On the stability of the stackelberg oligopoly equilibrium

The stability of the equilibrium in the Stackelberg undifferentiated oligopoly model where each oligopolist strives for leadership is analyzed for general demand and cost functions and for a discrete system with non-instantaneous adjustment.     

Coalition-proof supply function equilibria in oligopoly

In an industry where firms compete via supply functions, the set of equilibrium outcomes is large. If decreasing supply functions are ruled out, this set is reduced significantly, but remains large. Specifically, the set of prices that can be sustained by supply function equilibria is the interval between the competitive price and the Cournot price. In sharp contrast, when the number of firms is above a threshold we identify (e.g., three if demand is linear), only the Cournot outcome can be sustained by a coalition-proof supply function equilibrium.     

Price formation in heterogeneous oligopoly

Conclusion An equilibrium price vector has been defined and its existence and function in static price formation in cases of heterogeneous oligopoly has been established. Hitherto, authors have been misled, we think, by misinterpreting the role of reaction functions, which are central to our argument. After defining and proving the existence and uniqueness of an equilibrium price vector, it is possible to construct a generalization of the wellknown equilibrium concept of the Nash-Cournot game solution. It has previously been thought that the restriction to the Cournot solution is one of the main drawbacks to using game theoretical solutions to the problem of oligopoly. We have developed a generalization of the Nash solution in order to deal with a type of reaction hypothesis which cannot be expressed in terms of the Cournot reaction function. The new set of reaction functions here introduced demonstrate conclusively that the so-called identity on a Nash and a Cournot equilibrium is only of a formal character.We would like to thank F. Delbaen, R. A. Hirschfeld and K. Okuguchi for useful comments on an earlier draft of this paper.     

Long run equilibria in an asymmetric oligopoly

Summary. Consider an oligopolistic industry composed of two groups (or populations) of firms, the low cost firms and the high cost firms. The firms produce a homogeneous good. I study the finite population evolutionarily stable strategy defined by Schaffer (1988), and the long run equilibrium in the stochastic evolutionary dynamics based on imitation and experimentation of strategies by firms in each group. I will show the following results. 1) The finite population evolutionarily stable strategy (ESS) output is equal to the competitive (or Walrasian) output in each group of the firms. 2) Under the assumption that the marginal cost is increasing, the ESS state is the long run equilibrium in the stochastic evolutionary dynamics in the limit as the output grid step, which will be defined in the paper, approaches to zero. Received: September 19, 1997; revised: June 18, 1998     

Limited knowledge of demand and oligopoly equilibria

In a standard model of oligopoly with differentiated products, the existence of an equilibrium at which the first-order conditions for profit maximisation are simultaneously satisfied for all firms is proved and this is done without imposing any restrictions on the demand functions. This is an equilibrium in the following sense: although some firms may not necessarily be maximising their profits, nevertheless if each firm's knowledge of demand is limited to the linear approximation of its own demand curve, then it will believe that it is indeed maximising its profits.     

On the multiplicity of perfect equilibria in a dynamic oligopoly

In this paper we study the subgame perfect equilibrium solution of a dynamic oligopoly model. We show how there exists a family of perfect equilibria, since the perfectness concept does not encompass regularity conditions on the feedback rules.     

Existence and optimality of oligopoly equilibria in linear exchange economies

Summary. We consider a linear exchange economy and its successive replicas. We study the notion of Cournot-Walras equilibrium in which the consumers use the quantities of commodities put on the market as strategic variables. We prove that, generically, if the number of replications is large enough but finite, the competitive behaviour is an oligopoly equilibrium. Then, under a mild condition, which may be interpreted in terms of market regulation and/or market activity, we show that any sequence of oligopoly equilibria of successive replica economies converges to the Walrasian outcome and furthermore that every oligopoly equilibrium of large, but finite, replica is Pareto optimal. Consequently, under the same assumptions on the fundamentals of the economy, one has an asymptotic result on the convergence of oligopoly equilibria to the Walras equilibrium together with a generic existence result for the Cournot-Walras. Received: June 20, 2002; revised version: November 20, 2002 RID="*" ID="*" Part of this paper was written while the second author was visiting the Universidad de Vigo. The support of the department of mathematics is gratefully acknowledged. Correspondence to: J.M. Bonnisseau     

The cournot oligopoly and competitive equilibria as solutions to non-linear complementarity problems

The Cournot oligopoly and competitive equilibria are identified with solutions to non-linear complementarity problems, which leads to integration of the existence and stability proofs of these equilibria.     

