Scaling behavior in economics: The problem of quantifying company growth

Inspired by work of both Widom and Mandelbrot, we analyze the Computstat database comprising all publicly traded United States manufacturing companies in the years 1974-1993. We find that the distribution of company sizes remains stable for the 20 years we study, i.e., the mean value and standard deviation remain approximately constant. We study the distribution of sizes of the "new" companies in each year and find it to be well approximated by a lognormal. We find (i) the distribution 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 Gaussian, one would expect for a log-normal distribution, and (ii) the fluctuations in the growth rates - measured by the width of this distribution 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 bars, for several measures of the size of a company. In particular, we obtain beta = 0.20 +/- 0.03 for sales, beta = 0.18 +/- 0.03 for number of 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 company is affected by a tendency to retain an "optimal" size. That model leads to an exponential distribution of the logarithm of growth rate in agreement with the empirical results. Then, we study a hierarchical tree-like model of a company that enables us to relate beta to parameters 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 the probability that the lower levels follow the policy of higher levels in the hierarchy. We also study the output distribution of growth rates of this hierarchical model. We find that the distribution is consistent with the exponential form found empirically. We also discuss the time dependence of the shape of the distribution of the growth rates.