Gibrat's law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal

We address the general problem of testing a power law distribution versus a log-normal distribution in statistical data. This general problem is illustrated on the distribution of the 2000 US census of city sizes. We provide definitive results to close the debate between Eeckhout (2004, 2009) and Levy (2009) on the validity of Zipf's law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of the disagreement between Eeckhout and Levy stems from the limited power of their tests, we perform the {\em uniformly most powerful unbiased test} for the null hypothesis of the Pareto distribution against the lognormal. The $p$-value and Hill's estimator as a function of city size lower threshold confirm indubitably that the size distribution of the 1000 largest cities or so, which include more than half of the total U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be $1.4 \pm 0.1$, is equal to 1. For larger ranks, the $p$-value becomes very small and Hill's estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller cities. These two results reconcile the opposite views of Eeckhout (2004, 2009) and Levy (2009). We explain how Gibrat's law of proportional growth underpins both the Pareto and lognormal distributions and stress the key ingredient at the origin of their difference in standard stochastic growth models of cities \cite{Gabaix99,Eeckhout2004}.Comment: 7 pages + 2 figure

Learning, career paths, and the distribution of wages

© 2019 American Economic Association. We develop a theory of career paths and earnings where agents organize in production hierarchies. Agents climb these hierarchies as they learn stochastically from others. Earnings grow as agents acquire knowledge and occupy positions with more subordinates. We contrast these and other implications with US census data for the period 1990 to 2010, matching the Lorenz curve of earnings and the observed mean experience- earnings profiles. We show the increase in wage inequality over this period can be rationalized with a shift in the level of the complexity and profitability of technologies relative to the distribution of knowledge in the population