Introduction

Abstract

One of the broadly accepted universal laws of complex systems, particularly relevant in social sciences and economics, is that proposed by Zipf (1949). Zipf’s law usually refers to the fact that the probability P(s) = Pr{S > s} that the value S of some stochastic variable, usually a size or frequency, is greater than s, decays with the growth of s as P(s) ∼ s ^{− 1}. This in turn means that the probability density functions p(s) exhibits the power law dependence

$$p(s) \sim 1/{s}^{1+m}\ \ \ \mathrm{with}\ \ m = 1.$$

(1.1)

Perhaps the distribution most studied from the perspective of Zipf’s law is that of firm sizes, where size is proxied by sales, income, number of employees, or total assets. Many studies have confirmed the validity of Zipf’s law for firm sizes existing at current time t and estimated with these different measures (Simon and Bonini, 1958; Ijri and Simon, 1977; Sutton, 1997; Axtell, 2001; Okuyama et al., 1999; Gaffeo et al., 2003; Aoyama et al., 2004; Fujiwara et al., 2004a,b; Takayasu et al., 2008).