In economics, the
Contents 1 Definition 2 Calculation 2.1 Example: two levels of income 2.2 Alternate expressions 2.3 Discrete probability distribution 2.4 Continuous probability distribution 2.5 Other approaches 3 Generalized inequality indices 4 Gini coefficients of income distributions 4.1 Regional income Gini indices 4.2 World income Gini index since 1800s 5 Gini coefficients of social development 5.1
6 Features of Gini coefficient 7 Countries by Gini Index 8 Limitations of Gini coefficient 9 Alternatives to Gini coefficient 10 Relation to other statistical measures 11 Other uses 12 See also 13 References 14 Further reading 15 External links Definition[edit] Graphical representation of the Gini coefficient The graph shows that the
G = ∑ i = 1 n ∑ j = 1 n
x i − x j
2 ∑ i = 1 n ∑ j = 1 n x j = ∑ i = 1 n ∑ j = 1 n
x i − x j
2 n ∑ i = 1 n x i displaystyle G= frac displaystyle sum _ i=1 ^ n sum _ j=1 ^ n leftx_ i -x_ j right displaystyle 2sum _ i=1 ^ n sum _ j=1 ^ n x_ j = frac displaystyle sum _ i=1 ^ n sum _ j=1 ^ n leftx_ i -x_ j right displaystyle 2nsum _ i=1 ^ n x_ i When the income (or wealth) distribution is given as a continuous
probability distribution function p(x), where p(x)dx is the fraction
of the population with income x to x+dx, then the
G = 1 2 μ ∫ − ∞ ∞ ∫ − ∞ ∞ p ( x ) p ( y )
x − y
d x d y displaystyle G= frac 1 2mu int _ -infty ^ infty int _ -infty ^ infty p(x)p(y),x-y,dx,dy where μ is the mean of the distribution μ = ∫ − ∞ ∞ x p ( x ) d x displaystyle mu =int _ -infty ^ infty x,p(x),dx and the lower limits of integration may be replaced by zero when all incomes are positive. Calculation[edit] Example: two levels of income[edit] Richest u % of population (red) equally share f % of all income or wealth, others (green) equally share remainder: G = f − u. A smooth distribution (blue) with same u and f always has G > f − u. The most equal society will be one in which every person receives the
same income (G = 0); the most unequal society will be one in which a
single person receives 100% of the total income and the remaining N
− 1 people receive none (G = 1 − 1/N).
While the income distribution of any particular country need not
follow simple functions, these functions give a qualitative
understanding of the income distribution in a nation given the Gini
coefficient. The effects of minimum income policy due to
redistribution can be seen in the linear relationships.
An informative simplified case just distinguishes two levels of
income, low and high. If the high income group is u % of the
population and earns a fraction f % of all income, then the Gini
coefficient is f − u. An actual more graded distribution with these
same values u and f will always have a higher
Alternate expressions[edit] In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example, (taking y to mean the income or wealth of a person or household): For a population uniform on the values yi, i = 1 to n, indexed in non-decreasing order (yi ≤ yi+1): G = 1 n ( n + 1 − 2 ( ∑ i = 1 n ( n + 1 − i ) y i ∑ i = 1 n y i ) ) displaystyle G= frac 1 n left(n+1-2left( frac sum limits _ i=1 ^ n ;(n+1-i)y_ i sum limits _ i=1 ^ n y_ i right)right) This may be simplified to: G = 2 Σ i = 1 n i y i n Σ i = 1 n y i − n + 1 n displaystyle G= frac 2Sigma _ i=1 ^ n ;iy_ i nSigma _ i=1 ^ n y_ i - frac n+1 n This formula actually applies to any real population, since each person can be assigned his or her own yi.[17] Since the
G ( S ) = 1 n − 1 ( n + 1 − 2 ( Σ i = 1 n ( n + 1 − i ) y i Σ i = 1 n y i ) ) displaystyle G(S)= frac 1 n-1 left(n+1-2left( frac Sigma _ i=1 ^ n ;(n+1-i)y_ i Sigma _ i=1 ^ n y_ i right)right) is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like G, G (S) has a simpler form: G ( S ) = 1 − 2 n − 1 ( n − Σ i = 1 n i y i Σ i = 1 n y i ) displaystyle G(S)=1- frac 2 n-1 left(n- frac Sigma _ i=1 ^ n ;iy_ i Sigma _ i=1 ^ n y_ i right) . There does not exist a sample statistic that is in general an unbiased
estimator of the population Gini coefficient, like the relative mean
absolute difference.
