[下载]Theory of Zipf's Law and Beyond

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Zipf’s law is one of the few quantitative reproducible regularities found in economics.
It states that, for most countries, the size distributions of cities and of
firms (with additional examples found in many other scientific fields) are power
laws with a specific exponent: the number of cities and firms with a size greater
than S is inversely proportional to S. Most explanations start with Gibrat’s law of
proportional growth but need to incorporate additional constraints and ingredients
introducing deviations from it. Here, we present a general theoretical derivation of
Zipf’s law, providing a synthesis and extension of previous approaches. First, we
show that combining Gibrat’s law at all firm levels with random processes of firm’s
births and deaths yield Zipf’s law under a “balance” condition between a firm’s
growth and death rate. We find that Gibrat’s law of proportionate growth does not
need to be strictly satisfied. As long as the volatility of firms’ sizes increase asymptotically
proportionally to the size of the firm and that the instantaneous growth
rate increases not faster than the volatility, the distribution of firm sizes follows
Zipf’s law. This suggests that the occurrence of very large firms in the distribution
of firm sizes described by Zipf’s law is more a consequence of random growth
than systematic returns: in particular, for large firms, volatility must dominate over
the instantaneous growth rate. We develop the theoretical framework to take into
account (1) time-varying firm creation, (2) firm’s exit resulting from both a lack
of sufficient capital and sudden external shocks, (3) the coupling between firm’s
birth rate and the growth of the value of the population of firms. We predict deviations
from Zipf’s law under a variety of circumstances, for instance, when the
balance between the birth rate, the instantaneous growth rate and the death rate
is not fulfilled, providing 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. The results obtained here are general and provide an underpinning for
understanding and quantifying Zipf’s law and the power law distribution of sizes
found in many fields.
Nizhni Novgorod, Russia A. Saichev
Lyon & Saint-Etienne, France Y. Malevergne
Z¨urich, Switzerland D. Sornette

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关键词：Beyond Theory Zipf Theo zip 下载 Theory law Beyond Zipf