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The Emergence of Different Tail Exponents in the Distributions of Firm Size Variables

Author

Listed:
  • Atushi Ishikawa

    (Department of Informatics and Business, Faculty of Business Administration and Information Science, Kanazawa Gakuin University)

  • Shouji Fujimoto

    (Department of Informatics and Business, Faculty of Business Administration and Information Science, Kanazawa Gakuin University)

  • Tsutomu Watanabe

    (Graduate School of Economics,University of Tokyo)

  • Takayuki Mizuno

    (Doctoral Program in Computer Science, Graduate School of Systems and Information Engineering, University of Tsukuba)

Abstract

We discuss a mechanism through which inversion symmetry (i.e., invariance of a joint probability density function under the exchange of variables) and Gibrat’s law generate power-law distributions with different tail exponents. Using a dataset of firm size variables, that is, tangible fixed assets K, the number of workers L, and sales Y , we confirm that these variables have power-law tails with different exponents, and that inversion symmetry and Gibrat’s law hold. Based on these findings, we argue that there exists a plane in the three dimensional space (log K, log L, log Y ), with respect to which the joint probability density function for the three variables is invariant under the exchange of variables. We provide empirical evidence suggesting that this plane fits the data well, and argue that the plane can be interpreted as the Cobb-Douglas production function, which has been extensively used in various areas of economics since it was first introduced almost a century ago.

Suggested Citation

  • 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.
    • Handle: RePEc:upd:utppwp:003
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    References listed on IDEAS

    as
    1. 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.
    2. 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, December.
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