Discrete probability distribution[edit]
For a discrete probability distribution with probability mass function
f ( yi ), i = 1 to n, where f ( yi ) is the fraction of the population
with income or wealth yi >0, the
G = 1 2 μ ∑ i = 1 n ∑ j = 1 n f ( y i ) f ( y j )
y i − y j
displaystyle G= frac 1 2mu sum limits _ i=1 ^ n sum limits _ j=1 ^ n ,f(y_ i )f(y_ j )y_ i -y_ j where μ = ∑ i = 1 n y i f ( y i ) displaystyle mu =sum limits _ i=1 ^ n y_ i f(y_ i ) If the points with nonzero probabilities are indexed in increasing order (yi < yi+1) then: G = 1 − Σ i = 1 n f ( y i ) ( S i − 1 + S i ) S n displaystyle G=1- frac Sigma _ i=1 ^ n ;f(y_ i )(S_ i-1 +S_ i ) S_ n where S i = Σ j = 1 i f ( y j ) y j displaystyle S_ i =Sigma _ j=1 ^ i ;f(y_ j ),y_ j , and S 0 = 0 displaystyle S_ 0 =0, . These formulae are also applicable in the limit as n → ∞ displaystyle nrightarrow infty . Continuous probability distribution[edit]
When the population is large, the income distribution may be
represented by a continuous probability density function f(x) where
f(x) dx is the fraction of the population with wealth or income in the
interval dx about x. If F(x) is the cumulative distribution function
for f(x), then the
B = ∫ 0 1 L ( F ) d F . displaystyle B=int _ 0 ^ 1 L(F)dF. The
G = 1 − 1 μ ∫ 0 ∞ ( 1 − F ( y ) ) 2 d y = 1 μ ∫ 0 ∞ F ( y ) ( 1 − F ( y ) ) d y displaystyle G=1- frac 1 mu int _ 0 ^ infty (1-F(y))^ 2 dy= frac 1 mu int _ 0 ^ infty F(y)(1-F(y))dy The latter result comes from integration by parts. (Note that this
formula can be applied when there are negative values if the
integration is taken from minus infinity to plus infinity.)
The
G = 1 2 μ ∫ 0 1 ∫ 0 1
Q ( F 1 ) − Q ( F 2 )
d F 1 d F 2 . displaystyle G= frac 1 2mu int _ 0 ^ 1 int _ 0 ^ 1 Q(F_ 1 )-Q(F_ 2 ),dF_ 1 ,dF_ 2 . For some functional forms, the Gini index can be calculated explicitly. For example, if y follows a lognormal distribution with the standard deviation of logs equal to σ displaystyle sigma , then G = erf ( σ 2 ) displaystyle G=operatorname erf left( frac sigma 2 right) where erf displaystyle operatorname erf is the error function ( since G = 2 ϕ ( σ 2 ) − 1 displaystyle G=2phi left( frac sigma sqrt 2 right)-1 , where ϕ displaystyle phi is the cumulative standard normal distribution).[18] In the table below, some examples are shown. The Dirac delta distribution represents the case where everyone has the same wealth (or income); it implies that there are no variations at all between incomes.
