ENDOGENOUS TIMING AND STRATEGIC CHOICE: THE COURNOT‐BERTRAND MODEL

Cournot establishes a Nash equilibrium to a duopoly game under output competition; Bertrand finds a different Nash equilibrium under price competition. Both treat the strategic choice variable (output versus price) and the timing of play as exogenous. We investigate Cournot‐Bertrand models where one firm competes in output and the other competes in price in both static and dynamic settings. We also develop a general model where both the timing of play and the strategic choice variables are endogenous. Consistent with the conduct of Honda and Scion, we show that Cournot‐Bertrand behaviour can be a Nash equilibrium outcome.     

Self-enforcing collusion in large dynamic markets

The “folk theorem” in game theory implies that any outcome that is better for all players than some single period Nash outcome can be achieved through noncooperative equilibrium in repeated games with discounting. Whether the folk theorem holds for a repeated Cournot oligopoly as the number of firms, N, increases without bound, is investigated. It is shown that the folk theorem holds in the limit iff demand increases at the same rate as the number of firms and the Cournot price sequence is bounded strictly above by the supremum of marginal cost for large N.     

Bertrand vs. Cournot equilibrium with risk averse firms and cost uncertainty

Summary. In an oligopoly game with cost uncertainty and risk averse firms, we show that Bertrand and Cournot equilibrium have different convergence properties when the market is replicated. The Cournot equilibrium price converges to the competitive price. Under very typical and somewhat general conditions, the highest Bertrand equilibrium price converges to one higher than the competitive equilibrium. We also give examples to show how to compute the limit of the highest Bertrand equilibrium prices and illustrate the ideas of the proof. We explore conditions under which the supply curve is upward sloping, a useful condition for our results. Received: April 20, 2000; revised version: May 10, 2001     

Bilateral oligopoly and quantity competition

Bilateral oligopoly is a market game with two commodities, allowing strategic behavior on both sides of the market. When the number of buyers is large, bilateral oligopoly approximates a game of quantity competition played by sellers. We present examples which show that this is not typically a Cournot game. Rather, we introduce an alternative game of quantity competition (the market share game) and, appealing to results in the literature on contests, show that this yields the same equilibria as the many-buyer limit of bilateral oligopoly, under standard assumptions on costs and preferences. We also show that the market share and Cournot games have the same equilibria if and only if the price elasticity of the latter is one and investigate the differences in equilibria otherwise. These results lead to necessary and sufficient conditions for the Cournot game to be a good approximation to bilateral oligopoly with many buyers and to an ordering of total output when they are not satisfied.     

Learning to collude: An experiment in convergence and equilibrium selection in oligopoly

The paper considers a simple oligopoly model where firms know their own and the average pay-off in the industry. Firms choose decision rules for trading. The theory predicts that there are three types of Nash equilibria in this game (collusive, Cournot and Stackelberg). Our experiments test the selection process. We find that there is clear evidence of convergence to an equilibrium, and whilst both Cournot and collusive outcomes were selected, the collusive equilibrium is more common. The experimental results also give insights into the process of individual learning, confirming that subjects follow aspiration rules rather than reinforcement rules.     

Inferior factor in Cournot oligopoly

The condition for factor inferiority has been well known for a competitive firm facing perfect competition in both product and factor markets. Whether the factor inferiority is observable or not has been unknown for oligopoly facing perfect competition in factor market. We derive a general condition for the factor inferiority in Cournot oligopoly without product differentiation. We derive it on the basis of a novel technique for proving the existence of a unique Cournot equilibrium industry output.     

A Theory of Endogenous Coalition Structures

Consider an environment with widespread externalities, and suppose that binding agreements can be written. We study coalition formation in such a setting. Our analysis proceeds by defining on a partition function an extensive-form bargaining game. We establish the existence of a stationary subgame perfect equilibrium for such a game. Our main results are concerned with the characterization of equilibriumcoalition structures. We develop an algorithm that generates (under certain conditions) an equilibrium coalition structure. Our characterization results are especially sharp forsymmetricpartition functions. In particular, we provide a uniqueness theorem and apply our results to a Cournot oligopoly.Journal of Economic LiteratureClassification Numbers: C71, C72, C78, D62.     

