Operator theorems with applications to distributive problems and equilibrium models
type: Item
instance of: scholarly article
title: Operator theorems with applications to distributive problems and equilibrium models
zbMATH Open document ID: 0781.90016
author: Antonio Villar
published in: Lecture Notes in Economics and Mathematical Systems
publication date: 1992-09-18T00:00:00Z
review text: An operator is defined as a mapping from a topological vector space into itself. The operators in this book are continuous functions or upper hemi continuous correspondences (set valued functions). Most problems in economic theory (like existence of equilibrium, existence of optimal solutions, fair division, etc.) can be reduced to one out of eight problems, where operators are applied. These eight problems, presented in the first part of the book are: (1) equation systems, (2) optimization problems, (3) complementarity problems, (4) fixed points, (5) maximization of a binary relation, (6) quasi diagonal images, (7) variational inequalities and (8) nonempty intersection problems. (6) refers to a problem where the vector that solves it should satisfy the condition that a component is zero, if the value of the mapping, with the same index, is not the smallest. In (7) the inner product of the solution with its image should be minimal. That these problems are related is well known. The author systematically investigates the relationships and proves equivalences if the mapping satisfies certain conditions. He presents a number of theorems on the solvability of the eight problems under appropriate assumptions. The second part of the book presents some applications: fair division, equilibrium with nonconvex technologies, Lindahl equilibria, nonlinear input output. It appears that these problems of economic theory are directly equivalent to one of the eight problems. Using a theorem on the existence of solutions of problem (7) existence of a ``market'' equilibrium with nonconvex technologies is easily proved, where usually a detour via a fixed point is made. Similarly existence of a Lindahl equilibrium directly follows as a solution of problem (6). In economic theory usually theorems are proved by reducing the problem to a fixed point problem, say, and showing then that the fixed point corresponds to a solution of the problem at hand. This book contains a number of beautifu…
reviewed by: Claus Weddepohl
Mathematics Subject Classification ID: 91B50
Mathematics Subject Classification ID: 47N10
zbMATH DE Number: 53509
zbMATH Keywords: topological vector space
zbMATH Keywords: equation systems
zbMATH Keywords: optimization problems
zbMATH Keywords: complementarity problems
zbMATH Keywords: fixed points
zbMATH Keywords: variational inequalities
zbMATH Keywords: fair division
zbMATH Keywords: equilibrium with nonconvex technologies
zbMATH Keywords: Lindahl equilibria
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 53509