On the uniqueness of competitive equilibrium

Summary . The paper is concerned with the following question: in addition to local uniqueness, what other conditions must be imposed to ensure global uniqueness of competitive equilibrium? The answer is provided within a standard framework involving excess demand functions. Conditions are identified which have the nice property that they are true close to a regular equilibrium. Uniqueness is established by considering an adjustment process and showing that under the mentioned conditions, every equilibrium is locally asymptotically stable and moreover the process itself is globally stable; uniqueness follows from an Arrow and Hahn (1971) result. Alternatively, the paper may be seen as identifying conditions under which there is a vectorfield satisfying the requirements of an uniqueness result due to Dierker (1974). Received: July 1, 1996; revised version October 7, 1996     

Price and quantity regulation in general equilibrium

We consider a general equilibrium model with a production externality (e.g. pollution), where the regulator does not observe firm productivity shocks. We examine quantity (permit) regulation and price (tax) regulation. The quantity of permits issued by the regulator are independent of the productivity shock, since shocks are unobserved. Price regulation implies use of the regulated input is an increasing function of the productivity shock because firms take advantage of a good productivity shock by increasing input use. Thus price regulation generates higher average, but more variable, production. Therefore, we show that in general equilibrium the relative advantage of quantity versus price regulation depends not only on the slopes of marginal benefits and costs, but on general equilibrium effects such as risk aversion. The general equilibrium effects are often more important than the slopes of the marginal benefits and cost curves. In the simplest model, a reasonable risk aversion coefficient implies quantity regulation generates higher welfare regardless of the benefit function.     

Free entry and stability in a cournot model

In a Cournot model for a single homogeneous good, we study simultaneously two stability properties of a Cournot equilibrium: the stability of a continuous quantity adjustment process with a fixed number of firms and the stability relative to entry. Under usual assumptions, we show that these two properties are consistent—i.e., there exists an equilibrium that is stable in both senses—and determine quite sharply the number of active firms.     

On supply-constrained equilibria

The paper gives sufficient conditions for the existence of an equilibrium with price rigidities and quantity rationing where: (i) demand is never rationed; (ii) net trades of an a priori chosen numeraire are never rationed; and (iii) supply is rationed only when relative prices are downward rigid.     

Market allocation through search: Equilibrium adjustment and price dispersion

This essay discusses market allocation under uncertainty in a market for a homogeneous good. Market participants contact one another at random times to buy or sell single units of the good. Transactions are carried out at different prices simultaneously. Optimal search rules are employed to describe individuals behavior. Such models provide the framework of a market structure within which equilibrium adjustment processes are analyzed. These decentralized, nontâtonnement processes of price and quantity adjustment are utilized to examine equilibrium. Price dispersion, which characterizes equilibrium, is essentially due to finite lifetimes of market participants.     

SIZE,OPENNESS, AND MACROECONOMIC INTERDEPENDENCE

One common, simplifying assumption in open economy macroliterature is that the rest of the world can be thought of as a representative economy. This article formally investigates conditions under which this assumption can be justified using a multicountry general equilibrium model as a laboratory. We derive the conditions that ensure the existence of the equilibrium and study the properties of the equilibrium using large N asymptotics. Thereby, we show that the two‐country framework is a valid approximation only for economies that have diversified trade linkages and only when there is no globally dominant economy among the foreign economies.     

Tax Competition, Capital Mobility, and Public Good Provision Within a Trading Block

The authors examine a two–country general–equilibrium model of a two–country trading block where governments through tax policies attract mobile capital and provide an imported public consumption good. Within this framework the authors examine, among other things, how preferences over the public good and the size (population) of a country affect the Nash or cooperative equilibrium values of income tax rates in the two countries. The analysis identifies sufficient conditions under which (i) the Nash/cooperative equilibrium income tax rates are strategic substitutes or complements, and (ii) the Nash equilibrium income tax rates may be greater than the cooperative rates.     