On price taking behavior in a nonrenewable resource cartel–fringe game

We consider a nonrenewable resource game with one cartel and a set of fringe members. We show that (i) the outcomes of the closed-loop and the open-loop nonrenewable resource game with the fringe members as price takers (the cartel–fringe game à la Salant, 1976) coincide and (ii) when the number of fringe firms becomes arbitrarily large, the equilibrium outcome of the closed-loop Nash game does not coincide with the equilibrium outcome of the closed-loop cartel–fringe game. Thus, the outcome of the cartel–fringe open-loop equilibrium can be supported as an outcome of a subgame-perfect equilibrium. However the interpretation of the cartel–fringe model, where from the outset the fringe is assumed to be price taker, as a limit case of an asymmetric oligopoly with the agents playing Nash–Cournot, does not extend to the case where firms can use closed-loop strategies.     

An experiment on learning with limited information: nonconvergence, experimentation cascades, and the advantage of being slow

We present the results of an experiment on learning in a continuous-time low-information setting. For a dominance solvable version of a Cournot oligopoly with differentiated products, we find little evidence of convergence to the Nash equilibrium. In an asynchronous setting, characterized by players updating their strategies at different frequencies, play tends toward the Stackelberg outcome which favors the slower player. Convergence is significantly more robust for a “serial cost sharing” game, which satisfies a stronger solution concept of overwhelmed solvability. As the number of players grows, this improved convergence tends to diminish, seemingly driven by frequent and highly structured experimentation by players leading to a cascading effect in which experimentation by one player induces experimentation by others. These results have implications both for traditional oligopoly competition and for a wide variety of strategic situations arising on the Internet.     

N‐FIRM OLIGOPOLY WITH GENERAL ISO‐ELASTIC DEMAND

In this note oligopoly with iso‐elastic demand is analysed. Unlike previous studies we consider general iso‐elastic demand rather than the case of unit elasticity. An n‐firm Nash‐Cournot equilibrium for the case of heterogeneous constant marginal costs is derived. The main result is a closed‐form solution that shows the dependency of the equilibrium on the elasticity of demand and the share of industry costs. The result has applications to a wide range of areas in oligopoly theory by allowing comparisons across markets with different elasticities of demand.     

On the Effects of Entry in Cournot Markets

In the framework of symmetric Cournot oligopoly, this paper provides two minimal sets of assumptions on the demand and cost functions that imply respectively that, as the number of firms increases, the minimal and maximal equilibria lead to (i) decreasing industry price and increasing or decreasing per-firm output; and (ii) increasing industry price (and decreasing per firm output.) In both cases, per-firm profits are decreasing.
The analysis relies crucially on lattice-theoretic methods and yields general, unambiguous and easily interpretable conclusions of a global nature. As a byproduct of independent interest, new insight into the existence of Cournot equilibrium is developed.     

INTERNATIONAL TRADE AND COURNOT EQUILIBRIUM: EXISTENCE,UNIQUENESS AND COMPARATIVE STATICS *

This paper proves the existence and uniqueness of Cournot equilibrium in models of international trade under oligopoly. The existence of Cournot equilibrium is established without the usual assumption that profit functions are concave. Instead the proof uses a weaker‘aggregate concavity’condition. A simple proof is used to establish the uniqueness of the equilibrium. And, the paper considers the implications of the assumptions, used to prove the existence and uniqueness of the equilibrium, on the comparative static results.     

COLLUSIVE EQUILIBRIUM IN COURNOT OLIGOPOLIES WITH UNKNOWN COSTS*

The article studies collusive equilibria in an infinitely repeated Cournot oligopoly with unknown costs. I look at collusive pooled Perfect Bayesian equilibria and separating equilibria either with signaling or with communication. In the separating equilibria with signaling the firms play the strictly separating Bayesian Nash equilibrium in period 1 and produce the optimal incentive compatible collusive quantity vector from period 2 onwards. In the separating equilibrium with communication the optimal incentive compatible quantity vector is produced from period 1 onwards. Among these, the optimal separating equilibrium with communication gives the highest expected discounted joint profits.     

