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

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$K$, the number of workers L$L$, and sales Y$Y$, 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)$(logK,logL,logY)$, 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.