Poverty Inequality Measures- Tarun Das

8/14/2019 Poverty Inequality Measures- Tarun Das

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07/11/09 Poverty and Inequality- Prof. T. Das 1

Measures of Poverty andInequality

Prof. Tarun Das, IILM, New Delhiprepared for the ISS Training Program


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Contents of this presentation

1. Conceptual Issues2. Properties of An Ideal Measure of

Inequality3. Positive Measures of Inequalities4. Normative Measures of Inequalities5. Decomposition of Inequality6. Theoretical Income Distributions

7. Measures of Poverty


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1.1 Conceptual Issues- Equality andEquity

1. Inequality and Equity- Inequality is a

mathematical concept, whereas equityis a concept of justice andjurisprudence.

2. Measure of inequality is the degree bywhich existing income distribution(Y1,Y2 .Yn) differs from equaldistribution (, , ).

3. An equal income distribution may notbe an equitable distribution.

4. An equitable distribution may justifyunequal distribution due to differencesof needs, capabilities, skill, education,experience etc.

5. Even Gods are not equal. Hindu religionaccepts gradation of Gods and


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1.2 Conceptual Issues- Poverty

1. Similarly, poverty has manydimensions2. Poverty as measured by income3. Poverty measured by expenditure

4. Poverty measured by calorieintake5. Poverty measured by entitlements6. Poverty measured by human

development7. Poverty measured by social

welfare


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1.3 Four Basic Problems

1. (a) What should be the income

receiving/ demographic units?(b) How is income to be defined?(c) Over which period should we take

income?

(d) How can we measure the degree ofinequality and poverty?2. Who should be made equal?

In what respects?

Over which period?By which standards?3. First three questions are economic in

nature, while the last one is statistical.


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1.4 Choice of Income Receiving Unit

1. Various concepts- Individuals, adultequi-valent units, economically active

population (age group of 15-65), incomereceiving units, families and households2. More the heterogeneity, more the

inequality.Gini=0.784*0.050*logY12.184*/Y+

0.051*D1(0.016) (0.015) (4.144) (0.016)

+0.129*D20.033*D3 0.007 D4R2=0.37

(0.021) (0.018) (0.016)D1=1 for income recipients, 0 otherwiseD2=1 for economically active population, o

elseD3=1 for rural sector, 0 otherwise

D4=1 for urban sector, 0 otherwiseIf all Dis are zero e uation stands for all


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1.5 Simple Questions but no EasyAnswers

1. Income units should be easilyidentifiable, inclusive and distinctly

independent2. Concept of income- Expenditure andincomes surveys, Monetary and in-kind,pre-tax and post-tax, pre and posttransfer, tradable and non-tradable,home produced, capital gains, fringebenefits, current and life time income.

3. Time period- Week? Month? Year? LifeTime?

4. We need to make compromise betweenwhat is theoretically desirable and whatis empirically practicable. We simplyhave to take what we are given by thestatisticians, who have usually collected

data for other reasons (Harold Lydall1979 .


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1.6 Household Expenditureand Family Expenditure Surveys

1. Concepts of Households andFamilies2. While UK and India conduct households

expenditure surveys, USA conducts

family expenditure surveys3. Conversion of single person households

and higher household sizes into AdultEquivalent Units (AEUs)

4. Sample size- sampling and non-samplingerrors

5. Approximation of income fromexpenditure surveys


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2.1 Properties of an ideal inequalityindex

(a) It must depend on all income levels.

I = f (Y1, Y2, Y3 . Yn), Yi 0.(b) Income homogeneity: It is homogeneousof degree zero in all incomes i.e. if allincomes rise or fall equi-proportionately,then the inequality index remainsunchanged. It also implies that it isindependent of the units ofmeasurement, and depends on relativeincome .

I= I =f ( Y1, Y2, Y3 . Yn), >0(c)It ranges in between zero (in the case of

perfect equality) and unity (in the caseof absolute inequality, when one personhas all national income and others have

no income).


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2.2 Properties of Ideal InequalityIndex

(d)Population homogeneity: It remainsinvariant w.r.t. equi-proportionate change

in the number of units in all incomebrackets.(e)Pigou-Dalton criterion: A transfer of

income from the rich to the poor shouldreduce the degree of inequality and viceversa.

(f) Anonymity: It is unaffected by anypermutation of incomes i.e. We do notwant to know the identities of incomeearners. It is called the property ofanonymity or impartiality.

