Submitted to Econometrica IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY? Tom Bolton 1 and Christopher Zoppou 2 1 Health Economics and Policy Group, University of South Australia, Adelaide, Australia 2 Policy Modelling Branch, FaHCSIA, Canberra, Australia and Department of Applied Mathematics, The Australian National University, Canberra, Australia The use of Gini coefficient is well established as a measure of income in- equality and has been used for this purpose in a variety of fields including health economics, demography and income inequality studies. Using the Beta distribution as a particular functional form of an income distribution, the Gini coefficient is calculated for a variety of income distributions. The analysis re- veals that identical Gini coefficients are obtained for dramatically different in- come distributions. Similar or identical measures of the Gini coefficient, there- fore, may not necessarily indicate that two income distributions are similar. This investigation shows that the Gini coefficient should not be relied upon as the only measure of differences or similarity between income distributions, nor of the extent of equality of income. keywords: income distribution, Beta distribution, Lorenz curve, Gini coefficient, income inequality 1
Transcript
Submitted to Econometrica
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOMEINEQUALITY?
Tom Bolton1 and Christopher Zoppou2
1Health Economics and Policy Group, University of South Australia,Adelaide, Australia
2Policy Modelling Branch, FaHCSIA, Canberra, Australia andDepartment of Applied Mathematics, The Australian National
University, Canberra, Australia
The use of Gini coefficient is well established as a measure ofincome in-equality and has been used for this purpose in a variety of fields includinghealth economics, demography and income inequality studies. Using the Betadistribution as a particular functional form of an income distribution, the Ginicoefficient is calculated for a variety of income distributions. The analysis re-veals that identical Gini coefficients are obtained for dramatically different in-come distributions. Similar or identical measures of the Gini coefficient, there-fore, may not necessarily indicate that two income distributions are similar.This investigation shows that the Gini coefficient should not be relied upon asthe only measure of differences or similarity between income distributions, norof the extent of equality of income.
keywords: income distribution, Beta distribution, Lorenz curve, Ginicoefficient, income inequality
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1. INTRODUCTION
There is considerable interest in economic and social science litera-ture in describing income distributions as a measures of economic per-formance and income inequality. Whether there is inequality in societyis pertinent to issues such as the existence and operation ofincentives towork and save and has implications for issues such as social cohesion,health status and the effectiveness of safety nets for the under-privileged.
Economists and social scientists have sought to encapsulate the in-equality inherent in an income distribution with a single measure. Suchmeasures attempt to quantify the share of the total income accruing tovarious groups within society. The aim is to use a single metric, eitherto establish whether there has been a significant shift in inequality levelsover time or between countries or within an economy or society. Suchmeasures potentially provide a qualitative measure on how apolicy in-vention or other trend or event may have influenced the incomedistribu-tion.
One of the most commonly cited measures of income inequalityisthe Gini coefficient[17]. The Gini coefficient is used in a wide range ofeconometric and social policy studies. It has also been usedin many otherfields including, demography and health economics.
Examples of the myriad of the demographic and health economic appli-cations of the Gini coefficient include Bishop, Formy and James-Smith[6]who examined demographic change and income inequality in the UnitedStates, Shkolnikov, Andreev and Begun[31] in their examination of theGini coefficient as a life table function and Rodgers[28] in his exami-nation of income and inequality as a determinant of mortality. The Ginicoefficient has also been used in other econometric measuressuch as theTax Progressivity Measures (Reynolds-Smolensky Index[27] and Kak-wani index[21]) and Poverty indices (Sen’s Poverty Index[34], Wolfsonindex[38] and Foster, Geer and Thorbecke index [15]).
In order to test the suitability of the Gini coefficient as a measure ofincome inequality, an income distribution is required. Forthe purposesof this study, a number of an hypothetical income distributions have beengenerated to enable comparison of the Gini coefficient between differentdistributions. To facilitate the generation of these distributions, a func-tional form of the income distribution is required that is easy to manip-
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?3
ulate and evaluate, and that resembles a wide range of incomedistribu-tions.
The adequacy of the Gini coefficient as a measure of income inequalityis examined in this paper. On its own, the Gini coefficient wasfound to bean inadequate measure of income inequality, and that it is a poor indicatoras to whether two income distributions are similar.
