We discuss a mechanism through which inversion symmetry (i.e., invariance of a joint probability density function under the exchange of variables) and Gibrat\u27s 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\u27s 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.2012～2016年度科学研究費補助金[基盤研究(S)]「長期デフレの解明」(研究代表者 東京大学経済学研究科・渡辺努, 課題番号：24223003
