Relaxation and Nonoccurence of the Lavrentiev Phenomenon for Nonconvex Problems

Cataloged from PDF version of article.The paper studies a relaxation of the basic multidimensional variational problem, when the class of admissible functions is endowed with the Lipschitz convergence introduced by Morrey. It is shown that in this setup, the integral of a variational problem must satisfy a classical growth condition, unlike the case of uniform convergence. The relaxations constructed here imply the existence of a Lipschitz convergent minimizing sequence. Based on this observation, the paper also shows that the Lavrentiev phenomenon does not occur for a class of nonconvex problems. © 2013 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg

Characterization of spannability of functions

Cataloged from PDF version of article.Possibility of representation of a value of convexification of function at the given point as a convex combination of the values of function (spannability) is studied. Spannability of functions turned out to be important in different fields e.g. in the study of quasi-cores of monetary economies with nonconvex preferences (mathematical economics), in the theory of relaxation of variation problems (variational calculus). Apparently, Shapley and Shubik (Econometrica, 1966, 34, 805-827) were first to discuss it

Relaxation of multidimensional variational problems with constraints of general form

Cataloged from PDF version of article.A relaxation of multidimensional variational problems with constraint of rather general form on gradients of admissible functions is investigated. It is assumed that the gradient of an admissible function belongs to an arbitrary bounded set. This relaxation involves as a class of admissible function the closure of the class of admissible functions of the original problem in the topology of uniform convergence, and uses a theorem characterizing this closure

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

Cataloged from PDF version of article.It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Economettica 40, 581-583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometfca 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 smother 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. Econometfca 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. (c) 2008 Elsevier B.V. All rights reserved

Concave measures and the fuzzy core of exchange economies with heterogeneous divisible commodities

Cataloged from PDF version of article.The main purpose of this paper is to prove the existence of the fuzzy core of an exchange economy with a heterogeneous divisible commodity in which preferences of individuals are given by nonadditive utility functions defined on a sigma-algebra of admissible pieces of the total endowment of the commodity. The problem is formulated as the partitioning of a measurable space among finitely many individuals. Applying the Yosida-Hewitt decomposition theorem, we also demonstrate that partitions in the fuzzy core are supportable by prices in L-1. (c) 2012 Elsevier B.V. All rights reserved

A characterization of polyhedral convex sets

This paper describes a class of convex closed sets, S, in Rn for which the following property holds: for every correspondence defined on a probability space with relative open values in S its integral is a relative open subset of S. It turns out, that the only closed convex sets in R n having this property are generalized polyhedral convex sets. In particular, the only compact convex sets in Rn having this property are polytopes

Boundary behavior of excess demand and existence of equilibrium

Cataloged from PDF version of article.This paper presents a market equilibrium existence theorem that generalizes and unifies many well-known results. The importance of the theorem is illustrated by applications to large exchange economics. A further extension of Aumann's Existence theorem is obtained which dispenses with the monotony assumption on preferences. In addition the market equilibrium results of Grandmont and Neuefeind are compared. It is shown that the boundary condition of Grandmont's result is equivalent to some natural relaxation of Neuefeind's. In the case of two commodities they are equivalent but for a greater number of commodities Neuefeind's condition is stronger. (C) 1999 Academic Press

A theory of a heterogeneous divisible commodity echange economy

Cataloged from PDF version of article.In theoretical land economics the existence of a competitive equilibrium with an additive price is considered problematic. This paper studies the exchange and allocation of a heterogeneous divisible commodity such as land, which is modeled as a measurable space. In a 'land' trading economy with unordered convex preferences, the existence of a competitive equilibrium with an additive equilibrium price is proved. This paper demonstrates also the existence of a weak core and a fair allocation. (C) 2010 Elsevier B.V. All rights reserved

INTERPRETATION OF AUBIN FUZZY COALITIONS AND THEIR EXTENSION - RELAXATION OF FINITE EXCHANGE ECONOMICS

In this article a new interpretation of Aubin's fuzzy coalitions is given. This interpretation is based on associating each finite pure exchange economy with some large nonatomic economy. Then some connections between fuzzy core of finite economy and the core of associated nonatomic economy are established. From these results it is derived that for the case of convex preferences an allocation belonging to the fuzzy core is an equilibrium allocation. The assertion is not true in nonconvex case. We proceed by using the idea of J.P. Aubin - amplifying the class of coalitions in such a manner that an allocation unblocked by them is an equilibrium allocation also in the nonconvex case

Alpha-maxmin solutions to fair division problems and the structure of the set of Pareto Utility profiles

Cataloged from PDF version of article.A simple proof of the equivalence of Pareto optimality plus positiveness and alpha-maxmin optimality, dispensing with the assumption of closedness of the utility possibility set, is given. Also the structure of the set of Pareto optimal utility profiles is studied. (C) 2008 Elsevier B.V. All rights reserved.A simple proof of the equivalence of Pareto optimality plus positiveness and α-maxmin optimality, dispensing with the assumption of closedness of the utility possibility set, is given. Also the structure of the set of Pareto optimal utility profiles is studied