2. THE GINI COEFFICIENT
In 1912 the Italian statistician, Corrado Gini, disagreed with Pareto’sassertion that economic growth leads to less inequality. Tosubjectivelystudy inequality, Gini proposed a non-dimensional measureof incomeinequality, known as theGini coefficient, which is still in common usetoday. He published his work in1912 in Italian and later in1921 in En-glish[17].
The Gini coefficient is defined as
G(X) = 1− 2
∫
1
0
L(x) dx
in whichL(x) is theLorenz curvefor the income distribution. It is equalto twice the area between the Lorenz curve and the line ofperfect equal-ity, see Figure 1. This is the geometric interpretation of the Gini coeffi-cient. Other interpretations can be found in Xu[39].
The Lorenz curve was developed by Lorenz in1905[23]. It is a plotof the cumulative share of the total income earned by households rankedin ascending order of income in the income distribution. It is expressedmathematically as
L(F (x)) =
∫ x
−∞tf(t) dt
∫
∞
−∞tf(t) dt
in which the probability density functionf(x) has the cumulative distri-bution functionF (x). For the income vectorX, the Lorenz curve is apiecewise linear functionLi connecting the points(Fi, Li), i = 0, to n,with (F0 = 0, L0 = 0) and fori = 1 to n
A typical Lorenz curve is illustrated in Figure 1 along with the line ofperfect equality, which is the diagonal line in this figure corresponding toLi = i/n ∀ i, for example, everyone receives the same income. The lineof perfect inequality, corresponding toLi = 0 for i < n, for exampleonly one person receives all the income. The greater the deviation of theLorenz curve from the line of perfect equality. The Lorenz curve has
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
po
rtio
n o
f in
com
e
Perfe
ct eq
uality
Perfect inequality
Lorenz cu
rve
Figure 1: A typical Lorenz curve.
the following properties. It always starts at(0, 0) and ends at(1, 1). Ifnegative incomes are not possible in the income distribution, the Lorenzcurve;
• is always below the line of perfect equality;• is always above the line of perfect inequality;• is increasing; and• is a convex function.
If the income distribution contains negative incomes, thenthe Lorenzcurve will fall below the line of perfect inequality and it isa convex func-tion if the income distribution has a positive mean. If the income distri-bution has a negative mean and there are negative incomes in the distri-bution, then the Lorenz curve will be above the line of perfect inequality.
For a discrete income distribution whereX = (x1, x2, . . . , xn) repre-sents the income vector, sorted in ascending order,xi < xi+1, for a pop-ulation ofn individuals or income units, the Gini coefficient is defined
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?5
as
(1) G =1
n− 1
(
n + 1− 2
[∑n
i=1(n+ 1− i)xi∑n
i=1xi
])
.
A Gini coefficient of zero equates to perfect equality, whereeveryonereceives the same income. A Gini coefficient equal to one is perfect in-equality, only one person receives all the income, while everyone elsehas none. In general, the Gini coefficient for the income distribution of acountry is typically in the range0.2− 0.5. It is generally considered as asummary statistic of dispersion of a distribution[3].
Gini coefficients of income distributions by country provided by thethe United Nations[37] range from Denmark at0.25 to Namibia at0.74.Denmark (0.25), Japan (0.25) and Sweden (0.25) have the lowest Ginicoefficients. Most developed and developing countries fallin the0.30’s,e.g. Australia, United Kingdom, Italy and New Zealand have the sameGini coefficient (0.36). The more extreme examples tend to be in LatinAmerica and Africa countries, such as South Africa (0.58), Colombia at(0.59), Bolivia (0.60), Botswana (0.61), Sierra Leone (0.63) and Namibia(0.74).
The relationship between the Gini coefficient, the shape of the Lorenzcurve and income inequality is discussed in relation to ascertain whetherthe Gini coefficient is indeed a robust measure of income inequality.
3. INCOME DISTRIBUTION MODELS
For illustrative purposes, the Australian Unit Income distribution for2003 is used as an example of a typical income distribution, whichisshown in Figure 2. The distribution is highly skewed, which is a charac-teristic feature of most income distributions.
