Abstract

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.

Introduction

In econophysics, it is well-known that the cumulative distribution functions (CDFs) of capital K, labor L, and production Y of firms obey power laws in large scales that exceed certain size thresholds, which are given by K₀, L₀, and Y₀:

2014.01

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Abstract

We discuss a mechanism through which inversion symmetry (i.e., invariance of a joint probability density function under the exchange of variables) and Gibrat’s law generate power-law distributions with different tail exponents. Using a dataset of firm size variables, that is, tangible fixed assets K, the number of workers L, and sales Y , we confirm that these variables have power-law tails with different exponents, and that inversion symmetry and Gibrat’s law hold. Based on these findings, we argue that there exists a plane in the three dimensional space (log K, log L, log Y ), with respect to which the joint probability density function for the three variables is invariant under the exchange of variables. We provide empirical evidence suggesting that this plane fits the data well, and argue that the plane can be interpreted as the Cobb-Douglas production function, which has been extensively used in various areas of economics since it was first introduced almost a century ago.

Introduction

In various phase transitions, it is universally observed that physical quantities near critical points obey power laws. For instance, in magnetic substances, specific heat, magnetic dipole density, and magnetic susceptibility follow power laws of heat or magnetic flux. It is also known that the cluster-size distribution of the spin follows power laws. The renormalization group approach has been employed to confirm that power laws arise as critical phenomena of phase transitions [1].

2012.12

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Abstract

To explore the emergence of power laws in social and economic phenomena, the authors discuss the mechanism whereby reversal quasi-symmetry and Gibrat’s law lead to power laws with different powerlaw exponents. Reversal quasi-symmetry is invariance under the exchange of variables in the joint PDF (probability density function). Gibrat’s law means that the conditional PDF of the exchange rate of variables does not depend on the initial value. By employing empirical worldwide data for firm size, from categories such as plant assets K, the number of employees L, and sales Y in the same year, reversal quasi-symmetry, Gibrat’s laws, and power-law distributions were observed. We note that relations between power-law exponents and the parameter of reversal quasi-symmetry in the same year were first confirmed. Reversal quasi-symmetry not only of two variables but also of three variables was considered. The authors claim the following. There is a plane in 3-dimensional space (log K, log L, log Y ) with respect to which the joint PDF P_J (K, L, Y ) is invariant under the exchange of variables. The plane accurately fits empirical data (K, L, Y ) that follow power-law distributions. This plane is known as the Cobb-Douglas production function, Y = AK^αL^β which is frequently hypothesized in economics.

Introduction

In various phase transitions, it has been universally observed that physical quantities near critical points obey power laws. For instance, in magnetic substances, the specific heat, magnetic dipole density, and magnetic susceptibility follow power laws of heat or magnetic flux. We also know that the cluster-size distribution of the spin follows power laws. Using renormalization group methods realizes these conformations to power law as critical phenomena of phase transitions [1].

2011.08

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Abstract

We propose a new method for estimating the power-law exponents of firm size variables. 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.

Introduction

Power-law distributions are frequently observed in social phenomena (e.g., Pareto
(1897); Newman (2005); Clauset et al. (2009)). One of the most famous examples
in Economics is the fact that personal income follows a power-law, which was
first found by Pareto (1897) about a century ago, and thus referred to as Pareto
distribution. Specifically, the probability that personal income x is above x₀ is
given by

P>(x) ∝ x ^−µ  　for x > x₀

where µ is referred to as a Pareto exponent or a power-law exponent.

2011.07

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Abstract

本稿では生産関数の形状を選択する手法を提案する。世の中には数人の従業員で営まれる零細企業から数十万人の従業員を擁する超巨大企業まで様々な規模の企業が存在する。どの規模の企業が何社存在するかを表したものが企業の規模分布であり，企業の規模を示す変数である Y （生産）と K（資本）と L（労働）のそれぞれはベキ分布とよばれる分布に従うことが知られている。本稿では，企業規模の分布 関数 と生産 関数 という 2 つの関数の間に存在する関係に注目し，それを手がかりとして生産関数の形状を特定するという手法を提案する。具体的には，K と L についてデータから観察された分布の関数形をもとにして，仮に生産関数がある形状をとる場合に得られるであろう Y の分布関数を導出し，データから観察される Y の分布関数と比較する。日本を含む 25 カ国にこの手法を適用した結果，大半の国や産業において，Y ，K，L の分布と整合的なのはコブダグラス型であることがわかった。また，Y の分布の裾を形成する企業，つまり巨大企業では，K や L の投入量が突出して大きいために Y も突出して大きい傾向がある。一方，全要素生産性が突出して高くそれが原因で Y が突出して大きいという傾向は認められない。

Introduction

企業の生産関数の形状としてはコブダグラス型やレオンチェフ型など様々な形状がこれまで提案されており，ミクロやマクロの研究者によって広く用いられている。例えば，マクロの生産性に関する研究では，コブダグラス生産関数が広く用いられており，そこから全要素生産性を推計することが行われている。しかし，生産 Y と資本 K と雇用 L の関係をコブダグラス型という特定の関数形で表現できるのはなぜか。どういう場合にそれが適切なのか。そうした点にまで踏み込んで検討する研究は限られている。多くの実証研究では，いくつかの生産関数の形状を試してみて，回帰の当てはまりの良さを基準に選択するという便宜的な取り扱いがなされている。

2011.04

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