Firm Sizes: Facts, Formulae, Fables and Fantasies

Abstract

Recently discovered facts concerning the size distribution of U.S. firms are recapitulated - in short, these sizes are closely approximated by the Zipf distribution, a Pareto (power law) distribution with exponent of unity. Interesting consequences of this result are then developed, having primarily to do with formulae for the distribution's moments, and difficulties of reasonably characterizing a 'typical' firm. Then, a leading candidate explanation for these data - the Kesten random growth process - is assessed in terms of its realism vis-a-vis actual firm growth. Insofar as it has fluctuations that are quite different in character from actual firm size variability, the Kesten and related stochastic growth processes qualify more as fables of firm growth than as credible explanations. Finally, new explanations of the facts are proposed by considering firms to be partitions of the set of all workers. Assuming all partitions to be equally likely, the observed distribution of firm sizes is hypothesized to be the distribution of block sizes in the most likely partitions. An alternative derivation of this distribution as a constrained optimization problem is also described. Given that these calculations involve unimaginably vast magnitudes, it seems just short of fantastic to consider them relevant empirically.