IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v523y2019icp858-875.html
   My bibliography  Save this article

A tale of two tails: Do Power Law and Lognormal models fit firm-size distributions in the mid-Victorian era?

Author

Listed:
  • Montebruno, Piero
  • Bennett, Robert J.
  • van Lieshout, Carry
  • Smith, Harry

Abstract

The paper explores the frequency and size distributions of firm-size in a novel dataset for the mid-Victorian era from a recent extraction of the England and Wales population censuses of 1851, 1861, 1871, and 1881. The paper contrasts the hypothesis of the Power Laws against the Lognormal model for the tails of the distributions using maximum likelihood estimation, log likelihood ratio, clipped sample coefficient of variation UMPU-Wilks test, Kolmogorov–Smirnov statistic, among other state-of-the-art statistical methods. Our results show that the Power Law hypothesis is accepted for the size distribution for the years 1851 and 1861, while 1871 is marginally non-significant, but for 1881 the test is inconclusive. The paper discusses the process that generates these distributions citing recent literature that shows how after adding an i.i.d. noise to the Gibrat’s multiplicative model one can recreate a Power Law behaviour. Overall, the paper provides, describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a Pareto model and that the Lognormal model is firmly rejected.

Suggested Citation

  • Montebruno, Piero & Bennett, Robert J. & van Lieshout, Carry & Smith, Harry, 2019. "A tale of two tails: Do Power Law and Lognormal models fit firm-size distributions in the mid-Victorian era?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 858-875.
  • Handle: RePEc:eee:phsmap:v:523:y:2019:i:c:p:858-875
    DOI: 10.1016/j.physa.2019.02.054
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437119302079
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

