1

Measuring Poverty: Inequality Measures

Charting Inequality Share of Expenditure of Poor Dispersion Ratios Lorenz Curve Gini Coefficient Theil Index Comparisons Decomposition

2

Poverty in Lao PDR 1997/98

Lecs II

Percentage57.9 - 71.349.7 - 71.339.2 - 49.713.5 - 39.2

Dept of Poverty in Lao PDR 1997/98

Lecs II

Percentage17 to 24.711.9 to 17

9.5 to 11.92.8 to 9.5

Severity of Poverty in Lao PDR 1997/98

Lecs II

Percentage7.1 to 12.14.3 to 7.13.3 to 4.30.8 to 3.3

Poverty Measures, Lao PDR

3

Income Distribution

Types of analysis Functional distribution Size distribution

Functional distribution— income accrued to factors of production such as

land, labor, capital and entrepreneurship Size distribution—

income received by different households or individuals

4

What is Inequality?

Dispersion or variation of the distribution of income/consumption or other welfare indicator Equality– everyone has the same income Inequality– certain groups of the population have

higher incomes compared to other groups in the population

5

Why measure inequality? (1)

Indicator of well-being “Position” of individual relative to rest of population “Position” of subgroup relative to other subgroups

Different measures, different focus Poverty measures (HC, PGI, SPGI, etc) focus on the

situation of individuals who are below the poverty line– the poor.

Inequality is defined over the entire population, not only for the population below a certain poverty line.

6

Why measure inequality? (2)

Inequality is measured irrespective of the mean or median of a population, simply on the basis of the distribution (relative concept).

Inequality can be measured for different dimensions of well-being: consumption/expenditure and income, land, assets, and any continuous and cardinal variables.

7

Charting Inequality: Histogram

Divide population into expenditure categories

Example: 20% of households are in category 4 0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Expenditure categories

Per

cent

age

of p

opul

atio

n

8

Example: Income Classes

9

Example: Bar Chart, Income Classes

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

1 2 3 4 5 6 7 8 9 10 11 12Income Class

% o

f Fam

ilies

Percentage of families falling in each class

10

Example: CDF of Per Capita Expenditure0

.2.4

.6.8

1C

umu

lativ

e D

istr

ibu

tion

0 200000 400000 600000 800000 1000000Per capita Total Expenditure

11

Distribution: Quintile and Deciles

12

Expenditure/Income-iles

Divide population into ‘groups’ ranked from ‘poorest’ to ‘richest’ based on expenditure (or income)

Divide into 5 groups: income or expenditure quintiles Lowest 20% or first quintile– “poorest” Highest 20% or fifth quintile– “richest”

Divide into 10 groups: income or expenditure deciles

13

Expenditure per capita by Quintile, Viet Nam (1993)

Quintile Per Capita Expenditure

% of Total Expenditure

First: Lowest 518 8.4Second: Low-middle 756 12.3Third: Middle 984 16.0Fourth: Mid-upper 1,338 21.8Upper: Fifth 2,540 41.4All 1,227 100.0

Poorest

Richest

14

Share of Income of Poorest, KoreaIncome decile 2000 2001 2002 2003

1st 2.9 2.9 3.0 2.72nd 4.7 4.6 4.7 4.83rd 5.8 5.7 5.8 6.14th 6.9 6.8 6.9 7.15th 7.9 7.8 7.9 8.16th 9.1 9.1 9.2 9.37th 10.5 10.5 10.5 10.78th 12.2 12.3 12.4 12.59th 14.7 15.0 15.1 15.010th 25.4 25.4 24.6 23.8

15

Inequality Measures Based on -iles

Share of income/consumption of lowest –ile Dispersion ratios

16

Share of Consumption of the Poorest Definition: Total consumption/income of the poorest

group, as a share of total consumption/income in the population.

WhereN is the total populationm is the number of individuals in the lowest x %.

N

ii

m

ii

x

y

yC

1

1)(

17

Poorest Quintile’s Share in National Income or Consumption (UNSD, 2005)

18

Dispersion Ratio

Definition: measures the “distance” between two groups in the distribution of expenditure (or income or some other characteristic)

Distance: average expenditure of the “richest” group divided by the average expenditure of the “poorest” group

Example:average expenditure of fifth quintile

Dispersion ratio=average expenditure of first quintile

19

Dispersion Ratios: Examples

Expenditure decile Median

1st 37,3242nd 47,2893rd 54,3974th 62,9295th 74,7756th 89,4787th 108,6338th 129,8909th 172,01110th 267,214

(1) 10th:1st

(2) 10th :1st & 2d(Kuznet’s ratio)

20

Lorenz Curve and Gini Ratio

21

Lorenz Curve

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Cumulative % of population

Cum

ulat

ive

% o

f con

sum

ptio

n

22

Lorenz Curve: Interpretation (1) If each individual

had the same consumption (total equality), Lorenz curve would be the “line of total equality”.

