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Lorenz curve

Lorenz curve

Overview
In economics, the Lorenz curve is a graphical representation of the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 of the empirical probability distribution
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

 of wealth; it is a graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 showing the proportion of the distribution assumed by the bottom y% of the values. It is often used to represent income
Income
Income is the consumption and savings opportunity gained by an entity within a specified time frame, which is generally expressed in monetary terms. However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings...

 distribution, where it shows for the bottom x% of households, what percentage y% of the total income they have. The percentage
Percentage
In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...

 of households is plotted on the x-axis, the percentage of income on the y-axis.
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Encyclopedia
In economics, the Lorenz curve is a graphical representation of the cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 of the empirical probability distribution
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

 of wealth; it is a graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 showing the proportion of the distribution assumed by the bottom y% of the values. It is often used to represent income
Income
Income is the consumption and savings opportunity gained by an entity within a specified time frame, which is generally expressed in monetary terms. However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings...

 distribution, where it shows for the bottom x% of households, what percentage y% of the total income they have. The percentage
Percentage
In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...

 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 asset
Asset
In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset...

s. In such use, many economists consider it to be a measure of social inequality
Social inequality
Social inequality refers to a situation in which individual groups in a society do not have equal social status. Areas of potential social inequality include voting rights, freedom of speech and assembly, the extent of property rights and access to education, health care, quality housing and other...

. It was developed by Max O. Lorenz
Max O. Lorenz
Max Otto Lorenz was an American economist who developed the Lorenz curve in 1905 to describe income inequalities. He published this paper while a doctoral student at the University of Wisconsin–Madison...

 in 1905 for representing inequality of the wealth distribution.

The concept is useful in describing inequality among the size of individuals in ecology
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...

, and in studies of biodiversity
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...

, where cumulative proportion of species is plotted against cumulative proportion of individuals. 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.


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
Pareto principle
The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes.Business-management consultant Joseph M...

). 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
Gini coefficient
The Gini coefficient is a measure of statistical dispersion developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper "Variability and Mutability" ....

 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
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 piecewise linear function connecting the points ( Fi, Li ), i = 0 to n, where F0 = 0, L0 = 0, and for i = 1 to n:

For a discrete probability function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

 f(y), let yi, i = 1 to n, be the points with non-zero probabilities indexed in increasing order ( yi < yi+1). The Lorenz curve is the continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 piecewise linear function connecting the points ( Fi, Li ), i = 0 to n, where F0 = 0, L0 = 0, and for i = 1 to n:

For a probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 f(x) with the cumulative distribution function F(x), the Lorenz curve L(F(x)) is given by:


For a cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 F(x) with inverse x(F), the Lorenz curve L(F) is given by:


The inverse x(F) may not exist because the cumulative distribution function has jump discontinuities or intervals of constant values. However, the previous formula can still apply by generalizing the definition of x(F):
x(F1) = inf
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

 {y : F(y) ≥ F1}


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
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. 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
Gini coefficient
The Gini coefficient is a measure of statistical dispersion developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper "Variability and Mutability" ....

 and the Lorenz asymmetry coefficient
Lorenz asymmetry coefficient
The Lorenz asymmetry coefficient is a summary statistic of the Lorenz curve that measures the degree of asymmetry of the curve. The Lorenz asymmetry coefficient is defined asS = F+ L\,...

.

If the variable being measured cannot take negative values, the Lorenz curve:
  • cannot rise above the line of perfect equality,
  • cannot sink below the line of perfect inequality,
  • is increasing, and convex
    Convex function
    In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

    .


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 LX(F), then −X has the Lorenz curve:
LX = 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:

See also


  • Distribution (economics)
    Distribution (economics)
    Distribution in economics refers to the way total output, income, or wealth is distributed among individuals or among the factors of production .. In general theory and the national income and product accounts, each unit of output corresponds to a unit of income...

  • Distribution of wealth
    Distribution of wealth
    The distribution of wealth is a comparison of the wealth of various members or groups in a society. It differs from the distribution of income in that it looks at the distribution of ownership of the assets in a society, rather than the current income of members of that society.-Definition of...

  • Welfare economics
    Welfare economics
    Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...

  • Income inequality metrics
    Income inequality metrics
    The concept of inequality is distinct from that of poverty and fairness. Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income, and economic inequality among the participants in a particular economy, such as that of a specific...

  • Gini coefficient
    Gini coefficient
    The Gini coefficient is a measure of statistical dispersion developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper "Variability and Mutability" ....

  • Robin Hood index
    Robin Hood index
    The Hoover index is a measure of income inequality. It is equal to the portion of the total community income that would have to be redistributed for there to be perfect equality....

  • ROC analysis
  • Social welfare (political science)
  • Economic inequality
    Economic inequality
    Economic inequality comprises all disparities in the distribution of economic assets and income. The term typically refers to inequality among individuals and groups within a society, but can also refer to inequality among countries. The issue of economic inequality is related to the ideas of...

  • Zipf's law
  • Pareto distribution
  • Mean deviation

External links

  • WIID: World Income Inequality Database, the most comprehensive source of information on inequality, collected by WIDER
    WIDER
    The United Nations University World Institute for Development Economics Research ' is part of the United Nations University...

     (World Institute for Development Economics Research, part of United Nations University)
  • glcurve: Stata
    Stata
    Stata is a general-purpose statistical software package created in 1985 by StataCorp. It is used by many businesses and academic institutions around the world...

     module to plot Lorenz curve (type "findit glcurve" or "ssc install glcurve" in Stata prompt to install)
  • Free add-on to STATA to compute inequality and poverty measures
  • Free Online Software (Calculator) computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
  • Free Calculator: Online and downloadable scripts (Python
    Python (programming language)
    Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...

     and Lua
    Lua programming language
    Lua is a lightweight multi-paradigm programming language designed as a scripting language with extensible semantics as a primary goal. Lua has a relatively simple C API compared to other scripting languages.- History :...

    ) for Atkinson, Gini, and Hoover inequalities
  • Users of the R data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
  • A MATLAB Inequality Package, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.
  • A complete handhout� about the Lorenz curve including various applications, including an Excel spreadsheet graphing Lorenz curves and calculating Gini coefficients as well as coefficients of variation.
  • LORENZ 3.0 is a Mathematica
    Mathematica
    Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

     notebook which draw sample Lorenz curves and calculates Gini coefficient
    Gini coefficient
    The Gini coefficient is a measure of statistical dispersion developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper "Variability and Mutability" ....

    s and Lorenz asymmetry coefficient
    Lorenz asymmetry coefficient
    The Lorenz asymmetry coefficient is a summary statistic of the Lorenz curve that measures the degree of asymmetry of the curve. The Lorenz asymmetry coefficient is defined asS = F+ L\,...

    s from data in an Excel sheet.