It is common to seek to attempt to fit an analytical distribution functionto the income distribution as an analytical tool for such reasons as ease ofevaluation of statistical moments such as the mean,µ median, standarddeviation,σ variance, skewness,γ and kurtosis,κ. A parametric form isalso easily replicable for reasons of validation of results. Such a paramet-ric form is also capable of generating new distributions forcomparisonwith alternative distributions[7].
Figure 2: The Australian Unit Income distribution in2003 with the fitted Betadistribution,B[a, b, c, d] with a = 0, b = 10, 001, c = 1.437, andd = 15.244were determined using maximum likelihood estimates.
There have been numerous attempts to fit distributions to either empiri-cal or simulated income distributions. For example, Pareto[26] suggestedthat the log-Normal distribution was suitable for describing the upper tailof the income distribution. He assumed that income obeys a universalpower law which was valid for all times and countries. Gibrat[16] ex-tended these results to the whole distribution with mixed success. Othersuggested distributions include; the Beta distribution used by Thurow[36],the Gamma distribution used by Salem and Mount[29], the Wiebull dis-tribution used by Bartels and van Metelen[4] and by Singh-Madalla[32],the log-Logistic distribution used by Dagum[10], the Generalized Gammadistribution used by Taille[35], the four parameter generalized Beta dis-tribution used by McDonald[24], the five parameter generalized Beta dis-tribution used by McDonald and Xu[25], the Exponential distributionused by Dradulescu and Yakovenko[12], the Weibull, Dagum and gen-eralized Beta distributions used by Bandourianet al.[3] and the Paretodistribution used by di Guilmiet al.[11].
The relationship between some of the more common forms of thesedistributions is shown in Figure 3. Relationships with other distributions
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?7
Binomial
CauchyPoisson
LogNormal Normal
Gamma
Beta
ExponentialWeibull
Sp
ecial C
ase
Transfo
rm
Special Case
Transfo
rm
Transform
Limit
Lim
itLimit
Limit
Figure 3: Relationship between various common distributions and theNormaldistribution is illustrated in this figure. The Normal distribution is a limiting,special or transformation of other common distributions.
used in econometrics can be found in Cowell[9]. There have also beenattempts to fit composite distribution functions to the income distribu-tion, where an appropriate distribution is fitted to different parts of the in-come distribution. This was the case when Champerdowne[8] presentedtwo distribution functions which contained four and five parameters re-spectively. In both cases, the log-Normal distribution wasfitted to theupper tail of the distribution. Recently, a combination of log-Normaldistributions for the upper tail and a Bolzmann-Gibbs function for thelower levels of the income distribution was proposed by Dradulescu andYakovenko[13]. Although these models have been reported toproducebetter fits than a single distribution model, more parameters need to beestimated.
Recently, Campano and Salvatore[7] warned econometricians not todismiss many of the earlier models as fitting and flexibility,may be moreappropriate using one of the earlier simpler models. Unfortunately, a uni-
versal distribution function has not been found for the income distribu-tion.
4. THE BETA DISTRIBUTION
The Beta distribution is used in this paper because it is a reasonableapproximation of the Australian income distribution and because of thewide range of distributions that it can represent by simply changing itstwo shape parameters.
4.1. Beta Distribution,B[a, b, c, d]
The Beta distribution is commonly written as[14]
f(x; a, b, c, d) =1
(b− a)B(c, d)
(
x− a
b− a
)c−1(
b− x
b− a
)d−1
.
where theBeta function, B(c, d), is given by[1]
B(c, d) =
∫
1
0
tc−1(1− t)d−1 dt.
The two shape parameters,c andd, in the Beta distribution must be es-timated. The remaining two parameters,a andb, are easily established;they are the lower and upper limits of the distribution.
The types of distributions that can be produced using the Beta distri-bution can be found in Figure 4. Since the Beta distribution can assumemany shapes, it can be used to replace most of the common distributions.In addition, unlike many other distributions, its domain isbounded, orhascompact support, and can avoid negative or extreme values.
