Michael H R Stanley

Zipf plots and the size distribution of firms

We use a Zipf plot to demonstrate that the upper tail of the size distribution of firms is too thin relative to the log normal rather than too fat, as had previously been believed.

Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics

We discuss examples of complex systems composed of many interacting subsystems. We focus on those systems displaying nontrivial long-range correlations. These include the one-dimensional sequence of base pairs in DNA, the sequence of flight times of the large seabird Wandering Albatross, and the annual fluctuations in the growth rate of business firms. We review formal analogies in the models that describe the observed long-range correlations, and conclude by discussing the possibility that behavior of large numbers of humans (as measured, e.g., by economic indices) might conform to analogs of the scaling laws that have proved useful in describing systems composed of large numbers of inanimate objects.

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 size 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 log- normal. 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 σT — decrease with company size and increase with time T. We find that for annual growth rates (T = 1), σT ∼ S−β, and that the exponent β takes the same value, within the error bars, for several measures of the size of a company. In particular, we obtain β = 0.20 ± 0.03 for sales, β = 0.18 ± 0.03 for number of employees, β = 0.18±0.03 for assets, β = 0.18 ± 0.03 for cost of goods sold, and β = 0.20 ± 0.03 for propert, 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 β to parameters of a company structure. We find that β = −1n Π/1nz, where z defines the mean branching ratio of the hierarchical tree and Π 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.

Can statistical physics contribute to the science of economics

In recent years, a breakthrough in statistical physics has occurred. Simply put, statistical physicists have determined that physical systems which consist of a large number of interacting particles obey universal laws that are independent of the microscopic details. This progress was mainly due to the development of scaling theory. Since economic systems also consist of a large number of interacting units, it is plausible that scaling theory can be applied to economics. To test this possibility we study the dynamics of firm size. This may help to build a more complete characterization of the nature and processes behind firm growth. To date, the study of firm dynamics has primarily focused on whether small firms on average have higher growth rates than large firms. To a lesser extent, attention has been placed on the relationship between firm size and variation in growth rate. Our research goes beyond these questions by looking at the relationship between numerous firm characteristics and the entire distribution of growth rates. Thus, it may provide a better understanding of the mechanisms behind firm dynamics. In contrast to previous studies, this research analyzes data over many time scales, instead of just a single time interval. From a scientific standpoint, this work could be useful because it will affect the formulation of firm modeling—one of the basic building blocks of all economic analysis. In addition, this work will have practical applications. For example, there are Federal policies that are designed to encourage small businesses. While such policies might be justified on grounds other than their contribution to growth, any systematic difference in the growth rates of small and large firms might be relevant for evaluating such policies. Also, there has traditionally been a concern that an excessive amount of economic activity might become concentrated in a small number of firms. A more detailed understanding of the firm growth process will provide evidence for whether such concerns have any scientific foundation.

Scaling and universality in animate and inanimate systems

We illustrate the general principle that in biophysics, econophysics and possibly even city growth, the conceptual framework provided by scaling and universality may be of use in making sense of complex statistical data. Specifically, we discuss recent work on DNA sequences, heartbeat intervals, avalanche-like lung inflation, urban growth, and company growth. Although our main focus is on data, we also discuss statistical mechanical models.

Long-range correlations and generalized Lévy walks in DNA sequences

There is a mounting body of evidence suggesting that the noncoding regions of DNA are rather special for at least two reasons: 1. They display long-range power-law correlations, as opposed to previously-believed exponentially-decaying correlations. 2. They display features common to hierarchically-structured languages-specifically, a linear Zipf plot and a non-zero redundancy.