Robust characterization of multidimensional scaling relations between size measures for business firms

Yuh Kobayashi, Hideki Takayasu, Shlomo Havlin, Misako Takayasu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Although the sizes of business firms have been a subject of intensive research, the defini-tion of a “size” of a firm remains unclear. In this study, we empirically characterize in detail the scaling relations between size measures of business firms, analyzing them based on allometric scal-ing. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994–2015), we examined up to the trivariate relations between corporate size measures: annual sales, capital stock, total assets, and numbers of employees and trading partners. The data were examined using a multivariate generalization of a previously proposed method for analyzing bivariate scalings. We found that relations between measures other than the capital stock are marked by allometric scaling relations. Power–law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that ap-peared to correlate with national economic conditions. We established theoretical relations between the exponents. We expect these results to allow direct estimation of the effects of using alternative size measures of business firms in regression analyses, to facilitate the modeling of firms, and to enhance the current theoretical understanding of complex systems.

Original languageEnglish
Article number168
Pages (from-to)1-33
Number of pages33
Issue number2
StatePublished - Feb 2021

Bibliographical note

Funding Information:
Funding: This research was funded by Center for TDB Advanced Data Analysis and Modeling, Tokyo Institute of Technology, and partially funded by (i) the Grant-in-Aid for Scientific Research (B), JSPS, grant number 26310207; (ii) JST, Strategic International Collaborative Research Program (SICORP) on the topic of “ICT for a Resilient Society” by Japan and Israel; (iii) MEXT, as “Exploratory Challenges on Post-K computer (Study on multilayered multiscale spacetime simulations for social and economical phenomena)”.

Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.


  • Business firms
  • Companies
  • Econophysics
  • Multivariate scaling
  • Nontrivial power–law exponents
  • Size measures


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