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Lorenz curve

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File:Economics Gini coefficient2.svg
A typical Lorenz curve

In economics, the Lorenz curve is a graphical representation of the distribution of income or of wealth. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution.

The curve is a graph showing the proportion of overall income or wealth assumed by the bottom Template:Nowrap of the people, although this is not rigorously true for a finite population (see below). It is often used to represent income distribution, where it shows for the bottom Template:Nowrap of households, what percentage Template:Nowrap of the total income they have. The percentage of households is plotted on the Template:Mvar-axis, the percentage of income on the Template:Mvar-axis. It can also be used to show distribution of assets. In such use, many economists consider it to be a measure of social inequality.

The concept is useful in describing inequality among the size of individuals in ecology<ref name=EcolgyArticle>Template:Cite journal</ref> and in studies of biodiversity, where the cumulative proportion of species is plotted against the cumulative proportion of individuals.<ref name=natureArticle>Template:Cite journal</ref> It is also useful in business modeling: e.g., in consumer finance, to measure the actual percentage Template:Nowrap of delinquencies attributable to the Template:Nowrap of people with worst risk scores. Lorenz curves were also applied to epidemiology and public health, e.g., to measure pandemic inequality as the distribution of national cumulative incidence (y%) generated by the population residing in areas (x%) ranked with respect to their local epidemic attack rate.<ref>Template:Cite journal</ref>

ExplanationEdit

Template:Lorenz curve global income 2011.svg Data from 2005. Points on the Lorenz curve represent statements such as, "the bottom 20% of all households have 10% of the total income."

A perfectly equal income distribution would be one in which every person has the same income. In this case, the bottom Template:Nowrap of society would always have Template:Nowrap of the income. This can be depicted by the straight line Template:Math; called the "line of perfect equality."

By contrast, a perfectly unequal distribution would be one in which one person has all the income and everyone else has none. In that case, the curve would be at Template:Math for all Template:Math, and Template:Math when Template:Math. This curve is called the "line of perfect inequality."

The Gini coefficient is the ratio of the area between the line of perfect equality and the observed Lorenz curve to the area between the line of perfect equality and the line of perfect inequality. The higher the coefficient, the more unequal the distribution is. In the diagram on the right, this is given by the ratio Template:Math, where Template:Mvar and Template:Mvar are the areas of regions as marked in the diagram.

Definition and calculationEdit

File:Gini coefficient US 2016.svg
Lorenz curve for US wealth distribution in 2016 showing negative wealth and oligarchy

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 Template:Math, where Template:Mvar, the cumulative portion of the population, is represented by the horizontal axis, and Template:Mvar, the cumulative portion of the total wealth or income, is represented by the vertical axis.

The curve Template:Mvar need not be a smoothly increasing function of Template:Mvar, For wealth distributions there may be oligarchies or people with negative wealth for instance.<ref>Template:Cite journal</ref>

For a discrete distribution of Y given by values Template:Math, ..., Template:Math in non-decreasing order Template:Math and their probabilities <math>f(y_j) := \Pr(Y=y_j)</math> the Lorenz curve is the continuous piecewise linear function connecting the points Template:Math, Template:Math, where Template:Math, Template:Math, and for Template:Math: <math display="block">\begin{align} 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{align}</math>

When all Template:Math are equally probable with probabilities Template:Math this simplifies to <math display="block">\begin{align} F_i &= \frac i n \\ S_i &= \frac 1 n \sum_{j=1}^i \; y_j \\ L_i &= \frac{S_i}{S_n} \end{align} </math>

For a continuous distribution with the probability density function Template:Mvar and the cumulative distribution function Template:Mvar, the Lorenz curve Template:Mvar is given by: <math display="block"> L(F(x)) = \frac{\int_{-\infty}^x t\,f(t)\,dt}{\int_{-\infty}^\infty t\,f(t)\,dt} = \frac{\int_{-\infty}^x t\,f(t)\,dt}{\mu} </math> where <math>\mu</math> denotes the average. The Lorenz curve Template:Math may then be plotted as a function parametric in Template:Mvar: Template:Math vs. Template:Math. 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 Template:Math with inverse Template:Math, the Lorenz curve Template:Math is directly given by: <math display="block"> L(F) = \frac{\int_0^F x(F_1)\,dF_1}{\int_0^1 x(F_1)\,dF_1} </math>

The inverse Template:Math 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 Template:Math: <math display="block"> x(F_1) = \inf \{y : F(y) \geq F_1\}</math> where Template:Math is the infimum.

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

PropertiesEdit

File:Lorenz curve of Denmark, Hungary, and Namibia.png
A practical example of a Lorenz curve: the Lorenz curves of Denmark, Hungary, and Namibia

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.<ref name=EcolgyArticle />

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.<ref>Template:Cite journal</ref>

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 Template:Math is a random variable, for any positive number Template:Mvar the random variable Template:Mvar has the same Lorenz curve as Template:Math.

The Lorenz curve is flipped twice, once about Template:Math and once about Template:Math, by negation. If Template:Math is a random variable with Lorenz curve Template:Math, then Template:Math has the Lorenz curve:

Template:Math

The Lorenz curve is changed by translations so that the equality gap Template:Math changes in proportion to the ratio of the original and translated means. If Template:Math is a random variable with a Lorenz curve Template:Math and mean Template:Math, then for any constant Template:Math, Template:Math has a Lorenz curve defined by: <math display="block">F - L_{X+c}(F) = \frac{\mu_X}{\mu_X + c} ( F - L_X(F))</math>

For a cumulative distribution function Template:Math with mean Template:Mvar and (generalized) inverse Template:Math, then for any Template:Mvar with 0 < Template:Math :

  • If the Lorenz curve is differentiable:<math display="block">\frac{d L(F)}{d F} = \frac{x(F)}{\mu}</math>
  • If the Lorenz curve is twice differentiable, then the probability density function Template:Math exists at that point and: <math display="block">\frac{d^2 L(F)}{d F^2} = \frac{1}{\mu\,f(x(F))}\,</math>
  • If Template:Math is continuously differentiable, then the tangent of Template:Math is parallel to the line of perfect equality at the point Template:Math. This is also the point at which the equality gap Template:Math, 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: <math display="block">F(\mu) - L(F(\mu)) = \frac{\text{mean absolute deviation}}{2\,\mu}</math>

ExamplesEdit

Both Template:Math and Template:Math, for Template:Math, are well-known functional forms for the Lorenz curve.<ref>Template:Cite journal</ref>

See alsoEdit

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ReferencesEdit

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Further readingEdit

External linksEdit

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