Zipfâs 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 Gibratâs law of proportional growth but need to incorporate additional constraints and ingredients introducing deviations from it. Here, we present a general theoretical derivation of Zipfâs law, providing a synthesis and extension of previous approaches. First, we show that combining Gibratâs law at all rm levels with random processes of rmâs births and deaths yield Zipfâs law under a âbalanceâ condition between a rmâs growth and death rate. We nd that Gibratâs law of proportionate growth does not need to be strictly satis ed. As long as the volatility of rmsâ 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 Zipfâs law. This suggests that the occurrence of very large rms in the distri- tion of rm sizes described by Zipfâs law is more a consequence of random growth than systematic returns: in particular, for large rms, volatility must dominate over the instantaneous growth rate.
