Physica A 269 (1999) 125���131 www.elsevier.com/locate/physa Zipf���s law in income distribution of companies K. Okuyamaa , M. Takayasub , H. Takayasuc * aGraduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan bFaculty of Science and Technology, Keio University, Kawasaki 211-0985, Japan cSony CSL, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan Received 28 October 1998 received in revised form 20 January 1999 Abstract Distribution functions of annual income of companies are analyzed based on two company databases. A clear power law distribution consistent with the Zipf���s law can be conrmed for Japanese companies over more than three decades in income scale. Similar distributions can be conrmed in some other countries. It is conrmed that such power laws hold in most of job categories with slightly modied exponents. An annual income of a company is about two orders of magnitude smaller than its total assets, and the growth rate distribution of income is nearly independent of the income size in contrast to the case of growth rate of assets. c 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 05.90.+m 89.90.+n 05.45.DF Keywords: Zipf���s law Income distribution Growth rate of assets 1. Introduction Interactions of companies and complicated money ows may be viewed as the most typical real-world examples of complex systems. It is an important basic step of science to check whether there are any universal laws underlying such economical activities. Here we focus our attention on the distribution of company���s income and show evi- dences that the statistics of income is following a universal law. The history of study on income distribution is very long. More than one hundred years ago Pareto reported that personal income distribution follows a power law with a possibly universal exponent about 1.5 [1]. Gini studied income distributions of several countries in 1922 and found that the distributions can actually be approximated by * Corresponding author. E-mail address: takayasu@csl.sony.co.jp (H. Takayasu) 0378-4371/99/$-see front matter c 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0378-4371(99)00086-2

126 K. Okuyama et al. / Physica A 269 (1999) 125���131 power laws but the exponents are not universal [2]. In 1931 Gibrat proposed log-normal distributions of income based on a theoretical assumptions of multiplicative random processes [3]. Mandelbrot rediscovered the power law distributions of income in 1960 [5], and until now there are many open problems related to income distribution in economics [6]. With the increasing interest on power law behaviors in connection with critical be- haviors in physical sciences two physicists, Montroll and Shlesinger [4], pointed out in their physics paper that income of rich people follow a power law statistics while that of not rich people follows a log-normal distribution. The power law tails in income distribution have been introduced as an example of fractal behaviors in economics [7,8]. A relating topic is attracting physicists��� interest recently, that is, the company size distributions. Stanley et al. analyzed an American company���s database and found that the size distribution is closer to the lognormal law [9,10]. They also found a plausible scaling relation that the standard deviation of growth rate of company size is inversely proportional to a fractional power of its size, which is consistent with the intuitive empirical impression that uctuation of a large company is smaller than that of a small company. Okuyama and Takayasu analyzed an international company database and reported that the scaling law holds also in other countries but the company size distributions depend on country [11]. Company size can be measured by several quantities such as assets, number of em- ployee, or net sales. It is known that these quantities give consistent results in analysis of company size statistics [9,10]. On the other hand roughly speaking company���s in- come is dened by the dierence of the net sales minus net expenses, so it can take a negative value if the net expenses exceed the net sales. Obviously, the statistics of income should be dierent from assets or other quantities proportional to the company size. In the following section we describe our databases. Main results are listed in the Section 3 and we discuss the meaning of such income distributions in the nal section. 2. The databases In this study we analyze two databases. The rst one is a data CD, ���Japanese companies the best 85375��� which is published in 1998 by a publisher Diamond Inc. in Tokyo, Japan. This CD covers all companies having annual income (to be more precise, the income before taxes reported to the tax oce) more than 40,000,000 yen are listed. Although the number of companies is very large the information for each company is limited, only company���s name, its job category, and incomes for 4 years. We name this database as Data 1 in the following analysis. The other database is ���Company data��� distributed by Moody���s Investors Service Inc. A pair of CDs cover more than 10,000 companies in the USA and also more than 11,000 companies for other countries. The information for each company is rich