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Discussion Paper

No. 2008-45 | December 19, 2008
Power-Law and Log-Normal Distributions in Firm Size Displacement Data

Abstract

We have shown that firm size signed displacement data follow not only power-law in the large scale region but also the log-normal distribution in the middle scale one. In the analyses, we employ three databases: high-income data, high-sales data and positive-profits data of Japanese firms. It is particularly worth noting that the growth rate distributions of the firm size displacement have no wide tail which is observed in assets, sales of firms, the number of employees and personal income data. An extended-Gibrat’s law is also found in the growth rate distributions. This leads the power-law and the log-normal distributions of the firm size displacement under the detailed balance.

Paper submitted to the special issue “Reconstructing Macroeconomics

JEL Classification

D30 D31 D39

Cite As

Atushi Ishikawa (2008). Power-Law and Log-Normal Distributions in Firm Size Displacement Data. Economics Discussion Papers, No 2008-45, Kiel Institute for the World Economy. http://www.economics-ejournal.org/economics/discussionpapers/2008-45

Assessment



Comments and Questions


Anonymous - Referee Report
January 06, 2009 - 10:44

see attached file


Atushi Ishikawa - Response to Referee Report 1
January 12, 2009 - 11:18

I want to thank the referee for the careful reading and instructive comments and suggestions. My responses are as follows (I will upload the new version after revising my paper following the second and third referees.):

1. line 3 from the bottom in page 1
"the threshold of which ...[more]

... is denoted by xth"
which -> pdf?

Answer: Thank you for the comment.
I intended that this ``which'' indicates ``the large scale region''.
I have understood this sentence is misleading, so I have changed it.

2 - 4, 7, 9, 10 - 12.
Answer: Thank you for the comments very much. I have changed it.

5. line 2 from the top in page 3
"We exclude the negative data, because they are exclusive as profits data."
why is the negative profit exclusive?

Answer: Because the negative data are only gathered from high-sales data, and the number of the negative data is much less than that of the positive data. I have added this reason.

6. Eq.(8)
R^{-1} P_{1R} (x_2, R^{-1}) -> R^{-1} P_{2R} (x_2, R^{-1})

Answer: P_{1R} is a name of the function, so I think this expression is not wrong.

8. line 5 from bottom in page 4
"This threshold is coincident with the threshold in the Pareto's law."
Why do those thresholds coincide? Is there any theoretical reasoning?

Answer: Because there is no threshold in the detailed balance (Fig. 5).
I have added this reason.


Anonymous - Referee Report
January 08, 2009 - 11:54

see attached file


Atushi Ishikawa - Response to Referee Report 2
January 15, 2009 - 06:15

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 ...[more]

... 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.


Anonymous - Comment
January 08, 2009 - 11:56

Referee report on "Power-law and Log-normal..."
by A. Ishikawa (Discussion Paper Nr. 2008-45)

The paper deals with the distribution of firm's
financial quantities empirically. This is evidently
an important, almost essential subject in discussing
micro-foundation of macro economy. The authors
discusses not only the superficial empirical aspects, ...[more]

... />but goes a bit further to various relations of
phenomenological laws. As such, the present referee
recommends publication of this paper, but with
some further, possible improvements, which is listed below.

1. English. Some sentences are not grammatically
correct and/or are hard to understand. A typical case
is the first sentence on the page 3, "In this study,
....". A good reader could save most of them.

2. Their investigation is limited to Japanese data.
Although they briefly mention Fujiwara, et. al.'s
results on European data, it is quite narrow and
references needs to be expanded. How about Stanley
and his company's major research results? And
what about Gaix? It would be best the authors
could present a bit wider perspective.

3. Speaking of Gaix, I recall his paper "The Granular
Origins of Aggregate Fluctuations" and I wonder if
the authors could discuss a bit about implications of their
research with macro-economics.


