abdelkrim araar, sami bibi and jean-yves duclos

Inequality and equity PEP and UNDP June 2010 – 1 / 39

Inequality and equity

Abdelkrim Araar, Sami Bibi and Jean-Yves Duclos

Workshop on poverty and social impact analysisDakar, Senegal, 08-12 June 2010

Outline

Outline

Objectives

Thinking aboutinequality

Measuring inequality

Decomposinginequality

Conclusion


Thinking about inequality


Decomposing inequality

Conclusion

Objectives


� Understand basic notions of inequality measurement;� Review the most common principles used for the measurement of

inequality;� Distinguish absolute and relative concepts of inequality;� Define and estimate a Lorenz curve and a Gini index;� Link inequality to social welfare;� Define popular indices of inequality and their differences;� Think about the influence of inequality between and within groups on

inequality in the total population;� Assess the influence of income sources on total income inequality.

Thinking about inequality

Outline

Objectives


Basic notions

Income disparitiesand inequality

A fundamentalprinciple:The Pigou-Daltontransfer principle

The anonymityprinciple

The populationprinciple

Absolute and relativeinequality

Scale invariance



Conclusion


Basic notions


� Inequality refers to disparities between individuals.� Any reduction in disparity between two unequal individualswill reduce

inequality.

Income (y)

+a −a

y1 y1 + a y2 − a y2

Income disparities and inequality


Distributions y1 y2 y3 y4 y5A 2 6 10 14 18B 4 4 10 14 18C 2 8 8 14 18D 2 6 10 16 16

� A movement from distributionA to any ofB, C orD will decreaseinequality.

A fundamental principle:The Pigou-Dalton transfer principle


� The link between income disparities and inequality can be expressed by afundamental inequality principle, termed under the names ofPigou (1912) and Dalton (1920).

IA1 The Pigou-Dalton transfer principle: An income transfer from aricher person to a poorer person should register as a fall (orat leastnot as an increase) in inequality.

The anonymity principle


Distributions y1 y2 y3 y4 y5A 2 6 10 14 18B 14 6 10 2 18

� Inequality focuses on disparities across individuals; inequality inA andB should therefore be viewed the same. This is usually termed:

IA2 The anonymity principle: Inequality indices only depend on thedistribution of incomes, not on any other characteristics ofindividuals.

The population principle


Distributions y1 y2 y3 y4 y5 y6 y7 y8 y9 y10A 2 6 10 14 18 . . . . .B 2 6 10 14 18 2 6 10 14 18

� DistributionB with 10 individuals is a replicated distribution ofA with 5individuals. To make inequality indices comparable without having totake into account absolute population sizes, the indices must beinsensitive to such replications.

IA3 The population principle : Inequality should be invariant toreplications of a population.

Absolute and relative inequality


What should one compare across distributions when one thinks aboutinequality? Is it incomes (yi) or incomes relative to the mean,yi/µ?

Distributions y1 y2 y1/µ y2/µ µ Absolute Relativedifference difference

A 100 300 1/2 3/2 200 200 1B 300 500 3/4 5/4 400 200 1/2

� The absolute difference between individuals 1 and 2 is the same in bothdistributionsA andB; the difference in income shares is morepronounced inA.

� This suggests two main classes of inequality indices, theabsolute and therelative classes.

Absolute and relative inequality


1. Absolute inequality: refers to absolute income differences. Adding thesame amount to all incomes will not change absolute income differencesand will therefore not changeabsolute inequality.

2. Relative inequality: refers to relative income differences. Multiplyingincomes by the same scalar will not change income shares and willtherefore not change the level ofrelative inequality.

� Among the potential problems of theAbsolute approach is its ignoranceof the importance of “context”. Let two distributions be given byyA = {1, 3} andyB = {99, 101}. The two distributions have the same

level of absolute inequality, but the income difference represents 200% ofthe income of the poorer individual inA and only 2% inB.

� Another problem is sensitivity to the units of measurement.

Scale invariance



� VectorB is simplyA multiplied by 2.� This could be because incomes are in nominal terms inB and in real

terms inA (2 being the inflation factor or sampling weight).� Or it could be because the two vectors are expressed in different monetary

units, with 1 unit inA having the same purchasing power as 2 units inB.