If one individual had all the consumption, Lorenz curve would be the “curve of total inequality”.

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Cumulative % of population

Cum

ulat

ive

% o

f con

sum

ptio

nLi

ne of t

otal

equalit

y

Curve of total inequality

Lore

nz

curv

e

23

Lorenz Curve: Interpretation (2)

The further away from the line of total equality, the greater the inequality.

Example: Inequality is greater in country D than in country C.

0102030405060708090

100

0 50 100

C

D

24

Comparing Lorenz Curves

A

BC

D

1000

100

25

“Lorenz Criterion”

Whenever one Lorenz curve lies above another Lorenz curve the economy with the first Lorenz curve is more equal, and the latter more unequal e.g. A is more equal; D is more unequal

When 2 curves cross, the Lorenz criterion states that we “need more information (or additional assumptions) before we can determine which of the underlying economies are more equal” e.g. curves B and C

26

Constructing Lorenz Curve, Example (1)

Quintile Cumulative Share of

Population (p)

% of Total Expenditure

Cumulative share of

expenditure (e)

First 20 8.4 8.4

Second 40 12.3 20.7

Third 60 16.0 36.7

Fourth 80 21.8 58.5

Fifth 100 41.4 100.0

27

Constructing Lorenz Curve, Example (2)

0

20

40

60

80

100

0 20 40 60 80 100

p

e

28

Gini Coefficient: Definition

Measure of how close to or far from a given distribution of expenditure (or income) is to equality or inequality

Varies between 0 and 1 Gini coefficient 0 as the expenditure/income

distribution absolute equality Gini coefficient 1 as the expenditure/income

distribution absolute inequality

29

Gini Coefficient & Lorenz Curve (1)

Area between line of equality and Lorenz Curve (A)

If A=0 then G=0 (complete equality).

A

30

Gini Coefficient & Lorenz Curve (2)

Area below Lorenz Curve (B)

If B=0 then G=1 (complete inequality).

31

Gini Coefficient & Lorenz Curve (3)

Gini coefficient (G) is the ratio of the area between the line of total equality and the Lorenz curve (A) to the area below the line of total equality (A+B)

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Cumulative % of population

Cum

ulat

ive

% o

f con

sum

ptio

n

Line

of tota

l

equalit

y

Curve of total inequality

Lore

nz

curv

e

A

B

32

Lorenz Curve and Gini Coefficient

e

33

Gini Coefficient: A Formula

Here’s one. (There are other formulations.)

i iN

ii=1

Cov y ,fG = 2

1y

N

Where: N is population size y is expenditure of individual f is rank of individual in the distribution

34

Gini Coefficient: +’s and –’s (+) Easy to understand, in light of the Lorenz curve. (-) Not decomposable: the total Gini of the total

population is not equal to the sum of the Ginis for its subgroups.

(-) Sensitive to changes in the distribution, irrespective of whether they take place at the top, the middle or the bottom of the distribution (any transfer of income between two individuals has an impact, irrespective of whether it occurs among the rich or among the poor).

(-) Gives equal weight to those at the bottom and those at the top of the distribution.

35

Measures of Inequality, Example

36

Poor people in Senegal get bigger share of income than poor people in the US

Bottom 60%

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60

US

Senegal

37

General Entropy Indexes

represents the weight given to distances between incomes at different parts of the income distribution Sensitive to changes at the lower end of the distribution if α

is close to zero Equally sensitive to changes across the distribution if α is 1

(Theil index) Sensitive to changes at the top of the distribution if α takes a

higher value.

21

1 1( ) 1

Ni

i

yGE

N y

38

GE(1) and GE(0)

GE(1) is Theil’s T index

GE(0), also known as Theil’s L, is called mean log deviation measure :

N

i

ii

y

y

y

y

NGE

1

)ln(1

)1(

N

i iy

y

NGE

1

)ln(1

)0(

39

The Theil Index: Definition

Varies between 0 (total equality) and 1 (total inequality). The higher the index, the more unequal the distribution of expenditure (or income).