Dirac delta function δ ( x − x 0 ) , x 0 > 0 displaystyle delta (x-x_ 0 ),,x_ 0 >0 0 Uniform distribution 1 b − a a ≤ x ≤ b 0 o t h e r w i s e displaystyle begin cases frac 1 b-a &aleq xleq b\0&mathrm otherwise end cases b − a 3 ( b + a ) displaystyle frac b-a 3(b+a) Exponential distribution λ e − x λ , x > 0 displaystyle lambda e^ -xlambda ,,,x>0 1 / 2 displaystyle 1/2 Log-normal distribution 1 σ 2 π e − ( ln ( x ) − μ ) 2 σ 2 displaystyle frac 1 sigma sqrt 2pi e^ frac -(ln ,(x)-mu )^ 2 sigma ^ 2 erf ( σ / 2 ) displaystyle textrm erf (sigma /2) Pareto distribution α k α x α + 1 x ≥ k 0 x < k displaystyle begin cases frac alpha k^ alpha x^ alpha +1 &xgeq k\0&x<kend cases 1 0 < α < 1 1 2 α − 1 α ≥ 1 displaystyle begin cases 1&0<alpha <1\ frac 1 2alpha -1 &alpha geq 1end cases Chi-squared distribution 2 − k / 2 e − x / 2 x k / 2 − 1 Γ ( k / 2 ) displaystyle frac 2^ -k/2 e^ -x/2 x^ k/2-1 Gamma (k/2) 2 Γ ( 1 + k 2 ) k Γ ( k / 2 ) π displaystyle frac 2,Gamma left( frac 1+k 2 right) k,Gamma (k/2) sqrt pi Gamma distribution e − x / θ x k − 1 θ − k Γ ( k ) displaystyle frac e^ -x/theta x^ k-1 theta ^ -k Gamma (k) Γ ( 2 k + 1 2 ) k Γ ( k ) π displaystyle frac Gamma left( frac 2k+1 2 right) k,Gamma (k) sqrt pi Weibull distribution k λ ( x λ ) k − 1 e − ( x / λ ) k displaystyle frac k lambda ,left( frac x lambda right)^ k-1 e^ -(x/lambda )^ k 1 − 2 − 1 / k displaystyle 1-2^ -1/k Beta distribution x α − 1 ( 1 − x ) β − 1 B ( α , β ) displaystyle frac x^ alpha -1 (1-x)^ beta -1 B(alpha ,beta ) ( 2 α ) B ( α + β , α + β ) B ( α , α ) B ( β , β ) displaystyle left( frac 2 alpha right) frac B(alpha +beta ,alpha +beta ) B(alpha ,alpha )B(beta ,beta ) Other approaches[edit]
Sometimes the entire
Xk is the cumulated proportion of the population variable, for k = 0,...,n, with X0 = 0, Xn = 1. Yk is the cumulated proportion of the income variable, for k = 0,...,n, with Y0 = 0, Yn = 1. Yk should be indexed in non-decreasing order (Yk > Yk – 1) If the
G 1 = 1 − ∑ k = 1 n ( X k − X k − 1 ) ( Y k + Y k − 1 ) displaystyle G_ 1 =1-sum _ k=1 ^ n (X_ k -X_ k-1 )(Y_ k +Y_ k-1 )
is the resulting approximation for G. More accurate results can be
obtained using other methods to approximate the area B, such as
approximating the
k = A + N ( 0 , s 2 / y k ) displaystyle k=A+ N(0,s^ 2 /y_ k ) Ogwang showed that G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculation of the jackknife estimate for the standard error. Giles (2004) argued that the standard error of the estimate of A can be used to derive that of the estimate of G directly without using a jackknife at all. This method only requires the use of ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improving with increasing sample size.[19] However it has since been argued that this is dependent on the model's assumptions about the error distributions (Ogwang 2004) and the independence of error terms (Reza & Gastwirth 2006) and that these assumptions are often not valid for real data sets. It may therefore be better to stick with jackknife methods such as those proposed by Yitzhaki (1991) and Karagiannis and Kovacevic (2000). The debate continues.[citation needed] Guillermina Jasso[20] and Angus Deaton[21] independently proposed the following formula for the Gini coefficient: G = N + 1 N − 1 − 2 N ( N − 1 ) μ ( Σ i = 1 n P i X i ) displaystyle G= frac N+1 N-1 - frac 2 N(N-1)mu (Sigma _ i=1 ^ n ;P_ i X_ i ) where μ displaystyle mu is mean income of the population, Pi is the income rank P of person i, with income X, such that the richest person receives a rank of 1 and the poorest a rank of N. This effectively gives higher weight to poorer people in the income distribution, which allows the Gini to meet the Transfer Principle. Note that the Jasso-Deaton formula rescales the coefficient so that its value is 1 if all the X i displaystyle X_ i are zero except one. Note however Allison's reply on the need to
divide by N² instead.[22]
FAO explains another version of the formula.[23]
Generalized inequality indices[edit]
See also: Generalized entropy index
The
r j = x j / x ¯ displaystyle r_ j =x_ j / overline x , equals 1 for all j units in some population (for example, there is perfect income equality when everyone's income x j displaystyle x_ j equals the mean income x ¯ displaystyle overline x , so that r j = 1 displaystyle r_ j =1 for everyone). Measures of inequality, then, are measures of the average deviations of the r j = 1 displaystyle r_ j =1 from 1; the greater the average deviation, the greater the inequality. Based on these observations the inequality indices have this common form:[24] Inequality = Σ j p j f ( r j ) , displaystyle text Inequality =Sigma _ j ,p_ j ,f(r_ j ),, where pj weights the units by their population share, and f(rj) is a
function of the deviation of each unit's rj from 1, the point of
equality. The insight of this generalised inequality index is that
inequality indices differ because they employ different functions of
the distance of the inequality ratios (the rj) from 1.
