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Zipf law and the firm size distribution: a critical discussion of popular estimators

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  • Giulio Bottazzi

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  • Davide Pirino
  • Federico Tamagni

Abstract

The upper tail of the firm size distribution is often assumed to follow a Power Law. Several recent papers, using different estimators and different data sets, conclude that the Zipf Law, in particular, provides a good fit, implying that the fraction of firms with size above a given value is inversely proportional to the value itself. In this article we compare the asymptotic and small sample properties of different methods through which this conclusion has been reached. We find that the family of estimators most widely adopted, based on an OLS regression, is in fact unreliable and basically useless for appropriate inference. This finding raises doubts about previously identified Zipf behavior. Based on extensive numerical analysis, we recommend the adoption of the Hill estimator over any other method when individual observations are available. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Giulio Bottazzi & Davide Pirino & Federico Tamagni, 2015. "Zipf law and the firm size distribution: a critical discussion of popular estimators," Journal of Evolutionary Economics, Springer, vol. 25(3), pages 585-610, July.
  • Handle: RePEc:spr:joevec:v:25:y:2015:i:3:p:585-610 DOI: 10.1007/s00191-015-0395-7
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    References listed on IDEAS

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    1. Giulio Bottazzi & Alex Coad & Nadia Jacoby & Angelo Secchi, 2011. "Corporate growth and industrial dynamics: evidence from French manufacturing," Applied Economics, Taylor & Francis Journals, vol. 43(1), pages 103-116.
    2. de Wit, Gerrit, 2005. "Firm size distributions: An overview of steady-state distributions resulting from firm dynamics models," International Journal of Industrial Organization, Elsevier, vol. 23(5-6), pages 423-450, June.
    3. Xavier Gabaix & Augustin Landier, 2008. "Why has CEO Pay Increased So Much?," The Quarterly Journal of Economics, Oxford University Press, vol. 123(1), pages 49-100.
    4. Giulio Bottazzi & Angelo Secchi & Federico Tamagni, 2014. "Financial constraints and firm dynamics," Small Business Economics, Springer, vol. 42(1), pages 99-116, January.
    5. di Giovanni, Julian & Levchenko, Andrei A. & Rancière, Romain, 2011. "Power laws in firm size and openness to trade: Measurement and implications," Journal of International Economics, Elsevier, pages 42-52.
    6. Ahn, Tom & Arcidiacono, Peter & Wessels, Walter, 2011. "The Distributional Impacts of Minimum Wage Increases When Both Labor Supply and Labor Demand Are Endogenous," Journal of Business & Economic Statistics, American Statistical Association, pages 12-23.
    7. di Giovanni, Julian & Levchenko, Andrei A., 2013. "Firm entry, trade, and welfare in Zipf's world," Journal of International Economics, Elsevier, pages 283-296.
    8. Rafal Weron, 2001. "Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime," HSC Research Reports HSC/01/01, Hugo Steinhaus Center, Wroclaw University of Technology.
    9. Xavier Gabaix & Rustam Ibragimov, 2011. "Rank - 1 / 2: A Simple Way to Improve the OLS Estimation of Tail Exponents," Journal of Business & Economic Statistics, Taylor & Francis Journals, pages 24-39.
    10. Fujiwara, Yoshi & Di Guilmi, Corrado & Aoyama, Hideaki & Gallegati, Mauro & Souma, Wataru, 2004. "Do Pareto–Zipf and Gibrat laws hold true? An analysis with European firms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(1), pages 197-216.
    11. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    12. Nunes Amaral, Luís A & Buldyrev, Sergey V & Havlin, Shlomo & Maass, Philipp & Salinger, Michael A & Eugene Stanley, H & Stanley, Michael H.R, 1997. "Scaling behavior in economics: The problem of quantifying company growth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 244(1), pages 1-24.
    13. Okuyama, K & Takayasu, M & Takayasu, H, 1999. "Zipf's law in income distribution of companies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(1), pages 125-131.
    14. Erzo G. J. Luttmer, 2007. "Selection, Growth, and the Size Distribution of Firms," The Quarterly Journal of Economics, Oxford University Press, vol. 122(3), pages 1103-1144.
    15. Rafał Weron, 2001. "Levy-Stable Distributions Revisited: Tail Index> 2does Not Exclude The Levy-Stable Regime," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., pages 209-223.
    16. Gabaix, Xavier & Ibragimov, Rustam, 2011. "Rank − 1 / 2: A Simple Way to Improve the OLS Estimation of Tail Exponents," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(1), pages 24-39.
    17. Boris Podobnik & Davor Horvatic & Alexander M. Petersen & Branko Urov{s}evi'c & H. Eugene Stanley, 2010. "Bankruptcy risk model and empirical tests," Papers 1011.2670, arXiv.org.
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    Citations

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    Cited by:

    1. Metzig, Cornelia & Gordon, Mirta B., 2014. "A model for scaling in firms’ size and growth rate distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 398(C), pages 264-279.
    2. Zakaria Babutsidze, 2016. "Innovation, competition and firm size distribution on fragmented markets," Journal of Evolutionary Economics, Springer, pages 143-169.
    3. A.B. Atkinson, 2016. "Pareto and the upper tail of the income distribution in the UK: 1799 to the present," CASE Papers /198, Centre for Analysis of Social Exclusion, LSE.
    4. Giulio Bottazzi & Alessandro De Sanctis & Fabio Vanni, 2016. "Non-performing loans, systemic risk and resilience in financial networks," LEM Papers Series 2016/08, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
    5. repec:eee:phsmap:v:485:y:2017:i:c:p:35-47 is not listed on IDEAS
    6. Cortés, Lina M. & Mora-Valencia, Andrés & Perote, Javier, 2017. "Measuring firm size distribution with semi-nonparametric densities," Physica A: Statistical Mechanics and its Applications, Elsevier, pages 35-47.

    More about this item

    Keywords

    Firm size distribution; Hill estimator; Power-like distribution; Tail estimators; Zipf Law; L11; C15; C46; D20;

    JEL classification:

    • L11 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Production, Pricing, and Market Structure; Size Distribution of Firms
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • D20 - Microeconomics - - Production and Organizations - - - General

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