Generalized Pareto distribution and income inequality: an extension of Gibrat's law

Yong Tao; Yong Tao

doi:10.3934/math.2024730

AIMS Mathematics

2024, Volume 9, Issue 6: 15060-15075. doi: 10.3934/math.2024730

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Research article

Generalized Pareto distribution and income inequality: an extension of Gibrat's law

Yong Tao ^,

College of Economics and Management, Southwest University, Chongqing 400715, China

Received: 14 February 2024 Revised: 22 April 2024 Accepted: 23 April 2024 Published: 26 April 2024
MSC : 60H15, 60E05

Motivated by empirical observations, we proposed a possible extension of Gibrat's law. By applying it into the random growth theory of income distribution, we found that the income distribution is described by a generalized Pareto distribution (GPD) with three parameters. We observed that there is a parameter $ \eta $ in the GPD that plays a key role in determining the shape of income distribution. By using the Kolmogorov-Smirnov test, we empirically showed that, for typical market-economy countries, $ \eta $ is significantly close to 0, indicating that the income distribution is characterized by a two-class pattern: The bottom 90% of the population is approximated by an exponential distribution, while the richest 1%~3% is approximated by an asymptotic power law. However, we empirically found that in China during the period of the planned economy and the early stages of market reform (from 1978 to 1990), $ \eta $ deviated significantly from 0, indicating that the bottom of the population no longer conformed to an exponential distribution.
- random growth theory,
- Kolmogorov forward equation,
- generalized Pareto distribution,
- Kolmogorov-Smirnov test,
- generalized double-Pareto distribution
Citation: Yong Tao. Generalized Pareto distribution and income inequality: an extension of Gibrat's law[J]. AIMS Mathematics, 2024, 9(6): 15060-15075. doi: 10.3934/math.2024730

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Abstract

Motivated by empirical observations, we proposed a possible extension of Gibrat's law. By applying it into the random growth theory of income distribution, we found that the income distribution is described by a generalized Pareto distribution (GPD) with three parameters. We observed that there is a parameter $ \eta $ in the GPD that plays a key role in determining the shape of income distribution. By using the Kolmogorov-Smirnov test, we empirically showed that, for typical market-economy countries, $ \eta $ is significantly close to 0, indicating that the income distribution is characterized by a two-class pattern: The bottom 90% of the population is approximated by an exponential distribution, while the richest 1%~3% is approximated by an asymptotic power law. However, we empirically found that in China during the period of the planned economy and the early stages of market reform (from 1978 to 1990), $ \eta $ deviated significantly from 0, indicating that the bottom of the population no longer conformed to an exponential distribution.

