Transcript Document 7199691

Poverty Measurement
An Introduction
Paolo Verme
1
Basic Concepts
2
Measuring Welfare
Two Approaches:
Welfarist approach. This derives from orthodox economics and the theory of
revealed preferences. Individuals are the best judges of their needs and
express their needs through consumption. Measuring consumption and
consumption choices reveals individual preferences and utility.
Non Welfarist approaches. There are two approaches both developed by Sen
as a critique to orthodox economics.
– Basic needs or functionings. Individuals need to achieve a minimum
set of basic needs or functionings to be considered as non poor.
– Capabilities. Individuals need to have the capacity to reach a minimum
set of functionings irrespective of whether they actually make use of
these functionings or not (freedom of choice).
3
Welfare in Orthodox Economic Theory
Happiness = Utility = Welfare
How to measure utility? Revealed preferences.
I = C (present consumption) + S (future consumption)
There is no distinction between income, consumption or
expenditure
The objective is to maximise utility selecting the best
consumption bundle under a budget constraint.
4
Welfare in Orthodox Economic Theory
Budget Constraint
Quantity good A
100
Price Ratio = ΔA/ ΔB
ΔA
Indifference curve
ΔB
50
Quantitiy good B
5
Welfare According to Sen
Poverty is the inability to reach a Minimum Standard of Living.
6
Welfare According to Sen
What is the inability to reach…? It is the lack of capacity to obtain the minimum living
standards. However, people are different and have different needs, opportunities,
functionalities and capabilities.
Needs. Two persons may be have different needs. A person who weighs 150 kgs. has
different nutritional needs from a person that weighs 60 kgs.
Opportunities. Two persons may have the same, needs and abilities but different
opportunities because of discrimination or chance.
Functionings. Functionalities is when opportunities meet abilities. It is the capacity to
exploit opportunities. One disabled person and one able person have different
functionalities vis-à-vis the same opportunity of public transport.
Capabilities. Capabilities are the set of functionalities of which a person disposes of. The
more the better, even if some functionalities are not used.
=> How to adjust the welfare measure to individual functionalities is one of the objective
of this course (welfare adjustments).
7
Welfare According to Sen
What is a minimum standard of living? It is a concept generally defined
by countries and based on normative and positive criteria.
Value judgments about what is important for living (normative). Ex:
Including or excluding a TV set from the minimum consumption
basket.
Scientific notions about what is necessary for living (positive). Ex:
establishing a minimum amount of daily calories necessary for
survival.
=> How we define these minimum standards is one objective of
this course (Poverty Lines).
8
Welfarist or non Welfarist Approach?
In practice, when we measure poverty with HBSs, we use components of both
approaches.
We are interested in the notion of minimum standards (basic needs) and will
attempt to measure minimum standards with different methodologies.
We are also interested in functionings. We will adjust our measure of welfare to
individual needs.
We recognize that individuals have many functionings and that welfare is multidimentional. However, in this course, we limit ourselves to the study of one
dimension, income or consumption.
We measure welfare with consumption accepting de facto the notion of
revealed preferences put forward in orthodox economics.
9
The components of a poverty analysis
Who is poor (poverty measurement) => methodologies
What are the characteristics of the poor (poverty profile) => statistics
Where are the poor (poverty mapping) => spatial analysis
Why are the poor poor (causes of poverty) => econometrics
What can be done about the poor (pro-poor policies) => PRSP
=> This course focuses on poverty measurement
10
Tools for Poverty Measurement
Measurement instrument: Household Budget Survey
Welfare measure: Income, consumption, expenditure
Welfare adjustments: Prices, Household composition and
Economies of scale
Welfare threshold: Poverty lines
Welfare statistics: Poverty indexes and their decompositions
=> In the next sections, we explore these tools one by one.
11
The Household Budget Survey
A Note on Strata, Clusters and Weights
12
The Household Budget Survey
From Population to Sample:
Population
Strata
Clusters
Sample
From Sample to Population:
Population Weights
13
From Population to Sample
Population Sample (ex.
1% population)
Strata (ex. Urban)
Strata (ex. Rural)
Clusters (ex. Towns X, Y)
Clusters (ex: Villages X,
Y)
PSU (ex. 50 HH)
PSU (ex. 50 HH)
14
From Sample to Population
Population
Sample
Selec. Prob.
