Comments on Two Mathematical Formulations of the Theory of Differentiation in organizations

Abstract

THE aim in developing a deductive theory is to discover a few general principles that can account for many empirical relationships, some of which have already been observed and others are predicted to be observable in the future. Attempts to construct such a deductive theory inductively, starting with research findings to be explained, involve at least two steps. First, general concepts are introduced to subsume several empirical variables under a single concept. Thus, I used the concept of differentiation in organizations to encompass the division of labor, the number of hierarchical levels, the number of functional divisions, and some other variables (Blau, 1970). Many consistent findings are reduced to much fewer propositions by substituting general concepts. Second, the remaining propositions are organized into a deductive system by inferring some basic propositions from which all the others can be logically deduced. The two basic generalizations I inferred are (1) that the increasing size of organizations promotes structural differentiation at declining rates and (2) that differentiation enlarges the administrative component. The assumption is that all other propositions under consideration are logically implied by these two and that future, as well as all existing, empirical findings are consistent with the predictions implied by the theory. Further steps in refining the theory may entail explicating generalizations through new unmeasured concepts-for example, I used the concept of need for coordination to explain the influence of differentiation on the administrative component-and reformulating the propositions in mathematical terms. Hummon's Paper. An important advantage of mathematical formulations of a theory is that they make it possible to test its logical structure by showing whether the lower-order propositions can indeed be derived from the definitions and the higherorder ones without further assumptions. Hummon's analysis illustrates this. For instance, his discussion of elasticity under A1.2 proves mathematically that my proposition 1.2-the average size of structural components increases as the size of the entire organization does-follows strictly from the second part of the first generalization-that the positive influence of size on differentiation occurs at a declining rate. He thereby confirms my geometrical argument that the shape of the regression curves of size on various aspects of differentiation implies proposition 1.2 (Blau, 1970:207-208). In other instances, however, the mathematical translation reveals shortcomings of my verbal formulations. In my generalization that increasing size promotes differentiation, I implicitly assumed that differentiation depends not only on size but also on other factors. But assumptions must not be left implicit in formal theorizing lest misleading implications be drawn. In his discussion of V2.0, Hummon notes the need to make explicit the assumption that differentiation does not depend on size alone. He discovers a more important failure of mine to be fully explicit in his analysis of A2.2, in which he concludes that my proposition 2.2-the direct effect of size reducing the administrative ratio exceeds its indirect effect raising it-cannot be derived unconditionally. The question that I failed to answer is whether my proposition 1.5that there is an economy of scale in administration-is meant to imply that the gross effect of size reduces the administrative component or that only its net effect, controlling other conditions, produces such a reduction. If one assumes that size has a negative gross effect on the administrative component, * I am indebted to William H. Weber and Roland Wulbert for helping me interpret the mathematical formulations in the two preceding papers by Norman P. Hummon and Marshall W. Meyer.