# Lorenz curve

The Lorenz curve was developed by Max O. Lorenz in 1905 as a graphical representation of income distribution. It portrays observed income distributions and compares this to a state of perfect income equality. It can also be used to measure distribution of assets. Some doctrines (e.g. Socialism) consider it to be a graphical representation of social inequality as well as income inequality.

In discussions of personal income, we frequently make statements such as, "the bottom twenty percent of all households have ten percent of the total income". The Lorenz curve is based on such statements; every point on the curve represents one such statement.

The Lorenz curve is a graph that shows, for the bottom x% of households, the percentage y% of the total income which they have. The percentage of households is plotted on the x-axis, the percentage of income on the y-axis.

A perfectly equal income distribution in a society would be one in which every person has the same income. In this case, the bottom N% of society would always have N% of the income. Thus a perfectly equal distribution can be depicted by the straight line y = x; we call this line the line of perfect equality.

A perfectly unequal distribution, by contrast, would be one in which one person has all the income and everyone else has none. In that case, the curve would be at y = 0 for all x < 100, and y = 100 when x = 100. We call this curve the line of perfect inequality.

Note that providing that incomes (or whatever else is being measured) cannot be negative, it is impossible for the Lorenz curve to rise above the line of perfect equality, or sink below the line of perfect inequality. The curve must be increasing and convex to the y axis.

The Lorenz curve is used to calculate the Gini coefficient, which is the area between the line of perfect equality and the Lorenz curve, as a percentage of the area between the line of perfect equality and the line of perfect inequality.

A typical Lorenz curve looks like this:

Missing image
Lorenz-curve1.png
image:lorenz-curve1.png

Mathematically, a Lorenz curve is a technique for graphically displaying a probability distribution, in particular, the cumulative distribution function. For any distribution, the Lorenz curve L(F)  is written in terms of the probability density function (f(x)) or the cumulative distribution function (F(x)) as:

[itex]L(F)=\frac{\int_{-\infty}^{x(F)} xf(x)\,dx}{\int_{-\infty}^\infty xf(x)\,dx}

=\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}[itex]

where x(F) is the inverse of the cumulative distribution function F(x). (See, for example, the article on the Pareto distribution.)

## References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statisical Association. 9: 209-219.

[also Will Dawson's contributions]

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