Atomless Measure Space Research Articles

Walrasian equilibrium theory with and without free-disposal: theorems and counterexamples in an infinite-agent context

This paper provides four theorems on the existence of a free-disposal equilibrium in a Walrasian economy: the first with an arbitrary set of agents with compact consumption sets, the next highlighting the trade-offs involved in the relaxation of the compactness assumption, and the last two with a countable set of agents endowed with a weighting structure. The results generalize theorems in the antecedent literature pioneered by Shafer–Sonnenschein in 1975, and currently in the form taken in He–Yannelis 2016. The paper also provides counterexamples to the existence of non-free-disposal equilibrium in cases of both a countable set of agents and an atomless measure space of agents. One of the examples is related to one Chiaki Hara presented in 2005. The examples are of interest because they satisfy all the hypotheses of Shafer’s 1976 result on the existence of a non-free-disposal equilibrium, except for the assumption of a finite set of agents. The work builds on recent work of the authors on abstract economies, and contributes to the ongoing discussion on the modelling of “large” societies.

On two definitions of a narrow operator on Köthe–Bochner spaces

We prove that for a Kothe–Banach space E with an order continuous norm over a finite atomless measure space \((\Omega ,\Sigma ,\mu )\) and for Banach spaces X, Y, the classes of narrow and weakly functionally narrow operators from a Kothe–Bochner space E(X) to a Banach space Y are coincident. We also obtain that in the general case, without the assumption of order continuity of the norm of E, the definitions of narrow and weakly functionally narrow operators from a Kothe–Bochner space E(X) to a Banach space Y are equivalent if and only if the set of all simple elements is dense in E(X).

Points of narrowness and uniformly narrow operators

It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.

On the “function” and “lattice” definitions of a narrow operator

We prove that the function and lattice definitions of a narrow operator defined on a Kothe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).

The Daugavet property in rearrangement invariant spaces

We study rearrangement invariant spaces with the Daugavet property. The main result of this paper states that under mild assumptions the only nonseparable rearrangement invariant space X X over an atomless finite measure space with the Daugavet property is L ∞ L_{\infty } endowed with its canonical norm. We also prove that a uniformly monotone rearrangement invariant space over an infinite atomless measure space with the Daugavet property is isometric to L 1 L_1 . As an application we obtain that an Orlicz space over an atomless measure space has the Daugavet property if and only if it is isometrically isomorphic to L 1 L_1 .

Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L0(μ) such that \({L^\infty(\mu) \subset E \subset L^1(\mu)}\). We study absolutely continuous linear operators from E to a locally convex Hausdorff space \({(X, \xi)}\). Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators Tm : L∞(μ) → X. In particular, we characterize relatively compact sets \({\mathcal{M}}\) in caμ(Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology \({\mathcal{T}_s}\) of simple convergence in terms of the topological properties of the corresponding set \({\{T_m : m \in \mathcal{M}\}}\) of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L∞(μ) → X.

Price-dependent consumption externalities and non-existence of equilibria

We construct a concrete example of an exchange economy E which admits no equilibria in the presence of price-dependent consumption externalities. In our example, agents form a finite atomless measure space ( T , T , μ ) . A reference coalition of each agent t at a price system p is the subset C ( t , p ) ⊂ T consisting of those agents whose income falls into a certain income range associated with t. The interdependence of the taste of agent t operates through a reference consumption vector η , which represents the aggregated effect on t’s taste of the various consumption choices made by all agents in C ( t , p ) . We choose η = 1 / ( μ [ C ( t , p ) ] ) ∫ C ( t , p ) x ( s ) d μ ( s ) , the mean consumption bundle of all agents in C ( t , p ) , where x ( s ) denotes the assignment of a commodity bundle to agent s and where the measure ( 1 / ( μ [ C ( t , p ) ] ) ) μ may be interpreted as the relative frequencies of contact that t makes with other members in C ( t , p ) . In light of the law of large numbers, the above η can be viewed as the consumption trend of the agents in C ( t , p ) , which influences the taste of t.

On several results about convex set functions

In 1979, in an interesting paper, R.J. Morris introduced the notion of convex set function defined on an atomless finite measure space. After a short period this notion, as well as generalizations of it, began to be studied in several papers. The aim was to obtain results similar to those known for usual convex (or generalized convex) functions. Unfortunately several notions are ambiguous and the arguments used in the proofs of several results are not clear or not correct. In this way there were stated even false results. The aim of this paper is to point out that using some simple ideas it is possible, on one hand, to deduce the correct results by means of convex analysis and, on the other hand, to emphasize the reasons for which there are problems with other results.

A Model of Commodity Differentiation with Indivisibilities and Production

This paper presents an existence theorem in a general equilibrium model of a production economy with commodity differentiation and indivisibilities. The model is motivated by the existence of markets with indivisible commodities, such as the markets for automobiles and computers. As is standard in the literature, the space of commodity characteristics is described by a compact metric space and a commodity vector is described by an integer-valued Borel measure on the space of commodity characteristics. An atomless measure space of producers and consumers is assumed to overcome the problem of non-convexity of the production and consumption sets induced by indivisibilities.

Pure strategy Markov equilibrium in stochastic games with a continuum of players

We discuss stochastic games which are played by a continuum of players. The player space is given by the atomless measure space of a unit interval with the Lebesgue measure. The state space in period t is given by ( S, Σ, μ t ), where S is a complete, separable metric space, Σ is the Borel sigma algebra and μ t is a fixed probability measure. There is a common action space K which is either a finite set or the K-ball of an Euclidean space R n . The transition probabilities in period t are product measurable in the realizations of the state and average responses and norm continuous in the current average response of the players and absolutely continuous with respect to the measure μ t . The payoffs of the players in these infinite horizon stochastic games are the discounted sum of the payoffs received every period. We show that these stochastic games have pure strategy Markov equilibrium points. In the process we are able to provide a characterization of Markov equilibrium strategy as the equilibrium strategies of a sequence of single-period constituent games.

