The equilibrium set of infinite dimensional Walrasian economies and the natural projection

The natural projection plays a fundamental role to understand the behavior of the Walrasian economies. In this paper, we extend this method to analyze the behavior of infinite dimensional economies. We introduce the definition of the social equilibrium set, and we show that there exists a bijection between this set and the Walrasian equilibrium set of an infinite dimensional economy. In order to describe the main topological characteristics of both sets, we analyze the main differential characteristics of the excess utility function and then, we extend the method of the natural projection as suggested by Y. Balasko.     

Core and competitive equilibria of a coalitional exchange economy with infinite time horizon

The core and competitive equilibria of a large exchange economy on the commodity space ?^∞${?}^{\infty }$ will be discussed. We define the economy as a measure on the space of consumers’ characteristics following Hart and Kohlberg (1974), and prove the existence of competitive equilibria and their equivalence with the core without assuming the convexity of preferences.     

Private ownership economies with a finite number of equilibria

Interest in economies with a finite number of equilibria has been first drawn over by Debreu (1970). The present paper shows that such economies form a subset of dense interior of the set of private ownership economies and proves some continuity properties of the Walras correspondence. It generalizes similar results obtained for exchange economies by Dierker and Dierker (1972). Demonstrations also rest heavily on transversality theory but use somewhat stronger theorems.     

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 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.     

Edgeworth rejective core and dividends equilibria of satiated exchange economies

For an exchange economy, under assumptions which did not bring about the existence of equilibrium with dividends as yet, we prove the non-emptiness of the Edgeworth rejective core. Then, via  and ’s decentralization result, we solve the equilibrium with dividends existence problem. Adding to the same assumptions a weak non-satiation condition which differs from the weak non-satiation assumption introduced by Allouch and Le Van (2009), we show in the last section the existence of a Walrasian quasiequilibrium. This result, which fits with exchange economies whose consumers’ utility functions are not assumed to be upper semicontinuous, complements the one obtained by Martins-da-Rocha and Monteiro (2009).     

Computing equilibria of semi-algebraic economies using triangular decomposition and real solution classification

In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies, which may be expressed as semi-algebraic systems. The approach is based on triangular decomposition and real solution classification, two powerful tools of algebraic computation. Its effectiveness is illustrated by three examples of application.     

Uncertain equilibria and incomplete preferences

This paper studies the tendency for incomplete preference structures to be associated with equilibrium price indeterminacies in an Arrow–Debreu–McKenzie state-contingent setting. It is shown that the presence of a sufficiently smooth stochastic production technology is inconsistent with equilibrium price indeterminacies even if all individuals have incomplete preference structures. A particularly convenient characterization of Paretian equilibria in the presence of incomplete preferences, which allows Paretian equilibrium to be characterized using simple principles of convex optimization and (sub)differential analysis, is also developed.     

Quality of local equilibria in discrete exchange economies

This paper defines the notion of a local equilibrium of quality $(r,s)$, $0\le r,s$, in a discrete exchange economy: a partial allocation and item prices that guarantee certain stability properties parametrized by the numbers $r$ and $s$. The quality $(r,s)$ measures the fit between the allocation and the prices: the larger $r$ and $s$ the closer the fit. For $r,s\le 1$ this notion provides a graceful degradation for the conditional equilibria of Fu, Kleinberg and Lavi (2012) which are exactly the local equilibria of quality $(1,1)$. For $1<r,s$ the local equilibria of quality $(r,s)$ are more stable than conditional equilibria. Any local equilibrium of quality $(r,s)$ provides, without any assumption on the type of the agents’ valuations, an allocation whose value is at least $\frac{rs}{1+rs}$ the optimal fractional allocation. In any economy in which all agents’ valuations are $a$-submodular, i.e., exhibit complementarity bounded by $a\ge 1$, there is a local equilibrium of quality $(\frac{1}{a},\frac{1}{a})$. In such an economy any greedy allocation provides a local equilibrium of quality $(1,\frac{1}{a})$. Walrasian equilibria are not amenable to such graceful degradation.     

Continuous economies with a finite set of equilibria

This paper extends the results of G. Debreu's ‘Economies with a Finite Set of Equilibria’, within the context of the finite dimensional space of economies used therein, to the case where demand functions are assumed only to be continuous and to satisfy a certain technical condition. This condition, involving the concept of approximate limit, does not even require that the function be anywhere differentiable. Our main result is that the complement of the set of economies with a finite set of equilibrium prices has Lebesgue measure zero, although it need not be closed and in fact, may very well be dense.     

