On the non-existence of equilibria in differentiated product models

It is sometimes suggested that the non-existence of an equilibrium in the Hotelling spatial model pointed out by d'Aspremont, Gabszewicz and Thisse (1979) is due to demand discontinuity. In this paper we first show that demand continuity is a natural property in general spatial or differential product markets. However, we also claim that the non-existence of an equilibrium is also a general property of these models. Further the assumption of no mill-price undercutting suggested by Eaton and Lipsey (1978) and Novshek (1980) is shown to be not sufficient to restore existence.     

The non-existence of smooth demand in general banach spaces

We study the questions of existence and smoothness of demand functions with an infinite number of commodities. The main result obtained, under some hypothesis, is: if a C¹ demand exists in a commodity space B, then B can be given an inner product structure. For example, if B is L_p, 1p∞, and if there exists a C¹ demand function defined on B then p must be 2. Another result is: if a demand function exists, defined for all prices p and income, then the commodity space must be reflexive. For example, if B is L_p and a demand function exists on B, defined for all prices and incomes then 1<p<∞. We also study the cases L_∞ and L₁ with weaker assumptions. We finish the paper proving that the demand function is always defined for a dense set of prices and convenient incomes.     

Competitive equilibria in general choice spaces

This paper primarily demonstrates the existence of Arrow-Debreu equilibria in a general class of topological vector spaces of commodity bundles. Two conditions based on production possibilities, preferences, and the topological nature of bounded sets are shown to substitute, in any locally convex space, for the advantages of the Euclidean topology. Examples fulfilling these conditions are supplied. The approach is that of Bewley, demonstrating equilibria on finite dimensional sub-economies and establishing a net of these equilibria that converges to an equilibrium on the whole commodity space. An example of equilibrium with a storage technology is given. An auxiliary result concerns the price support of efficient allocations.     

Existence of maximal elements and equilibria in linear topological spaces

We present some mathematical theorems which are used to generalize previous results on the existence of maximal elements and of equilibrium. Our main theorem in this paper is a new existence proof for an equilibrium in an abstract economy, which is closely related to a previous result of Borglin–Keiding, and Shafer–Sonneschein, but allows for an infinite number of commodities and a countably infinite number of agents.     

On the uniqueness of local equilibria

The purpose of this paper is to define a new notion of local equilibrium in an exchange economy, where the consumers face lower bounds on net trades. Then, we show that the local equilibrium is unique if the lower bounds are closed enough to 0. By the way, we also provide a convergence result of local equilibrium price toward Walras equilibrium price of a suitable tangent linear economy.     

On the plausibility of sunspot equilibria

This study examines the plausibility of the emergence of sunspot equilibria in an agent-based artificial stock market. Using the agent-based model, we make the sunspots explicit so that we can test, e.g., by means of the Granger causality test, whether purely extrinsic uncertainty can influence price dynamics. In addition, through agent-based simulation, the coordination process, which is mainly driven by genetic programming, becomes observable, which enables us to analyze what agents perceive and whether they believe in sunspots. By manipulating different control variables, three series of experiments are conducted. Generally speaking, the chances of observing “sunspot equilibria” in this agent-based artificial stock market are small. However, the sunspot believers can never be driven out of the market. Nevertheless, they are always outnumbered by fundamental believers, which is evidence that the market as collective behavior is rational. We also find that lengthening the time horizon will make it difficult for sunspot believers to survive.     

Heterogeneous players in a Cournot model with differentiated products

In this article, we analyse a duopolistic Cournotian game with firms producing differentiated goods, marginal costs are constant and demand functions are microfounded. We consider firms adopting different decisional mechanisms which are based on a reduced degree of rationality. In particular, we assume that a firm adopts the local monopolistic approximation approach, while the rival adjusts its output level according to the gradient rule. We provide conditions for the stability of the Nash equilibrium and investigate some bifurcation scenarios as parameters vary. The main finding of the article is that both a high level and a low level in goods differentiation may have a destabilising role in the system.     

Quality-improving R&D and merger policy in a differentiated duopoly: Cournot and Bertrand equilibria

We combine a model of product research and development (R&D) and technological spillover with the concept of technological distance and examine horizontal mergers in a duopolistic market with R&D. The results are fourfold. First, a merger can better encourage R&D investment than the competition case. Second, with a small degree of product differentiation (PD), the merger criterion under the Cournot duopoly is stricter than that of the Bertrand case. By contrast, with a moderate or large degree of PD, the opposite is true. Third, with a small technological distance, a merger should be allowable. Finally, with a small degree of PD and moderate technological distance, a merger should be allowable.     

