The comparative statics of cumulative distribution function changes for the class of risk averse agents

We examine the connection between Walrasian equilibria of a limit economy (with infinitesimal firms) and the noncooperative (Cournot) equilibria of approximating finite economies (with significant firms). Following earlier work of Novshek and Sonnenschein we allow for set-up cost and permit a minimal form of mixed strategies. We depart from them by requiring that the aggregate production set exhibits some degree (however small) of decreasing returns. Contrasting with their results, it is shown that a (regular) Walrasian equilibrium of a limit economy can always be approximated by a sequence of noncooperative equilibria for the tail of the approximating (finite) economies. Thus, there is a surprising qualitative discontinuity when one passes from the Novshek-Sonnenschein case of aggregate constant returns to scale of the decreasing returns case of this paper.     

A note on allocations attainable through nash equilibria

This note reexamines the issue of attaining efficient resource allocations as noncooperative equilibria of a strategic game. Contrary to the spirit of earlier work on this subject, it is shown that there is an entire class of choice rules, aside from the Walrasian one, which are implementable in Nash equilibrium stategies.     

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     

Auctions of Heterogeneous Objects

Simultaneous sealed bid auctions of heterogeneous objects are analyzed. Each bidder's reservation value for an object depends upon the other objects he obtains. Bidders' reservation values are common knowledge. In simultaneous first-price auctions, the set of Walrasian equilibrium allocations contains the set of pure strategy Nash equilibrium allocations which in turn contains the set of strict Walrasian equilibrium allocations. Hence, pure strategy Nash equilibria (when they exist) are efficient. Mixed strategy Nash equilibria may be inefficient. In simultaneous second-price auctions, any efficient allocation can be implemented as a pure strategy Nash equilibrium outcome if a Walrasian equilibrium exists.Journal of Economic LiteratureClassification Numbers: D44, D51.     

On Nash implementation of Walrasian equilibria in a market for pure indivisibles without money

This paper deals with the Walrasian property of Nash and strong equilibria of a specific strategic market game which refers to a pure exchange economy involving purely indivisible commodities and no money. The game is of sealed-bid auction type and it is shown that any Nash equilibrium at which no agent is in status quo is a strong equilibrium and implements a Walrasian equilibrium. Moreover, it appears that two modifications of the game's rules ensure that any strong equilibrium outcome is Walrasian. These results are identical to those obtained by Svensson for markets involving purely indivisible goods and money.     

Ownership structure and efficiency in large economies

We analyze the limit behavior of sequences of oligopolistic equilibria in which firms follow objectives consistent with their shareholders?? interests. We show that convergence to a competitive outcome may fail for some distributions of firms?? shares across consumers and provide a characterization of the class of ownership structures that lead to Walrasian equilibrium allocations in the limit.     

Microeconomic uncertainty and macroeconomic indeterminacy

We propose a stylized intertemporal macroeconomic model wherein the combination of decentralized trading and microeconomic uncertainty (taking the form of privately observed and uninsured idiosyncratic shocks) creates an information problem between agents and generates indeterminacy of the macroeconomic equilibrium. For a given value of the economic fundamentals, the economy admits a continuum of equilibria that can be indexed by the sales expectations of firms at the time of investment. The Walrasian equilibrium is one of these possible equilibria but it is reached only if firms are optimistic enough. With a weaker degree of optimism, equilibrium output, employment and real wages will be lower than in the Walrasian equilibrium. Moreover, the range of possible equilibria will depend positively on the wage elasticity of the labour supply and on the magnitude of the information problem between buyers and sellers (in our case, the variance of the idiosyncratic shocks).Stochastic simulations performed on a calibrated version of the model show that pure demand expectation shocks may generate business cycle statistics that are not inconsistent with the observed ones.     

Continua of Hahn-unsatisfactory equilibria: Some examples

Quantity-constrained fix-price equilibria may occur even at Walrasian equilibrium prices. Structurally stable examples are given to support the claim that there may be a continuum of such equilibria, even in a neighborhood of the Walrasian one.     

On the existence of pure strategy Nash equilibria in two person discrete games

We construct a generalized two-person discrete strategy static game of complete information where continuity, convexity and compactness cannot be invoked to show the existence of pure strategy Nash equilibrium. We show that, when best responses are unique from both sides, a condition of Minimal Acyclicity is necessary and sufficient for the existence of pure strategy Nash equilibria.     

Mixed equilibria in runoff elections

We demonstrate that the Hotelling–Downs model with runoff voting always admits symmetric mixed strategy equilibria for any (even or odd) number of office-motivated candidates (provided they are at least four). In specific, (a) we show that the game does not admit any symmetric atomless equilibrium, (b) we fully characterize a class of symmetric atomic mixed equilibria which exist for any distribution of the voters' ideal policies and (c) we argue that these equilibria are more robust than pure strategy equilibria to introduction of uncertainty about the voters' preferences.     