The hybrid equilibria and core selection in exchange economies with externalities

This paper introduces a hybrid equilibrium concept that combines the elements of cooperative and non-cooperative behaviors in an exchange economy with externalities. For a fixed coalition structure (or a partition of traders), the hybrid equilibrium is a price and consumption bundle such that each coalition chooses a core solution from its budget set and the consumption bundles are feasible. It becomes the competitive equilibrium when the position is the finest, and it selects a core allocation when the partition is the coarsest. The paper provides sufficient conditions for the existence of a hybrid equilibrium for any coalition structure.     

Sequential regularity in smooth production economies

In general equilibrium models where agents trade sequentially in multiple periods, the equilibria that endogenously occur in later periods can be robustly indeterminate if production sets are not everywhere differentiable (as. for example, with linear activities). The present paper proves that if technology is smooth, then equilibria are sequentially regular. That is. the equilibria of the endogenously generated economies occurring in later periods that confirm the unanimous expectations formed by agents in earlier periods are regular (and thus, for example, isolated from other equilibria). The paper also proves and utilizes the standard result that the overall intertemporal equilibria are regular.     

Communication complexity and stability of equilibria in economies and games

The theory of decentralized allocation mechanisms introduced by Leonid Hurwicz enabled economists to analyze the implications of requiring an allocation mechanism to perform well over a range of possible economic environments. One of the earliest major results in this theory was the formalization and proof of Hayek’s famous assertion that the competitive market mechanism minimizes the communication needed to achieve Pareto efficient allocations. The informational efficiency theorem spawned a large literature on communication complexity in allocation mechanisms. This paper provides an exposition of some results on the communication required to make equilibria locally stable, and the possibility of eliciting the required communication in allocation mechanisms and games.     

Measuring the set of blocking coalitions in infinite dimensional economies

The classical core-Walras equivalence is investigated via the measurement of the set of blocking coalitions. In the framework of continuum economies with an infinite dimensional space of commodities, we find the measure of the set of coalitions that block a non-competitive Pareto optimal allocation. Then, from the relation between coalitions of a continuum economy with a finite number of types and fuzzy coalitions of an economy with finitely many agents, the previous results are translated as results on the measure of blocking fuzzy coalitions. Both results imply classically formulated core-Walras equivalence theorems. Finally, the measure of blocking "social communication structures" is determined.     

Cyclical equilibria in multi-sector productive economies: The role of substitution and factor intensity

The paper examines the existence of cyclical perfect foresight equilibria in a two-sector overlapping generations economy with production in which both goods enter consumption. Conditions for the existence of two types of cyclical equilibra are obtained in terms of elasticities of substitution in production and consumption, factor shares in production, and expenditure shares in consumption. The role of substitution possibilities and factor intensities is highlighted. It is shown that factor intensity assumptions are not enough to rule out cyclical equilibria and that limit cycles may exist for substantial amounts of substitution possibilities in production; what matters is the relative sizes of the consumption and production elasticities. The results are illustrated using CES functions. A dynamical simulation of an economy which exhibits a Hopf cycle is presented.     

Generic properties of the core and equilibria of pure exchange economies

We show that for an open dense set of markets with a continuum of traders the number of equilibrium allocations [which by the celebrated theorem of ^{Aumann (Econometrica. 1964, 32. 39–50)} coincide with the core allocations for such markets are finite. This presents a limiting case result that complements similar asymptotic theorems for cores of large economies proved by ^{Bewley (Econometrica 1973, 41, 425–454)}, and ^{Dierker (Journal of Mathematical Economics 1975, 2. 155–169)}. If we require that the measure on the space of agents be one with a finite number of atoms of equal weight, our reasoning recovers the classical results obtained by ^{Debreu (Econometrica. 1970. 38, 387–392)} for economies with a finite number of agents.     

Existence of marginal pricing equilibria in economies with externalities and non-convexities

We consider a general equilibrium model with externalities and non-convexities in production. The consumption sets, the preferences of the consumers and the production possibilities are represented by set-valued mappings to take into account possibility of external effects. There is no convexity assumption on the correspondences of production. We propose a definition of the marginal pricing rule, which generalizes the one used in the model without externality and, which satisfies a continuity assumption with respect to the external effect.We prove the existence of general equilibria under assumptions which allow us to encompass together the works on economies with externalities and convex conditional production sets, and those on marginal pricing equilibria in economies without externalities. We provide examples to illustrate the definition of the marginal pricing rule and to show the difference with the standard case.     

Existence of equilibria in atomless production economies with price dependent preferences

The existence of a competitive equilibrium for an economy with a measure space of agents is proved by reducing the economy to a three-person game and then applying Debreu's lemma (1952). In addition to the relative simplicity and shortness of the proof, we generalize known results [Aumann (1966), Hildenbrand (1970), Schmeidler (1969)] by allowing both price dependent preferences and satiation.