Gibrat's law (sometimes called Gibrat's rule of proportionate growth) is a rule defined by Robert Gibrat (1904–1980) in 1931 stating that the size of a firm and its relative rate of growth are independent. The law of proportionate growth gives rise to a distribution that is log-normal. Gibrat's law is also applied to cities size and growth rate, where proportionate growth process may give rise to a distribution of city sizes that is log-normal, as predicted by Gibrat's law. While the city size distribution is often associated with Zipf's law, this holds only in the upper tail, because empirically the tail of a log-normal distribution cannot be distinguished from Zipf's law. A study using administrative boundaries (places) to define cities finds that the entire distribution of cities, not just the largest ones, is log-normal. But this last claim that the lognormal distribution cannot be rejected has been shown to be the result of a statistics with little power: the uniformly most powerful unbiased test comparing the lognormal to the power law shows unambiguously that the largest 1000 cities are distinctly in the power law regime.
However, it has been argued that it is problematic to define cities through their fairly arbitrary legal boundaries (the places method treats Cambridge and Boston, Massachusetts, as two separate units). A clustering method to construct cities from the bottom up by clustering populated areas obtained from high-resolution data finds a power-law distribution of city size consistent with Zipf's law in almost the entire range of sizes. Note that populated areas are still aggregated rather than individual based. A new method based on individual street nodes for the clustering process leads to the concept of natural cities. It has been found that natural cities exhibit a striking Zipf's law  Furthermore, the clustering method allows for a direct assessment of Gibrat's law. It is found that the growth of agglomerations is not consistent with Gibrat's law: the mean and standard deviation of the growth rates of cities follows a power-law with the city size.
In general, processes characterized by Gibrat's law converge to a limiting distribution, which may be log-normal or power law, depending on more specific assumptions about the stochastic growth process.
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