Existence Of Walrasian Equilibrium Research Articles

The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences

The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent preferences, without free disposal, and with an infinite-dimensional commodity space

A new proof of the existence of a Walrasian equilibrium with an infinite dimensional commodity space is provided, which allows agents’ preferences to be discontinuous. The new theorems include as corollaries the existence results of Mas-Collel, Yannelis and Zame, Araujo and Monteiro, and Mas-Collel and Richard, among others.

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences, without free disposal, and without compact consumption sets

We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.

Existence of a Walrasian Equilibrium in Locally Convex Topological Vector Spaces, with Interdependent Utility Functions

In this paper we extend the well known result of existence of Walrasian equilibrium given in Debreu (1952) to economies modeled over topological vector spaces. Our work is closely related to the previous results of Yannelis-Pranhakar (1983), and Tan and Yuan (1999), but unlike these papers we prove the existence of a Walrasian equilibrium in infinite dimensional economies with interdependent and price-dependent utilities.

Relaxed large economies with infinite-dimensional commodity spaces: The existence of Walrasian equilibria

Whereas “convexification by aggregation” is a well-understood procedure in mathematical economics, “convexification by randomization” has largely been limited to theories of statistical decision-making, optimal control and non-cooperative games. In this paper, in the context of classical Walrasian general equilibrium theory, we offer a comprehensive treatment of relaxed economies and their relaxed Walrasian equilibria: our results pertain to a setting with a finite or a continuum of agents, and a continuum of commodities modeled either as an ordered separable Banach space or as an L∞-space. As a substantive consequence, we demonstrate that the convexity hypothesis can be removed from the original large economy under the saturation hypothesis, and that existing results in the antecedent literature can be effortlessly recovered.

On voluntary and efficient allocations

AbstractA set of sufficiency conditions is presented for Walrasian equilibrium to be equivalent to voluntary and efficient allocations. An important corollary of this equivalence is that voluntary trade in non‐clearing markets, which is omnipresent in all market economies, is inefficient. Moreover, our assumptions generalize all the results on the Pareto‐efficiency properties of disequilibrium. This generalization also shows that the inefficiency of disequilibrium is germane to the existence of Walrasian equilibrium.

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences

We generalize the classical equilibrium existence theorems by dispensing with the assumption of continuity of preferences. Our new existence results allow us to dispense with the interiority assumption on the initial endowments. Furthermore, we allow for non-ordered, interdependent and price-dependent preferences.

Distributed Channel Assignment in Cognitive Radio Networks: Stable Matching and Walrasian Equilibrium

We consider a set of secondary transmitter-receiver pairs in a cognitive radio setting. Based on channel sensing and access performances, we consider the problem of assigning channels orthogonally to secondary users through distributed coordination and cooperation algorithms. Two economic models are applied for this purpose: matching markets and competitive markets. In the matching market model, secondary users and channels build two agent sets. We implement a stable matching algorithm in which each secondary user, based on his achievable rate, proposes to the coordinator to be matched with desirable channels. The coordinator accepts or rejects the proposals based on the channel preferences which depend on interference from the secondary user. The coordination algorithm is of low complexity and can adapt to network dynamics. In the competitive market model, channels are associated with prices and secondary users are endowed with monetary budget. Each secondary user, based on his utility function and current channel prices, demands a set of channels. A Walrasian equilibrium maximizes the sum utility and equates the channel demand to their supply. We prove the existence of Walrasian equilibrium and propose a cooperative mechanism to reach it. The performance and complexity of the proposed solutions are illustrated by numerical simulations.

Compensated Lindahl Equilibrium: Efficiency Theorems and Gains from Trade with Individual Production and Public Goods

In general equilibrium theory for private good economies, possible discontinuities in budget correspondences create obstacles to decentralizing Pareto efficient allocations, as well as to existence of Walrasian equilibrium with gains from trade. Then McKenzie's (1957) notion of compensated demand, and the associated notion of compensated Walrasian equilibrium, are important tools for surmounting such obstacles. Here, for economies that include public goods along with both individual and public production, we provide analogous results for a compensated notion of Lindahl equilibrium.

Dynamic Auction: A Tractable Auction Procedure

Dynamic auctions are trading mechanisms for discovering market-clearing prices and efficient allocations based on price adjustment processes. This paper studies the computational issues of dynamic auctions for selling multiple indivisible items. Although the decision problem of efficient allocations in a dynamic auction in general is intractable, it can be solved in polynomial time if the economy under consideration satisfies the condition of Gross Substitutes and Complements, which is known as the most general condition that guarantees the existence of Walrasian equilibrium. We propose a polynomial algorithm that can be used to find efficient allocations and introduce a double-direction auction procedure to discover a Walrasian equilibrium in polynomial time.

Undecidability of Uzawa equivalence theorem and LLPO (Lesser limited principle of omniscience)

The Uzawa equivalence theorem [H. Uzawa, Walras’s Existence Theorem and Brouwer’s Fixed Point Theorem, Economic Studies Quarterly 8 (1962) 59–62] showed (classically) that the existence of Walrasian equilibrium in an economy with continuous excess demand functions is equivalent to Brouwer’s fixed point theorem, that is, the existence of a fixed point for any continuous function from an n -dimensional simplex to itself. We examine the Uzawa equivalence theorem from the point of view of constructive mathematics, and show that this theorem, properly speaking, the assumption of the existence of a Walrasian equilibrium price vector in this theorem, implies LLPO (Lesser limited principle of omniscience), and so it is non-constructive.