The stability of the cournot oligopoly model: A reassessment

The stability properties of the Cournot oligopoly model are examined for the continuous adjustment process. Sufficient conditions for instability are established which are shown to be of significance when the number of firms, n, is small. Local and global stability are shown under appropriate conditions which for global stability include the restriction n ? 5. The proof of global stability corrects a well-known proof by F. Hahn (Rev. Econ. Stud.29 (1962), 329–331). Results for the discrete time adjustment process are reported and the restrictiveness of the model's assumptions are discussed.     

On the rate of convergence of oligopoly equilibria in large markets: An example

In this note we explore the rate of convergence of Cournot-Nash equilibria with free entry and those with an entry deterring monopolist in a replicated market where the market demand functions is linear and firms face linear cost curves.     

Quantal response equilibria with heterogeneous agents

This paper introduces a model of quantal response equilibrium with heterogeneous agents and demonstrates the existence of a representative agent for such populations. Except in rare cases, the representative agent?s noise terms cannot be independently and identically distributed across the set of actions, even if that is assumed for the individual agents. This result demonstrates a fundamental difference between a representative agent and truly heterogeneous quantal responders and suggests that when fitting quantal response specifications to aggregate data from a population of subjects, the noise terms should be allowed to be jointly dependent across actions. Even though this introduces additional degrees of freedom, it makes the model well specified. The representative agent inherits a regular quantal response function from the actual agents, so this model does impose falsifiable restrictions.     

Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist

Summary. In infinite horizon economies only local equivalence of beliefs is needed to ensure the existence of an Arrow–Debreu equilibrium. In fact, agents can even disagree completely in the long run in the sense that asymptotically, their beliefs are singular. Received: November 3, 2000; revised version: February 13, 2002     

On the stability of non-Walrasian equilibria

We present a dynamic disequilibrium macro model which permits non-Walrasian or Keynesian equilibria as stationary states. Unlike the stability results obtained by Varian (1977), and Eckalbar (1979,1980,1981) in a series of papers in the same model, we show that such equilibria have the saddle point property when the real wage rate, the real money balances and expected output demand, all change in disequilibrium.     

Existence and stability of overconsumption equilibria

Growth models with endogenous mortality assume generally that life expectancy is increasing with output per capita and, thus, with individual consumption, whatever its level is. However, empirical evidence supports a U-shaped relationship between consumption and mortality, implying that the monotonicity of that relation is local but not global. This paper develops a two-period OLG model where life expectancy is a non-monotone function of consumption, and where agents form myopic anticipations about life expectancy. The existence, uniqueness and stability of steady-state equilibria are studied. It is shown that overconsumption equilibria — i.e. equilibria at which consumption exceeds the level maximizing life expectancy — exist in highly productive economies with a low impatience. We identify also conditions under which there exist long-run cycles in output and longevity around overconsumption equilibria.     

On the stability of nonsunspot equilibria

This paper analyzes the stability of the (Pareto efficient) nonsunspot equilibria as a function of the constraints faced by consumers in their ability to transfer wealth between states of nature. Equilibria that are stable for the certainty economy define nonsunspot equilibria that are stable in the following two polar cases: 1) All consumers are unconstrained; 2) All consumers are fully constrained. Furthermore, the stable certainty equilibria with small trade vectors define nonsunspot equilibria that are also stable independently of the constraint levels. Instability can develop only at intermediate constraint levels and for nonsunspot equilibria that feature sufficiently large trade vectors.     

The quadratic oil extraction oligopoly

Each extractor has a distinct initial endowment of oil and a distinct quadratic extraction cost and faces a linear industry demand schedule. We observe in a discrete-time model with a finite number of periods that the open loop and closed loop solutions are the same if initial stocks are such that each competitor is extracting in every period in which her competitors are extracting.     

The merger paradox in a mixed oligopoly

This paper examines the set of surplus maximizing mergers in a model of mixed oligopoly. The presence of a welfare maximizing public firm reduces the set of mergers for which two private firms can profitably merge. When a public firm and private firm merge, the changes in welfare and profit depend on the resulting extent of private ownership in the newly merged firm. When the government sets that share to maximize post merger welfare as assumed in the privatization literature, the merger paradox will often remain and the merger will not take place. Yet, we show there always exists scope for mergers that increase profit and increase (if not maximize) welfare. Interestingly, these mergers often include complete privatization.     

Learning stability in economies with heterogeneous agents

An economy exhibits structural heterogeneity when the forecasts of different agents have different effects on the determination of aggregate variables. We study the important case of economies in which agents' behavior depends on forecasts of aggregate variables and show how different forms of heterogeneity in structure, forecasts, and adaptive learning rules affect the conditions for convergence of adaptive learning towards rational expectations equilibrium. Results are applied to an overlapping generations model and a New Keynesian model of monetary policy.