(g)An equal absolute increase of all incomesleads to reduction of inequality, and anequal absolute reduction of incomes leads

to an increase of inequality. An impositionof head tax increases inequality, while


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2.3 Properties of Ideal InequalityIndex

(h)It should be subject to simple economicinterpretation.

(i)It should be easily calculated fromavailable data, computer programs andalgorithms.

(j) Should be subject to statistical tests ofsignificance.(k) It should be statistically decomposable.

Grand inequality = Weighted Mean ofgroup inequalities + Inequality of groupmean incomes

= Within Group Inequality + BetweenGroup Inequality


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3.1 Inequality Measures

1. Positive Measures (pure statistical

measures, no utility functions) - GiniLorenz ratio, CV, VARLOG, Theil EntropyMeasure

2. Normative Measures- Atkinson Index

(assumes utility functions)3. Let us assume that there are n personswith incomes (Y1, Y2, Y3, Yn) arrangedin ascending order i.e. Y1 Y2 Y3 .

Yn.Qj = Share of the j-th person in totalincome

=Yj/ TY = Yj / Yj

Where TY = Y1 + Y2+ Y3+.+Yn = Yj


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3.2Size Distribution of Income

Qj

=1

Pj

=1

N=

nj

Total

CQjCPjQjPjnjYj

CQ2CP2Q2P2n2Y2

CQ1CP1Q1P1n1Y1

CQjCPjQj =PjYj/ Pj= nj/NNo. ofrecipientAverageincome


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3.3 Lorenz Concentration Curve

Both CPj and CQj range in between 0and 1.

A Lorenz curve is drawn within a unit-square box diagram.

The 45 degree radius vector is called

the egalitarian line as on it CPj equalsCQi for each i=1, 2, 3 .. n. The area between the Lorenz curve

and the egalitarian line is called the

area of concentration (A). Gini-Lorenz ratio equals the area ofconcentration (A) divided by the areaof the triangle below the egalitarianline.


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3.4 Gini-Lorenz ratio

(1) A Lorenz curve is the locus of all points(CPj, CQj). It is positively sloped andconvex to the horizontal axis, where

CPj = Cumulative proportion of personsup to j-th income in ascending orderand

CQj = Cumulative shares of income ofthese individuals

(2) Lorenz ratio = 1 - Pj (CQj+ CQj-1)(3) Gini coefficient = Pi Pj IYi YjI / 2(4) It can be proved that LR = GC, so we

can call it Gini-Lorenz ratio.


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3.5 CV, Var (Log), RMAD

Coefficient of variation = / where is the standard deviation andis the arithmetic mean of incomes

= PjYj, =VAR = Pj (Yj-),2. Variance of logarithms- It is the

variance of the logarithms of incomes

of individuals= Pj(log Yj )= Pj(log Yj logGM)

= Pj [log (Yj/GM)]

where =AM(LogY)=Pj logYj = log (GM)


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3.6 TDM and Theil EntropyMeasure

3. Kuznets Total Disparity Measure (TDM)TDM = I Pj Qj I

4. RMAD = Pj I Yj- I / = IPjYj Pj I / = IQj PjI = TDM5. It can be proved that

[corrected]LR RMAD CV

6. Theil entropy measure= Qj ln (Qj / Pj) = Qj ln (Yj /)

arac er s cs o nequa y


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. arac er s cs o nequa yMeasures

All measures are ordinal in nature (1st,2nd,3rd), and not cardinal numbers (1, 2,

3). Lorenz curves provides many informationabout income distribution, while othersare summary measures. They concealmore facts than they reveal.

There does not exist an ideal measure ofinequality.

Due to differences in concepts onincome, demographic units, period,

sample designs and size, and theinequality index, it is very difficult todraw valid inter-country and inter-temporal differences of inequality.

So minor differences in inequalitiesshould be i nored.


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4.2 Various Normative Measures

1. Ye isthe generalized mean of order

=M(Y)where = (1 - ), is the inequalityaversion parameter. Properties of M(Y)indicated below:(a)M(Y) is an increasing function of.(b)M(Y) is homogeneous of degree unity inY.(c) M(Y) > M(Y) if Y > Y

(d) If>1, M( Yj) < M(Yj)(e) If M(Yj)2. Sens development index adjusted byinequality = (1- G)

3. Allingham distinguishes between income

i i f li


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5.1 Decomposition of InequalityMeasures

1. Statistically, there are two types ofdecomposition:

(a) Additive decomposition = Pj j, = Pj + Pj (Yj - )

(b) Multiplicative decomposition

GM = (GMj) where a =Pj2. Economically there are three types:(a) Sectoral/ regional decomposition(b) Functional decomposition by income

sources- wages, rents, interests, profits(c) Factor decomposition by incomegenerating factors such as age, sex,education, experience etc.


