Firm Growth Function and Extended-Gibrat’s Property

We analytically show that the logarithmic average sales of firms first follow power-law growth and subsequently follow exponential growth, if the growth-rate distributions of the sales obey the extended-Gibrat’s property and Gibrat’s law. Here, the extended-Gibrat’s property and Gibrat’s law are statistically observed in short-term data, which denote the dependence of the growth-rate distributions on the initial values. In the derivation, we analytically show that the parameter of the extended-Gibrat’s property is identical to the power-law growth exponent and that it also decides the parameter of the exponential growth. By employing around one million bits of exhaustive sales data of Japanese firms in the ORBIS database, we confirmed our analytic results.