To demonstrate its versatility, a Beta distribution has been fitted to theAustralian Unit Income distribution for2003 shown in Figure 2. Usingmaximum likelihood estimation[22] to estimate the two shape parame-ters, the fitted Beta distribution has been superimposed on the incomedistribution shown in Figure 2. In this case the Beta distribution providesa reasonable representation of the Australian Unit Income distribution.
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?9
B[0,1,1.5,8] B[0,1,8,1.5]
B[0,1,1.5,4] B[0,1,4,1.5]
00
1
2
3
4
5
6
fx
a,b
,c,d
(;
)
x
0.40.2 0.80.6 1 00
1
2
3
4
5
6
fx
a,b
,c,d
(;
)
x
0.40.2 0.80.6 1
B[0,1,1,1]
B[0,1,10,10]
B[0,1,5,5]
B[0,1,2,2]
(a) (b)
00
1
2
3
4
5
6
fx
a,b
,c,d
(;
)
x
0.40.2 0.80.6 1
B[0,1,0.8,2]
B[0,1,1,2] B[0,1,2,1]
B[0,1,2,0.8]
00
1
2
3
4
5
6
fx
a,b
,c,d
(;
)
x
0.40.2 0.80.6 1
B[0,1,0.8,0.2]B[0,1,0.2,0.8]
B[0,1,0.5,0.5]
(c) (d)
Figure 4: Examples of the Beta,B[a, b, c, d] distribution witha = 0 andb = 1and various values for the shape parametersc andd.
5. AN INVESTIGATION OF THE GINI COEFFICIENT
For a wide range of shape parameters,c andd, in the Beta distribution,B[a, b, c, d] with a = 0 andb = 1, ten thousand random variates weregenerated from the Beta distribution. For each generated income distri-bution several statistical moments were produced as well asa histogramof the income distribution, the probability density function, the cumula-tive probability distribution, the Lorenz curve and the Gini coefficient.By generating different income distributions using the Beta distribution,the limitations of the Gini coefficient as a measure of changes in a dis-
tribution is illustrated. The following discussion and results are based onthese generated income distributions.
Well established in the economic literature is the role ofLorenz domi-nancein the comparison of the income distributions. In comparingmea-sures of inequality, Lorenz dominance is used to discern themore equaldistribution between two different distributions. Barret, Crossly and Wor-swick[5] state that one distribution is classified as more equal than an-other distribution by the Lorenz dominance criteria if its correspondingLorenz curve lies everywhere above (or closer to the line of equality)than another. The relevance of Lorenz dominance in analysisof the in-come distribution has been stressed by authors such as Atkinson[2] andSen[33].
Atkinsons[2] seminal work on Lorenz dominance demonstrated that,under certain conditions, non-intersecting Lorenz curvescan be used toconstruct a welfare ordering for a broad class of social welfare functionswhich are equality preferring. Atkinson also stressed the importance ofexamination of the Lorenz curves when analysing income distributions.
It is suggested in the literature that establishing Lorenz dominance isa prerequisite for using inequality measures such as the Gini coefficient.In their article examining the United States income distribution, Bishop,Formby and James Smith[5] agree with Atkinson’s[2](p. 258)observa-tion that Lorenz dominance would prove to be of limited practical importbecause Lorenz curves intersect so often.
In many studies on income distributions that rely on the Ginicoeffi-cient, the implications of the form of the income distribution and the ex-istence or non-existence of Lorenz dominance are usually ignored. Thishas included research in Australia such as Harding and Greenwell[19]and Saunders[30] commenting on the relative stability of income inequal-ity in Australia, without reference to the changing shape orpattern of theincome distribution. Saunders, however, did comment on movements inthe mean of the distribution in his critique of Harding and Greenwells ap-proach. The pattern of researchers using the Gini coefficient as the onlymetric of inequality has also occurred in many overseas studies includ-ing in the context of research into income and health inequalities in theUnited Kingdom such as Gravelle and Sutton[18].