Endogenous rationing,price dispersion and collusion in capacity constrained supergames

This paper examines the feasibility of collusion in capacity constrained duopoly supergames. In each period firms simultaneously set a price–quantity pair specifying the price for the period and the maximum quantity the firm is willing to sell at this price. Under price–quantity competition firms are able to ration their output below capacity. For a wide range of capacity pairs, the equilibrium path providing the smaller firm with its highest stationary perfect equilibrium payoff requires that it undercut its rival’s price and ration demand. Furthermore, for some capacities and discount factors supporting security level punishments, price shading and rationing arise everywhere on the set of stationary perfect equilibrium paths yielding (constrained) Pareto optimal payoffs. That is, price shading may not only be consistent with optimal collusion, it may be a requirement of optimal collusion.     

Existence of equilibrium in the Helpman–Krugman model of international trade with imperfect competition

Helpman and Krugman (Market structure and foreign trade. Increasing returns, imperfect competition, and the international economy. MIT Press, Cambridge, 1985) provide a synthesis of the traditional factor proportions theory of international trade and the theory of international trade due to the exploitation of scale economies in imperfectly competitive markets. They derive illuminating results about trade patterns and gains from trade, among other things, leaving unanswered the question of existence of equilibrium, however. The central significance of their characterization of properties of free trade equilibria with inter-industry and intra-industry trade calls for an analysis of existence of equilibrium. This is the object of the present paper. We prove the existence of equilibrium for the integrated multi-sector multi-factor Helpman–Krugman economy without national borders. Well-known conditions ensure that the world economy under free trade reproduces this equilibrium and thus establishes existence of a free trade equilibrium. Since an equilibrium of the integrated economy is not necessarily unique, the same holds true for a free trade equilibrium.     

Destabilizing impacts of price and quantity adjustments to relative supply and demand

The static theories of the firm and household rest their practical validity on an assumption that disequilibrium conditions will rapidly be eliminated through price and quantity adjustments. This paper develops a model of stock management by producers to explore the time paths of production adjustment to disequilibrium. Model results suggest that typical corporate policies for responding to inventory and order backlog levels lead to continuing fluctuations in production and employment around equilibrium values. Moreover, fluctuations are acceptuated when price impacts on output and consumption decisions are included. Following Samuelson's Correspondence Principle, the results imply that equilibrium analyses for models containing equilibrium assumptions) have little explanatory power, especially over a short-run period of several years. Further, proper dynamic analysis requires comprehensive interrelating of stocks and flows, not merely incorporation of lagged adjustments to theoretical equilibrium conditions.     

Nonstandard foundations of equilibrium search models

We develop an equilibrium sequential search model which includes most of the literature as special cases. In particular, the model can accommodate heterogeneity in buyers’ search costs and demand functions and firms’ cost functions, with general demand and cost functions. We identify conditions which ensure existence of equilibrium in pure strategies, utilizing recent progress in the theory of large games by Khan and Sun. These conditions elucidate the essential structure of equilibrium search models. Although we focus on sequential search, our methodology can be used for other classes of equilibrium search models as well.     

Imperfect recall and the relationships between solution concepts in extensive games

Summary. In a game of imperfect recall, a sequential equilibrium may not be a Nash equilibrium, and a perfect equilibrium may not be a sequential equilibrium. Sufficiency conditions weaker than perfect recall are given to ensure the standard relationships hold between perfect equilibrium, sequential equilibrium and Nash equilibrium.Received: 22 October 2003, Revised: 18 November 2003, JEL Classification Numbers: C72.     

Non-parametric counterfactual analysis in dynamic general equilibrium

In this paper, we examine non-parametric restrictions on counterfactual analysis in a dynamic stochastic general equilibrium model. Under the assumption of time-separable expected utility and complete markets all equilibria in this model are stationary. The Arrow-Debreu prices uniquely reveal the probabilities and discount factor. The equilibrium correspondence, defined as the map from endowments to stationary (probability-free) state prices, is identical to the equilibrium correspondence in a standard Arrow-Debreu exchange economy with additively separable utility. We examine possible restriction on this correspondence and give necessary as well as sufficient conditions on profiles of individual endowments that ensure that associated equilibrium prices cannot be arbitrary. Although restrictions on possible price changes often exist, we show that results from a representative-agent economy usually do not carry over to a setting with heterogeneous agents.     

