Rabee Tourky

A Theory of Value with Non-linear Prices: Equilibrium Analysis beyond Vector Lattices

This paper presents a new theory of value with a personalized pricing system that naturally induces a family of non-linear prices. This affords a coordinate free theory of value in which the analysis is without any lattice theoretic considerations. When commodity bundles are perfectly decomposable the generalized prices become linear and the analysis specializes to the Walrasian model. This happens, for instance, whenever the commodity space is a vector lattice and consumption sets coincide with the positive cone. Our approach affords theorems on the existence of equilibrium and provides a value-based characterization of Pareto optimality and Edgeworth equilibrium where the Walrasian linear price-based characterization fails. The analysis has applications in the finite as well as the infinite dimensional setting. Journal of Economic Literature Classification Numbers: C62, C71, D46, D51, D61.

Markets with Many More Agents than Commodities: Aumann's “Hidden” Assumption

Abstract We address a question posed by J.-F. Mertens and show that, indeed, R. J. Aumanns classical existence and equivalence theorems depend on there being “many more agents than commodities.” We show that for an arbitrary atomless measure space of agents there is a fixed non-separable infinite dimensional commodity space in which one can construct an economy that satisfies all the standard assumptions but which has no equilibrium, a core allocation that is not Walrasian, and a Pareto efficient allocation that is not a valuation equilibrium. We identify the source of the failure as the requirement that allocations be strongly measurable. Our main example is set in a commodity–measure space pair that displays an “acute scarcity” of strongly measurable allocations—where strong measurability necessitates that consumer choices be closely correlated no matter the prevailing prices. This makes the core large since there may not be any strongly measurable improvements even though there are many weakly measurable strict improvements. Moreover, at some prices the aggregate demand correspondence is empty since disaggregated demand has no strongly measurable selections, though it does have weakly measurable selections. We note that our example can be constructed in any vector space whose dimension is greater than the cardinality of the continuum—that is, whenever there are at least as many commodities as agents . We also prove a positive core equivalence result for economies in non-separable commodity spaces. Journal of Economic Literature Classification Numbers: C62, C71, D41, D50.

Simple complexity from imitation games

We give simple proofs of refinements of the complexity results of Gilboa and Zemel (1989), and we derive additional results of this sort. Our constructions employ imitation games, which are two person games in which both players have the same sets of pure strategies and the second player wishes to play the same pure strategy as the first player.

Linear and non-linear price decentralization

Compendious and thorough solutions to the existence of a linear price equilibrium problem, the second welfare theorem, and the limit theorem on the core are provided for exchange economies whose consomption sets are the positive cone of arbitrary ordered Frechet-dispensing entirely with the assumption that the vector ordering of the commodity space is a lattice. The motivation comes from economic applications showing the need to bring within the scope of equilibrium theory vector orderings that are not lattices, which arise in the typical model of portfolio trading with missing options. The assumptions are on the primitives of the model. They are bounds on the marginals of non-linear prives and for omega-proper economies they are both sufficient and necessary.

Cone Conditions in General Equilibrium Theory

Abstract The modern convex-analytic rendition of the classical welfare theorems characterizes optimal allocations in terms of supporting properties of preferences by non-zero prices. While supporting convex sets in economies with finite dimensional commodity spaces is usually a straightforward application of the separation theorem, it is not that automatic in economies with infinite dimensional commodity spaces. In the last 30 years several characterizations of the supporting properties of convex sets by non-zero prices have been obtained by means of cone conditions. In this paper, we present a variety of cone conditions, study their interrelationships, and illustrate them with many examples. Journal of Economic Literature Classification Numbers: D46, D51.

Non-marketed options, non-existence of equilibria, and non-linear prices

Abstract This paper presents a surprising example that shows that the lattice theoretic properties in Mas-Colells (1986) seminal work are relevant to the existence of equilibrium even when the commodity space is finite dimensional. The example is a two-period securities model with a three-dimensional portfolio space and two traders. The paper identifies a non-marketed call option that fails to have a minimum cost super-replicating portfolio. Using this option, we construct an economy that satisfies all of Mas-Colells assumptions, except that the three-dimensional commodity space is not a vector lattice. In this economy, there is no Walrasian equilibrium and the second theorem of welfare economics fails . Our example has important finite- as well as infinite-dimensional implications. It is also an example of a “well behaved” economy in which optimal allocations that are not supported by linear Walrasian prices are decentralized by the non-linear prices studied in Aliprantis–Tourky–Yannelis (2001).

Truthful implementation and preference aggregation in restricted domains

In a setting where agents have quasi-linear utilities over social alternatives and a transferable commodity, we consider three properties that a social choice function may possess: truthful implementation (in dominant strategies); monotonicity in differences; and lexicographic affine maximization. We introduce the notion of a flexible domain of preferences that allows elevation of pairs and study which of these conditions implies which others in such domain. We provide a generalization of the theorem of Roberts (1979) [36] in restricted valuation domains. Flexibility holds (and the theorem is not vacuous) if the domain of valuation profiles is restricted to the space of continuous functions defined on a compact metric space, or the space of piecewise linear functions defined on an affine space, or the space of smooth functions defined on a compact differentiable manifold. We provide applications of our results to public goods allocation settings, with finite and infinite alternative sets.

The super order dual of an ordered vector space and the Riesz-Kantorovich formula

A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L is a vector lattice and f and g are order bounded linear functionals on L, then their supremum (least upper bound) f V g exists in L ∼ and for each x ∈ L + it satisfies the so-called Riesz-Kantorovich formula: [f ∨ g](x) = sup{f(y) + g(z): y,z ∈ L + and y + z = x}. Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f and g on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the abovementioned problem and to the properties of the Riesz-Kantorovich formula.

The Riesz-Kantorovich formula and general equilibrium theory

Abstract Let L be an ordered topological vector space with topological dual L′ and order dual L~. Also, let f and g be two order-bounded linear functionals on L for which the supremum f∨g exists in L. We say that f∨g satisfies the Riesz–Kantorovich formula if for any 0≤ω∈L we have f∨g(ω)= sup 0≤x≤ω [f(x)+g(ω−x)]. This is always the case when L is a vector lattice and more generally when L has the Riesz Decomposition Property and its cone is generating. The formula has appeared as the crucial step in many recent proofs of the existence of equilibrium in economies with infinite dimensional commodity spaces. It has also been interpreted by the authors in terms of the revenue function of a discriminatory price auction for commodity bundles and has been used to extend the existence of equilibrium results in models beyond the vector lattice settings. This paper addresses the following open mathematical question: ⋅ Is there an example of a pair of order-bounded linear functionals f and g for which the supremum f∨g exists but does not satisfy the Riesz–Kantorovich formula? We show that if f and g are continuous, then f∨g must satisfy the Riesz–Kantorovich formula when L has an order unit and has weakly compact order intervals. If in addition L is locally convex, f∨g exists in L~ for any pair of continuous linear functionals f and g if and only if L has the Riesz Decomposition Property. In particular, if L~ separates points in L and order intervals are σ(L,L~)-compact, then the order dual L~ is a vector lattice if and only if L has the Riesz Decomposition Property — that is, if and only if commodity bundles are perfectly divisible.

Economic Equilibrium: Optimality and Price Decentralization

Mathematical economics has a long history and covers many interdisciplinary areas between mathematics and economics. At its center lies the theory of market equilibrium. The purpose of this expository article is to introduce mathematicians to price decentralization in general equilibrium theory. In particular, it concentrates on the role of positivity in the theory of convex economic analysis and the role of normal cones in the theory of non-convex economies.