Abstract
Although the sizes of business firms have been a subject of intensive research, the definition 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 scaling. 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 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.
Highlights
An index of the size of a business firm has numerous implications, ranging from its method of corporate finance [1,2] and the quality of its CEO [3] to employee job satisfaction [4] and gender gaps in wages among employees [5]
We found that the number of employees has a higher relevance to sales than does the number of trading partners, and there was a mild variation in the trivariate scaling exponents in the period 1994–2015, which appeared to be correlated to the gross domestic product (GDP) variation (Section 3.6)
As small and medium enterprises are legally defined according to their capital stock in Japan [36], capital stock data are widely available for Japanese firms, which is the subject of the database used in our study
Summary
An index of the size of a business firm has numerous implications, ranging from its method of corporate finance [1,2] and the quality of its CEO [3] to employee job satisfaction [4] and gender gaps in wages among employees [5]. The question of what constitutes the “size” of a firm has rarely been addressed, and it is possible that statistical results regarding firm size depend on the measures used. Many of the effects previously found in empirical studies of corporate finance were not robust when different size measures were used as a control variable [19]. Researchers, who questioned the often implicit assumption that size measurements of firms are adequate indicators of their actual size [19,20,21], agree that the relations of various size measures commonly used have not been well-explored in the literature
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