Irving Fisher’s Ph.D. thesis, submitted to Yale University in 1891, contains a fully articulated general equilibrium model presented with the broad scope and formal mathematical clarity associated with Walras and his successors. In addition, Fisher presents a remarkable hydraulic apparatus for calculating equilibrium prices and the resulting distribution of society’s endowments among the agents in the economy. In this paper we provide an analytical description of Fisher’s apparatus, and report the results of simulating the mechanical/hydraulic “machine,” illustrating the ability of the apparatus to “compute” equilibrium prices and also to find multiple equilibria.
Given a generic m by n matrix A, a lattice point h in Z is a neighbor of the origin if the body {x : Ax < b}, with bi = max{0, aih}, i = 1, …, m, contains no lattice point other than 0 and h. The set of neighbors, N(A), is finite and Asymmetric. We show that if A’ is another matrix of the same size with the property that sign aih = sign ai’ h for every i and every h in N(A), then A’ has precisely the same set of neighbors as A. The collection of such matrices is a polyhedral cone, described by a finite set of linear inequalities, each such inequality corresponding to a generator of one of the cones Ci = pos(h in N(A) : aih < 0}. Computational experience shows that Ci has “few” generators. We demonstrate this in the first nontrivial case n = 3, m = 4.
Given a generic m x n matrix A, the simplicial complex K(A) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form {x : Ax < b}. The main result of this paper is that the topological space associated with K(A) is homeomorphic with Rm-1.
The simplicial complex K(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form {x : Ax < b}, with A a fixed (n + 1) × n matrix. The topological space associated with K(A) is shown to be homeomorphic to Rn, and the space obtained by identifying lattice translates of these simplices is homeomorphic to the n-torus.
In recent years many advances have been made in solution techniques for specially structured 0–1 integer programming problems. In contrast, very little progress has been made on solving general (mixed integer) problems. This, of course, is not true when viewed from the theoretical side: Lenstra (1981) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of Lovasz and Scarf (1988). This method allows us to avoid the ellipsoidal approximations required in Lenstra’s algorithm. We report on the solution of a number of small (but difficult) examples, up to 100 integer variables. Our computer code uses the linear programming optimizer CPlex as a subroutine to solve the linear programming problems that arise.
Keywords: Linear programming, mixed integer problems