Gibrat’s Law for Cities: Uniformly Most Powerful Unbiased Test of the Pareto Against the Lognormal

12 Pages Posted: 28 Sep 2009  

Yannick Malevergne

Université Paris I Panthéon-Sorbonne - Laboratoire PRISM

Vladilen Pisarenko

Russian Academy of Sciences (RAS) - International Institute of Earthquake Prediction Theory

Didier Sornette

Swiss Finance Institute; ETH Zürich - Department of Management, Technology, and Economics (D-MTEC)

Date Written: September 2009

Abstract

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).

Keywords: City sizes, Gibrat’s law, Zipf’s law

JEL Classification: D30, D51, J61, R12

Suggested Citation

Malevergne, Yannick and Pisarenko, Vladilen and Sornette, Didier, Gibrat’s Law for Cities: Uniformly Most Powerful Unbiased Test of the Pareto Against the Lognormal (September 2009). Swiss Finance Institute Research Paper No. 09-40. Available at SSRN: https://ssrn.com/abstract=1479481 or http://dx.doi.org/10.2139/ssrn.1479481

Yannick Malevergne

Université Paris I Panthéon-Sorbonne - Laboratoire PRISM ( email )

17 rue de la Sorbonne
Paris, 75005
France

Vladilen Pisarenko

Russian Academy of Sciences (RAS) - International Institute of Earthquake Prediction Theory ( email )

79, kor. 2
Moscow 113556
Russia

Didier Sornette (Contact Author)

Swiss Finance Institute ( email )

c/o University of Geneve
40, Bd du Pont-d'Arve
1211 Geneva, CH-6900
Switzerland

ETH Zürich - Department of Management, Technology, and Economics (D-MTEC) ( email )

Scheuchzerstrasse 7
Zurich, ZURICH CH-8092
Switzerland
41446328917 (Phone)
41446321914 (Fax)

HOME PAGE: http://www.er.ethz.ch/

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