Editing Lorenz curve

{{Short description|Graphical representation of the distribution of income or of wealth}}
{{More citations needed|date=May 2023}}
[[Image:Economics Gini coefficient2.svg|upright=1.2|thumb|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 [[Economic inequality|inequality]] of the [[wealth distribution]].

The curve is a [[graph of a function|graph]] showing the proportion of overall income or wealth assumed by the bottom {{nowrap|{{mvar|x}}%}} 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 {{nowrap|{{mvar|x}}%}} of households, what percentage {{nowrap|({{mvar|y}}%)}} of the total income they have. The [[percentage]] of households is plotted on the {{mvar|x}}-axis, the percentage of income on the {{mvar|y}}-axis.  It can also be used to show distribution of [[Asset (economics)|asset]]s.  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>{{cite journal
  | doi = 10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2
  | last = Damgaard
  | first = Christian
  | author-link = Christian Damgaard
  | author2 = Jacob Weiner
  | title = Describing inequality in plant size or fecundity
  | journal = Ecology
  | year = 2000
  | volume = 81
  | issue = 4
  | pages = 1139–1142
}}</ref> and in studies of [[biodiversity]], where the cumulative proportion of species is plotted against the cumulative proportion of individuals.<ref name=natureArticle>{{cite journal
 | doi = 10.1038/nature07840
 | last = Wittebolle
 | first = Lieven
 | pmid = 19270679
 | title = Initial community evenness favours functionality under selective stress
 | journal = [[Nature (journal)|Nature]]
 | year = 2009
 | volume = 458
 | issue = 7238
 | pages = 623–626
| bibcode = 2009Natur.458..623W | s2cid = 4419280
 | display-authors = etal}}</ref> It is also useful in [[business modeling]]: e.g., in [[consumer finance]], to measure the actual percentage {{nowrap|{{mvar|y}}%}} of [[debt#Individuals|delinquencies]] attributable to the {{nowrap|{{mvar|x}}%}} of people with worst [[credit score|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>{{Cite journal| last1=Nguyen|first1=Quang D.| last2=Chang|first2=Sheryl L.| last3=Jamerlan|first3=Christina M.| last4=Prokopenko|first4=Mikhail|date=2023| title=Measuring unequal distribution of pandemic severity across census years, variants of concern and interventions| journal=Population Health Metrics| volume=21| issue=17|page=17 | doi=10.1186/s12963-023-00318-6 |pmid=37899455 |doi-access=free | pmc=10613397}}</ref>

==Explanation==
{{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 {{nowrap|{{mvar|N}}%}} of society would always have {{nowrap|{{mvar|N}}%}} of the income.  This can be depicted by the straight line {{math|1=''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 {{math|1=''y'' = 0%}} for all {{math|''x'' < 100%}}, and {{math|1=''y'' = 100%}} when {{math|1=''x'' = 100%}}. 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 {{math|''A'' / (''A''+''B'')}}, where {{mvar|A}} and {{mvar|B}} are the areas of regions as marked in the diagram.

== Definition and calculation==
[[File:Gini coefficient US 2016.svg|upright=1.2|thumb|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 [[Random variable|variable]] against a hypothetical uniform distribution of that variable. It can usually be represented by a function {{math|''L''(''F'')}}, where {{mvar|F}}, the cumulative portion of the population, is represented by the horizontal axis, and {{mvar|L}}, the cumulative portion of the total wealth or income, is represented by the vertical axis.

The curve {{mvar|L}} need not be a smoothly increasing function of {{mvar|F}}, For wealth distributions there may be oligarchies or people with negative wealth for instance.<ref>{{cite journal |last1=Li |first1=Jie |last2=Boghosian |first2=Bruce M. |last3=Li |first3=Chengli |title=The Affine Wealth Model: An agent-based model of asset exchange that allows for negative-wealth agents and its empirical validation |date=14 February 2018|arxiv=1604.02370v2 }}</ref>