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?11
5.1. Beta Distributions with Identical Gini Coefficients
Figures 5, 7 and 8 illustrates the case of income distributions, hav-ing identical Gini coefficients, but vastly different features. In these fig-ures, the randomly generated income distribution, the probability densityfunction and the cumulative distribution that result in theGini coefficientequal to0.3 in Figure 5,0.2 in Figure 7 and0.15 in Figure 8. In manyof these examples, the Lorenz curve is identical. In others,the Lorenzcurve is symmetrical about the axisy = −x. For example, the Lorenzcurve forB[0, 1, 3, 45] andB[0, 1, 1.85, 3] has been plotted in Figure 6.The corresponding income distributions are shown in Figure5. Both ofthese income distributions produced a Gini coefficient equal to 0.3 andthe Lorenz curves are symmetrical about they = −x axis. The use of theRobin Hood index[20] or its position,r, would not assist in the interpreta-tion of the dramatic change in the income distributions. TheRobin Hoodor Hoover index, is the maximum distance between the Lorenz curve andthe line of perfect equality, see Figure 6. Additional metrics are requiredto reveal the dramatic change in these distribution. The standard devia-tion would indicate that there is a significant change in the shape of theincome distributions. In this case, they are;σ = 0.03 andσ = 0.20 forFigure 5(b) and 5(d) respectively. The order of magnitude variation in thestandard deviation, given in Tables I, II and III suggests that it is a betterindicator of changes in the income distribution than the Gini coefficient,which remains constant. Changes in other basic statisticalmoments, suchas the mean and skewness also reflect dramatic changes in the incomedistribution, see Tables I, II and III.
This is also the case in Figures 7 and 8 where the combination of in-creasing the dispersion of the income distribution and its location resultsin the same Lorenz curve but identical Gini coefficients. These are starkexamples of dramatic changes in the income distribution that have notbeen captured in the Gini coefficient. Other metrics or visual aids are re-quired. For example, it has been shown that the statistical moments mayprovide useful information. Plotting the cumulative distribution functionis another visual aid, other than the Lorenz curve, in identifying changesin the income distribution. Differences in the income distribution are re-flected in changes in the shape of the cumulative distribution function.
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?13
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(a)
0 0.2 0.4 0.6 0.8 10
200
400
600
800
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 10
2000
4000
6000
8000
10000
(b)
0 0.2 0.4 0.6 0.8 10
200
400
600
800
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 10
2000
4000
6000
8000
10000
(c)
0 0.2 0.4 0.6 0.8 10
200
400
600
800
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2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 10
2000
4000
6000
8000
10000
(d)
Figure 5: The Lorenz curve, (a) with a Gini coefficient equal to0.3 producedfrom several examples of the Beta,B[a, b, c, d] distribution with a = 0 andb = 1 and various values for the shape parametersc andd; (b) c = 3.00 andd = 45.00, (c) c = 2.50 andd = 9.50 and (d) c = 1.85 andd = 3.00
14
TABLE III
THE SHAPE PARAMETERS IN THEBETA DISTRIBUTION, B[0, 1, c, d] THAT IS USED
TO DEFINE THE INCOME DISTRIBUTION AND ITS STATISTICAL MOMENTS THAT
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?15
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion o
f in
com
e
y
x=
-
Robin Hood index
Perfe
ct eq
uality
r
Figure 6: Lorenz curve for the Beta probability distribution,B[0, 1, 1.85, 3] andB[0, 1, 3, 45] which have the same Gini coefficient,G = 0.3. The Lorenz curvesare symmetrical about the axisy = −x.
5.2. The Effect of the Shape Parameters on the Gini Coefficients
Examples where the shape parameters,c andd, are changed indepen-dently and wherec andd are changed in the same proportion are exam-ined. The income distribution, probability density function, the cumula-tive distribution function and the Lorenz curve for these studies are illus-trated in Figures 9, 10 and 11 respectively. Statistical measures and theGini coefficient for each study are given in Tables V, IV and VI.
5.2.1. The Effect ofc on the Gini Coefficients
The Gini coefficient and Lorenz curves are more sensitive to the shapeparameterc in the Beta distribution than they are to the other shape pa-rameter,d. This parameter dramatically influences the location of theincome distribution and, to a lesser extent on the re-distribution of theincome. This is illustrated in Figure 9. In Table IV, there isa dramaticchange in the mean of the income distribution asc changes, but only asmall change in the standard deviation or spread of the income distri-bution. In addition, the skewness coefficient also confirms that there hasbeen only a slight re-distribution of income in the income distribution,the skewness changing from being positive to slightly negative asc in-creases. The Gini coefficient in this case could be considered as a reliableindicator in the change or shift in the income distribution.
