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     

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     

Effect of credible quality investment with Bertrand and Cournot competition

We show how credible revelation and ability to commit to quality choice effect equilibrium qualities and welfare when product market is either Bertrand or Cournot competition. We show that results depend on the type of competition but not generally on the cost of quality function. We show that with Bertrand competition, the equilibrium qualities are lower with credible commitment. Competition is moderated and producer surplus is higher and consumer surplus lower. With Cournot competition, higher quality will be better but lower quality will be worse with credible commitment. Consumer surplus is always greater with credible commitment and if cost does not increase too quickly with quality, producer surplus will also increase. Thus credible commitment is a collusive device with Bertrand competition but it can improve social welfare with Cournot competition. Received: February 8, 2000; revised version: February 14, 2002 RID="*" ID="*" The idea of this paper originated in the weekly workshops of Mordecai Kurz at Stanford. I am forever in debted to Mordecai and fellow students – Luis Cabral, Peter DeMarzo, John Hillas, Michihiro Kandori, Steve Langois, Patrick McAllister, Steve Sharpe, Peter Streufert, Steve Turnbull and Gyu-Ho Wang – for their criticism and encouragement. I also benefited from comments from Yi-Heng Chen, Jin-Li Hu, Kala Krishna, Jinji Naoto, Thomas J. Prusa, and Shyh-Fang Ueng at various later stages of this work. Last but not least, I am grateful for the detailed comments of the referee.     

Winner-take-all price competition

Summary. We analyze an oligopoly model of homogeneous product price competition that allows for discontinuities in demand and/or costs. Conditions under which only zero profit equilibrium outcomes obtain in such settings are provided. We then illustrate through a series of examples that the conditions provided are “tight” in the sense that their relaxation leads to positive profit outcomes. Received: April 7, 2000; revised version: September 14, 2000     

Limit-pricing as Bertrand equilibrium

Summary. We consider a Bertrand duopoly model with increasing returns to scale where one of the firms have a cost advantage and prices vary over a grid. We find that typically more than one equilibria exist. However, there are only two perfect equilibria. Moreover, as the size of the grid becomes small, both these equilibria converge to the limit-pricing outcome. Received: February 25, 2000; revised version: January 9, 2001     

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     

The core in an oligopoly market with indivisibility

Summary. This paper studies the core in an oligopoly market with indivisibility. It provides necessary and sufficient conditions for core existence in a general m-buyer n-seller market with indivisibility. When costs are dominated by opportunity costs (i.e., a firm's variable costs are sufficiently small), the core condition can be characterized by the primitive market parameters. In a 3-2 market with opportunity cost, the core is non-empty if and only if the larger seller's opportunity cost is either sufficiently large or sufficiently small. Received: June 9, 1999; revised version: October 22, 1999     

Price equilibrium in address models of product differentiation: unit-elastic demand

Summary. I present a class of address models of product differentiation with unit-elastic individual demand and show the existence of Nash equilibrium in prices under assumptions on utility functions and the taste and income heterogeneity across consumers. This paper complements the work by Caplin and Nalebuff (1991, Econometrica), who analyze unit demand models of product differentiation. Received: December 28, 1998; revised version: September 5, 2001     

Boiteux's solution to the shifting-peak problem and the equilibrium price density in continuous time

Summary. Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production functions are Mackey continuous. A general form for production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. This gives a framework for settling Boiteux's conjecture on the shifting-peak problem. To make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges. The general equilibrium results are supplemented by results for prices supporting individual consumer or producer optima. Received: February 16, 2000; revised version: July 7, 2001     

Capacity precommitment as a barrier to entry: A Bertrand-Edgeworth approach

Summary. With few exceptions, the literature on the role of capacity as a strategic entry deterrent has assumed Cournot competition in the post-entry game. In contrast, this paper studies a model in which the incumbent and entrant sequentially precommit to capacity levels before competing in price. Interesting deterrence effects arise because firms need time to build, that is, cannot adjust capacity instantaneously in the post-entry game. This approach produces a simple and intuitive set of equilibrium behaviors and generates clear predictions about when these different outcomes are likely to arise. Our model also departs substantially from the existing literature in concluding that sunkness of capacity costs is neither necessary nor sufficient for capacity to have precommitment value. Received: August 25, 1999; revised version: October 15, 1999     

The option of joint purchase in vertically differentiated markets

Summary. Within the framework proposed by Mussa and Rosen (1978) for modelling quality differentiation, consumers are assumed to make mutually exclusive purchases. A unique pure strategy equilibrium exists in this case. In this note, we allow consumers to buy simultaneously different variants of the differentiated good. We call this the “joint purchase option”. The paper proposes a detailed analysis of price competition when this option is opened: first, we show that either uniqueness, or multiplicity, or absence of price equilibrium arise, depending on the utility derived from joint purchase relative to exclusive purchase. Second, we characterize these equilibria, whenever they exist. Received: July 25, 2001; revised version: October 21, 2002 RID="*" ID="*" The second author gratefully acknowledges the financial support from Interuniversity Attraction Pole Program- Belgian State- Federal Office for Scientific, Technical and Cultural Affairs under contract PAI 5/26. Correspondence to: X.Y. Wauthy     