Contributions to Cournot oligopoly theory

This work contributes to a number of questions concerning oligopoly models. In particular, uniqueness of the Cournot equilibrium point is demonstrated under the assumption that either the unit price function is differentiable and the derivative is strictly negative or the cost functions are strictly convex. Also, under the assumption of either strictly decreasing unit price function or strictly convex cost functions, it is shown that (a) the total production level at equilibrium increases with entry of additional players, (b) that cooperation between some of the players necessarily entails profit for the others, and (c) cooperative grouping causes decrease in production levels.     

Price discrimination in Cournot–Nash oligopoly

The standard Cournot–Nash oligopoly model is extended to encompass price discrimination, where firms charge more than one price for the same product based on willingness to pay. In the linear, symmetric case, the average price is not dependent on the extent of price discrimination.     

Reexamination of Strategic Public Policies

This paper attempts to reinterpret the familiar approach to strategic public policies from the viewpoint of inefficiencies involved in oligopoly where firms engage in Cournot competition. To this end, we introduce tools called “quasi‐reaction functions” and “quasi‐supply curves”. These tools allow us to conduct analyses through use of the standard partial‐equilibrium diagram, i.e. the quantity‐price plane. We can find the relationship between prices and quantities directly and, hence, deal with inefficiencies easily and also suggest policies to correct such inefficiencies. Specifically, we reexamine public policies related to mixed‐oligopoly, excess entry, technology choices with free entry and exit, and foreign oligopoly.     

Mixed duopoly: Differential game approach

This study formulates a dynamic mixed oligopoly model, in which a state‐owned public firm competes against a private firm over multiple periods. We adopt a differential game formulated by Fershtman and Kamien [Econometrica 55 (1987), pp. 1151–1164] and investigate how the dynamic competition affects the optimal privatization policy. We characterize the open‐loop Nash equilibrium (OLNE) and Markov‐perfect Nash equilibrium (MPNE). We show that in the MPNE, an increase in the degree of privatization has a nonmonotonic effect on the price, increasing it in a wide range of parameter spaces, which is in sharp contrast to the result in the OLNE or static analyses. We also find that the optimal degree of privatization is higher in the MPNE than that in the OLNE and static equilibrium. These results suggest that intertemporal strategic behavior changes the optimal privatization policy.     

Cournot equilibrium without apology: Existence and the Cournot inverse demand function

In a multi-commodity framework with absence of wealth effects, we prove the existence of equilibrium for Cournot oligopoly, and that the concept is completely non-ambiguous. We also obtain a uniquely defined endogenous inverse demand function, depending only on the competitive sector.     

Instability in the implementation of Walrasian allocations

The existence of game forms which implement Walrasian allocations as Cournot (Nash) equilibrium outcomes is well known. However, if the equilibria are also required to be locally dynamically stable, at least for environments with unique Walrasian allocations, this paper shows that the requisite game forms do not exist. Our definition of a game form entails certain regularity conditions, and requires the Cournot equilibrium to be unique when the Walrasian equilibrium is unique. The main result is that for such a game form, there does not exist a continuous-time strategy adjustment process which ensures the local stability of Cournot equilibria throughout a certain class of environments having unique Walrasian equilibria. Each trader adjusts his strategy in response to his own characteristics and the observed current strategies of others; but the direction and magnitude of adjustments are not constrained by any behavioral assumptions. The definitions permit the inclusion of an artificial player, such as an auctioneer, so the well-known tatonnement instability emerges as a special case.     

Cournot Oligopoly and the Theory of Supermodular Games

We reconsider the Cournot oligopoly problem in light of the theory of supermodular games. Invoking the recent ordinal version of this theory proposed by Milgrom and Shannon, we generalize Novshek's existence result, derive the associated uniqueness result, give an extension of a classical existence result under symmetry, and provide conditions making a Cournot oligopoly into a log-supermodular game (with the natural order on the action sets). We also provide extensive and precise insight as to why decreasing best-responses are widely regarded as being “typical” for the Cournot model with production costs. Several illustrative examples are provided.Journal of Economic LiteratureClassification Numbers: C72, L10, L13.