HORIZONTAL MERGERS: A SOLUTION OF THE INSIDERS' DILEMMA

We show that in a three‐firm infinitely repeated Cournot game, there exists a stick and carrot strategy equilibrium in which an exogenous bilateral horizontal merger is profitable and the incentive to remain out of the merger disappears. In this sub‐game perfect equilibrium, the merged entity produces the duopoly quantity and the outsider limits its production to half the duopoly quantity. Our stick and carrot strategy entails that the merged entity threatens to produce twice the triopoly quantity for two periods if the outsider does not produce half the duopoly quantity. In this equilibrium, the aggregate price remains high enough to make the merger profitable for the insiders. Also, the quantity produced by the outsider is sufficiently low to eliminate the difference between the profit of the outsider and the merging firm.     

Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players’ payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of e{\epsilon}-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.     

Multiple equilibrium under CES preferences

Under the simplifying assumptions that (1) there are two commodities and two agents, (2) production (endowment) of each commodity per period is fixed, (3) preferences of the two agents are of CES type, and (4) both the endowments and the CES preferences are mirror images of each other (called “supersymmetry”), it is shown that (a) there always exists a symmetric equilibrium with a unit price ratio, (b) a sufficient condition for the stability (hence uniqueness) of this symmetric equilibrium is that the elasticity of substitution satisfy ${\sigma\geqq 1/2}$ , and (c) multiple (triple) equilibrium is possible if and only if each agent has a relative preference for the good that is sold (“exported”) to the other agent. Further conditions for multiple equilibrium are derived.     

Over- and under-investment according to different benchmarks

In a two-stage oligopoly, with investment in the first stage and quantity or price competition in the second stage, there is a “Common Wisdom” Theorem which states that we find over-investment if the goods are substitutes and competition is in strategic substitutes, or if goods are complements and competition is in strategic complements, and that we find under-investment if we have complements and strategic substitutes or substitutes and strategic complements. The existing literature, however, lacks a proof of this theorem and, in particular, it lacks a systematic comparison of the different benchmarks for over- and under-investment. A “naive” benchmark is the cost efficient investment with respect to the subgame perfect (closed loop) equilibrium quantities. Alternative benchmarks (which are more often proposed) are the open loop equilibrium investment or the welfare maximizing investment. The chosen benchmark is critical because the Common Wisdom Theorem applies (under certain conventional conditions) only for the naive benchmark. The other two benchmarks give rise to subcases.     

Contribution to a public good under subjective uncertainty

This paper examines how voluntary contributions to a public good are affected by the contributors' heterogeneity in beliefs about the uncertain impact of their contributions. It assumes that contributors have Savagian preferences that are represented by a two‐state‐dependent expected utility function and different beliefs about the benefit that will result from the sum of their contributions. We establish general comparative statics results regarding the effect of specific changes in the distribution of beliefs on the (unique) Nash equilibrium provision of the public good, under certain conditions imposed on the preferences. We specifically show that the equilibrium public good provision is increasing with respect to both first‐ and second‐order stochastic dominance changes in the distribution of beliefs. Hence, increasing the contributors' optimism about the uncertain benefit of their contributions increases aggregate public good provision, as does any homogenization of these beliefs around their mean.     

Bilateral monopoly on a market for an intermediate good

In this paper we analyze a procedure through which price and quantity could be determined on a bilateral monopoly market for an intermediate good. The proposed solution concept is of a non-cooperative Nash equilibrium type. A dynamic process in which the bilateral monopolists repeatedly announce prices and quantities is studied. It is shown that, under particular assumptions, this process must converge to the proposed equilibrium solution. Finally, our solution concept leads to the competitive equilibrium if both sides of the market are ‘fractionated’ into a large number of ‘small’ buyers and ‘small’ sellers.     

A note on Stackelberg competition

This note investigates a Stackelberg–Nash competition model. We determine the conditions under which the leaders may achieve better profits than the followers when all firms compete on quantity in a two stage game. We focus on the properties of the followers’ best response functions. It is shown that the Stackelberg equilibrium may coincide with the Cournot equilibrium. In addition, the followers may achieve higher profits than the leaders. Such results put forward the working and the consequences of strategic complementarities. These results are illustrated with three examples.