{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T06:09:34Z","timestamp":1760854174359,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T00:00:00Z","timestamp":1611878400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["Grant-in-Aid for Scientific Research (B), 26310207"],"award-info":[{"award-number":["Grant-in-Aid for Scientific Research (B), 26310207"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002241","name":"Japan Science and Technology Agency","doi-asserted-by":"publisher","award":["Strategic International Collaborative Research Program (SICORP) on the topic of \u201cICT for a Resilient Society\u201d by Japan and Israel"],"award-info":[{"award-number":["Strategic International Collaborative Research Program (SICORP) on the topic of \u201cICT for a Resilient Society\u201d by Japan and Israel"]}],"id":[{"id":"10.13039\/501100002241","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001700","name":"Ministry of Education, Culture, Sports, Science and Technology","doi-asserted-by":"publisher","award":["Exploratory Challenges on Post-K computer (Study on multilayered multiscale spacetime simulations for social and economical phenomena)"],"award-info":[{"award-number":["Exploratory Challenges on Post-K computer (Study on multilayered multiscale spacetime simulations for social and economical phenomena)"]}],"id":[{"id":"10.13039\/501100001700","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Although the sizes of business firms have been a subject of intensive research, the definition of a \u201csize\u201d 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 scaling. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994\u20132015), 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\u2013law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that appeared 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.<\/jats:p>","DOI":"10.3390\/e23020168","type":"journal-article","created":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T11:34:33Z","timestamp":1611920073000},"page":"168","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Robust Characterization of Multidimensional Scaling Relations between Size Measures for Business Firms"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6409-8711","authenticated-orcid":false,"given":"Yuh","family":"Kobayashi","sequence":"first","affiliation":[{"name":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4452-5517","authenticated-orcid":false,"given":"Hideki","family":"Takayasu","sequence":"additional","affiliation":[{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Sony Computer Science Laboratories, Tokyo 141-0022, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shlomo","family":"Havlin","sequence":"additional","affiliation":[{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2467-614X","authenticated-orcid":false,"given":"Misako","family":"Takayasu","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan"},{"name":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1111\/j.1540-6261.1995.tb05184.x","article-title":"What Do We Know about Capital Structure? 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