Scale invariance



� Thescale invariance axiom says that relative inequality indices should bethe same inA andB.

IA4: Scale invariance principle I(y) = I(λy) with λ > 0.

� All of the inequality indices that obey the four axiomsIA1-IA4 belong tothe (conventional) class of relative inequality indices.


Outline

Objectives



The Lorenz curve

Properties of theLorenz curve

Constructing aLorenz curve

Example of Lorenzcurve

Inequality and theLorenz curve

The Gini index

The Gini index andthe Lorenz curve

Gini as a relativemean difference

Relative deprivationand the Gini

Social welfare andthe Gini

Social welfare andthe GiniIncome share and theGini index

Income share and theGini indexInequality and equity PEP and UNDP June 2010 – 12 / 39

The Lorenz curve


� For simplicity, we assume that the relative individual weight is defined asφi =

1n∀ i.

� The Lorenz curve is the most popular curve for visualizing inequality andcomparing distributions to see which one is more equal.

� The Lorenz curve at a percentilepi is:

L(pi) =

∑i

j=1Q (pj)∑n

j=1Q (pj). (1)

� Let the income share of individualj be given byψj =yj∑ni=1

yi; the Lorenz

curve is then:

L(pi) =i∑

j=1

ψj. (2)

Properties of the Lorenz curve


� L(pi) thus shows the share in total income of all those among the bottomp∗i proportion of the population.

0

1

Percentiles (p)

L(p∗)

p∗ 1

L(p)



� The slope of the Lorenz curve atpi = 0.5 equals the ratio between themedian and the average income. This is an indicator of whether thedistribution is skewed.

0

1

Percentiles (p)p = 0.5 1

Lorenz:L(p)

L(p = 0.5)

Median/Average



� The proportion of total income that one would need to reallocate from therich to the poor to achieve perfect income equality is given by:p = F (µ)− L(p = F (µ)) (this is theSchutz index).

0

1

Percentiles (p)p∗ = F (µ) 1

Lorenz:L(p)

L(p∗)

The Shultz index

Constructing a Lorenz curve


Individual Cumulative Income Cumul. incomeidentifier population Quantile share

i pi = F (yi) Q(pi) = yi L(pi)1 0.2 2 2/50 = 0.042 0.4 6 8/50 = 0.203 0.6 10 18/50 = 0.364 0.8 14 32/50 = 0.645 1.0 18 50/50 = 1.00

Example of Lorenz curve


Illustration with :y = {4, 6, 10, 20, 60} // Average(µ) = 20

Example of Lorenz curve


� The tangent atpi = 0.5 is(L(pi = 1.0)− L(pi = 0.2))/((pi = 1.0)− (pi = 0.2)) = 0.5, which isalsoQ(0.5)/µ, namely, 10/20.

� The Schutz index,F (µ)− L(F (µ)) = 0.8− 0.4 = 0.4, is the proportionof total income that would need to be redistributed to achieve perfectequality — we would have to transfer 40 from the richest two tothe threepoorest persons, and this represents 40/100 of total income.

Inequality and the Lorenz curve


� The closer the Lorenz curve is to the 45◦ line, the lower is inequality.� If a Lorenz curve for a distributionB is everywhere above that for a

distribution ofA, then, distributionB is unambiguously more equal thandistributionA.

0

1

Percentiles(p)

LA(p∗)

p∗ 1

Lorenz:L(p)

A

B

LB(p∗)

Inequality and the Lorenz curve


� Using the Theorem of Atkinson (1970), all inequality indices that obeythe Pigou-Dalton principle should show that inequality inA is higherthan inB.

� Lorenz curves may cross and we then cannot order distributionsaccording to them. The Lorenz curve ordering is thus a “partial” orderingof distributions.

� Using one inequality index can secure a “complete” orderingof twodistributions.

The Gini index


� The Gini index is one among many other examples of a syntheticindex ofinequality that compresses all information about inequality into onevalue.

� The values of the Gini index lie between zero (perfect equality) and one(perfect inequality) when incomes are non-negative.

� The expression of the Gini index can be manipulated algebraically toprovide different interpretations.

The Gini index


� The Gini index equals twice the expected share deficit (pi − L(pi)),which is twice the area between the 45◦ line and the Lorenz curve.

