Analytical Derivation of Power Laws in Firm Size Variables from Gibrat's Law and Quasi-inversion Symmetry: A Geomorphological Approach
Analytical Derivation of Power Laws in Firm Size Variables from Gibrat's Law and Quasi-inversion Symmetry: A Geomorphological Approach
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DOI：
10.7566/jpsj.83.034802
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发表时间：
2014-03-01
影响因子：
1.7
通讯作者：
Watanabe, Tsutomu
中科院分区：
文献类型：
--
作者：
Ishikawa, Atushi;Fujimoto, Shouji;Watanabe, Tsutomu
我们从Gibrat定律和三个企业规模变量的准反转对称性开始（即，有形固定资产K，雇员人数L和销售额Y），并推导出K和L的联合概率密度函数所满足的偏微分方程。然后，我们转换K和L，这是相关的，到两个独立的变量，通过应用表面开放度在地貌学和偏微分方程提供了一个解析解。使用世界范围内的公司规模变量的数据，我们确认，K，L和Y的幂律指数的估计满足理论所隐含的关系。
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.