SCALING BEHAVIOR IN ECONOMICS - THE PROBLEM OF QUANTIFYING COMPANY GROWTH

Inspired by work of both Widom and Mandelbrot, we analyze the Computst at database comprising all publicly traded United States manufacturing companies in the years 1974-1993. We find that the distribution of co mpany sizes remains stable for the 20 years we study, i.e., the mean v alue and standard deviation remain approximately constant. We study th e distribution of sizes of the ''new'' companies in each year and find it to be well approximated by a lognormal. We find (i) the distributi on of the logarithm of the growth rates, for a fixed growth period of T years, and for companies with approximately the same size S displays an exponential ''tent-shaped'' form rather than the bell-shaped Gauss ian, one would expect for a log-normal distribution, and (ii) the fluc tuations in the growth rates - measured by the width of this distribut ion sigma(T) - decrease with company size and increase with time T. We find that for annual growth rates (T = 1), sigma(T) similar to S-beta and that the exponent beta takes the same value, within the error bar s, for several measures of the size of a company. In particular, we ob tain beta = 0.20 +/- 0.03 for sales, beta = 0.18 +/- 0.03 for number o f employees, beta = 0.18 +/- 0.03 for assets, beta = 0.18 +/- 0.03 for cost of goods sold, and beta = 0.20 +/- 0.03 for property, plant, and equipment. We propose models that may lead to some insight into these phenomena. First, we study a model in which the growth rate of a comp any is affected by a tendency to retain an ''optimal'' size. That mode l leads to an exponential distribution of the logarithm of growth rate in agreement with the empirical results. Then, we study a hierarchica l tree-like model of a company that enables us to relate beta to param eters of a company structure. We find that beta = -ln Pi/ln z, where z defines the mean branching ratio of the hierarchical tree and Pi is t he probability that the lower levels follow the policy of higher level s in the hierarchy. We also study the output distribution of growth ra tes of this hierarchical model. We find that the distribution is consi stent with the exponential form found empirically. We also discuss the time dependence of the shape of the distribution of the growth rates.