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5.2 Purpose of Decompositions

It helps us to determine relativeimportance of factors, sources of incomeand regional pockets determininginequality or poverty.

Kuznets distinguishes betweenwarrantedand unwarrantedinequalities.

There is a secular trend of inequalityinduced by demographic factors,existing laws and regulations, which is

justifiable. Any thing above that isundesirable.

Public policy should deal withunwarranted inequality.

Decomposition helps to estimatewarranted and unwarrantedinequalities.


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5.3 Sectoral/regional Decompositions

ITotalIkkkk

Ijjjj

I2222I1111

Inequalit

y

VarAMSectors

5 4 S t l/ i l


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5.4 Sectoral/regionalDecompositions

Within Group inequality (WGI) = Weightedmean of group inequalities = Pj Ij

Between Group inequality (BGI) =Inequality in group means = I(1, 2, 3. k)

Grand Inequality = WGI + BGI + RSector CV Gini VarLN Theil tkinso TDMRural 0.26 0.28 0.32 0.13 0.87 0.40

Urban 0.27 0.29 0.42 0.14 0.93 0.44

All 0.27 0.29 0.36 0.14 0.89 0.42

WGI 0.26 0.28 0.35 0.13 0.89 0.42

BGI 0.01 0.04 0.01 0.00 0.04 0.08

WGI+BGI 0.27 0.32 0.36 0.14 0.93 0.50

R 0 0 0 0 0 0


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5.5 Sectoral/regional Decompositions

Theorem-1: Inequality of a group of sectors/regions cannot be less than weightedaverage of the inequalities of sectors/regions.

I (Yi) Qi IiCorollary-1: Grand inequality WGICorollary-2: Even when all sectors areequally unequal (Ii=),grand inequality isgreater than weighted average of sectoralinequalities ().

Corollary-3 If each sector is perfectly equali.e. Ii=0, then grand inequality equalsbetween group inequality.

Corollary-4 If there is no between groupinequality i.e. all sectors have same , then

grand inequality equals within group


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5.6 Functional Decompositions

Let Yi = Wi + Ri + INTi + Pi =1, 2, 3 ..n

Shares w r int pw = Wi/ Yi, r = Ri/ Yiint = INTi/ Yi p = Wi/ YiGrand I [w I(W)+ r I(R)

+ int I(INT)+ p I(W)]Theorem-1: Inequality of a sum of incomecomponents cannot exceed weightedaverage of the inequalities of components.

I(Yi) i IiTheorem-2: Equality holds good only when

all income components are equi-proportional for all income receiving units

Ri= Wi, INTi = Wi, Pi = Wi


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5.7 Functional Decompositions

Corollary-1: Even when all components areequally unequal,inequality of income isless than weighted average of inequalitiesof components.

Corollary-2: Inequality of a sum of incomecomponents is less than sum of inequalities

of components.Corollary-3:Yt = Past income + changeSo I(Yt) [ I(Yt-1) + I(change)]Corollary-4: I(Yt) - I(Yt-1) I(change)Change of inequality is less than inequalityof change.


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6.1 Theoretical Income Distributions

(1) f(y) = Probability distribution function ofy

(2) F(y) = Distribution functiony

= f(y) dy0

(6) (y) = incomplete first moment of yy

= y dF(y) / , = y f(y) dy0 0

1 1 1(4) LR= 1- 2 dF = F d - dF0 0 0

(5) d/dF = y/ >0,d / dF = 1/ [ f(y)]>0


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6.2Pareto DistributionFirst extensive study on income distributionwas done by Vilfredo Pareto (1948-1923), a

discipline of Leon Walras (1834-1910).Pareto Curve:Nx = A Xwhere a = -, 1.5Nx = Number of units having income y x is called Pareto Coefficient.Lorenz curve = 1 (1-F)^, = (-1)/Gini-Lorenz ratio = 1/(2-1)Paretos law: According to Pareto, 1.5and GLR =0.5 represents a stable macroeconomic situation. Any social reformattempting to reduce inequality below 0.5is doomed to be a failure.Harold Davis agreed and proposed theoriesof natural abilities and least efforts. Lange

did not agree and argued that Pareto lawdoes not hold ood for wa e incomes.

