Shape of Growth Rate Distribution Determines the Type of Non-Gibrat's Law

In this study, by employing exhaustive business data on Japanese firms that approximately fully cover the middle- and large-scale ranges in terms of firm size, the authors confirm the following findings. Detailed Balance is observed not only in profits data but also in sales data. The growthrate distribution of sales has wider tails than the linear growth-rate distribution of profits in log-log scale. On one hand, in the middle-scale range of profits, the probability of positive growth decreases and the probability of negative growth increases symmetrically as the initial value increases. This is called Non-Gibrat's First Law. On the other hand, in the middle-scale range of sales, the probability of positive growth decreases as the initial value increases, while the probability of negative growth hardly changes. This is called Non-Gibrat's Second Law. Under Detailed Balance, Non-Gibrat's First and Second Laws are analytically induced from the linear and quadratic growthrate distributions in log-log scale, respectively. In both cases, the log-normal distribution is deduced from Non-Gibrat's Laws and Detailed Balance. These analytic results are verified by empirical data. Consequently, it is clarified that the difference in shapes between growth-rate distributions of sales and profits is closely related to the difference between the two kinds of Non-Gibrat's Laws in the middle-scale range.