Lorenz curve

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In economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. It is often used to represent income distribution, where it shows for the bottom x% of households, what percentage y% of the total income they have.[1] The percentage of households is plotted on the x-axis, the percentage of income on the y-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. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution.

The concept is useful in describing inequality among the size of individuals in ecology [2], and in studies of biodiversity, where cumulative proportion of species is plotted against cumulative proportion of individuals.[3]. It is also useful in business modeling: e.g., in consumer finance, to measure the actual delinquency Y% of the X% of people with worst predicted risk scores.

Contents

Explanation

Every point on the Lorenz curve represents a statement like "the bottom 20% of all households have 10% of the total income." (see Pareto principle). A perfectly equal income distribution 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. This can be depicted by the straight line "y" = "x"; 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 "y" = 0 for all "x" < 100%, and "y" = 100% when "x" = 100%. This curve is called the "line of perfect inequality."

The Gini coefficient is the area between the line of perfect equality and the observed Lorenz curve, as a percentage of the area between the line of perfect equality and the line of perfect inequality. The higher the coefficient, the more unequal the distribution is.

Calculation

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

For a population of size n, with a sequence of values yi, i = 1 to n, that are indexed in non-decreasing order ( yiyi+1), the Lorenz curve is the continuous piecewise linear function connecting the points ( Fi , Li ), i = 0 to n, where F0 = 0, L0 = 0, and for i = 1 to n:

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