IDEAS home Printed from
   My bibliography  Save this paper

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



    (Université de Lyon – Université de Saint-Etienne, EMLYON Business School and ETH Zurich)


    (International Institute of Earthquake Prediction Theory and Mathematical Geophysics Russian Academy of Science)


    (ETH Zurich and Swiss Finance Institute)


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 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).

Suggested Citation

  • Y. Malevergne & V. Pisarenko & D. Sornette, "undated". "Gibrat’s law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal," Swiss Finance Institute Research Paper Series 09-40, Swiss Finance Institute.
  • Handle: RePEc:chf:rpseri:rp0940

    Download full text from publisher

    File URL:
    Download Restriction: no

    File URL:
    Download Restriction: no


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. repec:eee:phsmap:v:481:y:2017:i:c:p:265-275 is not listed on IDEAS
    2. Ichiki, Shingo & Nishinari, Katsuhiro, 2015. "Simple stochastic order-book model of swarm behavior in continuous double auction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 304-314.
    3. Bee, Marco & Riccaboni, Massimo & Schiavo, Stefano, 2017. "Where Gibrat meets Zipf: Scale and scope of French firms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 481(C), pages 265-275.
    4. Hisano, Ryohei & Mizuno, Takayuki, 2011. "Sales distribution of consumer electronics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(2), pages 309-318.
    5. Josep Roca & Blanca Arellano, 2011. "DOES THE SIZE MATTER? Zipf's Law for cities Revisited," ERSA conference papers ersa11p374, European Regional Science Association.
    6. Rafael GONZÀLEZ-VAL, 2012. "Zipf’S Law: Main Issues In Empirical Work," Region et Developpement, Region et Developpement, LEAD, Universite du Sud - Toulon Var, vol. 36, pages 147-164.
    7. Shingo Ichiki & Katsuhiro Nishinari, 2014. "Simple Stochastic Order-Book Model of Swarm Behavior in Continuous Double Auction," Papers 1411.2215,
    8. Massimo, Riccaboni & Jakub, Growiec & Fabio, Pammolli, 2011. "Innovation and Corporate Dynamics: A Theoretical Framework," MPRA Paper 30046, University Library of Munich, Germany.

    More about this item


    City sizes; Gibrat’s law; Zipf’s law.;

    JEL classification:

    • D30 - Microeconomics - - Distribution - - - General
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
    • J61 - Labor and Demographic Economics - - Mobility, Unemployment, Vacancies, and Immigrant Workers - - - Geographic Labor Mobility; Immigrant Workers
    • R12 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General Regional Economics - - - Size and Spatial Distributions of Regional Economic Activity; Interregional Trade (economic geography)


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:chf:rpseri:rp0940. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Marilyn Barja). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.