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

Author

Listed:
  • Y. MALEVERGNE

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

  • V. PISARENKO

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

  • D. SORNETTE

    (ETH Zurich and Swiss Finance Institute)

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 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
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    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, arXiv.org.
    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

    Keywords

    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)

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