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Firm Sizes: Facts, Formulae and Fantasies

Robert Axtell

Part of the CSED Working Paper Series


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 have to do primarily with formulae for the distribution’s moments, and difficulties of reasonably characterizing a ‘typical’ firm. By assessing the Kesten random growth process in terms of its realism vis-à-vis actual firm growth, the author finds 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.

A Brief History of the Firm Size Distribution

Gibrat (1931) inaugurated the systematic study of this subject, finding that the lognormal distribution well-described French industrial firms. Subsequently, similar analyses were carried out for other countries (e.g., Florence (1953) in the UK). Economic theorists joined the discussion with Simon’s attempts to explain the skew and leptokurtic size distributions obtaining among publicly-traded firms, by recourse to stochastic growth models (Simon 1955; Simon and Bonini 1958; see also Steindl 1965). Such models contain essentially no elements of conventional economics (e.g., prices, profits), yet so well rationalize these extremely regular data that Simon was led to inveigh caustically against the neoclassical U-shaped cost curve explanation of firm behavior and ‘optimal size’ (Ijiri and Simon 1977: 7-11). Given the undeniable regularity of these data, discussion of skew firm size distributions entered into the developing field of industrial organization (IO), although accompanied by little of Simon’s critique (e.g., Scherer 1970; 1980).


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