1

1ln

Ni i

i

y yT

N y y

iwhere y is expenditure of ith individual

y is average expenditure of population

40

Theil Index: +’s and –’s) (+) Gives more weight to those at the bottom of the

income distribution. (+) Can be decomposed into “sub-groups”: the

population Theil is the weighted average of the index for each sub-group where the weights are population shares of each sub-group

(-) Difficult to interpret (-) Sensitive to changes in the distribution, irrespective

of whether they take place at the top, the middle or the bottom of the distribution (any transfer of income between two individuals has an impact, irrespective of whether it occurs among the rich or among the poor).

41

Atkinson’s Index

This class also has a weighting parameter ε (which measures aversion to inequality)

The Atkinson class is defined as:

Ranges from 0 (perfect equality) to 1

)1(1

1

11

1

N

i

i

y

y

NA

42

Criteria for ‘Goodness’ of Measures

Mean independence– If all incomes are doubled, measure does not change.

Population size independence– If population size changes, measure does not change.

Symmetry– If two individuals swap incomes, the measure does not change.

Pigou-Dalton transfer sensitivity– Transfer of income from rich to poor reduces value of measure.

Decomposability– It should be possible to break down total inequality by population groups, income source, expenditure type, or other dimensions.

43

Checklist of Properties

Property Dispersion Gini Theil

Mean independence

Population size independence

Symmetry

Pigou-Dalton Transfer Sensitivity

Decomposability

44

Inequality Comparisons

Extent and nature of inequality among certain groups of households. This informs on the homogeneity of the various groups, an important element to take into account when designing interventions.

Nature of changes in inequality over time. One could focus on changes for different groups of the population to show whether inequality changes have been similar for all or have taken place, say, in a particular sector of the economy.

Other dimensions of inequality: land, assets, etc

46

Example: Inequality Changes over Time

Year Poverty Rate

Gini Coefficient

1985 48 0.44661988 40 0.44461991 40 0.46801994 36 0.45071997 32 0.48722000 34 0.4818

48

Example: Gini Ratios, Indonesia

49

Decomposition of Inequality

50

At One Point in Time (1)

Inequality decompositions are typically used to estimate the share of total inequality in a country which results from different groups, from different regions or from different sources of income.

Inequality can be decomposed into “between-group” components and “within-group” components. The first reflects inequality between people in different sub-groups (different educational, occupational, gender, geographic characteristics). The second reflects inequality among those people within the same sub-group.

51

Example, Viet Nam (1993)

52

Decomposition of Inequality, Egypt

53

At One Point in Time (2)

Inequality decompositions can be calculated for the General Entropy indices, but not for the Gini coefficient. For future reference, the formula is:

where fi is the population share of group j (j=1,2, … k), vj is the income share of group j; yj is the average income in group j.

1.1

)(..1

21

1k

j

jj

k

jjjjBW y

yfGEfvIII

54

Changes over Time (1)

Changes in the number of people in various groups or “allocation” effects

Changes in the relative income (expenditure) of various groups or “income” effects

Changes in inequality within groups or “pure inequality” effects.

55

Changes over Time (2)

The formula can get complicated, and is typically used for GE(0) only, as follows:

averages. represents bar over the and (y)),)/(y ( mean overall

the to relative j group of income mean the is operator, difference the is where

effects

effects Income effects ocation All inequality Pure

))(log()()log()()()(

jj

j

k

j

k

jj

k

jijjjjj

k

jjj yfvffGEGEfGE

1 1 11

000

56

Poverty Changes over Time (1)

Poverty is fully determined by the mean income or consumption of a population, and the inequality in income or consumption in the population.

Changes in poverty can result from changes in mean income/consumption – growth – or from changes in inequality.

57

Poverty Changes Over Time (2)

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140 160 180 200 220 240

Income

Sh

are

ind

ivid

ua

ls (

%)

OriginaldistributionHigher mean(grow th)

pove

rty

line

= 5

0

mea

n =

100

mea

n =

130

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140 160 180 200 220 240

Income

Sh

are

ind

ivid

ua

ls (

%)

OriginaldistributionLow erinequality

me

an

= 1

00

po

vert

y lin

e =

50

`

Growth effect Inequality effect

58

Poverty Changes Over Time (3)

Decomposition can be done as follows:

. curve Lorenz a and of period in income mean

to ingcorrespond measurepoverty the is ),( Where

Residual effect Inequality effect Growth

)],(),([)],(),([

tt

tt

rrrrr

Lt

LP

RLPLPLPLPP

1212

59

Conclusions & Recommendations

Inequality is a difficult concept to measure. For analysis, use several measures:

Lorenz curve Gini coefficient Dispersion ratios Share of expenditure of the poorest x% Theil Index

Analysis Comparisons across subgroups Comparisons over time