For a discrete distribution of Y given by values {{math|''y''{{sub|1}}}}, ..., {{math|''y''{{sub|''n''}}}} in non-decreasing order {{math|(''y''{{sub|''i''}} ≤ ''y''{{sub|''i''+1}})}} and their probabilities <math>f(y_j) := \Pr(Y=y_j)</math> the Lorenz curve is the [[continuous function|continuous]] [[piecewise linear function]] connecting the points {{math|(''F''{{sub|''i''}}, ''L''{{sub|''i''}})}}, {{math|1=''i'' = 0 to ''n''}}, where {{math|1=''F''{{sub|0}} = 0}}, {{math|1=''L''{{sub|0}} = 0}}, and for {{math|1=''i'' = 1 to ''n''}}:
<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 {{math|''y''{{sub|''i''}}}} are equally probable with probabilities {{math|1 / ''n''}} 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]] {{mvar|f}} and the [[cumulative distribution function]]  {{mvar|F}}, the Lorenz curve {{mvar|L}} 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 {{math|''L''(''F'')}} may then be plotted as a function parametric in {{mvar|x}}: {{math|''L''(''x'')}} vs. {{math|''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]] {{math|''F''(''x'')}} with inverse {{math|''x''(''F'')}}, the Lorenz curve {{math|''L''(''F'')}} 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 {{math|''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 {{math|''x''(''F'')}}:
<math display="block"> x(F_1) = \inf \{y : F(y) \geq F_1\}</math>
where {{math|inf}} is the [[infimum]].

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

==Properties==
[[File:Lorenz curve of Denmark, Hungary, and Namibia.png|thumb|A practical example of a Lorenz curve: the Lorenz curves of Denmark, Hungary, and Namibia|upright=1.2]]
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>{{Cite journal| last1=Bishop|first1=John A.| last2=Formby|first2=John P.| last3=Smith|first3=W. James| date=1991| title=Lorenz Dominance and Welfare: Changes in the U.S. Distribution of Income, 1967-1986| url=https://www.jstor.org/stable/2109695| journal=The Review of Economics and Statistics| volume=73| issue=1| pages=134–139| doi=10.2307/2109695| jstor=2109695| issn=0034-6535| url-access=subscription}}</ref>

If the variable being measured cannot take negative values, the Lorenz curve:
*cannot sink below the line of perfect inequality,
*is [[increasing function|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 {{math|'''''X'''''}} is a random variable, for any positive number {{mvar|c}} the random variable {{mvar|c'''X'''}} has the same Lorenz curve as {{math|'''''X'''''}}.

The Lorenz curve is flipped twice, once about {{math|1=''F'' = 0.5}} and once about {{math|1=''L'' = 0.5}}, by negation.  If {{math|'''''X'''''}} is a random variable with Lorenz curve {{math|''L''{{sub|'''''X'''''}}(''F'')}}, then {{math|−'''''X'''''}} has the Lorenz curve:
: {{math|1=''L''{{sub|−'''''X'''''}} = 1 − ''L''{{sub|'''''X'''''}}(1 − ''F'')}}

The Lorenz curve is changed by translations so that the equality gap {{math|''F'' − ''L''(''F'')}} changes in proportion to the ratio of the original and translated means.  If {{math|'''''X'''''}} is a random variable with a Lorenz curve {{math|1=''L''{{sub|'''''X'''''}}(''F'')}} and mean {{math|''μ''{{sub|'''''X'''''}}}}, then for any constant {{math|''c'' ≠ &minus;''μ''{{sub|'''''X'''''}}}}, {{math|'''''X''''' + ''c''}} 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 {{math|''F''(''x'')}} with mean {{mvar|μ}} and (generalized) inverse {{math|''x''(''F'')}}, then for any {{mvar|F}} with 0 &lt; {{math|''F'' &lt; 1}} :
*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 {{math|''f''(''x'')}} exists at that point and: <math display="block">\frac{d^2 L(F)}{d F^2} = \frac{1}{\mu\,f(x(F))}\,</math>
*If {{math|''L''(''F'')}} is continuously differentiable, then the tangent of {{math|''L''(''F'')}} is parallel to the line of perfect equality at the point {{math|''F''(''μ'')}}.  This is also the point at which the equality gap {{math|''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]]: <math display="block">F(\mu) - L(F(\mu)) = \frac{\text{mean absolute deviation}}{2\,\mu}</math>

==Examples==
Both {{math|1=''L''(''F'') = ''F''{{isup|''P''}}}} and {{math|1=''L''(''F'') = 1 − (1 − ''F''){{isup|1/''P''}}}}, for {{math|''P'' ≥ 1}}, are well-known functional forms for the Lorenz curve.<ref>{{cite journal |last1=Sitthiyot |first1=Thitithep |last2=Holasut |first2=Kanyarat |date=2021 |title=A simple method for estimating the Lorenz curve |url=https://www.nature.com/articles/s41599-021-00948-x |journal=Humanit Soc Sci Commun |volume=8 |issue=268 |doi=10.1057/s41599-021-00948-x |access-date=May 1, 2025|arxiv=2112.15291 }}</ref>