6 3 Lydalls Pyramidal Income


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6.3 Lydall s Pyramidal IncomeDistribution

People in any grade (except the bottom)control a fixed number of staff in the nextlower grade Ni / Ni+1 = K

Total income paid to any grade is

proportional to that in the next lowergrade.Ni+1 . Yi+1 / (Ni Xi ) = P

These assumptions lead to a Paretian tail

with Pareto coefficient= logK / logKP = logK / (logK + logP)Gini-Lorenz ratio = 1/(2-1)

= (logK + logP)/ (logK-logP)

= logKP/ log (K/P)

6 4 Stochastic Distribution and Markov


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6.4 Stochastic Distribution and MarkovProcess

Champernownes theory leading to Paretodistribution:

Income intervals follow a geometricprogression- y, ry, ry, ry Transitional probabilities Tij

(probability of a person in income

range i to move to income range j inthe next period) depends on thespread of classes Tij = f (i-j)

No individual can move up more thanone interval in a period, but can move

down several intervals. Incomes will not increase indefinitely. For every income recipient who dies or

exits for some reasons, there is asuccessor to his/her income in the next

period.


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6.5 Log normal distributionIf LogY is distributed normally with mean and variance , y is said to be log

normally distributed with the same meanand variance.Pdf f(Y)=1/[ y sqrt(2)] exp[-(lnY-)/2]Lorenz curve tcq = tcp -

Lorenz ratio= 2 (/sqrt(2)) -1Tcq is the value of z such that area underthe standard normal variate up to tcq isCQ, and (x) is the area under thestandard normal distribution up to x.

Special case-1: In the case of perfectequality, =0, so LR = 2 (0) 1 =1-1=0Special case-2. In the case of

extreme inequality, , LR = 2-1 =


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6.6 Theories of Random Walk1. Brownian Process and Gibrats law ofproportionate effect: Level of income ofany individual at any point of time is arandom multiple of income alreadyachieved.Yt = (1+ R1)(1+ R2)(1+ R3).. (1+ Rt)

Yo2. Atchison and Brown theory:

Income classes are in arithmetic

progressionTransition probabilities Tij depend onratio of classes. Tij = f (i/j)

Both these theories lead to log-normal

distribution.


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7.1 Measurement of PovertyIn September 2000, at the UN MillenniumSummit 189 nations adopted theMillennium Declaration to eradicateextreme poverty and hunger by 2015 andset targets on education, health,environment etc. These are called the

Millennium Development Goals (MDG).Multilateral agencies are providing fundsand helping the developing countries toprepare Poverty Reduction Strategy

Papers (PRSPs) for meeting MDG targets.Measurement and analysis of poverty hasthus become the central point forplanning and assume special significance.

7 2 Properties of an Ideal Measure of


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7.2 Properties of an Ideal Measure ofPoverty

Monotonocity criterion: Given otherthings, a poverty measure mustincrease (decrease) if the income of apoor family is reduced (increased).

Transfer criterion: Given other things,the poverty measure must increase(decrease) with any transfer of incomefrom a poor household to a non-poorhousehold (with reverse transfer)

Transfer sensitivity: Greater thetransfer from the richer to the poorpersons, greater the magnitudes ofpoverty reduction.


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7.3 Poverty Ratio and Sen Poverty IndexLet there are n individuals with incomesY1 Y2 Y3 . Yn. is Mean income, and Z = Poverty linem is the number of people below Z.Poverty gap = Z -Yi for people below thepoverty line.Headcount ratio (H) = m / nPoverty gap ratio (PG) = (Z Yi)/ m ZSen Poverty Index = H [ PG + (1 PG) Gini]where Gini is the Gini Lorenz ratio of the

poor. If everyone has income above thepoverty line, H = 0, so Sen Poverty Index=0.If all has no income,

H=PG=SenIndex=Gini=0


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7.4 Foster-Greer-Thorbecke Index (FGT)Foster-Greer-Thorbecke (FGT) family ofpoverty measures:

zP = [(1 Yi/Z] / ni=1

a=0 Headcount ratio = m / n

a=1 Poverty gap ratio = (Z Yi)/ nZa=2 FGT P2 measure

FGT satisfies all criteria of an ideal

measure of poverty. It also can bedecomposed.


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7.5 Hamada-Takayama Index

Actual income distribution Y1 Y2 . Yn.

Censored income distribution is one inwhich each income above the poverty lineis made equal to the poverty line.

Yi* = Yi if Yi < zYi* = z if Yi zThe Gini ratio, CV, Theil Entropy measureor any relative measure of dispersion of

the censored income distribution can betaken as measures of poverty.

Thank you


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Thank youHave a Good Day

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Poverty Inequality Measures- Tarun Das