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Gibrat's law, sometimes called Gibrat's rule of proportionate growth or the law of proportionate effect, is a rule defined by Robert Gibrat (1904–1980) in 1931 stating that the proportional rate of growth of a firm is independent of its absolute size.Gibrat R. (1931) ''Les Inégalités économiques'', Paris, France, 1931. The law of proportionate growth gives rise to a firm size distribution that is log-normal.Sutton, J. (1997), "Gibrat's Legacy", ''Journal of Economic Literature'' XXXV, 40–59. Gibrat's law is also applied to
cities A city is a human settlement of a substantial size. The term "city" has different meanings around the world and in some places the settlement can be very small. Even where the term is limited to larger settlements, there is no universally agree ...
size and growth rate,Bertaud, Alain. (2018), ''Order Without Design: How Markets Shape Cities'', The MIT Press. 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. When considering the entire size distribution, not just the largest cities, then the city size distribution is log-normal.Eeckhout J. (2004), "Gibrat's law for (All) Cities." ''American Economic Review'' 94(5), 1429–1451. The log-normality of the distribution reconciles Gibrat's law also for cities: The law of proportionate effect will therefore imply that the logarithms of the variable will be distributed following the log-normal distribution.Gibrat R. (1931) ''Les Inégalités économiques'', Paris, France, 1931. In isolation, the upper tail (less than 1,000 out of 24,000 cities) fits both the log-normal and the
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial scien ...
: the uniformly most powerful unbiased test comparing the lognormal to the power law shows that the largest 1000 cities are distinctly in the power law regime.Y. Malevergne, V. Pisarenko and D. Sornette (2011), "Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities," ''Physical Review E'' 83, 036111. However, it has been argued that it is problematic to define cities through their fairly arbitrary legal boundaries (the places method treats
Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...
and
Boston, Massachusetts Boston is the capital and most populous city in the Commonwealth (U.S. state), Commonwealth of Massachusetts in the United States. The city serves as the cultural and Financial centre, financial center of New England, a region of the Northeas ...
, 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.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. 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 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. 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 In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
of the growth rates of cities follows a power-law with the city size.Rozenfeld H, Rybski D, Andrade JS, Batty M, Stanley HE and Makse HA (2008), "Laws of Population Growth", ''Proc. Natl. Acad. Sci.'' 105, 18702–18707. In general, processes characterized by Gibrat's law converge to a limiting distribution, often proposed to be the log-normal, or a
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
, depending on more specific assumptions about the
stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...
growth process. However, the tail of the lognormal may fall off too quickly, and its
PDF Portable document format (PDF), standardized as ISO 32000, is a file format developed by Adobe Inc., Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, computer hardware, ...
is not monotonic, but rather has a
Y-intercept In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects ...
of zero probability at the origin. The typical power law is the Pareto I, which has a tail that cannot model fall-off in the tail at large outcomes size, and which does not extend downwards to zero, but rather must be truncated at some positive minimum value. More recently, the Weibull distribution has been derived as the limiting distribution for Gibrat processes, by recognizing that (a) the increments of the growth process are not independent, but rather correlated, in magnitude, and (b) the increment magnitudes typically have monotonic PDFs. The Weibull PDF can appear essentially log-log linear over orders of magnitude ranging from zero, while eventually falling off at unreasonably large outcome sizes. 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.


See also

* List of eponymous laws


References


External links


The New Palgrave Dictionary of Economics Online
{{DEFAULTSORT:Gibrat's Law Eponymous laws of economics 1931 in economic history