On the impossibility of insider trade in rational expectations equilibria

Existing no trade results are based on the common prior assumption (CPA). This paper identifies a strictly weaker condition than the CPA under which speculative trade is impossible in a rational expectations equilibrium (REE). As our main finding, we demonstrate the impossibility of speculative asset trade in an REE whenever an insider is involved who knows the asset's true value. To model insider trade as an equilibrium phenomenon an alternative equilibrium concept than the REE is thus required.     

Purification of Bayesian–Nash equilibria in large games with compact type and action spaces

We present a purification result for incomplete information games with a large but finite number of players that allows compact metric spaces for both actions and types. We then compare our framework and findings to the early purification theorems of Rashid (1983. Equilibrium points of non-atomic games: asymptotic results. Economics Letters 12, 7–10), Cartwright and Wooders (2002 On equilibrium in pure strategies in games with many players. University of Warwick Working Paper 686 (revised 2005)), Kalai (2004. Large robust games. Econometrica 72, 1631–1665) and Wooders, Cartwright and Selten (2006. Behavioral conformity in games with many players. Games and Economic Behavior 57, 347–360). Our proofs are elementary and rely on the Shapley–Folkman theorem.     

On the structure of the set of fixed price equilibria

Working in the framework suggested by Drèze, this paper studies the number of fixed price equilibria and their continuity with respect to the price system. In an exchange economy, the concept of a rationing scheme is introduced, which specifies how shortages are shared among agents. For given utility functions and a given rationing scheme, under standard assumptions, an existence theorem is recalled, and it is shown that the graph of the equilibrium correspondence, when prices and initial endowments vary, is a piecewise continuously differentiable manifold. Moreover, generically, the number of equilibria for an economy, at given prices, is finite and the set of equilibria varies continuously with the price system and the initial endowments.     

On the aubin-like characterization of competitive equilibria in infinite dimensional economies

For an exchange economy with finitely many agents and a commodity space which is infinite dimensional, it is proved that competitive equilibria are the same as allocations belonging to the fuzzy core of the economy. The latter is further characterized as a subset of the ordinary core of a suitably associated continuum economy.

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Riassunto Per un'economia di puro scambio con un numero finito di agenti e uno spazio dei beni di dimensione non necessariamente finita, si prova che gli equilibri Walrasiani possono essere caratterizzati come allocazioni che appartengono al nucleo fuzzy dell'economia. Inoltre, queste ultime allocazioni sono in corrispondenza biunivoca con allocazioni di tipo semplice che si trovano nel nucleo ordinario di un'opportuna economia che ha [0,1] come spazio rappresentativo dello spazio degli agenti.

     

On the optimality of differentiated SPR tests of composite hypotheses

Summary It is shown that differentiated SPR tests, which are limiting cases of ordinary SPR tests, are asymptotically optimal in a decision-theoretic formulation for composite hypotheses in one-parameter problems. It is also given some results connected with comparisons of different tests in this case.     

On the existence of equilibria in economies with an infinite dimensional commodity space

We prove an infinite dimensional extension of the Gale–Nikaido–Debreu lemma which includes all necessary limiting processes and allows a proof of the existence of equilibria under standard assumptions in an economy with infinitely many commodities which exactly parallels the proof of Debreu (1959) for the finite dimensional case.     

Inefficiency of credible strategies in oligopolistic resource markets with uncertainty

The implications of different information patterns for firms in oligopolistic resource markets are considered. The traditional open-loop Nash equilibrium with static information sets is one of many possible Nash equilibria and is not suitable for stochastic environments. When shocks to resource growth are serially uncorrelated, there are no gains from conditioning the harvest on past stock levels and the feedback or credible Nash equilibrium is the appropriate Nash equilibrium concept. This credible equilibrium assumes that firms have knowledge of current stocks of reserves, which typically leads to more rapid extraction of the resource and possibly extinction. Since the open-loop Nash equilibrium is efficient when demand is iso-elastic and extraction costs are zero, it is clear that an increase in information can be detrimental to firms in the industry.     

A simple proof of the equivalence theorem for oligopolistic mixed markets

In a rather long and complicated way Shitovitz proved that in mixed markets with at least two large traders all of the same ‘type’, all core allocations are competitive. We provide a simple and short proof for this important result. Unlike Shitovitz, we do not prove directly this equivalence. Rather, we first show that the core of an oligopolistic market coincides with the core of the atomless market derived by splitting the atoms into a continuum of traders. We then apply Aumann's Equivalence Theorem to this atomless market, whose set of competitive equilibria coincides with that of the original mixed market.     

On existence and stability of spatial equilibria and steady-states

This note discusses the existence and stability of two equilibrium concepts for a spatial economy in which the utility of an agent depends on the overall distribution of agents over space.