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Theory of Zipf's Law and Beyond

Author

Listed:
  • Yannick Malevergne

    (EM - EMLyon Business School)

  • Alex Saichev
  • Didier Sornette

Abstract

Zipf's law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of city sizes and of firms are power laws with a specific exponent: the number of cities and of firms with sizes greater than S is inversely proportional to S. Zipf's law also holds in many other scientific fields. Most explanations start with Gibrat's law of proportional growth (also known as "preferential attachment'' in the application to network growth) but need to incorporate additional constraints and ingredients introducing deviations from it. This book presents a general theoretical derivation of Zipf's law, providing a synthesis and extension of previous approaches. The general theory is presented in the language of firm dynamics for the sake of convenience but applies to many other systems. It takes into account (i) time-varying firm creation, (ii) firm's exit resulting from both a lack of sufficient capital and sudden external shocks, (iii) the coupling between firm's birth rate and the growth of the value of the population of firms. The robustness of Zipf's law is understood from the approximate validity of a general balance condition. A classification of the mechanisms responsible for deviations from Zipf's law is also offered.

Suggested Citation

  • Yannick Malevergne & Alex Saichev & Didier Sornette, 2010. "Theory of Zipf's Law and Beyond," Post-Print hal-02298139, HAL.
  • Handle: RePEc:hal:journl:hal-02298139
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    Cited by:

    1. Rujirutana Mandhachitara & Randall Shannon, 2016. "The Formation and Sustainability of same Product Retail Store Clusters in A Modern Mega City," Tijdschrift voor Economische en Sociale Geografie, Royal Dutch Geographical Society KNAG, vol. 107(5), pages 567-581, December.
    2. Ishikawa, Atushi & 石川, 温 & イシカワ, アツシ & Fujimoto, Shoji & 藤本, 祥二 & フジモト, ショウジ & Watanabe, Tsutomu & 渡辺, 努 & ワタナベ, ツトム & Mizuno, Takayuki & 水野, 貴之 & ミズノ, タカユキ, 2011. "Emergence of power laws with different power-law exponents from reversal quasi-symmetry and Gibrat’s law," Working Paper Series 9, Center for Interfirm Network, Institute of Economic Research, Hitotsubashi University.
    3. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243, arXiv.org.
    4. Cornelia Metzig & Mirta B. Gordon, 2013. "A Model for Scaling in Firms' Size and Growth Rate Distribution," Papers 1304.4311, arXiv.org, revised Nov 2013.
    5. Hideaki Aoyama & Hiroshi Yoshikawa & Hiroshi Iyetomi & Yoshi Fujiwara, 2010. "Productivity dispersion: facts, theory, and implications," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 5(1), pages 27-54, June.
    6. Frank Schweitzer & Giorgio Fagiolo & Didier Sornette & Fernando Vega-Redondo & Douglas R. White, 2009. "Economic Networks: What Do We Know And What Do We Need To Know?," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 12(04n05), pages 407-422.
    7. 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.
    8. A. Saichev & D. Sornette, 2011. "Time-Bridge Estimators of Integrated Variance," Papers 1108.2611, arXiv.org.
    9. Gilles Duranton & Diego Puga, 2023. "Urban Growth and Its Aggregate Implications," Econometrica, Econometric Society, vol. 91(6), pages 2219-2259, November.
    10. Kaldasch, Joachim, 2014. "Evolutionary model of the bank size distribution," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 8, pages 1-16.
    11. , & Lorenz, Jan & ,, 2016. "Innovation vs. imitation and the evolution of productivity distributions," Theoretical Economics, Econometric Society, vol. 11(3), September.
    12. D. Rybski & S. Buldyrev & S. Havlin & F. Liljeros & H. Makse, 2011. "Communication activity in social networks: growth and correlations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 84(1), pages 147-159, November.
    13. Gordon Mulligan & Mark Partridge & John Carruthers, 2012. "Central place theory and its reemergence in regional science," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 48(2), pages 405-431, April.
    14. Misako Takayasu & Hayafumi Watanabe & Hideki Takayasu, 2013. "Generalised central limit theorems for growth rate distribution of complex systems," Papers 1301.2728, arXiv.org, revised Jan 2014.
    15. Sandro Claudio Lera & Didier Sornette, 2017. "Quantification of the evolution of firm size distributions due to mergers and acquisitions," PLOS ONE, Public Library of Science, vol. 12(8), pages 1-16, August.
    16. V. I. Yukalov & D. Sornette, 2012. "Statistical Outliers and Dragon-Kings as Bose-Condensed Droplets," Papers 1205.1364, arXiv.org.
    17. Kaldasch, Joachim, 2012. "Evolutionary model of the growth and size of firms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(14), pages 3751-3769.
    18. Atushi Ishikawa & Shouji Fujimotoa & Tsutomu Watanabe & Takayuki Mizuno, 2012. "The Emergence of Different Tail Exponents in the Distributions of Firm Size Variables," CARF F-Series CARF-F-297, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    19. Atushi Ishikawa & Shouji Fujimoto & Tsutomu Watanabe & Takayuki Mizuno, 2012. "The Emergence of Different Tail Exponents in the Distributions of Firm Size Variables," UTokyo Price Project Working Paper Series 003, University of Tokyo, Graduate School of Economics, revised Dec 2012.

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