Abstract
Let X be a continuous or discrete random variable with values in [0,M] and consider all functions (here called transformations) \(q:[0,M]\to [0,\infty )\) that are increasing and have given bounded rates \(B \le \frac {q(v)-q(u)}{v-u} \le A\) for u < v. We prove that among such transformations, there is a transformation q that minimizes the Gini index of q(X), and such a q can be chosen as piecewise linear with only two rates, namely A and B. In the motivation for the study, X represents the incomes of a population. Our results imply that among all such tax policies with fixed allowable minimum and maximum tax rates, there is a tax policy that minimizes the Gini index of the disposable incomes of the population and such a tax policy has only two brackets with the given minimum and maximum rates.
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References
Arnold, B.C.: The Lorenz curve : Evergreen after 100 Years. In: Betti, G., Lemmi, A. (eds.) Advances on income inequality and concentration measures, Routledge, London, pp 12–24 (2008)
Arnold, B.C., Villaseñor, J.A.: Lorenz ordering of order statistics. In: Mosler, K., Scarsini, M. (eds.) Stochastic Orders and Decision under Risk, Lecture Notes-Monograph Series, vol. 19, pp 38–47. Institute of Mathematical Statistics, Hayward (1991)
Artstein, Z.: Discrete and continuous bang-bang and facial spaces or: Look for the extreme points. SIAM Rev. 22(2), 172–185 (1980)
Atkinson, A.B.: On the measurement of inequality. J. Econom. Theory 2(3), 244–263 (1970)
Carbonell-Nicolau, O., Llavador, H.: Inequality reducing properties of progressive income tax schedules: The case of endogenous income. Theor. Econ. 13, 39–60 (2018)
Evans, L.: An introduction to mathematical optimal control theory, ver 0.2, https://math.berkeley.edu/evans/control.course.pdf, accessed June 1, 2021 (2021)
Farris, F.A.: The Gini index and measures of inequality. Amer. Math. Monthly 117(10), 851–864 (2010)
Fellman, J.: The effect of transformations on Lorenz curves. Econometrica 44(4), 823–824 (1976)
Fellman, J.: Mathematical properties of classes of income redistributive properties. Eur. J. Polit. Econ. 17, 179–192 (2001)
Fellman, J.: Properties of Lorenz curves for transformed income distributions. Theor. Econ. Lett. 2(5), 487–493 (2012). https://doi.org/10.4236/tel.2012.25091
Fellman, J.: Properties of non-differentiable tax policies. Theor. Econ. Lett. 3(3), 142–145 (2013). https://doi.org/10.4236/tel.2013.33022
Fellman, J., Jäntti, M., Lambert, P.J.: Optimal tax-transfer systems and redistributive policy. Scand. J. Econ. 101(1), 115–126 (1999)
Feng, C., Wang, H., Tu, X.M., Kowalski, J.: A note on generalized inverses of distribution function and quantile transformation. Appl. Math. 3(12A), 2098–2100 (2012)
Fleming, W., Rishel, R.: Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975)
Gale, W.G., Kearney, M.S., Orszag, P.R.: A significant increase in the top income tax rate wouldn’t substantially alter income inequality, Brookings Report, Monday, September 28, 2015, https://www.brookings.edu/research/would-a-significant-increase-in-the-top-income-tax-rate-substantially-alter-income-inequality/, accessed June 1, 2021 (2015)
Gastwirth, J.: A general definition of the Lorenz curve. Econometrica 39(6), 1037–1039 (1971)
Gini, C.: Variabilità e mutatilità, 1912, reprinted. In: Pizetti, E., Salvemini, T. (eds.) Statistica, Memorie di Metodologica, Libreria Eredi Virgilio Veschi, Rome (1955)
Goldberg, R.R.: Methods of Real Analysis. Xerox College Publishing, Lexington MA (1964)
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer-Verlag, New York (1969)
Jakobsson, U.: On the measurement of the degree of progression. J. Public Econ. 5(1-2), 161–168 (1976)
Kakwani, N.C.: Applications of Lorenz curves in economic analysis. Econometrica 45(3), 719–728 (1977)
König, J., Schröder, C.: Inequality-minimization with a given public budget. J. Econ. Ineq. 16, 607–629 (2018)
Lambert, P.J.: The Distribution and Redistribution of Income, 3rd. Manchester University Press, Manchester and New York (2001)
Pham-Gia, T., Turkkan, N.: Determination of the beta distribution from its Lorenz curve. Math. Comput. Model. 16(2), 73–84 (1992)
Piketty, T.: Capital in the Twenty-First Century. Belknap Press: An Imprint of Harvard University Press, Cambridge MA (2014)
Prete, V., Sommacal, A., Zoli, C.: Optimal non-welfarist income taxation for inequality and polarization reduction. University of Verona Department of Economics Working Papers (2017)
Savaglio, E., Vannucci, S.: On Lorenz preorders and opportunity inequality in finite environments. In: Betti, G., Lemmi, A. (eds.) Advances on Income Inequality and Concentration Measures, Routledge, London, pp 117–128 (2008)
Xu, K.: How has the literature on Gini’s index evolved in the past 80 years? https://papers.ssrn.com/sol3/papers.cfm?abstract_id=423200 accessed June 1 (2021)
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McAsey, M., Mou, L. Transformations that minimize the Gini index of a random variable and applications. J Econ Inequal 20, 483–502 (2022). https://doi.org/10.1007/s10888-021-09508-4
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DOI: https://doi.org/10.1007/s10888-021-09508-4