There are relatively few empirical laws in sociology…Several empirical laws—the size-density law, the rank-size rule, the urban density law, the gravity model, and the urban area-population law—have been reported in the ecological or social-demographic literature. They have also been derived from the theory of time-minimization (1979).
The purpose of this paper is to examine a non-ecological law, 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 “square-cube law”, a law which has stirred considerable interest and controversy since its introduction. Haire examined longitudinal data from 4 firms. He divided the employees of these firms into “external employees”, those who interact with others outside the firm, and “internal 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 (pg286–287) show regression lines of the form
I1⁄3 = a + bE1⁄2 (Equation 1)
where I and E are the number of internal and external employees and a and b are the intercept and slope of the regression line (see Figure 1 for an example). His explanation of the square-cube law is based on certain mathematical properties of physical objects, extended to an explanation of biological form and analogically applied by Haire to the shape of formal organizations. For a given physical object, say a cube, an increase in the length of a side results in an increase of the surface area and also of the volume. If the new length is 13× the old, the area will be 102 or 103× the old, and the new volume will be 103 or 1,003× the old. Thus, the cube-root of the volume will be proportional to the square-root of the surface area.
…1962 tested Haire’s law with cross-sectional data for 62 firms in 8 industries. They conclude that the square-cube law “is a reasonable and consistent description of the industrial organizational composition among firms of varying size in different industries” (pg342). A second study, by 1963, examined data for a single educational organization over a 45-year period. They showed that regression analysis of the untransformed data produced nearly as good a fit as did the square-cube transformation in Equation 1…Carlisle analyzed data for 7 school districts using both the square-cube transformations and the raw data. He found, supporting Draper-Strother, that the correlation coefficients were about equally good under the 2 tests.
…Derivation of the Square-Cube Law: …As 1965’s own scatter diagram shows (pg345), all 3 fit the data fairly well. Under such conditions, when the data themselves do not provide conclusive evidence favoring one model over another, the best criterion is often a logical one: Can one of the models be derived from some general theory?
…We now proceed to suggest a theoretical derivation of the square-cube law, not by analogy but by a direct consideration of the underlying processes involved. The general theory from which the derivation will proceed is the theory of time-minimization mentioned above (Stephan). Its central assumption is that social structures evolve in such a way as to minimize the time which must be expended in their operation.
Assume a firm specified by a boundary which separates it from its environment, and which includes people who spend some of their time as its employees. Assume 2 measurements made on the firm, measurements which produce the numbers E (the number of “external employees, those who interact with others outside the firm) and I (the number of “internal employees”, those who interact only with others inside the firm). Finally, from the general theory of time-minimization, assume that social structures, including the firm, evolve in such a way as to minimize the time which must be expended in their operation.
All the employees of the firm must be supported or compensated from the total pool of benefits held within the firm. Since this pool of benefits is brought in through the time-expenditures of the external employees, we may say that they in effect support themselves. At least on average, a portion of what they bring in is consumed by them. In contrast, the internal employees represent a special time-cost to the firm. The internal employees, by definition, do not bring the means of their own support into the firm. They must be supported, ultimately, through the time expenditures of the external employees. The average support time will be directly proportional to the number of internal employees and inversely proportional to the number of external employees. Thus
Ts = aI / E (Equation 6)
where a is the constant of proportionality.
If the internal employees thus appear parasitical, as a cost factor, they also contribute to reducing other costs of the firm. The benefit factor is that internal employees contribute by coordinating the work of the external employees. If there were no internal structure, if the external employees had to spend time coordinating their own activities by themselves, the amount of time spent would detract from the time they could spend at their primary assignment, bringing resources into the firm.
How much time would be spent in coordination? Assuming that each one potentially could interact with all others, the time spent should be proportional to E(E − 1)/2, the number of pairwise interactions in a group of E individuals; thus, as E becomes modestly large, the coordination time should be proportional to E2. Since this work is actually done by the internal employees, we have an average coordination time which is directly proportional to E2 and inversely proportional to I. Thus,
Tc = bE2/I (Equation 7)
where b is the constant of proportionality.
These 2 cost/benefit ratios represent the time expenditures of the internal and the external employees relative to one another. Their sum should give the overall time expenditure, the expenditure which the theory of time-minimization says will be minimized.
…The values of E and I can never be negative, so the second derivative must be positive; Equation 10 therefore represents the condition when T is a minimum. Rearranging terms, and taking the 6th root of both sides, we obtain
I1⁄3 = kE1⁄2 (Equation 11)