Abstract
Employing profits data of Japanese firms in [2003][2004][2005], we kinematically exhibit the static log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed balance is adopted together with following two approximations. Firstly, the probability density function of profits growth rate is described as a tent-shaped exponential function. Secondly, the value of the origin of the growth rate distribution divided into bins is constant. The derivation is confirmed in the database consistently.
Abstract
This static procedure is applied to a quasi-static system. We dynamically describe a quasi-static log-normal distribution in the middle scale region. In the derivation, a NonGibrat's law under the detailed quasi-balance is adopted together with two approximations confirmed in the static system. The resultant distribution is power-law with varying Pareto index in the large scale region and the quasi-static log-normal distribution in the middle scale region. In the distribution, not only the change of Pareto index but also the change of the variance of the log-normal distribution depends on the parameter of the detailed quasibalance. As a result, Pareto index and the variance of the log-normal distribution are related to each other.
Introduction
Log-normal distributions are frequently observed not only in natural phenomena but also in social phenomena. For representative example, the probability density function P (x) of personal income or firm size x is considered to obey the log-normal distribution [1] - [2] P LN (x) = 1
Introduction
in the middle scale region. 1 Herex is a mean value and σ 2 is a variance.
Introduction
The simplest model which describes the log-normal distribution is the pure multiplicative stochastic process defined by
Introduction
where R(t) is a positive random variable. By applying this process iteratively, we obtain x(t) = R(t − 1) R(t − 2) · · · R(0) x(0) .
Introduction
The logarithm of this equation is log x(t) = log R(t − 1) + log R(t − 2) + · · · + log R(0) + log x(0) .
Introduction
If log x(0) is negligible compare to log x(t) in the limit t → ∞ and log R(i) (i = 0, 1, · · · , t − 1) are independent probability variables, log x(t) follows the normal distribution in the limit. As a result, this model derives the stationary log-normal distribution
Introduction
Eq. (2) means that the distribution of the growth rate does not depend on x(t). This is known as Gibrat's law [1] that the conditional probability density function of the growth rate Q(R|x 1 ) is independent of the initial value x 1 :
Introduction
Here x 1 and x 2 are two successive incomes, assets, sales, profits, the numbers of employees and so forth. The growth rate R is defined as R = x 2 /x 1 and Q(R|x 1 ) as
Introduction
by using the probability density function P (x 1 ) and the joint probability density function P 1R (x 1 , R).
Introduction
As far as firm sizes in the middle scale region, however, it is reported that the growth rate distributions do not follow the Gibrat's law (5) (see Refs. [6] - [8] for instance). The pure multiplicative stochastic process model (2) cannot be applied for explaining the log-normal distribution in the middle scale region. Instead, we have shown that the log-normal distribution can be derived by using no model such as the pure multiplicative stochastic process [9] . In the derivation, two laws are employed which are observed in profits data of Japanese firms. One is the law of detailed balance which is observed in a stable economy [10] . The other is a NonGibrat's law which describes a statistical dependence in the growth rate of the past value [9] .
Introduction
In Ref. [9] , we kinematically derived the static log-normal distribution under the detailed confirm the derivation more firmly. After that, we dynamically derive the log-normal distribution by replacing the detailed balance with the detailed quasi-balance proposed in Ref. [11] . By this procedure, the log-normal distribution is described as quasi-static.
Static log-normal distribution under the detailed balance
In this section, we briefly review the study in Ref. [9] . After that, we empirically confirm the analytic result by added data analyses. We employ profits data of Japanese firms in 2003,   2004 and 2005 which are available on the database "CD Eyes 50" published in 2005 and 2006 by TOKYO SHOKO RESEARCH, LTD. [12] . Figure 1 shows the joint probability density function P 12 (x 1 , x 2 ) of all firms in the database, 
Static log-normal distribution under the detailed balance
Figure 4 shows probability density functions of profits in [2003] [2004] [2005] . The distributions are almost stable and following power-law is observed in the large scale region
Static log-normal distribution under the detailed balance
where x 0 is a certain threshold. This power-law is called Pareto's law [3] and the exponent µ is named Pareto index. Notice that Pareto's law does not hold below the threshold x 0 . The purpose of this section is to exhibit the distribution in the middle scale region under the detailed balance (7).