Gini coefficients of income distributions[edit]
See also: List of countries by income equality
Gini coefficients of income are calculated on market income as well as
disposable income basis. The
Regional income Gini indices[edit] According to UNICEF, Latin America and the Caribbean region had the highest net income Gini index in the world at 48.3, on unweighted average basis in 2008. The remaining regional averages were: sub-Saharan Africa (44.2), Asia (40.4), Middle East and North Africa (39.2), Eastern Europe and Central Asia (35.4), and High-income Countries (30.9). Using the same method, the United States is claimed to have a Gini index of 36, while South Africa had the highest income Gini index score of 67.8.[30] World income Gini index since 1800s[edit] Growth spells last longer in more equal countries. A 10 percentile
increase in equality (represented by a change in the Gini index value
from 40 to 37) increases the expected length of a growth spell by 50
percent. Percentage changes in GDP growth spell length are shown as
each factor moves from 50th to 60th percentile and all other factors
are held constant.
The table below presents the estimated world income Gini coefficients over the last 200 years, as calculated by Milanovic.[32] Taking income distribution of all human beings, the worldwide income inequality has been constantly increasing since the early 19th century. There was a steady increase in the global income inequality Gini score from 1820 to 2002, with a significant increase between 1980 and 2002. This trend appears to have peaked and begun a reversal with rapid economic growth in emerging economies, particularly in the large populations of BRIC countries.[33]
Year World Gini coefficients[10][30][34] 1820 0.43 1850 0.53 1870 0.56 1913 0.61 1929 0.62 1950 0.64 1960 0.64 1980 0.66 2002 0.71 2005 0.68 More detailed data from similar sources plots a continuous decline
since 1988. This is attributed to globalization increasing incomes for
billions of poor people, mostly in India and China. Developing
countries like Brazil have also improved basic services like health
care, education, and sanitation; others like
Year World Gini coefficient[36] 1988 .80 1993 .76 1998 .74 2003 .72 2008 .70 2013 .65 Gini coefficients of social development[edit]
Anonymity: it does not matter who the high and low earners are.
Scale independence: the
Countries by Gini Index[edit] Countries' income inequality according to their most recent reported Gini index values (often 10+ years old) as of 2014: red = high, green = low inequality Main article: List of countries by income equality
A Gini index value above 50 is considered high; countries including
Brazil, Colombia, South Africa, Botswana, and Honduras can be found in
this category. A Gini index value of 30 or above is considered medium;
countries including Vietnam, Mexico, Poland, The United States,
Argentina, Russia and Uruguay can be found in this category. A Gini
index value lower than 30 is considered low; countries including
Austria, Germany, Denmark, Slovenia, Sweden and Ukraine can be found
in this category.[51]
Limitations of Gini coefficient[edit]
The
Table A. Different income distributions with the same Gini Index[23] Household
Group
Country A
Annual
1 20,000 9,000 2 30,000 40,000 3 40,000 48,000 4 50,000 48,000 5 60,000 55,000 Total Income $200,000 $200,000 Country's Gini 0.2 0.2 Different income distributions with the same Gini coefficient Even when the total income of a population is the same, in certain
situations two countries with different income distributions can have
the same Gini index (e.g. cases when income Lorenz Curves cross).[23]
Table A illustrates one such situation. Both countries have a Gini
coefficient of 0.2, but the average income distributions for household
groups are different. As another example, in a population where the
lowest 50% of individuals have no income and the other 50% have equal
income, the
Extreme wealth inequality, yet low income Gini coefficient A Gini index does not contain information about absolute national or
personal incomes. Populations can have very low income Gini indices,
yet simultaneously very high wealth Gini index. By measuring
inequality in income, the Gini ignores the differential efficiency of
