  In this study, the authors examine exhaustive business data on Japanese
firms, which cover nearly all companies in the mid- and large-scale ranges in
terms of firm size, to reach several key findings on profits/sales distribution
and business growth trends. First, detailed balance is observed not only in
profits data but also in sales data. Furthermore, the growth-rate distribution
of sales has wider tails than the linear growth-rate distribution of profits in
log-log scale. On the one hand, in the mid-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 Property. On the other hand, in the mid-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 Property. Under detailed balance, Non-Gibrat's First
and Second Properties are analytically derived from the linear and quadratic
growth-rate distributions in log-log scale, respectively. In both cases, the
log-normal distribution is inferred from Non-Gibrat's Properties and detailed
balance. These analytic results are verified by empirical data. Consequently,
this clarifies the notion that the difference in shapes between growth-rate
distributions of sales and profits is closely related to the difference between
the two Non-Gibrat's Properties in the mid-scale range.
