Gibrat’s Law for Cities: Uniformly Most Powerful Unbiased Test of the Pareto Against the Lognormal
University of Saint Etienne - Graduate School of Economics and Business Administration (ISEAG); EM Lyon (Ecole de Management de Lyon) - Department of Economics, Finance, Control
Russian Academy of Sciences (RAS) - International Institute of Earthquake Prediction Theory
Swiss Finance Institute; ETH Zurich
Swiss Finance Institute Research Paper No. 09-40
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 performthe 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 includemore than half of the total U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be 1.4 ± 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) 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 (Gabaix 1999, Eeckhout 2004).
Number of Pages in PDF File: 12
Keywords: City sizes, Gibrat’s law, Zipf’s law
JEL Classification: D30, D51, J61, R12working papers series
Date posted: September 28, 2009
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