use of household income. By ignoring wealth (except as it contributes
to income) the Gini can create the appearance of inequality when the
people compared are at different stages in their life. Wealthy
countries such as Sweden can show a low
Table B. Same income distributions but different Gini Index Household
number
Country A
Annual
1 20,000 1 & 2 50,000 2 30,000 3 40,000 3 & 4 90,000 4 50,000 5 60,000 5 & 6 130,000 6 70,000 7 80,000 7 & 8 170,000 8 90,000 9 120,000 9 & 10 270,000 10 150,000 Total Income $710,000 $710,000 Country's Gini 0.303 0.293 Small sample bias – sparsely populated regions more likely to have low Gini coefficient Gini index has a downward-bias for small populations.[59] Counties or
states or countries with small populations and less diverse economies
will tend to report small Gini coefficients. For economically diverse
large population groups, a much higher coefficient is expected than
for each of its regions. Taking world economy as one, and income
distribution for all human beings, for example, different scholars
estimate global Gini index to range between 0.61 and 0.68.[10][11] As
with other inequality coefficients, the
Table C. Household money income distributions and Gini Index, USA[62]
Under $15,000 14.6% 13.7% $15,000 – $24,999 11.9% 12.0% $25,000 – $34,999 12.1% 10.9% $35,000 – $49,999 15.4% 13.9% $50,000 – $74,999 22.1% 17.7% $75,000 – $99,999 12.4% 11.4% $100,000 – $149,999 8.3% 12.1% $150,000 – $199,999 2.0% 4.5% $200,000 and over 1.2% 3.9% Total Households 80,776,000 118,682,000 United States' Gini on pre-tax basis 0.404 0.469
Expanding on the importance of life-span measures, the Gini
coefficient as a point-estimate of equality at a certain time, ignores
life-span changes in income. Typically, increases in the proportion of
young or old members of a society will drive apparent changes in
equality, simply because people generally have lower incomes and
wealth when they are young than when they are old. Because of this,
factors such as age distribution within a population and mobility
within income classes can create the appearance of inequality when
none exist taking into account demographic effects. Thus a given
economy may have a higher
Inability to value benefits and income from informal economy affects
Some countries distribute benefits that are difficult to value.
Countries that provide subsidized housing, medical care, education or
other such services are difficult to value objectively, as it depends
on quality and extent of the benefit. In absence of free markets,
valuing these income transfers as household income is subjective. The
theoretical model of
A U C = ( G + 1 ) / 2 displaystyle AUC=(G+1)/2 .
1 / λ displaystyle 1/lambda is used to quantify diversity, and this should not be confused with the Simpson index λ displaystyle lambda . These indicators are related to Gini. The inverse Simpson index increases with diversity, unlike Simpson index and Gini coefficient which decrease with diversity. The Simpson index is in the range [0, 1], where 0 means maximum and 1 means minimum diversity (or heterogeneity). Since diversity indices typically increase with increasing heterogeneity, Simpson index is often transformed into inverse Simpson, or using the complement 1 − λ displaystyle 1-lambda , known as Gini-Simpson Index.[74]
Other uses[edit]
Although the
G 1 displaystyle G_ 1 , in calculation section above, may be used for the final model and also at individual model factor level, to quantify the discriminatory power of individual factors. It is related to accuracy ratio in population assessment models. See also[edit] Diversity index
Economic inequality
Great Gatsby curve
Human
References[edit] ^ Gini (1912).
^ Gini, C. (1909). "Concentration and dependency ratios" (in Italian).
English translation in Rivista di Politica Economica, 87 (1997),
769–789.
^ "Current Population Survey (CPS) – Definitions and Explanations".
US Census Bureau.
^ Note:
Further reading[edit] Amiel, Y.; Cowell, F. A. (1999). Thinking about Inequality. Cambridge.
ISBN 0-521-46696-2.
Anand, Sudhir (1983). Inequality and
External links[edit] Wikimedia Commons has media related to Gini coefficient. Deutsche Bundesbank: Do banks diversify loan portfolios?, 2005 (on
using e.g. the
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