Theory of Zipf's Law and Beyond (Lecture Notes in Economics and Mathematical Systems, 632)

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 €œbalance€ 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.