measures-of-inequality.ppt

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
Percentage
57.9 - 71.3
49.7 - 71.3
39.2 - 49.7
13.5 - 39.2
Dept of Poverty in Lao PDR 1997/98
Lecs II
Percentage
17 to 24.7
11.9 to 17
9.5 to 11.9
2.8 to 9.5
Severity of Poverty in Lao PDR 1997/98
Lecs II
Percentage
7.1 to 12.1
4.3 to 7.1
3.3 to 4.3
0.8 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
Percentage
of
population

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 12
Income Class
%
of
Families
 Percentage of families falling in each class

10
Example: CDF of Per Capita Expenditure
0
.2
.4
.6
.8
1
0 200000 400000 600000 800000 1000000
Per 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 8.4
Second: Low-middle 756 12.3
Third: Middle 984 16.0
Fourth: Mid-upper 1,338 21.8
Upper: Fifth 2,540 41.4
All 1,227 100.0
Poorest
Richest

14
Share of Income of Poorest, Korea
Income decile 2000 2001 2002 2003
1st 2.9 2.9 3.0 2.7
2nd 4.7 4.6 4.7 4.8
3rd 5.8 5.7 5.8 6.1
4th 6.9 6.8 6.9 7.1
5th 7.9 7.8 7.9 8.1
6th 9.1 9.1 9.2 9.3
7th 10.5 10.5 10.5 10.7
8th 12.2 12.3 12.4 12.5
9th 14.7 15.0 15.1 15.0
10th 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.
 Where
N is the total population
m is the number of individuals in the lowest x %.




 N
i
i
m
i
i
x
y
y
C
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 37,324
2nd 47,289
3rd 54,397
4th 62,929
5th 74,775
6th 89,478
7th 108,633
8th 129,890
9th 172,011
10th 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
Cumulative
%
of
consumption

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
consumption
Curve of total inequality

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.
0
10
20
30
40
50
60
70
80
90
100
0 50 100
C
D

24
Comparing Lorenz Curves
A
B
C
D
100
0
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
Area below Lorenz
Curve (B)
If B=0 then G=1
(complete inequality).

31
 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
consumption
Curve of total inequality
A
B

32
Lorenz Curve and Gini Coefficient
e

33
Gini Coefficient: A Formula
 Here’s one. (There are other formulations.)
 
i i
N
i
i=1
Cov y ,f
G = 2
1
y
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.
2
1
1 1
( ) 1
N
i
i
y
GE
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
i
i
y
y
y
y
N
GE
1
)
ln(
1
)
1
(



N
i i
y
y
N
GE
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
1
ln
N
i i
i
y y
T
N y y

 
   
 

i
where 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
1
1
1























 
N
i
i
y
y
N
A

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 0.4466
1988 40 0.4446
1991 40 0.4680
1994 36 0.4507
1997 32 0.4872
2000 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.

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
2
1
1
k
j
j
j
k
j
j
j
j
B
W
y
y
f
GE
f
v
I
I
I







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(
)
(
)
(
)
(
j
j
j



















   
   

k
j
k
j
j
k
j
i
j
j
j
j
j
k
j
j
j y
f
v
f
f
GE
GE
f
GE
1 1 1
1
0
0
0

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
Share
individuals
(%)
Original
distribution
Higher mean
(grow th)
poverty
line
=
50
mean
=
100
mean
=
130
0
2
4
6
8
10
12
14
0 20 40 60 80 100 120 140 160 180 200 220 240
Income
Share
individuals
(%)
Original
distribution
Low er
inequality
mean
=
100
poverty
line
=
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
ing
correspond
measure
poverty
the
is
)
,
(
Where
Residual
effect
Inequality
effect
Growth
)]
,
(
)
,
(
[
)]
,
(
)
,
(
[
t
t
t
t
r
r
r
r
r
L
t
L
P
R
L
P
L
P
L
P
L
P
P





 




 1
2
1
2

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

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