Equilibria in large economies with a separable Banach commodity space and non-ordered preferences

The purpose of this paper is to provide an existence result of equilibria for economies with a measure space of agents, a non-trivial production sector and an infinite dimensional commodity space. The commodity space is modeled by an ordered separable Banach space whose positive cone has a non-empty interior. The discretization approach proposed in this paper, allows us to extend the existence results in Khan and Yannelis [Equilibrium in markets with a continuum of agents and commodities. In: Khan, M.A., Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces. Springer, Berlin, 1991] and Podczeck [Economic Theory 9 (1997) 585] to economies with a non-trivial production sector and with possibly non-ordered but convex preferences as well as partially ordered (possibly incomplete) but non-convex preferences.     

Equilibria in Banach lattices without ordered preferences

This paper establishes a very general result on the existence of competitive equilibria for exchange economies (with a finite number of agents) with an infinite-dimensional commodity space. The commodity spaces treated are Banach lattices, but no interiority assumptions on the positive cone are made; thus, the commodity spaces covered by this result include most of the spaces considered in economic analysis. Moreover, this result applies to preferences which may not be monotone, complete, or transitive; preferences may even be interdependent. Since preferences need not be monotone, the result allows for non-positive prices, and an exact equilibrium is obtained, rather than a free-disposal equilibrium.     

Arbitrage and equilibrium in unbounded exchange economies with satiation

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (1987) proved the existence of a competitive equilibrium, under a price no-arbitrage condition, without assuming either local or global nonsatiation. Werner’s existence result contrasts sharply with classical existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. First, we show that in unbounded exchange economies, even if some agents’ preferences are satiated, the absence of arbitrage is sufficient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades– that is, as long as agents’ preferences satisfy weak nonsatiation. Second, we provide a new approach to proving existence in unbounded exchange economies. The key step in our new approach is to transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy are equilibria of the original economy. What our approach makes clear is that it is precisely the condition of weak nonsatiation – a condition considerably weaker than local or global nonsatiation – that makes possible this transformation.     

The index theorem for a GEI economy when the degree of incompleteness is even

The purpose of this paper is to present a method to compute the index in an incomplete market economy. We show that generically in endowments and the asset structure the index theorem holds when the deficiency of markets, S−J, is even. The result is based on the indices of homotopies which have been presented for computation of an equilibrium in incomplete market economies.     

Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space

We prove an equilibrium existence theorem for economies with externalities, general types of non-convexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L_∞(M,M,μ), the space of all μ-essentially bounded M-measurable functions on M.As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough.Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.     

The Aubin private core of differential information economies

Abstract We establish characterizations of Radner equilibrium allocations via private core notions in the framework of differential information economies with a finite number of states of nature and a measure space of agents that may have atoms. The commodity space is an ordered separable Banach space whose positive cone admits an interior point. We introduce the notion of Aubin private core and prove that it provides a characterization of Radner equilibrium allocations. We show that the Aubin private core is equivalent to Edgeworth private equilibria and to privately non-dominated allocations of suitable associated economies. Mathematics Subject Classification (2000): 91B50, 91B44 Journal of Economic Literature Classification: D51, D82, D11     

Animal spirits and unemployment: a disequilibrium analysis

The past decade has seen a number of advances in modelling disequilibrium dynamics. This paper draws on separate approaches to disequilibrium dynamics to demonstrate a Keynesian result concerning the formal relevance of “animal spirits” in production economies. Specifically, it is shown that a parameter that can be associated with the “animal spirits” of firms is crucial to the stability of full employment equilibrium in a production economy. This approach to “animal spirits” is different to that taken by recent New Keynesian DSGE-type models, but similar in spirit to “Old Keynesian” approaches, including that of the General Theory. The corollary of the main conclusion is that price flexibility is not a sufficient condition for convergence on full employment equilibrium.     

Constrained inefficiency in GEI: A geometric argument

In this paper we use global analysis to study the welfare properties of general equilibrium economies with incomplete markets (GEI). Our main result is to show that constrained Pareto optimal equilibria are contained in a submanifold of the equilibrium set. This result is explicitly derived for economies with real assets and fixed aggregate resources, of which real numéraire assets are a special case. As a by product of our analysis, we propose an original global parametrization of the equilibrium set that generalizes to incomplete markets the classical one, first, proposed by Lange [Lange, O., 1942. The foundations of welfare economics. Econometrica 10, 215–228].     

