Beyond the Power Law: Log-Normal Bodies, Pareto Tails, and the Limits of Industry Statistics

Background: The claim that a power law governs the distribution of value in a digital economy is usually asserted about sectors, while the evidence for heavy tails concerns firms. The two are different objects, and the claim is rarely tested at the level at which it is made. Methods: We test it at three levels using two censuses: every US employer firm from 1978 to 2023 in ten published size classes and 71 industries in the national accounts from 1997 to 2025. Distributions are fitted to interval-censored counts by maximum likelihood and compared by Vuong’s test; the tail is located by a threshold sweep; concentration from grouped data is bounded rather than estimated. Results: A log-normal is preferred over a Pareto in all forty-six years, yet underpredicts the number of firms with 10,000 employees or more by a factor of twenty-one, so the distribution is a mixture whose tail alone is consistent with a power law. When refitted with more than twenty employees, the exponent is 1.056, within three thousandths of a published microdata estimate obtained by a different method. Between sectors, the claim fails on two tests: industry scale carries no information about the share of value an industry retains, with correlations of 0.009 (Pearson) and −0.047 (Spearman), and the rise in the large-firm employment share is not compositional, since employment moved toward less concentrated sectors throughout. Conclusions: The power law operates on firms within industries, in the tail, which is where every instrument used here stops resolving differences. Rejecting a sectoral power law does not refute firm-level winner-take-all; it places that phenomenon below the resolution of the public statistics in which the debate is usually conducted.