Journal of Evolutionary Economics

, Volume 25, Issue 3, pp 585–610 | Cite as

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

  • Giulio Bottazzi
  • Davide Pirino
  • Federico Tamagni
Regular Article


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.


Firm size distribution Hill estimator Power-like distribution Tail estimators Zipf Law 

JEL Classification

L11 C15 C46 D20 


  1. Amaral L, Buldyrev S, Havlin S, Maass P, Salinger M, Stanley H, Stanley M (1997) Scaling behavior in economics: The problem of quantifying company growth. Physica A 244:1–24CrossRefGoogle Scholar
  2. Axtell RL (2001) Zipf Distribution of U.S. Firm Sizes. Sci 293:1818–1820CrossRefGoogle Scholar
  3. Beirlant J, Dierckx G, Goegebeur Y, Matthys G (1999) Tail Index Estimation and an Exponential Regression Model. Extremes 2:177–200CrossRefGoogle Scholar
  4. Beirlant J, Vynckier P, Teugels J (1996) Tail Index Estimation, Pareto Quantile Plots, and Regression Diagnostics. J Am Stat Assoc 91:1659–1667Google Scholar
  5. Bottazzi G, Coad A, Jacoby N, Secchi A (2011) Corporate Growth and Industrial Dynamics: Evidence from French Manufacturing. Appl Econ:43Google Scholar
  6. Bottazzi G, Secchi A (2006) Explaining the Distribution of Firms Growth Rates. The RAND J Econ 37:235–256CrossRefGoogle Scholar
  7. Bottazzi G, Secchi A, Tamagni F (2014) Financial constraints and firm dynamics. Small Bus Econ 42:99–116CrossRefGoogle Scholar
  8. Danielsonn J, Haan LD, Peng L, Vries CGD (2001) Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation. J Multivar Anal 76:226–248CrossRefGoogle Scholar
  9. De Haan L, Peng L (1998) Comparison of tail index estimators. Statistica Neerlandica 52:60–70CrossRefGoogle Scholar
  10. de Wit G (2005) Firm size distributions: An overview of steady-state distributions resulting from firm dynamics models. Int J Ind Organ 23:423–450CrossRefGoogle Scholar
  11. di Giovanni J, Levchenko AA (2010) Firm Entry, Trade, and Welfare in Zipf’s World, NBER Working Papers 16313, National Bureau of Economic ResearchGoogle Scholar
  12. di Giovanni J, Levchenko AA, Ranciére R (2011) Power laws in firm size and openness to trade: Measurement and implications. J Int Econ 85:42–52CrossRefGoogle Scholar
  13. DuMouchel WH (1983) Estimating the Stable Index α in Order to Measure Tail Thickness: A Critique. The Ann Stat 11:1019–1031Google Scholar
  14. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2003) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European Firms, Quantitative Finance Papers. arXiv:cond-mat/0310061
  15. Gabaix X (2009) Power Laws in Economics and Finance. Ann Rev Econ 1:255–293. also available as NBER Working Paper n. 14299CrossRefGoogle Scholar
  16. Gabaix X, Ibragimov R (2011) Rank-1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents. J Bus Econ Stat 29:24–39CrossRefGoogle Scholar
  17. Gabaix X, Landier A (2008) Why Has CEO Pay Increased So Much?. The Q J Econ 123:49–100CrossRefGoogle Scholar
  18. Hall P (1982) On Some Simple Estimates of an Exponent of Regular Variation. J Royal Stat Soc. Series B (Methodological) 44:37–42Google Scholar
  19. Hall P, Welsh A (1985) Adaptive Estimates of Regular Variation. The Ann Stat 13:331–341CrossRefGoogle Scholar
  20. Hill B (1975) A Simple General Approach to Inference About the Tail of a Distribution. The Ann Stat 3:1163–1174CrossRefGoogle Scholar
  21. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions. Wiley, New YorkGoogle Scholar
  22. Kleiber C, Kotz S (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley, New YorkCrossRefGoogle Scholar
  23. Luttmer EGJ (2007) Selection, growth and the size distribution of firms. The Q J Econ 122:1103–1144CrossRefGoogle Scholar
  24. Mason DM (1982) Laws of Large Numbers for Sums of Extreme Values. The Ann Probab 10:754–764CrossRefGoogle Scholar
  25. Newman M (2005) Power Laws, Pareto Distributions and Zipf’s Law. Contemp Phys 46:323–351CrossRefGoogle Scholar
  26. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Physica A 269:125–131CrossRefGoogle Scholar
  27. Pareto V (1886) Sur la courbe de la répartition de la richesse, Université de Lousanne English translation:. Rivista di Politica Economica 87(1997):645–700Google Scholar
  28. Pictet OV, Dacorogna MM, Muller UA (1998) Hill, Bootstrap and Jacknife Estimator for Heavy Tails in. In: Feldman ARJRE, Taqqu MS (eds) A Practical Guide to Heavy Tails. Birkhauser, Boston, pp 283–310Google Scholar
  29. Podobnik B, Horvatic D, Petersen AM, Urosevic B, Stanley EH (2010) Bankruptcy risk model and empirical tests. Proc National Acad Sci USA 107:18325–18330CrossRefGoogle Scholar
  30. Resnick S, Starica C (1997) Smoothing the Hill estimator. Adv Appl Probab 29:273–293CrossRefGoogle Scholar
  31. Resnick S, Starica C (1998) Tail Index Estimation for Dependent Data. The Ann Appl Probab 8:1156–1183CrossRefGoogle Scholar
  32. Silverman B (1986) Density estimation for statistics and data analysis. Chapman and Hall, LondonCrossRefGoogle Scholar
  33. Weron R (2001) Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime. Int J Mod Phys C 12:209–223CrossRefGoogle Scholar
  34. Zipf GK (1932) Selected Studies of the Principle of Relative Frequency in Language. Librairie du Recuil Sirey, ParisCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Giulio Bottazzi
    • 1
  • Davide Pirino
    • 2
  • Federico Tamagni
    • 1
  1. 1.IE and LEM, Scuola Superiore Sant’AnnaPisaItaly
  2. 2.Scuola Normale SuperiorePisaItaly

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