Local public goods as indivisible commodities

A sharper version of Ellickson's (1979) result on the existence of an approximate Tichout equilibrium is shown to be true under much weaker assumptions. It is also shown that there exists a second-best policy regarding the bundle of public goods each local government offers to a resident. The existence result can also be seen as an extension of a result of Anderson, Khan and Rashid (1982) to economies with production. In this context, the main technical difficulty in that the approximate equilibrium allocation does not lie in the attainable and standard techniques of truncation are not adequate.     

A simple proof of the existence of an equilibrium when the weak axiom holds

An elementary proof of the existence of a competitive equilibrium is given for economies where the weak axiom holds. It is shown by an induction argument on the number of goods that the problem is reduced to the question of the existence of an equilibrium in a two-good economy.     

Equilibria and core of large economies

In this paper we apply results on regular economies to study equilibria and core in a non-differentiable framework. We show that the distributions of agents' characteristics of regular economies form a dense subset of all distributions of agents' characteristics. Therefore ‘most’ economies have equilibria which are contained in finitely many ?-balls. And the core of ‘most’ sufficiently large economies is contained in finitely many ?-balls centered at equilibrium allocations of these economies.     

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.     

Existence of smooth utilities on Banach lattices

In this paper, we focus our attention on the representability of a preference relation by differentiable utility functions when the consumption sets belong to an infinite dimensional commodity space. We obtain sufficient conditions for the existence of a C^r function representing a preference relation defined on an open subset of a Banach lattice.     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.     

Some results on the existence of utility functions on path connected spaces

This paper concerns the existence of utility representations for preferences defined on a path connected space X. This includes any convex set. A classical result of Eilenberg (1941) proves the existence of utility representations when the consumption set is connected and separable. In an infinite dimensional space the above result may not be useful, because we lack, in general, the separability of the space. The non-separable spaces L^∞ and ca(K) are typical examples in mathematical economics. In this paper we show that a continuous preference relation ≽, on X has a continuous utility representation if and only if it is countably bounded, i.e., there is some countable subset F of X such that for all x in X there exist y and z in F with y≽x≽z. An easy corollary states that any continuous preference which has a best and a worst point has a continuous representation. We also obtain a convex continuous preference on a Banach lattice that has not a utility representation, because it is not countably bounded.     

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.     

On the structure of moving average processes

This paper uses Hilbert space methods to develop a rigorous proof that the sum of two uncorrelated moving average processes of order q₁ and q₂ is an MA process of order q ≦ max (q₁, q₂). The methods establish the existence of suitable random shocks for the summed process, they illuminate relationships between the coefficients of such processes and their random shocks, and they provide means for proving that the random shocks of the summed processes are normal when the shocks of the underlying processes are normal. The role of the Wold decomposition is examined in terms of multiple representations of an MA process.     

Refinements and incentive efficiency in Walrasian models of insurance economies

The literature on Walrasian markets in large economies with adverse selection has used various equilibrium refinements, but has obtained no general incentive efficiency of equilibrium, namely when cross-subsidies are needed for efficiency. We show that the same refined equilibria may also be incentive inefficient even when general mechanisms that allow for such cross-subsidies are priced and can be traded. In the process, we also prove existence of some type of forward induction equilibria in this context.     

Equilibrium in abstract economies without ordered preferences

We extend a theorem of G. Debreu on the existence of equilibrium in a generalized N-person game (‘A Social Equilibrium Existence Theorem’). Applying standard techniques the result can be used to prove the existence of equilibrium in economies with interdependent preferences, price-dependent preferences and preferences which may be both non-transitive and non-complete. This generalizes a recent theorem of A. Mas-Colell.     

On the existence of walrasian equilibrium in large-square economies

An existence theorem for Walrasian equilibrium is demonstrated for an economy with a continuum of consumers and an infinite-dimensional commodity space, such as l₁ or c₀, having an ‘order-compatible’ basis.