� Denoting the area below the Lorenz curve byA , we have:

IGini = 1− 2A andA =1

2n

n∑

i=1

(L(pi) + L(pi−1)) (3)

The Gini index


� Denoting the area below the Lorenz curve byA , we have:

IGini = 1− 2A andA =1

2n

n∑

i=1

(L(pi) + L(pi−1)) (3)

0

1

Percentiles (p)1

Lorenz:L(p)0.5*Gini

SurfaceA

The Gini index and the Lorenz curve


� Let y = {1, 2, 3, 4}. As shown in the figure below, the Gini index equals2(0.5− A) with A being the areaa+ b+ c+ d.

ab

c

d

0.5 Gini

0.2

.4.6

.81

Lore

nz: L

(p)

0 .25 .5 .75 1Pecentiles (p)

The Gini index and the Lorenz curve


i Q(pi) = yi pi = F (yi) L(pi) 1/4 ∗ (L(pi) + L(pi−1))– 0 0.00 0/10 = 0.00 -1 1 0.25 1/10 = 0.10 0.0252 2 0.50 3/10 = 0.30 0.1003 3 0.75 6/10 = 0.60 0.2254 4 1.00 10/10 =1.00 0.400

Total - - 1.00 0.750 (= 2A)The Gini index equals 1.00 - 0.75 =0.25

Gini as a relative mean difference


� The Gini coefficient may be defined as the half of therelative meandifference (RMD).

� Themean difference (MD) is the expected distance between tworandomly selected incomes in a population.

� Therelative mean difference is themean difference divided by averageincome:

IGini(y) =MD

2µand MD =

1

n2

n∑

i=1

n∑

j=1

|yi − yj| (4)

Gini as a relative mean difference


j 1 2 3 4i y1 = 1 y2 = 2 y3 = 3 y4 = 4

∑n

i=1 |yi − yj|1 y1 = 1 0 1 2 3 62 y2 = 2 1 0 1 2 43 y3 =3 2 1 0 1 44 y4 =4 3 2 1 0 6

Total 6 4 4 6 20The Gini index then equals20/(2 ∗ (n = 4)2 ∗ (µ = 2.5)) = 0.25

Relative deprivation and the Gini


� The deprivation of individuali relative toj is:

δi,j =

{yj − yi if yi < yj0 otherwise.

(4)

The deprivation of individuali relative to all other individuals thenequals:

δ̄i =

n∑

j=1

δi,j

n(5)

The Gini index can be expressed as:

IGini =1

nµ

n∑

i=1

δ̄i. (6)

Relative deprivation and the Gini


j 1 2 3 4i y1 = 1 y2 = 2 y3 = 3 y4 = 4 δ(yi) =

∑ni=1 (yi − yj)+ /n

1 y1 = 1 0 1 2 3 1.502 y2 = 2 0 0 1 2 0.753 y3 =3 0 0 0 1 0.254 y4 =4 0 0 0 0 0

Total 2.5The Gini index equals2.5/((n = 4) ∗ (µ = 2.5)) = 0.25

Social welfare and the Gini


� It is usually assumed that social welfareW increases with averageincomeµ but decreases with inequalityI(y).

� An equally distributed equivalent (EDE) income,ξ, can also be defined as

W (y1, y2, ..., yn) ≡W (ξ, ξ, ..., ξ︸︷︷︸

n times

) ≡ ξ(y) (4)

� I(y) is then usually defined as:

I(y) = 1−ξ(y)

µ. (5)

andW (y) then equals:

ξ(y) = µ (1− I(y)) . (6)



� Let y1 ≤ y2 ≤ · · · ≤ yi ≤ · · · ≤ yn; the Gini social welfare function isdefined as:

WGini(y) =n∑

i=1

υiyi with υi =2(n− i) + 1

n2(4)

� Note thatn∑

i=1

υi = 1 (5)



� Hence, theEDE income for the Gini social welfare function equals

ξGini(y) =n∑

i=1

υiyi (4)

� One can therefore rewrite the Gini index as:

IGini(y) = µ−1

n∑

i=1

υi(µ− yi). (5)



