Explaining Zipf's Law by Rapid Growth

The fact that the distributions of firm sales and individual incomes have a Pareto tail is one of the important statistical regularities, and numerous theoretical models have been proposed to explain it. However, recent studies have pointed out a difficulty with these existing models: they predict that the time required for firms to become giants or individuals to be super-rich is excessively long compared to what is observed in empirical data. Furthermore, our empirical data show that Zipf's law holds within the size distributions of younger firms and younger individuals, contradicting existing models that predict that Zipf's law is primarily driven by older firms and older individuals. This paper provides an alternative explanation for Zipf's law to address the inconsistencies observed in empirical data. We focus on the heavy-tailed nature of the distribution of growth rates for firm sales and individual incomes, showing that their growth dynamics are characterized by rapid growth over short periods. We show that the emergence of giant firms and the super-rich results from this rapid growth, leading to the formation of Zipf's law. Zipf's law reflects the shared growth dynamics underlying firm sales and individual incomes.