Operator theorems with applications to distributive problems and equilibrium models
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 beautiful theorems.
scientific article; zbMATH DE number 53509