Gibrat's law, sometimes called Gibrat's rule of proportionate growth or the law of proportionate effect, is a rule defined by
Robert Gibrat
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honou ...
(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
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a 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 notable size.Goodall, B. (1987) ''The Penguin Dictionary of Human Geography''. London: Penguin.Kuper, A. and Kuper, J., eds (1996) ''The Social Science Encyclopedia''. 2nd edition. London: Routledge. It can be def ...
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 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 university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...
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.
[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 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
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a 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 proportional relative change in the other quantity, inde ...
, depending on more specific assumptions about the
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
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 in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
is not monotonic, but rather has a Y-intercept 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
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Ren ...
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
This list of eponymous laws provides links to articles on laws, principles, adages, and other succinct observations or predictions named after a person. In some cases the person named has coined the law – such as Parkinson's law. In other ...
References
External links
The New Palgrave Dictionary of Economics Online
{{DEFAULTSORT:Gibrat's Law
Economics laws
Eponyms
1931 in economics