Static log-normal distribution under the detailed balance
In order to identify a statistical dependence in the growth rate of the past value, we examine the probability density faction of the profits growth rate in 2003-2004 firstly. In the database, we divide the range of x 1 into logarithmically equal bins as x 1 ∈ 4×[10 1+0.2(n−1) , 10 1+0.2n ] thousand yen with n = 1, 2, · · · , 20. Figure 5 shows the probability density functions for r = log 10 R in the case of n = 1, · · · , 5, n = 6, · · · , 10, n = 11, · · · , 15 and n = 16, · · · , 20. The number of the firms in Fig. 5 is "22, 005", "89, 507", "85, 020" and "24, 203", respectively.
Static log-normal distribution under the detailed balance
From Fig. 5 , we approximate log 10 q(r|x 1 ) by linear functions of r:
Static log-normal distribution under the detailed balance
log 10 q(r|x 1 ) = c(x 1 ) + t − (x 1 ) r for r < 0 .
Static log-normal distribution under the detailed balance
Here q(r|x 1 ) is the probability density function for r, which is related to that for R by Q(R|x 1 ) = log 10 q(r|x 1 ) − r − log 10 (ln 10). These approximations (9)-(10) are expressed as tent-shaped exponential forms as follows
Static log-normal distribution under the detailed balance
where d(x 1 ) = 10 c(x 1 ) /ln 10. Furthermore, figure 5 shows that the dependence of c(x 1 ) on x 1 is negligible for n = 9, · · · , 20 . We assess the validity of these approximations against the results. Figure 6 represents the dependence of t ± (x 1 ) on the lower bound of each bin x 1 = 4 × 10 1+0.2(n−1) . For n = 17, · · · , 20, t ± (x 1 ) hardly responds to x 1 . This means that Gibrat's law (5) holds only in the large scale region of profits (x > x 0 ). 2 In contrast, t + (x 1 ) linearly increases and t − (x 1 ) linearly decreases symmetrically with log 10 x 1 for n = 9, · · · , 13 . From Fig. 6 , the slops are described as [13] 
Static log-normal distribution under the detailed balance
The parameters are estimated as follows:
Static log-normal distribution under the detailed balance
∼ 63, 000 thousand yen and x min = 4 × 10 1+0.2(9−1) ∼ 1, 600 thousand yen. Notice that approximations (9)- (10) uniquely fix the expression of t ± (x 1 ) under the detailed balance [9] . This derivation is included in the proof in the next section. We call Eqs. (11)- (13) Non-Gibrat's law.
Static log-normal distribution under the detailed balance
For n = 9, · · · , 20, the dependence of d(x 1 ) on x 1 is negligible. In this case, the Non-Gibrat's law determines the probability density function of profits as follows
Static log-normal distribution under the detailed balance
where
Static log-normal distribution under the detailed balance
. This is the power-law in the large scale region (x 1 > x 0 ) and the log-normal distribution in the middle scale region (x min < x 1 < x 0 ). The relations between parameters σ,x in Eq. (1) and α, µ, x 0 are given by Figure 7 shows that the distribution (14) fits with the empirical data consistently. Notice that the distribution cannot fit with the empirical data, if α is different from the value estimated in (14) is static, because the derivation is based on the detailed balance (7) which is static time-reversal symmetry ( Fig. 1-3 ).
Static log-normal distribution under the detailed balance
On the other hand, we have derived Pareto's law with annually varying Pareto index under the detailed quasi-balance [11] :
Static log-normal distribution under the detailed balance
It is assumed that, in the quasi-static system, the joint probability density function has "a x 1 θ ↔ x 2 " symmetry where θ is a slope of a regression line:
Static log-normal distribution under the detailed balance
log 10 x 2 = θ log 10 x 1 + log 10 a .
Static log-normal distribution under the detailed balance
The detailed balance (7) has the special symmetry θ = a = 1. Because the detailed quasi-balance (16) is imposed on the system, θ is related to a as follows:
Static log-normal distribution under the detailed balance
Here 10 Γ is a sufficient large value compared to the upper bound in which θ and a are estimated.
Static log-normal distribution under the detailed balance
In Ref. [11] , these results have been empirically confirmed by employing data on the assessed value of land [15] - [16] in 1983-2005. In the derivation, we have used the Gibrat's law (5) valid only in the large scale region without linear approximations (9)-(10). The purpose of this section is to show that the approximations uniquely fix a Non-Gibrat's law under the detailed quasibalance. After that, we identify the quasi-static distribution not only in the large scale region but also in the middle scale region.
