Atomless Economy Research Articles

A note on bargaining sets in atomless economies

A note on bargaining sets in atomless economies

Fairness and formation rules of coalitions

AbstractIn this paper, we study the problem of a fair redistribution of resources among agents of an exchange economy and how certain limitations imposed on coalition formation may impact the set of allocations judged fair. The study is conducted in atomless economies as well as in the so‐called mixed markets.

Absence of Envy among “Neighbors” Can Be Enough

Abstract We assume that the set of agents is decomposed into several classes containing individuals related each other in some way, for example groups of neighbors. We propose a new definition of fairness by requiring efficiency and envy-freeness only within each group. We identify conditions under which absence of envy among “neighbors” is enough to ensure fairness in the entire society. We also show that equal-income Walrasian equilibria are the only fair allocations according to our notion, deriving as corollaries the equivalence theorems of Zhou (1992) and Cato (2010). The analysis is conducted in atomless economies as well as in mixed markets.

Core equivalence with differentiated commodities

This paper presents improved core equivalence results for atomless economies with differentiated commodities in the framework of Ostroy and Zame (1994). Commodity bundles are elements of the space M(K) of signed Borel measures on a compact space K of commodity characteristics. Ostroy and Zame provide two sufficient conditions for core equivalence: It is sufficient that markets are “physically thick”, so that there are many suppliers of every commodity, or that markets are “economically thick”, so that consumers are sufficiently willing to substitute commodities with a similar composition for each other. The sufficient conditions in Ostroy and Zame (1994) all imply that there are “many more agents than commodities”, an idea of Aumann that was formalized and discussed in Tourky and Yannelis (2001) and Greinecker and Podczeck (2016). We generalize the framework in Ostroy and Zame (1994) and weaken their sufficient conditions to not imply the presence of “many more agents than commodities”. In particular, we drop the requirement that K is metrizable from the basic model, the requirement of an uniform bound on endowments from the condition of “physically thick markets”, and the requirement that preferences are weak∗-continuous from the condition of “economically thick markets”. Core equivalence still holds, showing that “many more agents than commodities” are not needed for core equivalence in models of commodity differentiation.

Multiple markets: new perspective on nonlinear pricing

We discuss how linear equilibrium pricing in certain competitive market structures may represent nonlinear equilibrium pricing of Aliprantis et al. (J Econ Theory 100:22–72, 2001, J Econ Theory 121:51–74, 2005). Their work extends the theory of value beyond the scope of the Walrasian single market linear price model. Our arguments include a new and general result on the existence of linear price equilibrium with multiple markets. Each market has its own price vector (linear functional), and agents’ involvement in various markets is heterogeneous. As a result, price differences across markets may prevail in equilibrium. We present an example in which single market linear price equilibrium does not exist, but certain corresponding equilibrium with two markets does. This example is a particular instance of a prevalent nonexistence problem in atomless economies with differential information. Bypassing the nonexistence problem is one of the achievements of the nonlinear equilibrium theory. Our equilibrium with multiple markets, on the other hand, offers a solution with a more standard economic interpretation. Besides, our general framework is a model of multiple markets in their own right, and our results are related to the role of economic intermediation and bilateral trade.

Replica core limit theorem for economies with satiation

Dividend equilibrium, defined by Aumann and Dreze (1986), is one of the most general competitive equilibrium concepts for markets that include satiated consumers. Konovalov (2005) shows a core equivalence theorem to the dividend equilibrium using the concept of rejective core. Konovalov’s argument, however, is based on the setting of a continuum atomless economy with a finite number of types, and the core limit problem for dividend equilibrium remains unsolved. In a previous paper, Urai and Murakami (2016), we provided a generalization of the Debreu–Scarf (1963) core limit theorem for monetary overlapping generations economies under an extended concept of replica core allocation. In this paper, we show that the concept and method also provide rejective core limit and equivalence theorems for economies with satiation.

Edgeworth’s conjecture and the number of agents and commodities

We answer the question asked by Robert Aumann as to whether core equivalence depends on there being “many more agents than commodities.” We show that for a large class of commodity spaces, which might be infinite-dimensional and even non-separable, core equivalence is indeed equivalent to the presence of “many more agents than commodities” when allocations are Bochner integrable. By contrast, we show that in a classical model of an atomless economy with an infinite-dimensional commodity space, the model where the commodity space is $$L_\infty (\mu )$$ with the Mackey topology and allocations are Gelfand integrable, core equivalence holds in full generality, even when there are “many more commodities than agents.” The assumptions we make on economies are much weaker than what is commonly used in core equivalence results for infinite-dimensional commodity spaces and reduce to Aumann’s original assumptions when there are finitely many commodities.

On Vind’s theorem for an economy with atoms and infinitely many commodities

We extend Vind’s classical theorem on the measure of blocking coalitions valid in finite dimensional atomless economies (see Vind (1972)), to include the possibility of infinitely many commodities as well as the presence of atoms. The commodity space is assumed to be an ordered Banach space which has possibly the empty positive cone. The lack of interior points is compensated by an additional assumption of a cone of arbitrage that allows us to use Lyapunov’s convexity theorem in its weak form. The measure space of agents involves both negligible and non negligible traders. The extension is proved in the general class of Aubin coalitions for which a suitable version of Grodal’s result (Grodal (1972)) is also formulated. Our results wish to point out the relevance of cone conditions dealing with blocking coalitions of arbitrary measure or weight.

