blake Allison for assistance running the experiments and the anonymous reviewers and conference participants at the North American Economic Science Association Meetings for helpful comments.1. IntroductionThis article reports the results of duopoly market experiments where firms first commit to capacities and then compete in prices. The theoretical literature pertaining to these duopoly models suggests that the way residual demand is rationed is fundamental to the character of equilibrium outcomes. The goal of this study is to test the impact of residual demand rationing on the behavior of subjects in an experimental environment.In a two-stage game, where firms first commit to capacities and then compete in prices, Kreps and Scheinkman (1983; hereafter K&S) prove that the unique Nash equilibrium outcome coincides with the Cournot equilibrium outcome with efficient rationing of residual demand. Efficient rationing means that the higher-priced firm is left with the worst possible residual demand curve. If demand is composed of many similar consumers that each demand multiple units and we construct aggregate demand by way of a representative consumer, then efficient rationing is imposed by the model. On the other hand, if demand is instead composed of a mass of consumers with unit demand and varied willingness to pay, then the efficient rationing rule imposes the restriction that the highest-valued consumers buy from the firm with the lower price. Davidson and Deneckere (1986) make a compelling argument that proportional rationing is a more appropriate assumption in the latter case. Proportional rationing means that a uniform distribution of possible consumers buy from the firm with the lower price. Davidson and Deneckere show that under proportional rationing (as well as any rationing rule other than the efficient rule), the Cournot outcome is not necessarily the unique equilibrium of the two-stage game. The Davidson and Deneckere result is problematic for the K&S model because it deflates the generality of their theoretical result.1Lepore (2009) further clarifies the relationship between proportional rationing and the Cournot outcome. In the K&S game with proportional rationing, the Cournot outcome is the unique Nash equilibrium only when capacity cost is greater than a threshold level. The purpose of our experiments is to assess if the play of experimental subjects in the two-stage game is sensitive to the rationing scheme in the way that theory predicts. To do this, we construct an experimental design with four treatments. The treatments vary by two controls: the demand-rationing scheme and the cost of capacity. We conduct two efficient rationing treatments, one with a high capacity cost and one with a low cost. For both costs, theory predicts that the outcome coincide with the Cournot model. We also conduct two treatments using proportional rationing, one with each of the capacity costs used in the efficient rationing treatments. With a high cost, theory predicts the outcome should coincide with the Cournot model, as in the case of efficient rationing. However, theory predicts that the Cournot outcome should not occur when the cost of capacity is low, and prices should not clear the market.Recently several authors have used laboratory experiments as a means to empirically test the predictions of the K&S model. …

Revealed Preference Tests of the Cournot Model

The aim of this paper is to develop revealed preference tests for Cournot equilibrium. The tests are akin to the widely used revealed preference tests for consumption, but have to take into account the presence of strategic interaction in a game-theoretic setting. The tests take the form of linear programs, the solutions to which also allow us to recover cost information on the firms. To check that these nonparametric tests are sufficiently discriminating to reject real data, we apply them to the market for crude oil. [PUBLICATION ABSTRACT]

Quantum Cournot equilibrium for the Hotelling–Smithies model of product choice

This paper demonstrates the quantization of a spatial Cournot duopoly model with product choice, a two stage game focusing on non-cooperation in locations and quantities. With quantization, the players can access a continuous set of strategies, using a continuous variable quantum mechanical approach. The presence of quantum entanglement in the initial state identifies a quantity equilibrium for each location pair choice with any transport cost. Also higher profit is obtained by the firms at Nash equilibrium. Adoption of quantum strategies rewards us by the existence of a larger quantum strategic space at equilibrium.