5.2.2. The Effect ofd on the Gini Coefficients
There is a dramatic change in the re-distribution of the income asdincreases. This is indicated by the change in the skewness inthe incomedistribution shown in Figure 10 and given in Table V. The re-distributionis towards the lowest income. The Lorenz curves only change slightly asd changes. However, there is a dramatic change in the cumulative distri-bution function and probability density function. This is not evident inthe Lorenz or Gini coefficients. The standard deviation is also a bettermeasure of changes in the distribution than the Gini coefficient, see Ta-ble V, changing by an order of magnitude compared to one-halffor theGini coefficient.
As d increases, the negatively skewed distribution becomes positivelyskewed and the Gini coefficient increases as a greater proportion of thepopulation have a lower income. However, the Gini coefficient and the
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?17
Lorenz curve have not changed as dramatically as the mean andstandarddeviation of the income distributions. There are also dramatic changes inthe cumulative distribution plots asd increases. In this case other indica-tors might assist in identifying changes or shifts in the income distribu-tion.
5.2.3. The Effect of changingc andd in the same proportion on the GiniCoefficients
It has been demonstrated that decreasingd leads to a greater incomedispersion than increasingc. The shape variablec dominatesd in termsof growth in absolute income. In the case wherec = d, increases inthe same proportion results in a symmetrical income distribution with aconstant mean, see Figure 11 and Table VI. In this case the twoshapeparameters counteract each other and the absolute income remains staticbecause the mean remains constant. The major change is in thestandarddeviation of the income distribution. Sinced is primarily responsible fordispersion in the income distribution, then it dominatesc.
These figures indicate that if there is no shift in the absolute income,but there is a compression of the income distribution, then the Gini co-efficient is sensitive to these changes. As the shape parameters increase,there is an increase in income equality. In addition, if there is very littlechange in the overall movement in the income distribution, there will bea corresponding small change in the Gini coefficient. However, the Ginicoefficient is more sensitive to changes in the absolute income and lessto the redistribution of income.
6. DISCUSSION
In the Beta distributionB[a, b, c, d], increasingc leads to greater in-come dispersion and increases ind leads to less income dispersion. Whilethe distribution impacts ofc andd offset each other,d dominatesc in
18
TABLE IV
THE SHAPE PARAMETERS IN THEBETA DISTRIBUTION, B[0, 1, c, d], WITH d = 10.0REMAINING CONSTANT, THAT IS USED TO DEFINE THE INCOME DISTRIBUTION
AND ITS STATISTICAL MOMENTS AND THE CORRESPONDINGGINI COEFFICIENT.
Shape Parameters Statistical Moments Gini Coefficientc d µ σ κ γ G
terms of growth in absolute income. The Gini coefficient is more sensi-tive to c, shifts in the income distribution, and not on the re-distributionof income, which is dictated by the parameterd.
By using the Beta distribution to generate income distributions, thisanalysis demonstrates that radically different income distributions canproduce identical Gini coefficients and significant transformations of in-come distributions may give rise to counterintuitive results. Income dis-
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?19
TABLE VI
THE SHAPE PARAMETERS IN THEBETA DISTRIBUTION, B[0, 1, c, d], WITH c AND d
REMAINING INCREASING IN THE SAME PROPORTION, THAT IS USED TO DEFINE
THE INCOME DISTRIBUTION AND ITS STATISTICAL MOMENTS AND THE
CORRESPONDINGGINI COEFFICIENT.
Shape Parameters Statistical Moments Gini Coefficientc d µ σ κ γ G
tributions with identical Gini coefficients can have radically differentmeans, variances, kurtosis and skewness measures.
When comparing income distributions, the implications of Lorenz dom-inance are often ignored. The fact that different distributions can generatethe same Gini coefficient, as demonstrated in this paper reinforces con-cerns about the limitations of the Gini coefficient as the sole metric ofincome inequality.
This study thus illustrates the severe limitations in the use of the Ginicoefficient to measure income inequality. This work suggests that alter-native measures and techniques in combination with the Ginicoefficientand the Lorenz curve should be employed when examining changes toempirical or simulated income distributions.
