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Analytical Derivation of Power Laws in Firm Size Variables from Gibrat’s Law and Quasi-inversion Symmetry: A Geomorphological Approach

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)

  • Takayuki Mizuno

    (National Institute of Informatics.)

  • Tsutomu Watanabe

    (Graduate School of Economics, University of Tokyo)

Abstract

We start from Gibrat’s law and quasi-inversion symmetry for three firm size variables (i.e., tangible fixed assets K, number of employees L, and sales Y) and derive a partial differential equation to be satisfied by the joint probability density function of K and L.We then transform K and L, which are correlated, into two independent variables by applying surface openness used in geomorphology and provide an analytical solution to the partial differential equation. Using worldwide data on the firm size variables for companies, we confirm that the estimates on the power-law exponents of K, L, and Y satisfy a relationship implied by the theory.

Suggested Citation

  • Atushi Ishikawa & Shouji Fujimoto & Takayuki Mizuno & Tsutomu Watanabe, 2014. "Analytical Derivation of Power Laws in Firm Size Variables from Gibrat’s Law and Quasi-inversion Symmetry: A Geomorphological Approach," UTokyo Price Project Working Paper Series 019, University of Tokyo, Graduate School of Economics.
    • Handle: RePEc:upd:utppwp:019
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    References listed on IDEAS

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    Cited by:

    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. Atushi Ishikawa & Shouji Fujimoto & Takayuki Mizuno, 2021. "Why does production function take the Cobb–Douglas form?," Evolutionary and Institutional Economics Review, Springer, vol. 18(1), pages 79-102, April.

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    Keywords

    econophysics; power law; Gibrat’s law;
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