==See also==
{{Commons category|Lorenz curve}}
* [[Distribution (economics)]]
* [[Distribution of wealth]]
* [[Welfare economics]]
* [[Income inequality metrics]]
* [[Gini coefficient]]
* [[Hoover index]] (a.k.a. Robin Hood index)
* [[ROC analysis]]
* [[Social welfare (political science)]]
* [[Economic inequality]]
* [[Zipf's law]]
* [[Pareto distribution]]
* [[Mean deviation (disambiguation)|Mean deviation]]
* [[The Elephant Curve]]

==References==
{{Reflist}}

==Further reading==
*{{cite journal | author=Lorenz, M. O. | title=Methods of measuring the concentration of wealth | journal=Publications of the American Statistical Association | year=1905 | volume=9 | pages=209–219 | doi = 10.2307/2276207 | jstor=2276207 | issue=70 | publisher=Publications of the American Statistical Association, Vol. 9, No. 70 | bibcode=1905PAmSA...9..209L| s2cid=154048722 }}
*{{cite journal | author=Gastwirth, Joseph L. | title=The Estimation of the Lorenz Curve and Gini Index | journal=The Review of Economics and Statistics | year=1972 | volume=54 | pages=306–316 | doi = 10.2307/1937992 | jstor=1937992 | issue=3 | publisher=The Review of Economics and Statistics, Vol. 54, No. 3}}
*{{cite book | first=S. R. | last=Chakravarty | year=1990 | title=Ethical Social Index Numbers | publisher=Springer-Verlag | location=New York | isbn=0-387-52274-3 }}
*{{cite book | first=Sudhir | last=Anand | year=1983 | title=Inequality and Poverty in Malaysia | publisher=Oxford University Press | location=New York | isbn=0-19-520153-1}}

==External links==
* [http://www.wider.unu.edu/research/Database/en_GB/database/ WIID] {{Webarchive|url=https://web.archive.org/web/20110313002049/http://www.wider.unu.edu/research/Database/en_GB/database/ |date=2011-03-13 }}: World Income Inequality Database, a source of information on inequality, collected by [[WIDER]] (World Institute for Development Economics Research, part of United Nations University)
* [https://ideas.repec.org/c/boc/bocode/s366302.html glcurve]: [[Stata]] module to plot Lorenz curve (type "findit glcurve" or "ssc install glcurve" in Stata prompt to install)
* [http://dasp.ecn.ulaval.ca/ Free add-on to STATA to compute inequality and poverty measures]
* [https://archive.today/20121204174230/http://www.wessa.net/co.wasp Free Online Software (Calculator)] computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
* Free Calculator: [http://www.poorcity.richcity.org/calculator.htm Online] and [https://web.archive.org/web/20041012085224/http://luaforge.net/project/showfiles.php?group_id=49 downloadable scripts] ([[Python (programming language)|Python]] and [[Lua programming language|Lua]]) for Atkinson, Gini, and Hoover inequalities
* Users of the [http://www.r-project.org/ R] data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
* A [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=19968 MATLAB Inequality Package] {{Webarchive|url=https://web.archive.org/web/20081004090028/http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=19968 |date=2008-10-04 }}, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.
* A [https://docs.google.com/Doc?docid=0AXe2E1Mm09WIZGhzazhxaDRfMjUzZ25nMjdkZzY&hl=en complete handout] about the Lorenz curve including various applications, including an [https://docs.google.com/uc?id=0B3e2E1Mm09WIMzQ1ODg5MDgtZjgwNi00NmU1LTgyNmMtZDRhZTYyMTRiYzlk&export=download&hl=en Excel spreadsheet] graphing Lorenz curves and calculating Gini coefficients as well as coefficients of variation.
* [http://pure.au.dk/portal/en/cfd@dmu.dk LORENZ 3.0 ] is a [[Mathematica]] notebook which draw sample Lorenz curves and calculates [[Gini coefficient]]s and [[Lorenz asymmetry coefficient]]s from data in an Excel sheet.

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{{DEFAULTSORT:Lorenz Curve}}
[[Category:Economics curves]]
[[Category:Welfare economics]]
[[Category:Statistical charts and diagrams]]
[[Category:Income inequality metrics]]