    File URL: https://libkey.io/10.1016/j.physa.2019.02.054?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. P. Allanson, 1992. "Farm Size Structure In England And Wales 1939‐89," Journal of Agricultural Economics, Wiley Blackwell, vol. 43(2), pages 137-148, May.
    2. Michael Batty, 2006. "Rank clocks," Nature, Nature, vol. 444(7119), pages 592-596, November.
    3. Zhang, Jianhua & Chen, Qinghua & Wang, Yougui, 2009. "Zipf distribution in top Chinese firms and an economic explanation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(10), pages 2020-2024.
    4. S. Redner, 1998. "How popular is your paper? An empirical study of the citation distribution," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 4(2), pages 131-134, July.
    5. Robson, Geoffrey B & Gallagher, Colin C, 1994. "Change in the Size Distribution of U.K. Firms," Small Business Economics, Springer, vol. 6(4), pages 299-312, August.
    6. Pascoal, Rui & Augusto, Mário & Monteiro, A.M., 2016. "Size distribution of Portuguese firms between 2006 and 2012," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 342-355.
    7. Sutton, John, 2002. "The variance of firm growth rates: the ‘scaling’ puzzle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 312(3), pages 577-590.
    8. Jan Eeckhout, 2009. "Gibrat's Law for (All) Cities: Reply," American Economic Review, American Economic Association, vol. 99(4), pages 1676-1683, September.
    9. Yannick Malevergne & Alex Saichev & Didier Sornette, 2010. "Theory of Zipf's Law and Beyond," Post-Print hal-01892766, HAL.
    10. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    11. Gatto, Riccardo & Jammalamadaka, S. Rao, 2002. "A saddlepoint approximation for testing exponentiality against some increasing failure rate alternatives," Statistics & Probability Letters, Elsevier, vol. 58(1), pages 71-81, May.
    12. 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.
    13. Heinrich, Torsten & Dai, Shuanping, 2016. "Diversity of firm sizes, complexity, and industry structure in the Chinese economy," Structural Change and Economic Dynamics, Elsevier, vol. 37(C), pages 90-106.
    14. A. Saichev & Y. Malevergne & D. Sornette, 2008. "Theory of Zipf's Law and of General Power Law Distributions with Gibrat's law of Proportional Growth," Papers 0808.1828, arXiv.org.
    15. Gaffeo, Edoardo & Gallegati, Mauro & Palestrini, Antonio, 2003. "On the size distribution of firms: additional evidence from the G7 countries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 117-123.
    16. 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.
    17. Pasquale Cirillo, 2009. "Some evidence about the evolution of the size distribution of Italian firms by age," Economics Bulletin, AccessEcon, vol. 29(3), pages 1723-1730.
    18. Arshad, Sidra & Hu, Shougeng & Ashraf, Badar Nadeem, 2018. "Zipf’s law and city size distribution: A survey of the literature and future research agenda," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 75-92.
    19. Aaron Clauset & Maxwell Young & Kristian Skrede Gleditsch, 2007. "On the Frequency of Severe Terrorist Events," Journal of Conflict Resolution, Peace Science Society (International), vol. 51(1), pages 58-87, February.
    20. Jeff Alstott & Ed Bullmore & Dietmar Plenz, 2014. "powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions," PLOS ONE, Public Library of Science, vol. 9(1), pages 1-11, January.
    21. Philip Lund & Roger Price, 1998. "The Measurement of Average Farm Size," Journal of Agricultural Economics, Wiley Blackwell, vol. 49(1), pages 100-110, March.
    22. Gillespie, Colin S., 2015. "Fitting Heavy Tailed Distributions: The poweRlaw Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 64(i02).
    23. 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.
    24. Xavier Gabaix, 1999. "Zipf's Law for Cities: An Explanation," The Quarterly Journal of Economics, Oxford University Press, vol. 114(3), pages 739-767.
    25. Alexander Shapiro & Jos Berge, 2002. "Statistical inference of minimum rank factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 67(1), pages 79-94, March.
    26. Alex Saichev & Yannick Malevergne & Didier Sornette, 2010. "Theory of Zipf's Law and Beyond," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02946-2, June.
    27. Xavier Gabaix, 1999. "Zipf's Law and the Growth of Cities," American Economic Review, American Economic Association, vol. 89(2), pages 129-132, May.
    28. Bee, Marco & Riccaboni, Massimo & Schiavo, Stefano, 2013. "The size distribution of US cities: Not Pareto, even in the tail," Economics Letters, Elsevier, vol. 120(2), pages 232-237.
    29. Kang, Sang Hoon & Jiang, Zhuhua & Cheong, Chongcheul & Yoon, Seong-Min, 2011. "Changes of firm size distribution: The case of Korea," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(2), pages 319-327.
    30. Gan, Li & Li, Dong & Song, Shunfeng, 2006. "Is the Zipf law spurious in explaining city-size distributions?," Economics Letters, Elsevier, vol. 92(2), pages 256-262, August.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Campolieti, Michele, 2020. "The distribution of union size: Canada, 1913–2014," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).
    2. Asif, Muhammad & Hussain, Zawar & Asghar, Zahid & Hussain, Muhammad Irfan & Raftab, Mariya & Shah, Said Farooq & Khan, Akbar Ali, 2021. "A statistical evidence of power law distribution in the upper tail of world billionaires’ data 2010–20," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 581(C).
    3. Tomaschitz, Roman, 2020. "Multiply broken power-law densities as survival functions: An alternative to Pareto and lognormal fits," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lyócsa, Štefan & Výrost, Tomáš, 2018. "Scale-free distribution of firm-size distribution in emerging economies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 508(C), pages 501-505.
    2. Pascoal, Rui & Augusto, Mário & Monteiro, A.M., 2016. "Size distribution of Portuguese firms between 2006 and 2012," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 342-355.
    3. Segarra, Agustí & Teruel, Mercedes, 2012. "An appraisal of firm size distribution: Does sample size matter?," Journal of Economic Behavior & Organization, Elsevier, vol. 82(1), pages 314-328.
    4. Wang, Yuanjun & You, Shibing, 2016. "An alternative method for modeling the size distribution of top wealth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 443-453.
    5. 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, vol. 485(C), pages 35-47.
    6. Heinrich, Torsten & Dai, Shuanping, 2016. "Diversity of firm sizes, complexity, and industry structure in the Chinese economy," Structural Change and Economic Dynamics, Elsevier, vol. 37(C), pages 90-106.
    7. Gao, Baojun & Chan, Wai Kin (Victor) & Li, Hongyi, 2015. "On the increasing inequality in size distribution of China's listed companies," China Economic Review, Elsevier, vol. 36(C), pages 25-41.
    8. Sebastien TERRA, 2009. "Zipf's Law for Cities: On a New Testing Procedure," Working Papers 200920, CERDI.
    9. Hernán D. Rozenfeld & Diego Rybski & Xavier Gabaix & Hernán A. Makse, 2011. "The Area and Population of Cities: New Insights from a Different Perspective on Cities," American Economic Review, American Economic Association, vol. 101(5), pages 2205-2225, August.
    10. Asif, Muhammad & Hussain, Zawar & Asghar, Zahid & Hussain, Muhammad Irfan & Raftab, Mariya & Shah, Said Farooq & Khan, Akbar Ali, 2021. "A statistical evidence of power law distribution in the upper tail of world billionaires’ data 2010–20," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 581(C).
    11. Pengfei Li & Ming Lu, 2021. "Urban Systems: Understanding and Predicting the Spatial Distribution of China's Population," China & World Economy, Institute of World Economics and Politics, Chinese Academy of Social Sciences, vol. 29(4), pages 35-62, July.
    12. Arshad, Sidra & Hu, Shougeng & Ashraf, Badar Nadeem, 2019. "Zipf’s law, the coherence of the urban system and city size distribution: Evidence from Pakistan," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 87-103.
    13. Marcus Berliant & Hiroki Watanabe, 2015. "Explaining the size distribution of cities: Extreme economies," Quantitative Economics, Econometric Society, vol. 6(1), pages 153-187, March.
    14. 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.
    15. Hernández-Pérez, R., 2010. "An analogy of the size distribution of business firms with Bose–Einstein statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3837-3843.
    16. Da Silva, Sergio & Matsushita, Raul & Giglio, Ricardo & Massena, Gunther, 2018. "Granularity of the top 1,000 Brazilian companies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 68-73.
    17. Gualandi, Stefano & Toscani, Giuseppe, 2019. "Size distribution of cities: A kinetic explanation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 221-234.
    18. Malevergne, Y. & Saichev, A. & Sornette, D., 2013. "Zipf's law and maximum sustainable growth," Journal of Economic Dynamics and Control, Elsevier, vol. 37(6), pages 1195-1212.
    19. Campolieti, Michele, 2020. "The distribution of union size: Canada, 1913–2014," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).
    20. Katahira, Kei & Chen, Yu & Akiyama, Eizo, 2021. "Self-organized Speculation Game for the spontaneous emergence of financial stylized facts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 582(C).

    Corrections

    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:eee:phsmap:v:523:y:2019:i:c:p:858-875. 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: . General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.