Walrasian Allocations Research Articles

Some equivalence results for a bargaining set in finite economies

AbstractWe present a bargaining set for finite economies using Aubin's () veto and show its coincidence with the set of Walrasian allocations, providing a discrete approach to the characterization of competitive equilibria obtained by Mas‐Colell () for continuum economies. We also study how the restriction on the formation of coalitions affects the bargaining set. In the last part of the work, using our equivalence result along with some known characterizations of Walrasian allocations, we state additional interpretations of the bargaining set.

Bargaining set with endogenous leaders: A convergence result

We provide a notion of bargaining set for finite economies where the proponents of objections (leaders) are endogenous. We show its convergence to the set of Walrasian allocations when the economy is replicated.

Bargaining sets in finite economies

We provide a notion of bargaining set for a finite production economy based on a two-step veto mechanism à la Aubin (1979). We show that this bargaining set and the set of Walrasian allocations coincide. At the light of our result we refine Mas-Colell’s bargaining set for replicas of a finite economy. Our main result shows the persistence of Anderson et al.’s (1997) non-convergence of the bargaining sets to the set of Walrasian allocations. In addition, we analyze how the restriction on the formation of coalitions affects the bargaining set.

A limit result on bargaining sets

We introduce a notion of bargaining set for finite economies and show its convergence to the set of Walrasian allocations.

Externalities in the Edgeworth Box

It is shown how, in a general equilibrium framework, the presence of an externality can be illustrated in a standard Edgeworth box. The key to this is the introduction of what I call objective indifference curves. Contract curves with and without the externality can be depicted in the same box which makes it evident why standard Walrasian equilibrium allocations - corresponding to the contract curve without the externality - are inefficient. I further illustrate how the introduction of a market for the right to generate the externality renders possible to obtain any Pareto-efficient allocation as a competitive general equilibrium solution, possibly after a reallocation of initial endowments, thus extending the Second Welfare Theorem to hold in the presence of an externality, too.

Indivisible commodities and an equivalence theorem on the strong core

We consider a pure exchange economy with finitely many indivisible commodities that are available only in integer quantities. We prove that in such an economy with a sufficiently large number of agents, but finitely many agents, the strong core coincides with the set of expenditure-minimizing Walrasian allocations. Because of the indivisibility, the preference maximization does not imply the expenditure minimization. An expenditure-minimizing Walrasian equilibrium is a state where, under some price vector, all agents satisfy both the preference maximization and the expenditure minimization.

Indivisible Commodities and an Equivalence Theorem on the Strong Core

We consider a pure exchange economy with finitely many indivisible commodities that are available only in integer quantities. We prove that in such an economy with a sufficiently large number of agents, but finitely many agents, the strong core coincides with the set of expenditure-minimizing Walrasian allocations. Because of the indivisibility, the preference maximization does not imply the expenditure minimization. An expenditure-minimizing Walrasian equilibrium is a state where, under some price vector, all agents satisfy both the preference maximization and the expenditure minimization.

Fair allocations in large economies with unequal production skills

This paper considers the problem of fair allocation among individuals with unequal production skills. We introduce the concept of productivity‐adjusted average no‐envy. It is shown that equal‐income Walrasian allocations are the only surviving allocations that are productivity‐adjusted average envy‐free and efficient when the original economy is infinitely replicated. We also examine local versions of productivity‐adjusted average no‐envy and other equity concepts.

Designing stable mechanisms for economic environments

We study the design of mechanisms that implement Lindahl or Walrasian allocations and whose Nash equilibria are dynamically stable for a wide class of adaptive dynamics. We argue that supermodularity is not a desirable stability criterion in this mechanism design context, focusing instead on contractive mechanisms. We provide necessary and sufficient conditions for a mechanism to Nash-implement Lindahl or Walrasian allocations, show that these conditions are inconsistent with the contraction property when message spaces are one-dimensional, and then show how to use additional dimensions to achieve dynamic stability while gaining budget balance out of equilibrium.

Multilateral bargaining and Walrasian equilibrium

A class of bargaining games in which agents bargain over prices and maximum trading constraints is considered: It is proved that all the Stationary Subgame Perfect Equilibria of these games implement Walrasian allocations as the bargaining frictions vanish. The result holds for any number of agents and is robust to different specifications of the bargaining process.

A Note on the Private Core and Coalitional Fairness under Asymmetric Information

We study relations among Walrasian expectations allocations, coalitional fair allocations and the private core of economies with uncertainty and asymmetric information. Our analysis covers finite exchange models, as well as models of mixed markets consisting of some large traders and an ocean of small traders. The adopted notion of coalitional fairness requires that: 1. Under a “c-fair” allocation, no coalition could benefit from achieving the net trade of some other disjoint coalition; 2. Coalition bargaining takes place without information sharing among agents. We introduce a notion of restricted Walrasian expectations allocation and examine its relations with c-fairness.