Atushi Ishikawa - Response to Referee Report 3
January 22, 2009 - 02:42

I want to thank the referee for the careful reading and instructive comments and suggestions. My responses are as follows (I will upload the new version.):

1. English. Some sentences are not grammatically correct and/or are hard to understand. A typical case is the first sentence on the ...[more]

... page 3, "In this study, ....". A good reader could save most of them.

Answer: Thank you for the suggestion, and I am sorry for my poor English.
I have asked an English speaker to correct my English.
I think I can revise my English.

2. Their investigation is limited to Japanese data. Although they briefly mention Fujiwara, et. al.'s results on European data, it is quite narrow and references needs to be expanded. How about Stanley and his company's major research results? And what about Gaix? It would be best the authors could present a bit wider perspective.

Answer: Thank you for the suggestion.
I think the referee's opinion is completely right.
I have added the references which employ European and/or North American data.

3. Speaking of Gaix, I recall his paper "The Granular Origins of Aggregate Fluctuations" and I wonder if the authors could discuss a bit about implications of their research with macro-economics.

Answer: Thank you for the suggestion.
I have read his paper.
I think the mechanism in this paper might be useful for understanding aggregate phenomena in macro-economics.
I have added this, and want to discuss with a macro-economist about this point.


Patrick McCloughan - Review
February 10, 2009 - 11:25

I read the paper and enjoyed it.

Just a few small comments.

Gibrat book was originally 1931 (not 1932)

Gibrat’s Law general form may be written as X(t) = X(t-1)^beta.exp{u(t)} where u(t) is the proportionate growth distribution of firms in a market (Gibrat’s Law applies to well-defined ...[more]

... markets and that is important in its application in economics and business). In particular, u(t) has mean and variance and may be normal (in which case exp{u(t)} is lognormal).
Gibrat’s Law means that each firm in the market has the same chances of growing at any rate as any other firm in the market and thus the distribution is the same for all firms in the market. Also beta=1.
Your figs 1-5 suggest beta=1 (45 degree line plot) but that the variance of u(t) is higher for smaller firms than larger firms – which is commonly found in economics and business (i.e. smaller firms exhibit more volatile growth than their larger counterparts).
Figs 6-8 suggest that u(t) is normal (as commonly assumed).
Zipf’s power law may be studied by plotting size of firm against rank of firm and fitting an equation like y=ax^-alpha (alpha commonly estimated to be 1.5-2) but again this really only makes sense when studied in the context of markets, where firms compete with each other (like environments in the natural sciences).


You might be interested in some earlier work I did in this field (the latter papers relate to new estimators of market concentration and the Gini coefficient given grouped data).



McCloughan, P. (1995) ‘Simulation of Concentration Development from Modified Gibrat Growth-Entry-Exit Processes’, The Journal of Industrial Economics, Vol. XLIII, No. 4, pp. 405-433.

McCloughan, P. (1999) ‘Lognormality of the Size Distribution of Firms: Review and New Evidence from Computer Simulations’, in The Current State of Economic Science (Vol. 4), pp. 2,135-2,155, ed. Professor Dahiya, S. B., Spellbound Publications, ISBN 81-7600-042-6.

McCloughan, P. and Abounoori, E. (2003) ‘How to Estimate Market Concentration given Grouped Data’, Applied Economics, Vol. 35, pp. 973-983.

Abounoori, E. and McCloughan, P. (2003) ‘A Simple Way to Calculate the Gini Coefficient for Grouped as well as Ungrouped Data, Applied Economics Letters, Vol. 10, pp. 505-509.

McCloughan, P. (2004) ‘Construction Sector Concentration: Evidence from Britain’, Construction Management and Economics, Vol. 22 (Number 9), pp. 979-990.

McCloughan, P. (2005) ‘What’s been Happening to Concentration in Irish Industry 1991-2001’, The Economic and Social Review, Vol. 36, No. 2, Summer/Autumn, pp. 127-56.


Atushi Ishikawa - Revised Version
February 11, 2009 - 09:56

see attached file