Lorenz curve

Definition and calculation

The Lorenz curve is a probability plot (a P–P plot) comparing the distribution of a variable against a hypothetical uniform distribution of that variable. It can usually be represented by a function L(F), where F, the cumulative portion of the population, is represented by the horizontal axis, and L, the cumulative portion of the total wealth or income, is represented by the vertical axis.

The curve L need not be a smoothly increasing function of F, For wealth distributions there may be oligarchies or people with negative wealth for instance.

For a discrete distribution of Y given by values y₁, ..., y_n in non-decreasing order ( y_i ≤ y_i+1) and their probabilities ${\displaystyle f(y_j):=Pr(Y=y_j)}$ the Lorenz curve is the continuous piecewise linear function connecting the points ( F_i, L_i ), i = 0 to n, where F₀ = 0, L₀ = 0, and for i = 1 to n: $${\displaystyle \begin{array}{cc}{F}_i& :=\sum _{j=1}^{i}f(y_j)\\ S_i& :=\sum _{j=1}^{i}f(y_j)y_j\\ L_i& :=\frac{S_i}{S_n}\end{array}}$$

When all y_i are equally probable with probabilities 1/n this simplifies to $${\displaystyle \begin{array}{cc}{F}_i& =\frac{i}{n}\\ S_i& =\frac{1}{n}\sum _{j=1}^i{y}_j\\ L_i& =\frac{S_i}{S_n}\end{array}}$$

For a continuous distribution with the probability density function f and the cumulative distribution function F, the Lorenz curve L is given by: $${\displaystyle L(F(x))=\frac{{\int }_{-\mathrm{\infty }}^{x}tf(t)dt}{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}tf(t)dt}=\frac{{\int }_{-\mathrm{\infty }}^{x}tf(t)dt}{\mu }}$$ where ${\displaystyle \mu }$ denotes the average. The Lorenz curve L(F) may then be plotted as a function parametric in x: L(x) vs. F(x). In other contexts, the quantity computed here is known as the length biased (or size biased) distribution; it also has an important role in renewal theory.

Alternatively, for a cumulative distribution function F(x) with inverse x(F), the Lorenz curve L(F) is directly given by: $${\displaystyle L(F)=\frac{{\int }_0^{F}x(F_1)d{F}_1}{{\int }_0^{1}x(F_1)d{F}_1}}$$

The inverse x(F) may not exist because the cumulative distribution function has intervals of constant values. However, the previous formula can still apply by generalizing the definition of x(F): $${\displaystyle x(F_1)=inf\{y:F(y)\ge F_1\}}$$ where inf is the infimum.

For an example of a Lorenz curve, see Pareto distribution.

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.

A Lorenz curve that never falls beneath a second Lorenz curve and at least once runs above it, has Lorenz dominance over the second one.

If the variable being measured cannot take negative values, the Lorenz curve:

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

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: $${\displaystyle F-L_{X+c}(F)=\frac{{\mu }_X}{{\mu }_X+c}(F-L_X(F))}$$

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:$${\displaystyle \frac{dL(F)}{dF}=\frac{x(F)}{\mu }}$$
• If the Lorenz curve is twice differentiable, then the probability density function f(x) exists at that point and: $${\displaystyle \frac{d^{2}L(F)}{d{F}^2}=\frac{1}{\mu f(x(F))}}$$
• 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 absolute deviation: $${\displaystyle F(\mu )-L(F(\mu ))=\frac{\text{mean absolute deviation}}{2\mu }}$$