Static and Dynamic Equilibria in Games With Continuum of Players

This paper is a study of a general class of deterministic dynamic games with an atomless measure space of players and an arbitrary time space. The payoffs of the players depend on their own strategy, a trajectory of the system and a function with values being finite dimensional statistics of static profiles. The players' available decisions depend on trajectories of the system. The paper deals with relations between static and dynamic open-loop equilibria as well as their existence. An equivalence theorem is proven and theorems on the existence of a dynamic equilibrium are shown as consequences.

On the robustness of factor structures to asset repackaging

The paper provides a framework to study asset repackaging in a large asset economy, modeled as an atomless measure space of assets. The main theorem shows that, given an initial economy with strict factor space F, any economy obtained by repackaging these assets has a unique factor space F′⊂ F. Thus, in contrast to earlier literature, factor structures are robust to the repackaging of asset.

On the strongest locally convex Lebesgue topology on Orlicz spaces

Let Lφ be an Orlicz space defined by an Orlicz function φ taking only finite values with \({{\rm lim\ inf}\atop {u\rightarrow \infty}}{\varphi(u)\over u} >0\) (not necessarily convex) over a complete, σ-finite and atomless measure space and let Lφ)n ∼ stand for the order continuous dual of Lφ. Then the strongest locally convex Lebesgue topology τ on Lφ (= the Mackey topology τ(Lφ, (Lφ)n ∼) is equal to the restriction of the strongest Lebesgue topology η on \(L^{\overline\varphi}\), where \(\overline\varphi\) is the convex minorant of φ and τ is generated by a family of norms defined by some convex Orlicz functions.

Markets with infinitely many commodities and a continuum of agents with non-convex preferences

Contrary to the finite dimensional set-up, the hypothesis of an atomless measure space of traders does not entail convexity of aggregate demand sets if there are infinitely many commodities. In this paper an assumption is introduced which sharpens the non-atomicity hypothesis by requiring that there are “many agents of every type.” When this condition holds, aggregate demand in an infinite dimensional setting becomes convex even if individual preferences are non-convex. This result is applied to prove the existence of competitive equilibria in such a context.

Continuous Nash equilibria

A topological space of players is introduced (as opposed to a measure space of players). We prove the existence of continuous Nash equilibria for continuous games. Intuitively this result says that there is an equilibrium where players with similar characteristics choose similar actions. Conceptual and technical issues are discussed in relation to non-cooperative games in general and large games (games with an atomless measure space of players) in particular.

Edgeworth's conjecture in atomless economies with a non-separable commodity space

This paper contains results for economies with infinite dimensional commodity space. (1) We establish a convexity of closure of aggregate demand set for economies with an atomless measure space of agents and non-separable Dedekind complete Riesz commodity space. (2) Under the assumption of boundedness of the set of all individual endowments, we establish the core-Walras equivalence theorem. This result includes as a special case Aumann and Hildenbrand finite dimensional theorems.

Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF:\(F:u \to \smallint _{\rm T} f(u(t),t)d\mu (t)\) and byDom(F) its domain, it is the set {ueL p(T,E):ū(t)=f(u),t)eL 1(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL p(T, E).

Strong barrelledness properties in Lebesgue-Bochner spaces

If (›; §;„) is a flnite atomless measure space and X is a normed space, we prove that the space Lp(„;X), 1• p•1is a barrelled space of class@0, regardless of the barrelledness of X: That enables us to obtain a localization theorem of certain mappings deflned in Lp(„;X): By \space we mean a \real or complex Hausdorfi locally convex space. Given a dual pair (E;F), as usual ae(E;F) denotes the weak topology on E: If B is a subset of a linear space E,hBi will denote its linear hull.

Consistent Solutions in Atomless Economies

We consider the problem of allocating a bundle of commodities among a group of agents who are collectively entitled to them. We prove: For an atomless economy with possibly satiated preferences, any solution that is efficient, equitable, and consistent must select allocations that are supported by equal-budget Walrasian equilibria with slack. WE CONSIDER THE PROBLEM of allocating a bundle of commodities among a group of agents who are collectively entitled to them. We search for systematic methods, or solutions, of resolving this problem. For a solution to be considered desirable, we propose that it should satisfy several requirements. First, it should be efficient. Second, it should be equitable. A variety of distributional require- ments have been proposed in the literature. The requirement we will impose here is that it select allocations at which no agent would prefer the equal division of the social endowment to his own consumption. In addition to these two basic efficiency and equity conditions, we require that the solution be consistent: the recommendations it makes for any economy are never contra- dicted by the recommendations it makes for any reduced economy obtained by allowing some subgroup of agents to leave the scene with their allotted consumptions. The requirement of consistency is very natural. Here is a motivation for it, offered by Thomson (1988): consider all the resources available on the planet earth; one would like not only that they be distributed fairly when the planet is considered as a whole but also that whatever resources end up in each continent, country, city,... be fairly distributed among the members of that subpopulation, when considered in isolation. In this paper we study economies with a large number of agents modelled as a complete atomless measure space. Our main result is a characterization of the solutions that satisfy all three requirements of efficiency, equity, and con- sistency. It follows from our result that, under standard assumptions on preferences, any such solution must select allocations that are supported by equal-income Walrasian equilibria. However, our characterization holds for more general economies since it applies to economies in which preferences are possibly satiated. Mas-Colell (1988) extended traditional general equilibrium theory to the class of such economies by introducing the notion of Walrasian

COMPLEMENTED COPIES OFc0IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.