Gibrat's law (sometimes called Gibrat's rule of proportionate growth) is a rule defined by Robert Gibrat (1904–1980) stating that the size of a firm and its growth rate 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.
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.
In the study of the firms (business), the scholars do not agree that the foundation and the outcome of Gibrat's law are empirically correct.
- Gibrat R. (1931) "Les Inégalités économiques", Paris, France, 1931.
- Sutton, J. (1997), "Gibrat's Legacy", Journal of Economic Literature XXXV, 40–59.
- Eeckhout J. (2004), Gibrat's law for (All) Cities. American Economic Review 94(5), 1429–1451.
- Rozenfeld, Hernán D., Diego Rybski, Xavier Gabaix, and Hernán A. Makse. 2011. "The Area and Population of Cities: New Insights from a Different Perspective on Cities." American Economic Review, 101(5): 2205-25.
- Jiang B, Jia T (2011),"Zipf's law for all the natural cities in the United States: a geospatial perspective", International Journal of Geographical Information Science 25(8), 1269-1281.
- Rozenfeld H, Rybski D, Andrade JS, Batty M, Stanley HE and Makse HA (2008), "Laws of Population Growth", Proc. Nat. Acad. Sci. 105, 18702–18707.