Identifier: i Income:yi υi υi(1− yi/µ)1 1 0.4375 0.26252 2 0.3125 -0.03753 3 0.1875 -0.03754 4 0.0625 0.0625

Total — 1.000 0.250

Income share and the Gini index


� The Gini index is simply an average weighted income shares and one canrewrite the Gini index as:

IGini(y) =n∑

i=1

κiψi andκi =

(

1−υiφi

)

(6)

� Note thatn∑

i=1

φiκi = 0 (7)

Income share and the Gini index


Identifier: i Income:yi υi κi ψi κiψi

1 4 0.0625 0.75 0.40 0.3002 3 0.1875 0.25 0.30 0.0753 2 0.3125 -0.25 0.20 -0.0504 1 0.4375 -0.75 0.10 -0.075

Total — — — — 0.250

The Atkinson inequality index


� Atkinson (1970) proposes a relative inequality based on an additive socialwelfare function:

WAtkinson(y) = n−1

n∑

i=1

U(yi; ǫ) (8)

where utilityU with incomeyi, denoted asU(yi, ǫ), is defined as:

U(yi; ǫ) =

{y1−ǫi

(1−ǫ), when ǫ 6= 1

ln(yi), when ǫ = 1.(9)



� This imposes a concavity assumption on the social evaluation ofindividuals incomes. The increase in utility fromU(yi) toU(yi + a) islower than fromU(yi + a) toU(yi + 2a) (ǫ > 0).

Income (y)yi + 2ayi yi + a

U(yi)

U(yi + a)U(yi + 2a)

U(y)



� The equally distributed equivalent income is given by

ξ(y; ǫ) =

(1n

∑n

i=1 y(1−ǫ)i

) 1

1−ǫ

, when ǫ 6= 1,

exp(1n

∑n

i=1 ln(yi)), when ǫ = 1,

(8)

� and the Atkinson index is defined by:

I(y; ǫ) = 1−ξ(y; ǫ)

µ. (9)

� I(y; ǫ) is the loss in social welfare (relative to the mean) entailedbyinequality in the distribution of incomes.



� Let y = {100, 400, 900, 1600} andǫ = 0.5.

Identifier: i Income:yi U(yi; ǫ = 0.5)1 100 202 400 403 900 604 1600 80

Average 750 50

� The Atkinson EDE is 625 sinceU(ξ(ǫ)) = 2 ∗ 625(1−0.5) = 50 = WAtkinson and the Atkinson index ofinequality then equals1− (ξ(ǫ)/µ) = 1− (625/750) = 0.166.

The generalized entropy indices


� The generalized entropy indicesI(y; θ) are defined as follows:

I(y; θ) =

1θ(θ−1)

(

1n

∑n

i=1

(yiµ

)θ

− 1

)

if θ 6= 0, 1

1n

∑n

i=1 ln(

µ

yi

)

if θ = 0,

1n

∑n

i=1yiµln(

yiµ

)

if θ = 1.

(8)

� For the casesθ = 0 andθ = 1, we have the two Theil indicesI(θ = 0)andI(θ = 1), which are special measures provided by thegeneralizedentropy class.

The generalized entropy indices


� If all individuals have the same incomeµ, these indices are zero.� If a transfera is made from a richer person with incomeyj to a poorer

person with incomeyi, such thatyj − yi > a, then the change inI(θ = 0)

is given by ln(yiyj)−ln((yi+a)(yj−a))

n.

� Since(yi + a)(yj − a) > yiyj, then any such transfer will decrease thisindex of inequality.

� This is true for all other members of that class.

Differences across inequality indices


� Can the ordering of inequality differ with indices?� The sensitivity of inequality indices to transfers in different parts of the

distribution differ.� Let y = {2, 6, 10, 12, 20}. Let us see the proportional change in

inequality implied by locally equalizing transfers that replace the incomesof individualsi andi+ 1 by 0.5(yi + yi+1).



−15

−10

−5

0T

he r

elat

ive

chan

ge in

ineq

ualit

y in

(%

)

(2,6) (6,10) (10, 12) (12,20)Equalized incomes

Gini Atkinson (ε=0.5)Entropy (θ=1.0 )



� Is this difference in sensitivity to local inequality sufficient to reverse theorder of distributions?

