The Lorenz Curve

Figure 1: Lorenz Curve Example

In this example, the population is represented as households and plotted on the x axis from 0% to 100%, and the variable income, is plotted on the y axis, also from 0% to 100%. The line of perfect equality is the baseline function, and displayed in this figure as the green line. The red curve is composed of discrete points because the amount of data is finite. In cases where the curve does not lie on the line of perfect equality, every point on the curve represents a statement like: "the bottom 20% of households has 10% of the total income".

The Lorenz curve can be represented by a function $L(F)$, where $F$ is the horizontal axis, and $L$ is the vertical axis.

For a population of size n, with a sequence of values $y_{i}$ $i = 1$ to $n$ that are indexed in non-decreasing order $(y_{i} <= y_{i+1})$ the Lorenz curve is the continuous piecewise linear function connecting the points $( F_{i} , L_{i} )$, $i = 0$ to $n$, where $F_{0} = 0, L_{0} = 0$, and for $i = 1$ to $n$:

$F_{i} = i/n \\ S_{i} = \Sigma_{j=1}^i \; y_{j} \\ L_{i} = S_{i} / S_{n}$

The amount of inequality in two societies, or in two scenarios can be compared based on their Lorenz curves. If the curve in one case is farther away from the line of perfect equality for every value along the horizontal axis, then that case is considered to have less equality than a case with a curve nearer to the equality line.

The line of perfect equality in the Lorenz curve is $L(F) = F$ and represents a uniform distribution, or equality.

Properties:

1. A Lorenz curve always starts at (0,0) and ends at (1,1).
2. The Lorenz curve is not defined if the mean of the probability distribution is zero or infinite.
3. The Lorenz curve for a probability distribution is a continuous function. However, Lorenz curves representing discontinuous functions can be constructed as the limit of Lorenz curves of probability distributions, the line of perfect inequality being an example.
4. Lorenz curves are scale invariant. They do not take into account whether distributions have different total values, rather they are based on how these values are distributed.
5. Lorenz curves can be used to look at a distribution of a single year, or the changes in the Lorenz curve over time can by analyzed.
6. If the variable being measured cannot take negative values, the Lorenz curve: it cannot rise above the line of perfect equality, cannot sink below the line of perfect inequality, is increasing function, and is a convex function.