Zipf's Law without the Stationarity Assumption

Author Name ARATA Yoshiyuki (Fellow, RIETI)
Creation Date/NO. December 2023 23-E-085
Research Project Study group on corporate finance and firm dynamics
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First draft: December 2023
Revised: May 2024


This paper analyzes one of the classic empirical regularities in the literature on firm growth: Zipf's law of the firm size distribution. Firstly, using firm-level data and decomposing the sample by the age of the firms, I found that the Pareto tail is observed within each age cohort. In particular, Zipf's law is observed only among younger firms (e.g., firms under 50 years of age). This empirical finding contradicts previous research which assumes that Zipf's law is observed only when the size distribution of firms from each age cohort is aggregated. To address this empirical inconsistency, this paper provides another explanation for Zipf's law. Specifically, Zipf's law is explained by two assumptions: the random walk assumption (i.e., the log of a firm’s sales follows a random walk) and the heavy-tailed assumption that the growth rate distribution has a heavier tail than an exponential. In my analysis, the stationarity assumption (i.e., the firm size distribution is at the stationary state) is not needed. This new explanation resolves the empirical inconsistency and implies that a large firm arises from a few large jumps in size within a short period. Zipf's law reflects this property of firm growth dynamics.