Equilibria in incomplete assets economies with infinite dimensional spot markets

The paper studies the two period incomplete markets model where assets are claims on state contingent commodity bundles and there are no bounds on portfolio trading. The important results on the existence of equilibrium in this model assume that there is a finite number of commodities traded in each spot market and that preferences are given by smooth utility functions. With these assumptions an equilibrium exists outside an “exceptional” set of assets structures and initial endowments. The present paper extends these results by allowing for general infinite dimensional commodity spaces in each spot market. These include all the important commodity spaces studied in the literature on the existence of Walrasian equilibrium—in each spot market the consumption sets are the positive cone of an arbitrary locally solid Riesz space or of an ordered topological vector space with order unit or of a locally solid Riesz space with quasi-interior point. The paper establishes that even with our very general commodity spaces there exists an equilibrium for a “very” dense set of assets structures. Our approach is in the main convex analytic and the results do not require that preferences be smooth or complete or transitive. The concepts and techniques studied in this paper have important finite as well as infinite dimensional applications.

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.

Equilibria in Incomplete Assets Economies with Infinite Dimensional Spot Markets

The paper studies the two period incomplete markets model where assets are claims on state contingent commodity bundles and there are no bounds on portfolio trading. The important results on the existence of equilibrium in this model assume that there is a infnite number of commodities traded in each spot market and that preferences are given by smooth utility functions. With these assumptions an equilibrium exists outside an exceptional set of assets structures and initial endowments. The present paper extends these results by allowing for general infinite dimensional commodity spaces in each spot market. These include all the important commodity spaces studied in the literature on the existence of Walrasian equilibrium - in each spot market the consumption sets are the positive cone of an arbitrary locally solid Riesz space or of an ordered topological vector space with order unit or of a locally solid Riesz space with quasi-interior point.

On complexity of single-minded auction

We consider complexity issues for a special type of combinatorial auctions, the single-minded auction, where every agent is interested in only one subset of the commodities. First, we present a matching bound on the communication complexity for the single-minded auction under a general communication model. Next, we prove that it is NP-hard to decide whether Walrasian equilibrium exists in a single-minded auction. Finally, we establish a polynomial size duality theorem for the existence of Walrasian equilibrium for the single-minded auction.

Walrasian Equilibrium with Gross Substitutes

We study economies with indivisibilities that satisfy the gross substitutes (GS) condition. The simplest example of GS preferences are unit demand preferences. We prove that the set of GS preferences is the largest set containing unit demand preferences for which the existence of Walrasian equilibrium is guaranteed. We show that if a GS economy is replicated sufficiently many times, the equilibrium payment of any agent in the Vickrey–Clarke–Groves mechanism is equal to the value of the allocation he receives at the smallest Walrasian prices. The model extended to include production. Journal of Economic Literature Classification Numbers: D4, D44, D5, D51.

Random equilibra of random generalized games with applications to non-compact random quasi-variational inequalities

In the last two decades, the classical Arrow and Debreu result [3] on the existence of Walrasian equilibria has been generalized in many directions. MasColell [35] has first shown that the existence of an equilibrium can be established without assuming preferences to be total or transitive. Next, by using an existence theorem of maximal elements, Gale and Mas-Colell [20] gave a proof of the existence of a competitive equilibrium without ordered preferences. By using Kakutani’s fixed point theorem, Shafer and Sonnenschein [43] proved a powerful result on the “Arrow and Debreu Lemma” for abstract economies in which preferences may not be total or transitive but have open graphs. Meanwhile, Borglin and Keiding [8] proved a new existence theorem for a compact abstract economy with KF -majorized preference correspondences. Following their ideas, there have been a number of generalizations of the existence of equilibria for compact abstract economies (see e.g. Aliprantis et al. [1], Border [7], Chang [11], Debreu [13], Ding et al. [15]–[16], Flam [18], Florenzano [19], Hildenbrand and Sonnenschein [25], Kajii [27], Keiding [28], Mehta and Tarafdar [37], Shafer [42], Khan and Yannelis [29], Mas-Colell and Zame [36], Tian [53], Tan and Yuan [48]–[49], Tarafdar [51], Tarafdar and Mehta [52], Tulcea [54]–[55] etc.). These

On the existence of equilibria in economies with increasing returns

We provide a result on the existence of equilibria in economies with non-convex production sets. The principal assumption is a boundary condition and an income hypothesis on a subset of the production sets. As corollaries of our main results, we obtain the results of Beato and Mas-Colell (1985), Brown et al. (1986) and the classical result on the existence of Walrasian equilibria in convex economies. The result of Bonnisseau and Cornet (1988) is similar, but not directly comparable, to ours.

Existence of equilibria with infinitely many commodities: Banach lattices reconsidered

The purpose of this paper is twofold. The first aim is to present an extension of the results on the existence of Walrasian equilibrium to the infinite dimensional setting. The result depends on two crucial assumptions. These are the compactness of the collection of feasible allocations and the non-emptiness of the interior of the production set. The proof is a direct generalization of Bewley's (1972) proof for the L ∞ case. The second purpose of this paper is to show that the recent result of Mas-Colell (1986) on the existence of equilibrium for exchange economies on Banach lattices can be obtained through an argument based on the result outlined above. That is, exchange economies on Banach lattices with ‘uniformly proper’ preferences behave as though they were production economies in which the production sets have non-empty interior.