Static log-normal distribution under the detailed balance
By using the relation P 12 (x 1 , x 2 )dx 1 dx 2 = P 1R (x 1 , R)dx 1 dR under the change of variables (x 1 , x 2 ) ↔ (x 1 , R), these two joint probability density functions are related to each other
Static log-normal distribution under the detailed balance
where we use a modified growth rate R = x 2 /(a x 1 θ ) . From this relation, the detailed quasibalance (16) is rewritten as
Static log-normal distribution under the detailed balance
Substituting P 1R (x 1 , R) for Q(R|x 1 ) defined in Eq. (6), the detailed quasi-balance is reduced to be
Static log-normal distribution under the detailed balance
forms (11)- (12) . Under the approximations, the detailed quasi-balance is expressed as
Static log-normal distribution under the detailed balance
for R > 1. Here we use the notationP (x) = P (x)d(x). By expanding Eq.(22) around R = 1, the following differential equation is obtained:
Static log-normal distribution under the detailed balance
where x denotes x 1 . The same equation is obtained for R < 1. The solution is expressed as
Static log-normal distribution under the detailed balance
where
Static log-normal distribution under the detailed balance
In order to make the solution (24) around R = 1 satisfy Eq. (22), the following equation must be valid for all R:
Static log-normal distribution under the detailed balance
The derivative of this equation with respect to x is
Static log-normal distribution under the detailed balance
By expanding Eq. (26) around R = 1, the following differential equations are obtained:
Static log-normal distribution under the detailed balance
The solutions are given by
Static log-normal distribution under the detailed balance
In order to make these solutions satisfy Eq. (25), the coefficients C −2 and C +1 must be 0 .
Static log-normal distribution under the detailed balance
As a result, t ± (x) under the detailed quasi-balance is uniquely fixed as
Static log-normal distribution under the detailed balance
This is the same expression under the detailed balance.
Static log-normal distribution under the detailed balance
By using the Non-Gibrat's law (31), probability density functions P 1 (x 1 ), P 2 (x 2 ) are uniquely reduced to be
Static log-normal distribution under the detailed balance
with
Static log-normal distribution under the detailed balance
in the large or middle scale region where the dependence of d(x 1 ) on x 1 is negligible.
Static log-normal distribution under the detailed balance
Here we consider two log-normal distributions in the middle scale region:
Static log-normal distribution under the detailed balance
By comparing Eqs. (32)- (33) to (35)- (36), we identify
Static log-normal distribution under the detailed balance
Consequently, the relation between σ 1 , σ 2 and θ is expressed as
Static log-normal distribution under the detailed balance
This is the equation which quasi-static log-normal distributions satisfy.
Conclusion
In this study, by employing profits data of Japanese firms, we have exhibited the static log-normal distribution of profits in the middle scale region from a Non-Gibrat's law under the detailed balance. In the derivation, we have adopted two approximations observed in the database. One is that the probability density function of the profits growth rate is described as a tent-shaped exponential function. The other is that the value of the origin of the growth rate distribution divided into bins is constant. This static discussion has been applied to a quasi-static system. We have derived a quasistatic log-normal distribution in the middle scale region from a Non-Gibrat's law under the detailed quasi-balance. In the derivation, we have also assumed two approximations confirmed in the database. Even in the quasi-static system, the approximations uniquely fix the NonGibrat's law to be the same expression under the detailed balance. The resultant distribution is power-law with varying Pareto index in the large scale region and the quasi-static log-normal distribution in the middle scale region. Notice that not only the change of Pareto index µ but also the change of the variance of the log-normal distribution σ depends on a parameter of the detailed quasi-balance θ. In other words, µ and σ are related to each other. This phenomenon is suggested in Ref. [17] .
Conclusion
First issue in the future is to confirm this quasi-static derivation by data analyses. For this aim, long-term economic data must be investigated in the middle scale region where dynamical transitions are observed. We believe that the analytic results in this study are confirmed in such a database, if two approximations assumed in this paper are applicable. Probability Profit Distribution of Firms P (x1) Profits x 1 (in thousand yen )
Conclusion
profits data in 2003 Pareto law log-normal type α=0.10 log-normal type α=0.14 log-normal type α=0.20 Probability Profit Distribution of Firms P
(x)
Profits x (in thousand yen )
(x)
profits data in 2004 Pareto law log-normal type α=0.10 log-normal type α=0.14 log-normal type α=0.20 Probability Profit Distribution of Firms P
(x)
Profits x (in thousand yen ) profits data in 2003 Pareto law log-normal type α=0.10 log-normal type α=0.14 log-normal type α=0.20 

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