Statistical Laws in the Income of Japanese Companies

TL;DR

This paper confirms that Japanese companies' income, capital, sales, and employees follow a consistent Zipf's law over 30 years, and models their growth using stochastic processes with additive and multiplicative noise.

Contribution

It demonstrates the persistent power law distribution in Japanese companies' financial data and introduces a stochastic growth model based on Langevin equations to explain this phenomenon.

Findings

Income distribution follows Zipf's law with exponent -1 over 30 years

Growth rate of companies is approximately independent of company size

The stochastic model fits observed power law exponents reasonably well

Abstract

Following the work of Okuyama, Takayasu and Takayasu [Okuyama, Takayasu and Takayasu 1999] we analyze huge databases of Japanese companies' financial figures and confirm that the Zipf's law, a power law distribution with the exponent -1, has been maintained over 30 years in the income distribution of Japanese companies with very high precision. Similar power laws are found not only in income distribution of company's income, but also in the distributions of capital, sales and number of employees. From the data we find an important time evolutionary property that the growth rate of income is approximately independent of the value of income, namely, small companies and large ones have similar statistical chances of growth. This observational fact suggests the applicability of the theory of multiplicative stochastic processes developed in statistical physics. We introduce a discrete version of Langevin equation with additive and multiplicative noises as a simple time evolution model of company's income. We test the validity of the Takayasu-Sato-Takayasu condition [Takayasu, Sato and Takayasu 1997] for having an asymptotic power law distribution as a unique statistically steady solution. Directly estimated power law exponents and theoretically evaluated ones are compared resulting a reasonable fit by introducing a normalization to reduce the effect of gross economic change.