Pricing, learning, and strategic behavior in a single-sale model

Summary. We analyze an infinite horizon model where a seller who owns an indivisible unit of a good for sale has incomplete information about the state of the world that determines not only the demand she faces but also her own valuation for the good. Over time, she randomly meets potential buyers who may have incentives to manipulate her learning process strategically. We show that i) the seller's incentives to post a high price and to experiment are not necessarily monotonic in the information conveyed by a buyer's rejection; and ii) as the discount factors tend to one, there are equilibria where the seller always ends up selling the good at an ex-post individually rational price. Received: January 6, 1999; revised version: July 15, 2000     

Uncertainty and insurance in strategic market games

Summary. For perfectly competitive economies under uncertainty, there is a well-known equivalence between a formulation with contingent goods and one with state-specific securities followed by spot markets for goods. In this paper, I examine whether this equivalence carries over to a particular form of imperfect competition. Specifically, I look at three Shapley-Shubik strategic market games: one with contingent commodities, one with Arrow securities traded under imperfect competition and one with Arrow securities traded under perfect competition. First I compare the feasibility constraints of these three games. Then I compare their equilibrium sets. As in Peck and Shell (1989), the only common equilibria between the first and the second game are those which involve no transfer of income across states. However, if the securities markets are competitive, then the set of equilibria of the contingent commodities game and the securities game coincide. Received: June 16, 1997; revised version: April 30, 1998     

Delegation and polarization of platforms in political competition

Summary. We consider a model of political competition among two ideological parties who are uncertain about the distribution of voters. The distinguishing feature of the model is that parties can delegate electoral decisions to candidates by nomination. It is shown that if the credible platform commitments of the candidates is feasible, then at least one of the parties nominates in equilibrium to a candidate who has an ideology that is more radical than the delegating party's ideology. In a variety of circumstances, this, in turn, yields a polarization of equilibrium policy choices of the candidates. It is thus argued formally here that strategic nomination of the candidates may well be one of the major reasons behind the well documented observation that the platforms associated with the political parties in two-party democracies are often surprisingly polarized. Received: January 10, 2002; revised version: May 8, 2002 RID="*" ID="*" We thank Alberto Alesina, Levent Ko?kesen, Antonio Merlo, Ronny Razin, Vijay Krishna, Alessandro Lizzeri, and seminar participants at Alicante, Columbia, Copenhagen, and NYU for helpful comments. We also thank an anonymous referee for its useful suggestions. A good fraction of this research was conducted while Ok was a visitor in the Department of Economics at University of Alicante; he thanks for the kind hospitality of this institution. We gratefully acknowledge the financial support from the Spanish Ministry of Education through grant CICYT BEC2001-0535 (Faulí-Oller) and BEC2001-0980 (Ortu?o-Ortín). Correspondence to:I. Ortu?o-Ortin     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

A strategic market game with seigniorage costs of Fiat money

Summary. A model that includes the cost of producing money is presented and the nature of the inefficient equilibria in the model are examined. It is suggested that if one acknowledges that transactions are a form of production, which requires the consumption of resources, then the concept of Pareto optimality is inappropriate for assessing efficiency. Instead it becomes necessary to provide an appropriate comparative analysis of alternative transactions mechanisms in the appropriate context. Received: September 5, 2000; revised version: May 3, 2001     

A simple proof of Ekeland and Scheinkman's result on the necessity of a transversality condition

Summary. Ekeland and Scheinkman (1986) prove the necessity of a standard transversality condition under certain technical conditions. Their result is one of the most powerful on the necessity of a transversality condition currently available in the literature, and their proof involves numerous estimations and relies on Ekeland's variational principle and Fatou's lemma. This note relaxes some of their assumptions and provides a simple proof that uses neither Ekeland's principle nor a convergence result like Fatou's lemma. Received: April 24, 1998; revised version: September 8, 1998     

Profitability of price and quantity strategies in a duopoly with vertical product differentiation

Using a model according to Mussa and Rosen (1978) and Bonanno and Haworth (1998) we consider a sub-game perfect equilibrium of a two-stage game in a duopolistic industry in which the products of the firms are vertically differentiated. In the industry, there are a high quality firm and a low quality firm. In the first stage of the game, the firms choose their strategic variables, price or quantity. In the second stage, they determine the levels of their strategic variables. We will show that, under an assumption about distribution of consumers' preference, we obtain the result that is similar to Singh and Vives (1984)' proposition (their Proposition 3) in the case of substitutes with nonlinear demand functions. That is, in the first stage of the game, a quantity strategy dominates a price strategy for both firms. Received: April 23, 1999; revised version: May 31, 2000     

On Lipschitz continuity of policy functions in continuous-time optimal growth models

Summary. This paper proves the C ^1,1 differentiability of the value function for continuous time concave dynamic optimization problems, under the assumption that the instantaneous utility is C ^1,1 and the initial segment of optimal solutions is interior. From this result, the Lipschitz dependence of optimal solutions on initial data and the Lipschitz continuity of the policy function are derived, by adding an assumption of strong concavity of the integrand. Received: July 29, 1996; revised version: November 25, 1997