4. Extension of Non-Gibrat’s Property

Similar to current profits discussed in Chap. 2, operating revenues and total assets also follow a power-law distribution in the large-scale range and a log-normal distribution in the mid-scale range in a certain year. Also, observing such short-term changes as two consecutive years, there is a time-reversal symmetry in the joint probability density function of such firm-size variables over “a certain year” and “next year.” Furthermore, Gibrat’s law also holds, which states that the conditional growth-rate distribution of firm-size variables does not depend on the initial values in a large-scale range. However, unlike the positive current profits discussed in Chap. 2, the distribution of such growth rates as operating revenues and total assets is not linear on the logarithmic axis; it has a downward curvature. Even in this case, the initial dependence of the conditional growth-rate distribution in the mid-scale range is regular. Here we extend our discussion in Chap. 2 to present a non-Gibrat’s property that describes its regularity. We show that log-normal distribution is derived from time-reversal symmetry and the extended non-Gibrat’s property and conclude that the results are consistent with the empirical data.