Distributions y1 y2 y3 y4 IGini IAtkinson(ǫ = 1)A 3 3 10 16 .3594 .2300B 1 5 13 13 .3438 .3261

Difference – – – – -.0156 .0961

� Differences in inequality between two distributions can thus havedifferent signs according to the choice of inequality index.

Transfer sensitivity principle


� To discriminate between indices that put more weights in thelower tail ofthe distribution, such as the Atkinson index, it is useful toconsider thefollowing principle:

A5: Transfer sensitivity principle A beneficial Pigou-Dalton transferwithin the lower part of the distribution accompanied by an adversePigou- Dalton transfer within the upper part of the distribution thatkeeps the variance unchanged should not increase inequality.

� Let 0 < a < y2 − y1 = y4 − y3. Inequality indices obey theTransfersensitivity principle if I{y1, y2, y3, y4} > I{y1+ a, y2− a, y3− a, y4+ a}.



� Let y = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}� Let four composite transfers (A, B, C andD) be as shown in the

following diagram. For instance, with (a = 1) the composite transferAwill generate the income vector:y = {10, 20, 30, 41, 49, 59, 71, 80, 90, 100}.

10 20 30 40 50 60 70 80 90 100

ABCD



−2

−1.

5−

1−

.50

The

rel

ativ

e ch

ange

in in

equa

lity

in (

%)

(A) (B) (C) (D)Equalized incomes

Gini Atkinson (ε=0.5)Entropy (θ=0.0 ) Entropy (θ=1.0 )

� As illustrated above, the Gini index can fail to satisfy thatprinciple, butmany of the other indices obey it.

Decomposing inequality

Outline

Objectives




Decomposinginequality bypopulation groups

Gini index and groupdecomposition

Decomposition byincome components

Gini index andincome sources

Conclusion


Decomposing inequality by population groups


� We first wish to assess the contribution of within-group andbetween-group inequality to total inequality.

� Between-group inequality is inequality if each individualhas the averageincome of his group.

IA6: Subgroup decomposability principle Total inequality can beassessed from within-group and between-group inequality alone.

� It can be shown that the decomposable inequality indices that obey thescale invariance axiom are just a transformation of the generalizedentropy indices.



� The generalized entropy indices can be decomposed as:

I(y; θ) =L∑

l=1

φl

(µl

µ

)θ

I(yl; θ)

︸︷︷︸

Within-groupinequality

+ I(µ1, ..., µl, ...µL; θ)︸︷︷︸

Between-groupinequality

(8)



� Let two distributionsA andB be given by

DistributionA DistributionBGroup (l) y µl

y µl

1 2 4 1 21 6 4 3 22 12 16 18 182 20 16 18 18




y µl

1 2 4 1 21 6 4 3 22 12 16 18 182 20 16 18 18

� When inequality within-group is nil, like the case of group 2indistributionB, within-group contribution of this group to total inequalityis nil.




y µl

1 2 4 1 21 6 4 3 22 12 16 18 182 20 16 18 18

� The more average incomes of groups are distance the more is thecontribution of between group inequality.



DistributionA DistributionBGroup (l) y µl y µl

1 2 4 1 21 6 4 3 22 12 16 18 182 20 16 18 18



� For a given group, its within-group inequality is not sufficient to judgeabout the importance of its contribution to total inequality. Indeed, onemust take into account the importance of population share and that of itsaverage income.

� The between-group inequality increases with the increase in number ofgroups (L). For instance, if we are interested in inequality according toeducation, one can construct groups by number of educated years (forinstance, between zero and twenty years (L = 21), by of education level(without education, primary, secondary and tertiary,L = 4)), etc.

Gini index and group decomposition


� The Gini index can be decomposed as follows:

IGini(y) =L∑

l=1

φlψlIGini(yl)

︸︷︷︸

Within-groupinequality

+ IGini(µ1, ..., µL)︸︷︷︸

Between-groupinequality

+ Residual︸︷︷︸

Groupoverlap

(8)

whereIGini(yl) is the Gini coefficient for groupl.

� TheResidual exists when the incomes of the different groups overlap.This occurs when the maximum income of a given group is higherthanthe minimum income of another group.

Decomposition by income components


� This shows the importance of the contribution of different incomesources to total inequality.

