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Theory of Zipf's Law and Beyond Alexander I. Saichev

Theory of Zipf's Law and Beyond By Alexander I. Saichev

Theory of Zipf's Law and Beyond by Alexander I. Saichev


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Summary

First, we show that combining Gibrat's law at all rm levels with random processes of rm's births and deaths yield Zipf's law under a balance condition between a rm's growth and death rate.

Theory of Zipf's Law and Beyond Summary

Theory of Zipf's Law and Beyond by Alexander I. Saichev

Zipf's law is one of the few quantitative reproducible regularities found in e- nomics. It states that, for most countries, the size distributions of cities and of rms (with additional examples found in many other scienti c elds) are power laws with a speci c exponent: the number of cities and rms with a size greater thanS is inversely proportional toS. 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 rm levels with random processes of rm's births and deaths yield Zipf's law under a balance condition between a rm's growth and death rate. We nd that Gibrat's law of proportionate growth does not need to be strictly satis ed. As long as the volatility of rms' sizes increase asy- totically proportionally to the size of the rm and that the instantaneous growth rate increases not faster than the volatility, the distribution of rm sizes follows Zipf's law. This suggests that the occurrence of very large rms in the distri- tion of rm sizes described by Zipf's law is more a consequence of random growth than systematic returns: in particular, for large rms, volatility must dominate over the instantaneous growth rate.

Table of Contents

Continuous Gibrat#x2019;s Law and Gabaix#x2019;s Derivation of Zipf#x2019;s Law.- Flow of Firm Creation.- Useful Properties of Realizations of the Geometric Brownian Motion.- Exit or #x201C;Death#x201D; of Firms.- Deviations from Gibrat#x2019;s Law and Implications for Generalized Zipf#x2019;s Laws.- Firm#x2019;s Sudden Deaths.- Non-stationary Mean Birth Rate.- Properties of the Realization Dependent Distribution of Firm Sizes.- Future Directions and Conclusions.

Additional information

NLS9783642029455
9783642029455
3642029450
Theory of Zipf's Law and Beyond by Alexander I. Saichev
New
Paperback
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
20091118
171
N/A
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