Quantifying an income distribution using a single measure is problem-atic. An alternative single measure, that might be worth considering asan alternative to the Gini coefficient, is the Wolfson index[38]. It uses theGini coefficient and also includes a measure of the skewness of the dis-tribution, that is, it is in fact a composite measure. Another measure mayinvolve difference in the area between cumulative distribution functions.Fitting a theoretical distribution to the empirical or simulated income dis-tribution and examining the effect of the reforms on the fitted parameters
might also be an alternative method of testing the impact of reforms onthe income distribution. Other visual aids, other than the Lorenz curve,may be useful in identifying changes in the income distribution, such asexamination of the cumulative distribution function, for example, or sim-ply plotting the income distribution.
TOM I HAVE REMOVED THE FOLLOWING
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[27] Reynolds, M. and Smolensky, E.,Public Expenditure, Taxes andthe Distribution of Income: The United States, 1950, 1961, 1970,Academic Press, New York, 1977. (cited on page 2)
[28] Rodgers, G.B., Income and inequality as determinants of mortal-ity: an international cross-section analysisa,International Journalof Epidemiology, 31(3), 533-538, 2002. (cited on page 2)
[29] Salem, A.B.Z. and T.D. Mount, A convenient descriptivemodel ofincome distribution,Economertica, 42, 1115-1127, 1974. (citedon page 6)
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[31] Shkolnikov, V.M, E.E. Andreev and A.Z. Begun, Gini coefficientas a life table function: Computation from discrete data, decom-position of differences and empirical examples,Demographic Re-search, 8(11), 305-358, 2003. (cited on page 2)
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[35] Taille, C., Lorenz ordering within the Generalized Gamma familyof income distributions,Statistical distributions in Scientific Work,Eds., C. Taille, G.P. Patil and B. Balderssari, Reidel, Boston, 6,181-192, 1981. (cited on page 6)
[36] Thurow, L.C., Analyzing the American Income distribution, TheAmerican Economic Review, 60, 261-269, 1970. (cited on page 6)
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?23
[37] United Nations, UN Human Development Report 2007-8, 2007,http://hdr.undp.org/en/reports/global/hdr2007-2008/, Accessed 11March 2008. (cited on page 5)
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[39] Xu, K., How has the literature on Gini’s index evolved inthe past80 years?,Department of Economics, Dalhousie University, Hali-fax, Nova Scocia, 2007. (cited on page 3)
Figure 7: The Lorenz curve, (a) with a Gini coefficient equal to0.2 producedfrom several examples of the Beta,B[a, b, c, d] distribution with a = 0 andb = 1 and various values for the shape parametersc andd; (b) c = 5.00 andd = 10.00, (c) c = 5.90 andd = 21.00 and (d) c = 6.90 andd = 68.00
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?25
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
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of in
com
e
(a)
0 0.2 0.4 0.6 0.8 10
200
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800
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(b)
0 0.2 0.4 0.6 0.8 10
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(c)
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0 0.2 0.4 0.6 0.8 10
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(d)
Figure 8: The Lorenz curve, (a) with a Gini coefficient equal to0.15 producedfrom several examples of the Beta,B[a, b, c, d] distribution with a = 0 andb = 1 and various values for the shape parametersc andd; (b) c = 9.50, (c)c = 8.0 andd = 12.0 and (d) c = 11.0 andd = 45.0
26
0 0.2 0.4 0.6 0.8 10
200
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
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12
14
Proportion of maximum income
Pro
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lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(a)
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
1200
1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
4000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(b)
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
1200
1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of populationP
ropo
rtio
n of
inco
me
(c)
0 0.2 0.4 0.6 0.8 10
200
400
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(d)
0 0.2 0.4 0.6 0.8 10
200
400
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(e)
0 0.2 0.4 0.6 0.8 10
200
400
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(f )
Figure 9: The effect of changes inc only on the Beta distributionB[a, b, c, d]and the Gini coefficient where; (a) B[0, 1, 1.5, 10.0] resulting in Gini= 0.3954(b) B[0, 1, 3.0, 10.0] resulting in Gini= 0.2716, (c) B[0, 1, 5.0, 10.0] resultingin Gini = 0.2004, (d) B[0, 1, 10.0, 10.0] resulting in Gini= 0.1241, and (e)B[0, 1, 20.0, 10.0] resulting in Gini= 0.0715, (f ) B[0, 1, 30.0, 10.0] resultingin Gini = 0.0502.