Blocking efficiency in an economy with asymmetric information

In this paper, we study a pure exchange atomless economy with asymmetric information and having an ordered Banach space with an interior point in its positive cone as the commodity space. An extension of the main theorem in Vind (1972) to the private core without free disposal is established. As a particular case of this result, a solution to a problem mentioned in Pesce (2010) is derived.

The bargaining set in strategic market games

This paper presents a hybrid equilibrium notion that blends together the ‘cooperative’ and the ‘noncooperative’ theories of competition. In particular, the Mas-Colell bargaining set has been modified in order to accommodate the features of strategic market games. In other words, allocations, objections and counter objections of the standard bargaining set theory are described for an economy, where trades among groups of individuals are conducted via the Shapley–Shubik mechanism. In the main part of the paper, it is proved that in atomless economies the allocations resulting from this equilibrium notion are competitive.

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.

A three way equivalence

In view of the well known core equivalence results in atomless economies, coincidence of market game equilibrium allocations with competitive allocations is tantamount to a three way equivalence between market game mechanisms, competitive equilibria and the core. Based on this idea we study equilibrium refinements of market games, which allow us to use the core equivalence machinery in order to provide an exact market game characterization of competitive equilibria.

The measure of blocking coalitions in differential information economies

We provide an evaluation of the measure of privately blocking coalitions in differential information economies. In the case of atomless economies, it is proved that for a Pareto optimal allocation that is not a Walrasian expectations equilibrium, to any symmetric profile there corresponds a ball such that "almost half” of the profiles it contains are privately blocking. Analogous results are proved in the case of finite differential information economies for generalized coalitions and social coalition structures. From a different point of view, the paper can be considered as a contribution showing private core equivalence theorems under restrictions on coalition formation.

Organization of production in a competitive economy

In the present paper we introduce a general equilibrium model of the organization of production and analyze the optimality and existence of an equilibrium. We introduce the production technology of the economy as a collection of production processes and allow each potential producer to choose the production processes to operate. We model organization and transaction costs by means of an organizational technology that determines the production set of a producer as a function of the production processes that the producer chooses to operate. For a competitive economy, we show the equilibrium organization of production is optimal and then prove a general existence theorem for an atomless economy.

Erratum to "Do pure indivisibilities prevent core equivalence? core equivalenc theorem in an atomless economy with purely indivisible commodities only

Erratum to "Do pure indivisibilities prevent core equivalence? core equivalenc theorem in an atomless economy with purely indivisible commodities only

Do pure indivisibilities prevent core equivalence? Core equivalence theorem in an atomless economy with purely indivisible commodities only

We consider a pure exchange economy with a continuum of agents and finitely many indivisible commodities. Every commodity can be consumed only in integer amounts. Thus, agents’ preferences are locally satiated and no commodity bundle has necessarily local cheaper points. We introduce a core which is an intermediate concept between the strong core and the weak core. In our economy, this core is the most natural concept in the sense that it coincides with the set of all exactly feasible Walras allocations.

Discrete representations of a continuum economy

We consider an atomless economy in which the continuum of agents is represented by a real interval. By dividing the interval and associating to every agent in each subinterval the same initial endowments and preferences, we define sequences of discrete economies as approximations to the initial continuum economy. We obtain convergence results for the core (or, alternatively, for the set of Walrasian allocations) of the continuum economy in terms of the cores of the approximating discrete economies. Finally, we state some counterexamples which provide a boundary for more general results in this framework.

Symmetric von Neumann–Morgenstern stable sets in pure exchange economies

We show that in a (finite) neoclassical pure exchange economy (with a weaker monotonicity condition than the usual one): if the number of traders of each type is the same and every type has a corner on one of the commodities, then the set of all symmetric Pareto-optimal allocations in the economy (i.e., Pareto-optimal allocations which assign indifferent commodity bundles to traders of the same type) is a von Neumann–Morgenstern stable set. Moreover, this is the unique stable set of symmetric allocations. A similar result holds in atomless economies with a finite number of types without the convexity assumption on the preference relations. It is also not necessary to assume (in the atomless case) that the members of the types' partition have the same measure.

Manipulation-proof equilibrium in atomless economies with commodity differentiation

In this paper, we introduce a perfect competition test which checks the incentives of arbitrarily small coalitions to behave strategically in endowments and preferences. We apply this coalitional incentive compatibility test to atomless economies with a continuum of differentiated commodities. We show that, under thickness conditions, economies with a finite number of types and economies whose set of agents' preferences is compact, pass this perfect competition test. Limiting results for replica economies are also presented.

Equitable nature of core allocations in atomless economies

The purpose of this paper is to prove the equal treatment property for the $\star$ -core allocations of an atomless economy without any condition on the data of economy. This result prompts the same property for the core allocations.