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Zipf's Law for Atlas Models

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  • Ricardo T. Fernholz
  • Robert Fernholz

Abstract

A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

Suggested Citation

  • Ricardo T. Fernholz & Robert Fernholz, 2017. "Zipf's Law for Atlas Models," Papers 1707.04285, arXiv.org, revised Jun 2020.
    • Handle: RePEc:arx:papers:1707.04285
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    References listed on IDEAS

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    1. Xavier Gabaix, 2016. "Power Laws in Economics: An Introduction," Journal of Economic Perspectives, American Economic Association, vol. 30(1), pages 185-206, Winter.
    2. Ricardo T. Fernholz & Christoffer Koch, 2016. "Why are big banks getting bigger?," Working Papers 1604, Federal Reserve Bank of Dallas.
    3. Anthony B. Atkinson & Thomas Piketty & Emmanuel Saez, 2011. "Top Incomes in the Long Run of History," Journal of Economic Literature, American Economic Association, vol. 49(1), pages 3-71, March.
    4. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
    5. Ricardo T. Fernholz, 2017. "The distributional effects of progressive capital taxes," Journal of Economic Policy Reform, Taylor & Francis Journals, vol. 20(2), pages 99-112, April.
    6. 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, vol. 29(1), pages 24-39, January.
    7. Banner, Adrian D. & Ghomrasni, Raouf, 2008. "Local times of ranked continuous semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1244-1253, July.
    8. Ioannides, Yannis & Skouras, Spyros, 2013. "US city size distribution: Robustly Pareto, but only in the tail," Journal of Urban Economics, Elsevier, vol. 73(1), pages 18-29.
    9. Robert Fernholz, 2001. "Equity portfolios generated by functions of ranked market weights," Finance and Stochastics, Springer, vol. 5(4), pages 469-486.
    10. Fernholz, Ricardo & Fernholz, Robert, 2014. "Instability and concentration in the distribution of wealth," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 251-269.
    11. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    12. 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.
    13. Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
    14. Ricardo T. Fernholz, 2017. "Nonparametric methods and local†time†based estimation for dynamic power law distributions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(7), pages 1244-1260, November.
    15. Xavier Gabaix, 1999. "Zipf's Law for Cities: An Explanation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 739-767.
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    Cited by:

    1. Ricardo T. Fernholz & Caleb Stroup, 2018. "Asset Price Distributions and Efficient Markets," Papers 1810.12840, arXiv.org.
    2. Robert Fernholz, 2018. "Numeraire markets," Papers 1801.07309, arXiv.org.
    3. Fernholz, Ricardo & Kramer, Rory, 2021. "Racing to Zipf's Law: Race and Metro Population Size 1910-2010," SocArXiv p5tuh, Center for Open Science.
    4. Steven Campbell & Qien Song & Ting-Kam Leonard Wong, 2024. "Macroscopic properties of equity markets: stylized facts and portfolio performance," Papers 2409.10859, arXiv.org, revised Oct 2024.

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