A Research Note on Deriving the Square-Cube Law of Formal Organizations from the Theory of Time-Minimization

Abstract

There are relatively few empirical laws in sociology, at least in the sense in which that word is applied in the physical sciences: empirically supported equations which precisely describe relations between variables. Most of us are satisfied to find reasonably strong associations between variables. It is therefore a matter of some interest when an exact law is proposed. It becomes a matter of some importance to derive such a law theoretically since the more general theoretical statement may permit derivation of still other laws. Several empirical laws-the size-density the rank-size rule, the urban density the gravity model, and the urban area-population law-have been reported in the ecological or socialdemographic literature. They have also been derived from the theory of time-minimization (Stephan). The purpose of this paper is to examine a nonecological one developed from the study of formal organizations, and to derive that law from the theory of time-minimization. The law is Mason Haire's squarecube law, a law which has stirred considerable interest and controversy since its introduction. Haire examined longitudinal data from four firms. He divided the employees of these firms into employees, those who interact with others outside the firm, and employees, those who interact only with others inside the firm. His finding was that, over time, the cube-root of the number of internal employees was directly proportional to the square-root of the number of external employees. The scatter diagrams he presented (286-7) show regression lines of the form