� The sum of theK income sources equals total income:

yi = yi,1 + ...+ yi,k + ...+ yi,K (9)

whereyi,k is the income level of individuali from the sourcek.

Gini index and income sources


� The Gini index may be defined as follows:

IGini(y) = 1−

n∑

i=1

υiyi

µ

=∑K

k=1µk

µ

1−

n∑

i=1

υiyi,k

µk

=∑K

k=1µk

µIC(yk)

(10)

whereIC(yk) is the concentration coefficient of sourcek.



� The concentration coefficient of sourcek is defined as

IC(yk) =n∑

i=1

κiψi,k (10)



� Assume that the population is composed of two individuals with incomes20 and 40. The Gini index is 1/6. Also, assume that there are threeincome sources (K = 3):

Component Individual 1 Individual 2 µk

µIC(yk)

k = 1 6 4 1/6 -0.1k = 2 4 16 2/6 0.3k = 3 10 20 3/6 1/6

Income 20 40 – –

� The poor person has a relatively large share of the first source. Thissource helps reduce total inequality; the concentration index is negative.





µIC(yk)

k = 1 6 4 1/6 -0.1k = 2 4 16 2/6 0.3k = 3 10 20 3/6 1/6

Income 20 40 – –

� Inequality in the second income source is more pronounced than for totalincome. Its concentration index is higher than the Gini; this shows thatthis source tends to increase inequality.





µIC(yk)

k = 1 6 4 1/6 -0.1k = 2 4 16 2/6 0.3k = 3 10 20 3/6 1/6

Income 20 40 – –

� Inequality in the distribution of this third source is similar to that of totalincome. Removing it would not change total inequality. It isthereforeneutral.

Conclusion

Outline

Objectives




Conclusion

Summary

Relevant DASPcommands

Exercises with Stataand DASP

References


Summary


� A fundamental principle is that an income transfer from a richer person toa poorer person should decrease inequality;

� Inequality measures are usually population and scale invariant;� The Lorenz curve shows the share in total income of those at the bottom

of the income distribution;� The Gini index is a (popular) measure of the difference between

population and income shares;� Social welfare is usually assumed to increase with average income and to

decrease with inequality.

Summary


� Inequality indices differ in their sensitivity to transfers in different partsof the distribution.

� Some inequality indices allow total inequality to be expressed as a sum ofwithin-group and between-group inequality alone.

Relevant DASP commands


� Gini and concentration indices (igini)� Difference between Gini/concentration indices (digini)� Generalised entropy index (ientropy)� Difference between generalized entropy indices (diengtropy)� Atkinson index (iatkinson)� Difference between Atkinson indices (diatkinson)� Lorenz and concentration curves (clorenz)� Lorenz/concentration curves with confidence intervals (clorenzs)� Differences between Lorenz curves with C.I. (clorenzs2d)� Inequality: decomposition by income sources (diginis)� Gini index: decomposition by population subgroups (diginig)� Entropy indices: decomposition by population subgroups (dentropyg)

http://132.203.59.36/DASP/help/DASP_MANUAL_V2.1.htm#_Toc247593937












Exercises with Stata and DASP


� Part II: Exercises 1.1, 1.2, 1.3� Part II: Exercises 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8� Part II: Exercise 5.1� Part II: Exercises 6.3, 6.4, 6.5

http://dasp.ecn.ulaval.ca/exercises/exercises.htm#_Toc255461640




References


AABERGE, R. (2007): “Gini’s Nuclear Family,”Journal of EconomicInequality, 5, 305–322.

ATKINSON, A. (1970): “On the Measurement of Inequality,”Journal ofEconomic Theory, 2, 244–63.

BOURGUIGNON, F. AND C. MORRISSON(2002): “Inequality among WorldCitizens: 1820-1992,”American Economic Review, 92, 727–44.

DALTON , H. (1920): “The Measurement of the Inequality of Incomes,”TheEconomic Journal, 30, 348–61.

M ILANOVIC , B. (2002): “True World Income Distribution, 1988 and 1993:First Calculation Based on Household Surveys Alone,”Economic Journal,112, 51–92.

PIGOU, A. (1912):Wealth and welfare, London: Macmillan.

abdelkrim araar, sami bibi and jean-yves duclos

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