IS THE GINI COEFFICIENT AN ADEQUATE MEASURE OF INCOME INEQUALITY?27
0 0.2 0.4 0.6 0.8 10
200
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(a)
0 0.2 0.4 0.6 0.8 10
200
400
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(b)
0 0.2 0.4 0.6 0.8 10
200
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of populationP
ropo
rtio
n of
inco
me
(c)
0 0.2 0.4 0.6 0.8 10
200
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
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Proportion of maximum income
Pro
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lity
dens
ity
0 0.2 0.4 0.6 0.8 10
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Proportion of maximum income
Cum
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ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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1
Proportion of population
Pro
port
ion
of in
com
e
(d)
0 0.2 0.4 0.6 0.8 10
200
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Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
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14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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1
Proportion of population
Pro
port
ion
of in
com
e
(e)
0 0.2 0.4 0.6 0.8 10
200
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Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
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Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
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Proportion of maximum income
Cum
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ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
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0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(f )
Figure 10: The effect of changes ind only on the Beta distributionB[a, b, c, d]and the Gini coefficient where; (a) B[0, 1, 1.5, 1.0] resulting in Gini= 0.2525(b) B[0, 1, 1.5, 1.5] resulting in Gini= 0.2908, (c) B[0, 1, 1.5, 2.0] resultingin Gini = 0.3148, (d) B[0, 1, 1.5, 5.0] resulting in Gini= 0.3713, and (e)B[0, 1, 1.5, 10.0] resulting in Gini= 0.3954, (f ) B[0, 1, 1.5, 20.0] resulting inGini = 0.4101.
28
0 0.2 0.4 0.6 0.8 10
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Fre
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cy
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Pro
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dens
ity
0 0.2 0.4 0.6 0.8 10
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Cum
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e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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1
Proportion of population
Pro
port
ion
of in
com
e
(a)
0 0.2 0.4 0.6 0.8 10
200
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Proportion of maximum income
Fre
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0 0.2 0.4 0.6 0.8 10
2
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Pro
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dens
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0 0.2 0.4 0.6 0.8 10
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Cum
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e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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1
Proportion of population
Pro
port
ion
of in
com
e
(b)
0 0.2 0.4 0.6 0.8 10
200
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Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
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14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
4000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of populationP
ropo
rtio
n of
inco
me
(c)
0 0.2 0.4 0.6 0.8 10
200
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Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
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Pro
babi
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dens
ity
0 0.2 0.4 0.6 0.8 10
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Cum
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e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
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1
Proportion of population
Pro
port
ion
of in
com
e
(d)
0 0.2 0.4 0.6 0.8 10
200
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Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
4000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Proportion of population
Pro
port
ion
of in
com
e
(e)
0 0.2 0.4 0.6 0.8 10
200
400
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1400
Proportion of maximum income
Fre
quen
cy
0 0.2 0.4 0.6 0.8 10
2
4
6
8
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12
14
Proportion of maximum income
Pro
babi
lity
dens
ity
0 0.2 0.4 0.6 0.8 10
2000
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Proportion of maximum income
Cum
mul
ativ
e fr
eque
ncy
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
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1
Proportion of population
Pro
port
ion
of in
com
e
(f )
Figure 11: The effect of c and d rising in the same proportion on theBeta distributionB[a, b, c, d] distribution and the Gini coefficient where; (a)B[0, 1, 1.0, 1.0] resulting in Gini= 0.3359 (b) B[0, 1, 2.0, 2.0] resulting in Gini= 0.2557, (c) B[0, 1, 3.0, 3.0] resulting in Gini= 0.1921, (d) B[0, 1, 4.0, 4.0]resulting in Gini= 0.1921, (e) B[0, 1, 5.0, 5.0] resulting in Gini= 0.1728 and(f ) B[0, 1, 10.0, 10.0] resulting in Gini= 0.1241.
