Lorenz Curve - Properties

Properties

A Lorenz curve always starts at (0,0) and ends at (1,1).

The Lorenz curve is not defined if the mean of the probability distribution is zero or infinite.

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.

The information in a Lorenz curve may be summarized by the Gini coefficient and the Lorenz asymmetry coefficient.

The Lorenz curve cannot rise above the line of perfect equality. If the variable being measured cannot take negative values, the Lorenz curve:

• cannot sink below the line of perfect inequality,
• is increasing, and convex.

Note however that a Lorenz curve for net worth would start out by going negative due to the fact that some people have a negative net worth because of debt.

The Lorenz curve is invariant under positive scaling. If X is a random variable, for any positive number c the random variable c X has the same Lorenz curve as X.

The Lorenz curve is flipped twice, once about F = 0.5 and once about L = 0.5, by negation. If X is a random variable with Lorenz curve L_X(F), then −X has the Lorenz curve:

L _{− X} = 1 − L _X (1 − F)

The Lorenz curve is changed by translations so that the equality gap F − L(F) changes in proportion to the ratio of the original and translated means. If X is a random variable with a Lorenz curve L _X (F) and mean μ _X, then for any constant c ≠ −μ _X, X + c has a Lorenz curve defined by:

For a cumulative distribution function F(x) with mean μ and (generalized) inverse x(F), then for any F with 0 < F < 1 :

• If the Lorenz curve is differentiable:

• If the Lorenz curve is twice differentiable, then the probability density function f(x) exists at that point and:

• If L(F) is continuously differentiable, then the tangent of L(F) is parallel to the line of perfect equality at the point F(μ). This is also the point at which the equality gap F − L(F), the vertical distance between the Lorenz curve and the line of perfect equality, is greatest. The size of the gap is equal to half of the relative mean deviation: