The Pareto distribution with scale parameter \(k\), shape parameter \(\alpha\) and denoted as \(Pareto(k, \alpha)\), where \(k>0\) and \(\alpha>0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)
$$f(y)=\displaystyle \frac{\alpha k^{\alpha}}{y^{\alpha +1}}, $$
and a cumulative distribution function given by
$$F(y) = \displaystyle 1 - \left(\frac{k}{y}\right)^{\alpha},$$
where \(y \geq k\).
The Gini index can be computed as
$$G = \left\{
\begin{array}{cl}
1 , & 0<\alpha <1; \\
\displaystyle \frac{1}{2\alpha-1}, & \alpha \geq 1.
\end{array}
\right.
$$
References
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.
Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.
# Gini index for the Pareto distribution with 'shape = 2'.gpareto(shape = 2)
# Gini indices for the Pareto distribution and different shape parameters.gpareto(shape = 1:5)