References

[1]	A. B. Atkinson, T. Piketty, E. Saez, Top incomes in the long run of history, J. Econ. Lit., 49 (2011), 3–71. https://doi.org/10.1257/jel.49.1.3 doi: 10.1257/jel.49.1.3
[2]	C. I. Jones, Pareto and Piketty: the macroeconomics of top income and wealth inequality, J. Econ. Perspect., 29 (2015), 29–46. https://doi.org/10.1257/jep.29.1.29 doi: 10.1257/jep.29.1.29
[3]	V. Pareto, Cours d' economie politique, L' Universite de Lausanne, 1897.
[4]	D. G. Champernowne, A model of income distribution, Econ. J., 63 (1953), 318–351. https://doi.org/10.2307/2227127 doi: 10.2307/2227127
[5]	X. Gabaix, Zipf's law for cities: an explanation, Q. J. Econ., 114 (1999), 739–767. https://doi.org/10.1162/003355399556133 doi: 10.1162/003355399556133
[6]	X. Gabaix, Power laws in economics and finance, Annu. Rev. Econ., 1 (2009), 255–294. https://doi.org/10.1146/annurev.economics.050708.142940 doi: 10.1146/annurev.economics.050708.142940
[7]	X. Gabaix, Power laws in economics: an introduction, J. Econ. Perspect., 30 (2016), 185–206. https://doi.org/10.1257/jep.30.1.185 doi: 10.1257/jep.30.1.185
[8]	J. Benhabib, A. Bisin, S. Zhu, The distribution of wealth and fiscal policy in economies with finitely lived agents, Econometrica, 79 (2011), 123–157. https://doi.org/10.3982/ECTA8416 doi: 10.3982/ECTA8416
[9]	C. I. Jones, J. Kim, A Schumpeterian model of top income inequality, J. Polit. Econ., 126 (2018), 1785–1826.
[10]	R. Gibrat, Les inegalites economiques, Paris: Librairie du Receuil Sirey, 1931.
[11]	Y. Malevergne, A. Saichev, D. Sornette, Zipf's law and maximum sustainable growth, J. Econ. Dyn. Control, 37 (2013), 1195–1212. https://doi.org/10.1016/j.jedc.2013.02.004 doi: 10.1016/j.jedc.2013.02.004
[12]	S. Aoki, M. Nirei, Zipf's Law, Pareto's Law, and the evolution of top incomes in the United States, Am. Econ. J.: Macroecon., 9 (2017), 36–71. https://doi.org/10.1257/mac.20150051 doi: 10.1257/mac.20150051
[13]	D. H. Autor, Skills, education, and the rise of earnings inequality among the other 99 percent, Science, 344 (2014), 843–851. https://doi.org/10.1126/science.1251868 doi: 10.1126/science.1251868
[14]	A. Drăgulescu, V. M. Yakovenko, Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States, Phys. A, 299 (2001), 213–221. https://doi.org/10.1016/S0378-4371(01)00298-9 doi: 10.1016/S0378-4371(01)00298-9
[15]	M. Nirei, W. Souma, A two factor model of income distribution dynamics, Rev. Income Wealth, 53 (2007), 440–459. https://doi.org/10.1111/j.1475-4991.2007.00242.x doi: 10.1111/j.1475-4991.2007.00242.x
[16]	Y. Tao, Spontaneous economic order, J. Evol. Econ., 26 (2016), 467–500. https://doi.org/10.1007/s00191-015-0432-6 doi: 10.1007/s00191-015-0432-6
[17]	Y. Tao, X. Wu, T. Zhou, W. Yan, Y. Huang, H. Yu, et al., Exponential structure of income inequality: evidence from 67 countries, J. Econ. Interact. Coord., 14 (2019), 345–376. https://doi.org/10.1007/s11403-017-0211-6 doi: 10.1007/s11403-017-0211-6
[18]	Y. Tao, L. Lin, H. Wang, C. Hou, Superlinear growth and the fossil fuel energy sustainability dilemma: evidence from six continents, Struct. Change Econ. Dyn., 66 (2023), 39–51. https://doi.org/10.1016/j.strueco.2023.04.006 doi: 10.1016/j.strueco.2023.04.006
[19]	M. Almus, Testing "Gibrat's Law" for young firms–empirical results for West Germany, Small Bus. Econ., 15 (2000), 1–12. https://doi.org/10.1023/A:1026512005921 doi: 10.1023/A:1026512005921
[20]	L. Becchetti, G. Trovato, The determinants of growth for small and medium sized firms. The role of the availability of external finance, Small Bus. Econ., 19 (2002), 291–306. https://doi.org/10.1023/A:1019678429111 doi: 10.1023/A:1019678429111
[21]	S. O. Daunfeldt, N. Elert, When is Gibrat's law a law? Small Bus. Econ., 41 (2013), 133–147. https://doi.org/10.1007/s11187-011-9404-x doi: 10.1007/s11187-011-9404-x
[22]	T. Blanchet, J. Fournier, T. Piketty, Generalized Pareto curves: theory and applications, Rev. Income Wealth, 68 (2022), 263–288. https://doi.org/10.1111/roiw.12510 doi: 10.1111/roiw.12510
[23]	X. Gabaix, J. M. Lasry, P. L. Lions, B. Moll, The dynamics of inequality, Econometrica, 84 (2016), 2071–2111. https://doi.org/10.3982/ECTA13569 doi: 10.3982/ECTA13569
[24]	J. Pickands Ⅲ, Statistical inference using extreme order statistics, Ann. Statist., 3 (1975), 119–131. https://doi.org/10.1214/aos/1176343003 doi: 10.1214/aos/1176343003
[25]	J. R. M. Hosking, J. R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, 29 (1987), 339–349. https://doi.org/10.1080/00401706.1987.10488243 doi: 10.1080/00401706.1987.10488243
[26]	N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, 2 Eds., Vol. 1, New York: Wiley, 1994.
[27]	B. C. Arnold, Pareto distributions, 2 Eds., New York: Chapman and Hall/CRC, 2015. https://doi.org/10.1201/b18141
[28]	N. Unnikrishnan Nair, P. G. Sankaran, N. Balakrishnan, Quantile-based reliability analysis, New York: Birkhä;user, 2013. https://doi.org/10.1007/978-0-8176-8361-0
[29]	I. Karatzas, S. Shreve, Brownian motion and stochastic calculus, 2 Eds., Berlin: Springer-Verlag, 1991.
[30]	N. Stokey, The economics of inaction: stochastic control models with fixed costs, Princeton University Press, 2009.
[31]	M. Perc, The Matthew effect in empirical data, J. R. Soc. Interface, 11 (2014), 20140378. https://doi.org/10.1098/rsif.2014.0378 doi: 10.1098/rsif.2014.0378
[32]	A. J. Bowlus, J. M. Robin, An international comparison of lifetime inequality: how continental Europe resembles North America, J. Eur. Econ. Assoc., 10 (2012), 1236–1262. https://doi.org/10.1111/j.1542-4774.2012.01088.x doi: 10.1111/j.1542-4774.2012.01088.x
[33]	Oxfam, Survival of the richest: how we must tax the super-rich now to fight inequality, 2023. Available form: https://www.oxfamamerica.org/explore/research-publications/survival-of-the-richest/

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