Optimal Location and Production: A Profit Function Analysis in Cartesian Space

Abstract

This paper specifies a dual profit function which facilitates the generalization and extension of the analysis of the Hoover–Isard–Moses problem of integration of optimal location and production decisions. The optimal location of a multi-input, multi-market firm in two-dimensional space is analyzed. The central question is, which combinations of the economic characteristics will be sufficient to induce concavity in the dual profit function with respect to the Cartesian coordinate space over which it is defined? We show that for any configuration of input and market sites which are not all collinear, concavity of the dual profit function depends upon a sufficiently small elasticity of input substitution. The model is sufficiently general so that previous results derived under restrictive conditions of zero substitution, city-block metric, or one-dimensional space can be obtained as special cases. The use of vectors and Cartesian coordinates facilitate not only the straightforward generalization to multiple inputs and markets but also allow an intuitively appealing representation of the location optimality condition. Furthermore, we demonstrate how our approach simplifies and facilitates comparative static analysis, the introduction of the profit-improving location direction, and the economic and geometric interpretations of the results.