Zipf's Law for Firms: Relevance of Birth and Death Processes

Zipf's law states that, for most countries, the number of firms with size greater than S is inversely proportional to S. Most explanations start with Gibrat's rule of proportional growth but need to incorporate additional constraints and ingredients introducing deviations from it. We show that combining Gibrat's rule at all firm levels with random processes of firms' births and deaths yield Zipf's law under a "balance" condition between firm growth and their death rate. We predict deviations from Zipf's law under a variety of circumstances, which provides a framework for identifying the possible origin(s) of the many reports of deviations from the pure Zipf's law. Reciprocally, deviations from Zipf's law in a given economy provides a diagnostic, suggesting possible policy corrections.