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Equilibrium distribution of labor productivity: a theoretical model


We construct a theoretical model for equilibrium distribution of workers across sectors with different labor productivity, assuming that a sector can accommodate a limited number of workers which depends only on its productivity. A general formula for such distribution of productivity is obtained, using the detail-balance condition necessary for equilibrium in the Ehrenfest–Brillouin model. We also carry out an empirical analysis on the average number of workers in given productivity sectors on the basis of an exhaustive dataset in Japan. The theoretical formula succeeds in explaining the two distinctive observational facts in a unified way, that is, a Boltzmann distribution with negative temperature on low-to-medium productivity side and a decreasing part in a power–law form on high productivity side.

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  1. This covers all the sectors, except for the finance and insurance (95 firms in 2008), the deep-sea foreign transport of freight (9 firms in 2008), and the holding companies (138 firms in 2008), all of which show abnormally high value of productivity compared with other sectors.


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The authors would like to thank Yoshi Fujiwara, Yuichi Ikeda and Wataru Souma for helpful discussions and comments during the course of this work. and Toshiyuki Masuda, Chairman of The Kyoto Shinkin Bank for his comments on Japanese small to medium firms. We would also thank the Credit Risk Database for the data used in this paper. This work is supported in part by the Program for Promoting Methodological Innovation in Humanities and Social Sciences by Cross-Disciplinary Fusing of the Japan Society for the Promotion of Science.

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Correspondence to Hideaki Aoyama.

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Aoyama, H., Iyetomi, H. & Yoshikawa, H. Equilibrium distribution of labor productivity: a theoretical model. J Econ Interact Coord 10, 57–66 (2015).

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  • Labor productivity
  • Equilibrium
  • Detailed balance
  • Firm theory

JEL Classification

  • C02
  • E20
  • O40