Von Neumann Models Defined by Transformations and by Production Cones

Abstract

In the economic literature two formulations of the von Neumann models are considered: the classical von Neumann definition as a pair of non-negative matrices A and B, and the second one as a production cone, i.e., a cone S in the product of the commodity spacep Y x Y = Y². It is intuitively clear how to get a production cone from a matrix model. To this end, matrices should be taken as transformations of the intensity space X into the commodity space Y and S to be a set of pairs (xA, xB) for x > 0. It is not so obvious, how ever, how to get a matrix model from a given production cone. In general, it is not always possible to do so, unless S is a polyhedral cone, since only a polyhedral cone can be presented as a set of pairs (xA,xB) with x¹ s non-negative in all coordinates. This observation seems to confirm the common belief that models defined by production cones are more general than the matrix ones.