Local envy-freeness and equal-income Walrasian allocations

This paper introduces a local version of envy-freeness and investigates its implications in a continuum agent economy with connected preferences. We show that the set of locally envy-free and Pareto efficient allocations coincides with the set of equal-income Walrasian allocations.

Local strict envy-freeness in large economies

This paper proposes a concept of local strict envy-freeness ( LS-envy-freeness), which is a local version of Zhou’s ( 1992) strict envy-freeness, and investigates its implications in large economies. In spite of the weakness of this concept, it works effectively by combining with efficiency. It is shown that an LS-envy-free and efficient allocation is a strict envy-free allocation. That is, efficiency expands the local version of strict envy-freeness into strict envy-freeness. Therefore, the set of LS-envy-free and efficient allocations coincides with the set of equal-income Walrasian allocations.

Assignment Messages and Exchanges

“Assignment messages” are maximally general messages to describe substitutable preferences by means of a linear program. With “integer assignment messages,” there exist integer-valued Walrasian allocations, extending a result of Lloyd S. Shapley and Martin Shubik (1971). Any pure Nash equilibrium profile of the Walrasian mechanism with participants limited to assignment messages is also a Nash equilibrium of the unrestricted Walrasian mechanism. Assignment exchanges are generalizations of single-product double auctions and are related to ascending multi-product clock auctions and the Vickrey mechanism. Assignment messages also have additional applications in mechanism design. (JEL D44, D82)

Equilibrium Blocking in Large Quasilinear Economies

We study information transmission in large interim quasilinear economies using the theory of the core. We concentrate on the core with respect to equilibrium blocking, a core notion in which information is transmitted endogenously within coalitions, as blocking can be understood as an equilibrium of a communication mechanism used by players in coalitions. We consider independent, ex-post and signal-based replicas of the basic economy. For each, we offer an array of negative and positive convergence results as a function of the complexity of the mechanisms used by coalitions. We identify conditions under which asymmetric information remains as an externality and non-market outcomes stay in the core, as well as those for the core to converge to the set of incentive compatible ex-post Walrasian allocations. Further, all the results are robust to the relaxation of the incentive constraints, and hence suggest a process through which information may get incorporated into a fully revealing equilibrium price function.

Walrasian social orderings in exchange economies

We study social ordering functions in exchange economies. We show that if a social ordering function satisfies certain Pareto, individual rationality, and local independence conditions, then (i) the set of top allocations of the chosen social ordering is contained in the set of Walrasian allocations and is typically non-empty, and (ii) all individually rational but non-Walrasian allocations are typically ranked indifferently. Thus, such a social ordering function is quite similar to the Walrasian correspondence, which can be regarded as the social ordering function whose associated indifference classes are the set of Walrasian allocations and the set of other allocations.

Information Transmission and Core Convergence in Quasilinear Economies

We study core convergence in interim quasilinear economies with asymmetric information, concentrating on core notions in which information is transmitted endogenously within coalitions and the incentive constraints are relevant. Specifically, we shall focus on the credible core and randomized mediated core concepts. We consider independent replicas of the basic economy: independent copies of the economy in which each individual's utility only depends on the information of the individuals who belong to the same copy. We provide an example in which core convergence does not obtain for the Dutta-Vohra credible core and for Myerson's randomized mediated core. On the other hand, we establish a positive convergence result for a refinement of Myerson's core for which information disseminates across coalitions within a given random blocking mechanism. Under some conditions, this core converges to the set of incentive compatible ex-post Walrasian allocations.

Core, equilibria and incentives in large asymmetric information economies

We consider a perfectly competitive ex ante economy with a continuum of agents and negligible asymmetric information. For such an economy we recast the basic classical results on the existence of Walrasian equilibrium, core equivalence, and the blocking size of coalitions. Moreover, we examine the incentive compatibility of the ex ante Pareto, core and Walrasian allocations.

Perfect competition in asymmetric information economies: compatibility of efficiency and incentives

The idea of perfect competition for an economy with asymmetric information is formalized via an idiosyncratic signal process in which the private signals of almost every individual agent can influence only a negligible group of agents, and the individual agents’ relevant signals are essentially pairwise independent conditioned on the true states of nature. Thus, there is no incentive for an individual agent to manipulate her private information. The existence of incentive compatible, ex post Walrasian allocations is shown for such a perfectly competitive asymmetric information economy with or without “common values”. Consequently, the conflict between incentive compatibility and Pareto efficiency is resolved exactly, and its asymptotic version is derived for a sequence of large, but finite private information economies.

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.