Do Pareto–Zipf and Gibrat laws hold true? An analysis with European firms

Abstract

By employing exhaustive lists of large firms in European countries, we show that the upper-tail of the distribution of firm size can be fitted with a power-law (Pareto-Zipf law), and that in this region the growth rate of each firm is independent of the firm's size (Gibrat's law of proportionate effect). We also find that detailed balance holds in the large-size region for periods we investigated; the empirical probability for a firm to change its size from a value to another is statistically the same as that for its reverse process. We prove several relationships among Pareto-Zipf's law, Gibrat's law and the condition of detailed balance. As a consequence, we show that the distribution of growth rate possesses a non-trivial relation between the positive side of the distribution and the negative side, through the value of Pareto index, as is confirmed empirically.2 See [2] for modern and high-quality personal-income data in Japan. 3 Another interesting and related quantity is flow, e.g. profits, rather than stock. See [7] for growth of individual personal-income and [8] for firms tax-income growth, and validity of Gibrat's law.

The social architecture of capitalism

“…al. [18] found that the Zipf law characterised the size distribution of about 260,000 large firms from 45 European countries during the years 1992-2001. A Zipf law implies that a majority of small firms coexist with a decreasing number of disproportionately large firms.…”

Transition Law of Firms’ Activity and the Deficit Aspect of Non-Gibrat’s law

“…We denote financial data (assets, sales, number of employees, and so forth) in year T and T + 1 as x T and x T +1 , respectively. Inverse symmetry denotes that joint PDF P J (x T , x T +1 ) is symmetric under the inverse exchange of variables x T ↔ x T +1 [23,24]:…”

“…Interestingly, these short-term laws lead to a power-law and a lognormal distribu-tion observed at a point of a certain time. Under inverse symmetry, Gibrat's law and non-Gibrat's law respectively lead to power-law [23,24] and log-normal distribution [29]- [32]. Furthermore, under quasi-inverse symmetry, Gibrat's law and non-Gibrat's law respectively lead to a quasi-varying power-law and quasi-varying lognormal distribution [25,26].…”

“…In short-term financial data as two successive years, the following laws have been also observed: inverse symmetry [23,24], quasi-inverse symmetry [25,26], Gibrat's law [27,28], and non-Gibrat's law [29]- [32]. Inverse symmetry is observed in the joint probability density function (PDF), which is symmetric under the inverse exchange of financial variables for two consecutive years.…”

“…Quasi-inverse symmetry is quasi-statically modified inverse symmetry. Gibrat's law means that the conditional PDF of the growth rate of financial variables does not depend on the initial value over some size threshold [23,24]. Non-Gibrat's law refers to the dependence law of conditional PDFs on the initial value under the size threshold.…”