More on economies with a finite set of Walrasian equilibria

Proposition. The graph of the Walrasian equilibrium correspondence is a piecewise continuously differentiable manifold. Furthermore, there exists an open dense set of economies, Ω, such that for all w in Ω (a) the number of equilibria of the economy w is finite; and (b) there exists a neighborhood V(w) on which the set of equilibria is represented by a finite family of piecewise continously differentiable functions.     

Imperfectly competitive cycles with Keynesian and Walrasian features

We consider a multi-sector overlapping generations model with oligopolistic firms in the output markets and wage-setting trade unions in the labour markets. A coordination problem between firms creates multiple temporary equilibria which are either Walrasian or of the Keynesian unemployment type. There exist many deterministic and stochastic equilibrium cycles fluctuating between Keynesian recession and Walrasian boom periods with arbitrarily long phases in each regime. The cycles are in accordance with certain empirical regularities. Money is neutral and superneutral, but appropriate countercyclical fiscal policies stabilize the cycles in a textbook Keynesian way.     

Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives

In a game of a finite number of repetitions of a Cournot-type model of an industry, if firms are satisfied to get close to (but not necessarily achieve) their optimal responses to other firms' sequential strategies, then in the resulting noncooperative “equilibria” of the sequential market game, (1) if the lifetime of the industry is large compared to the number of firms, there are equilibria corresponding to any given duration of the cartel, whereas (2) if the number of firms is large compared to the industry's lifetime, all equilibria will be close (in some sense) to the competitive equilibrium.     

On the existence of pure strategy Bertrand equilibrium

Summary The paper analyses the existence of pure strategy Nash equilibrium in price competition (or Bertrand equilibrium) in a homogeneous product market when costs are strictly convex and proves that if output is demand determined such equilibrium always exists. This paper also characterises such equilibria and shows that if firms are identical such equilibria are necessarily non-unique. However for firms with asymmetric costs it can be unique or non-unique.I am greatly indebted to Anjan Mukherji and Kunal Sengupta for this paper. I also express my profound gratitude to a referee of this journal for a very helpful set of suggestions.     

Coalition-proof Bertrand equilibria

Summary. This paper examines the coalition-proof Nash equilibria of a Bertrand model of price competition where firms supply all demand. When firms are asymmetric we prove existence and provide a sufficient condition for uniqueness. For symmetric firms, we show that an equilibrium is necessarily unique. We also examine whether this unique equilibrium outcome is implementable through a sequential move game where the firms take turns at announcing prices. Finally we examine the limiting property of such equilibria as the number of firms go to infinity.Received: 20 March 2002, Revised: 5 August 2003JEL Classification Numbers: D43, D41, L13.Correspondence to: Kunal SenguptaWe are deeply indebted to an anonymous referee for very helpful and incisive comments that led to substantial improvements in the paper. We also gratefully acknowledge the hospitality of the Department of Finance, Hong Kong University of Science and Technology where much of the work on this paper was carried out.     

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     

First-price auction symmetric equilibria with a general distribution

In this paper I obtain the mixed strategy symmetric equilibria of the independent private values, first-price auction for any distribution of valuations. The equilibrium strategy is a pure strategy at the continuity points of the distribution and bids a mixed strategy at the discontinuity points of the distribution.     

A graphic representation of the Nash and Lindahl equilibria in an economy with a public good

In an economy with a public good the noncooperative Nash equilibrium and the cooperative Lindahl equilibrium are represented graphically. The Nash equilibrium is shown to be non-optimal, while the Lindahl equilibrium is shown to be optimal. Simple stability analyses are undertaken. Finally, the two equilibria are compared. It appears that more public good will be allocated in the Lindahl than in the Nash equilibrium, and that in a sufficiently large economy a transition from a Nash to a Lindahl equilibrium will be advantageous for everyone.     

ENDOGENOUS TIMING IN QUALITY CHOICES AND PRICE COMPETITION

We modify the price‐setting version of the vertically differentiated duopoly model by Aoki (2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Our results show that there are multiple equilibria in pure strategies, whereby firms always select sequential play at the quality stage. We also investigate the mixed‐strategy equilibrium, revealing that the probability of generating outcomes out of equilibrium is higher than the probability of playing one Nash equilibria in pure strategies. In the alternative case with full market coverage, we show that the quality stage is solved in dominant strategies and therefore the choice of roles becomes irrelevant as the Nash and Stackelberg solutions coincide. With full market coverage and corner solution, the results show that the game has a unique subgame perfect equilibrium in pure strategies, where the high‐quality firm takes the lead in the quality stage.     

Costly Search, Capacity Constraints, and Bertrand Equilibrium Price Dispersion

This article analyzes the impact of transaction (search) costs and capacity constraints in an almost competitive market with homogeneous firms that compete on price. We characterize conditions under which Nash equilibria with price dispersion exist; in equilibrium, firms play pure strategies in prices and consumers adopt a symmetric mixed search strategy. Price dispersion is possible even though consumers all have the same search cost and valuation for the item and prices charged by all firms are common knowledge.