Comments and Questions

I want to thank the referee for the careful reading and significant comments and suggestions. My responses are as follows (I will upload the new version after revising my paper following the third referee.): (A) The author rely on statistical tests with little extent. For example, the detailed-balance condition stated in Eq. (3) “is obviously confirmed”. I am sure to say that it would not be possible to claim its statistical significance only by taking a look at scatter-plots. Why doesn’t one need to take any trends in the average growth of sales, incomes and profits into account? Answer: Thank you for the comment.I think the referee's opinion is completely right.I have taken the one-dimensional Kolmogorov-Smirnov test to confirm the detailed balance of high-income, high-sales, positive-profits and temporal changes of sales. I have added the data analyses in the Appendix. In these data analyses, we should take into account the trends in the average growth truthfully.However, there are many same data at round figures in the database originally.If we subtract the trends from the data, the values are displaced.As a result, $p$ value is underestimated by the displacements.In the databases, the effect of the trends in the average growth is not too large, so we can confirm the detailed balance approximately without subtracting the trends. I have written this reason in the Appendix. (B) The author's contribution (i) does not seem to have been published in the author's previous papers, although seemingly related things had already been shown (Refs. [8][9][10]) with some contradiction. I followed the equations in Section 2.2, and could see that the solutions (21) and (22) satisfy the detailed balance in (18). This might not be trivial, because the derivation is based on perturbative argument, that is, by expanding the equation (18) with respect to R around R = 1. The author could mention about the fact that the solutions satisfy the equation (18) beyond perturbation under the restricted assumption of Eqs. (14) and (15). In addition, I wonder how the solutions are compatible with the previous results in Refs. [8][9][10]. Answer: Thank you for the comment.I think this calculation is a simple version essentially equivalent to the calculation in Refs. [8][9][10]. Therefore, I should mention about the fact that the solutions satisfy the Eq. (18).I have added the sentence after Eq. (22). (C) When one studies the temporal change of increment or decrement of firm-size, one must be careful about the trends in average growth. Such effects might be actually observed in Figure 14, for instance. As mentioned in (A), one would need some kind of careful statistical tests or, at least, some operation that does subtraction of trends. So how much significance in statistical sense do the results in Section 3 (the contribution (ii)) have? Answer: Thank you for the comment.I think the referee's opinion is completely right.The effect of the trends in average growth is observed in Figure 14 especially for n=1.At the same time, from the reason in the answer of (A), it is difficult to take K-S test after subtracting the trends from the data.We can approximately confirm the detailed balance by K-S test without subtracting the trends for n=2-5.So I think the result (ii) is valid above $x_{min} \sim 10^{4+0.5(2-1)} = 10^{4.5}$.I have written this in the Appendix and have changed the value of $x_{min}$ from $10^4$ to $10^{4.5}$. (D) The word ``displacement'' may not be comprehensible in the title and also in the main body of texts. The author is presumably meaning by it temporal changes of growth, or a kind of ``second-derivative'' of firm-size. More appropriate word could be employed. The author may want to ask for assistance by a native English speaker. Answer: Thank you for the suggestion.I have asked an English speaker and have confirmed that the word ``displacement'' should be replaced by ``temporal change''.I have replaced the word ``displacement'' by ``temporal change'' in the title and also in the main body of texts.