Shouji Fujimoto

A new method for measuring tail exponents of firm size distributions

We propose a new method for estimating the power-law exponent of a firm size variable, such as annual sales. Our focus is on how to empirically identify a range in which a firm size variable follows a power-law distribution. As is well known, a firm size variable follows a power-law distribution only beyond some threshold. On the other hand, in almost all empirical exercises, the right end part of a distribution deviates from a power-law due to finite size effect. We modify the method proposed by Malevergne et al. (2011) so that we can identify both of the lower and the upper thresholds and then estimate the power-law exponent using observations only in the range defined by the two thresholds. We apply this new method to various firm size variables, including annual sales, the number of workers, and tangible fixed assets for firms in more than thirty countries.

Non-Gibrat's Law in the Middle Scale Region

By using numerical simulation, we confirm that Takayasu--Sato--Takayasu (TST) model which leads Paretos law satisfies the detailed balance under Gibrats law. In the simulation, we take an exponential tent-shaped function as the growth rate distribution. We also numerically confirm the reflection law equivalent to the equation which gives the Pareto index

Shape of growth-rate distribution determines the type of Non-Gibrat’s Property

\mu

Firm Age Distributions and the Decay Rate of Firm Activities

in TST model. Moreover, we extend the model modifying the stochastic coefficient under a Non-Gibrats law. In this model, the detailed balance is also numerically observed. The resultant pdf is power-law in the large scale Gibrats law region, and is the log-normal distribution in the middle scale Non-Gibrats one. These are accurately confirmed in the numerical simulation.

The difference of growth rate distributions between sales and profits

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. Here, profits denote net profits. 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.

Firm Growth Function and Extended-Gibrat’s Property

In this study, we investigated around one million pieces of Japanese firm-size data, which are included in the database ORBIS, and confirmed that the age distribution of firms approximately obeys an exponential function. We estimated the decay rate of firms by comparing their activities in 2008 and 2013 and found that it does not depend on firm age and can be regarded to be constant. Here, decay rate of firms denotes the state transition probability of firm activities. These two observations are qualitatively consistent when the number of newly founded firms is nearly constant. This phenomenon is analogous to nuclear decay. We quantitatively confirmed this consistency by comparing the parameters of exponential age distribution with the decay rate of firm activities. At the same time, using this result, we estimated the number of firms founded annually and the decay rate of firm activities in Japan before World War II.

Analytical Derivation of Power Laws in Firm Size Variables from Gibrat’s Law and Quasi-inversion Symmetry: A Geomorphological Approach

Using numerical simulations, the authors exhibit the difference between two types of the growth rate distributions, the one of which is observed in both positive and negative data such as profits, and the other of which is in non-negative data such as sales. In the simulation, firstly the Langevin equation generates both positive and negative variables, the growth rate distributions of which are linear functions of the logarithmic growth rate. By superposing the variables not to be negative, we find that the growth rate distributions of the non-negative variables have wider tails than line shape on a log-log scale. At the same time, two types of Non-Gibrats Laws in the middle scale range are also confirmed as observed in real economic data.

The relation between firm age distributions and the decay rate of firm activities in the united states and Japan

We analytically show that the logarithmic average sales of firms first follow power-law growth and subsequently follow exponential growth, if the growth-rate distributions of the sales obey the extended-Gibrat’s property and Gibrat’s law. Here, the extended-Gibrat’s property and Gibrat’s law are statistically observed in short-term data, which denote the dependence of the growth-rate distributions on the initial values. In the derivation, we analytically show that the parameter of the extended-Gibrat’s property is identical to the power-law growth exponent and that it also decides the parameter of the exponential growth. By employing around one million bits of exhaustive sales data of Japanese firms in the ORBIS database, we confirmed our analytic results.

Geographic Dependency of Population Distribution

We start from Gibrat’s law and quasi-inversion symmetry for three firm size variables (i.e., tangible fixed assets K, number of employees L, and sales Y) and derive a partial differential equation to be satisfied by the joint probability density function of K and L.We then transform K and L, which are correlated, into two independent variables by applying surface openness used in geomorphology and provide an analytical solution to the partial differential equation. Using worldwide data on the firm size variables for companies, we confirm that the estimates on the power-law exponents of K, L, and Y satisfy a relationship implied by the theory.

Nowcast of firm sales using POS data toward stock market stability

In this study, we investigated firm activity data in 2008 and 2014 in the United States and Japan. We reconfirmed that the decay rate of firm activity does not depend on firm age in Japan. This is consistent with the observation that firm age distribution follows an exponential function in Japan. At the same time, in the United States, we found that the decay rate of young firms is high and it becomes lower and settle a constant value as firms age. From these observations, we proposed the model that follows the property observed in the decay rate of firm activity. The constant decay rate in Japan is included as a particular case in this model. The model analytically leads to firm age distribution, the young firm in which deviates from an exponential function. This analytical feature is observed in empirical data in the United States. We confirmed the consistency of this analysis comparing parameters estimated from the decay rate with those from firm age distribution.