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The emergence of different tail exponents in the distributions of firm size variables

机译：企业规模变量分布中不同尾指数的出现

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摘要

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 (logK,logL,logY), 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.
机译：我们讨论了一种机制，通过该机制，反演对称性（即在变量交换下联合概率密度函数的不变性）和吉布拉特定律可以生成具有不同尾指数的幂律分布。使用企业规模变量的数据集，即有形固定资产K，工人人数L和销售额Y，我们确认这些变量具有幂指数尾巴且指数不同，并且反对称性和吉布拉特定律成立。基于这些发现，我们认为在三维空间（logK，logL，logY）中存在一个平面，对于该平面而言，三个变量的联合概率密度函数在变量交换下是不变的。我们提供的经验证据表明该平面与数据非常吻合，并认为该平面可以解释为Cobb-Douglas生产函数，该函数自将近一个世纪以来首次被引入经济学的各个领域。

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