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Gini in a bottle

Posted by ubpdqn on September 18, 2011

Gini Index by country

The relationship between distribution of wealth and distribution of income are usually  visualised using Lorenz curve and the Gini index (or coefficient).

The above graphic is inferior to that from Wikipedia or WolframAlpha but illustrates the range of Gini indices by country (white: data missing).

The Lorenz curve plots the $\int_0^c F^{-1} (u) f(F^{-1}(u)) du/\int_0^\infty F^{-1} (u) f(F^{-1} (u)) du$ against $u$.  $f(x)$ is the probability density function of the income distribution, $F(x)$ is the corresponding  cumulative distribution  function. Here the existence of the inverse is achieved using the infimum method to deal with piecewise constant segments and point discontinuities. The distributions of interest are obviously  discrete. Insights are obtained using  continuous functions.

The Gini index is $\frac {0.5-\int_0^1 L(F) dF }{0.5}$ where $L(F)$ is the Lorenz function.

The following animated gif provides insight into the relationship between the Lorenz Curve and the Gini index.  The Lorenz curves were generated using Pareto distribution with minimum value 10000 and  increasing shape parameters (1.2 to 2.5). yielding a range of Gini indices similar to those observed. This is purely illustrative and makes NO pretense  of representing a real income distribution.  The Gini index is the ratio of the dark purple area to the combined dark and light purple area.

Lorenz curve and Gini index