Weight
Population
Urban (Towns)
6000
300
0.05
20
6000
Rural (Villages)
4000
800
0.2
5
4000
Total
10000
1100
10000
A Poverty analysis which ignores population weights, is an analysis
of the sample, not the population. Even if we calculate only a mean,
we need to use weights. Stata provides simple ways to do it with
most commands (not all). In stata, population weights are described
as ‘pweight’ or ‘fweight’. Remember: A statistics from the sample is
only an estimate of the population statistics.
15
Choosing a Measure of Welfare
16
Measuring Welfare with HBSs
HBSs measure income, expenditure and sometimes savings, they do not
measure consumption.
W=Welfare
I=Income
S=Savings
E=Expenditure
C=Consumption
I=C+S
W=I?
C=E?
W=C?
W=E?
W=S?
Self-production
Self-consumption
Durable goods
Inter-temporal consumption
17
How Good is Income?
It varies according to:
– Seasons
– Life-cycle
Likely to be poorly reported because:
– Illicit activities
– Informal income (tax evasion)
– Fear
– Gross Vs. Net income
– Recall bias
18
How Good is Expenditure?
Better than income because:
Less vulnerable to seasonality
Less vulnerable to life-cycle
Closer to the utility that people effectively extract from income
Less vulnerable to measurement errors because respondents
have less reasons to lie
For the poor, by definition, most of income is consumed (little
savings, access to credits, little capital goods)
19
How Good is Expenditure?
However:
The list of expenditure items is much larger than incomes
Households do not report properly certain consumption items
such as alcohol, cigarettes, gambling, prostitution, drugs which
may consume considerable amounts of income
Some consumption items are often neglected such as small,
recurrent or irrelevant purchases.
Seasonality also affects consumption, especially the structure of
consumption
Consumption includes durable goods such as a TV set or a stock
of flour for the winter
Some purchases are very rare such as a flat and last long periods
20
of time
From Expenditure to Consumption
Expenditure and consumption are different:
We do not consume everything that we buy. Some purchased goods
are wasted before we consume them. Very few HBS measure HH
waste. Some other goods are bought and donated to other people.
Donations are often but not always measured in HBS. We buy some
goods that we may consume in subsequent periods.
We consume some of the things that we do not buy. We consume
some goods that we produce. Self-consumption is usually measured in
HBS. We consume some goods that are donated to us. Donations are
often but not always measured in HBS. We consume goods that we
bought in previous periods such as durable goods.
21
From Expenditure to Consumption
In substance, we measure consumption by measuring expenditure and
adjust this measure with self-production, donations, amortization, and
other information that may be present in the HBS. In particular, we
adjust for:
– Self-consumption (HH diary)
– Rents (Estimates)
– Durable goods (Depreciation)
– Net donations in kind (Received donations-donations made)
22
Rent Imputation
Rent is part of expenditure and contributes to people’s welfare. Rents are
registered for those who pay rent but do not appear for the owners of flats or
houses. We need to calculate a fictitious rent for the owners of properities.
There are many techniques to do this. One is with a simple econometric
procedure as follows:
A) Estimate how rents vary according to the property characteristics of tenants:
Rent of tenants=a+b*(Properties characteristics)+u
B) Predict rent for all:
Rent of all=E(a)+E(b)*(Properties characteristics)
C) Use the predicted rent for the owners as imputed rent.
23
Imputation of Durable Goods
Durable goods are purchsed occasionally but provide welfare for extended periods of
time. We need to spread the cost of th durable good over the life of the good. This is the
same process such as amortization of durable goods in companies. In principle we
should consider:
– Change of value of the durable good (depreciation)
– Opportunity cost of investment (foregone income of alternative investment)
Example:
• I bought a TV set for 250 USD last year and this year is worth 190$.
• 250 USD in a bank have a return of 10%
• The total cost of the televisor for this year has been:
250-190=60+(250*0.1)=85 USD
Alternatively, we could simply attribute to the purchased good a standard duration in
years and divide the cost of the good for its duration taking inflation into account.
The problem with such imputation is not about methodology but information. We need
to know the purchase cost, inflation, duration of the good, present value and similar
information which are not always available in the HBS we dispose of.
24
Estimates of self-consumption
Especially in rural areas, self-consumption can represent an important share of the total
HH consumption.
First, we need to have a HBS which measures self-consumption.
Second, we need to attribute a value to self-consumption via market prices. The issue is
what are the relevant market prices to consider given that food prices may be very
variable across seasons and locations.
Third self-consumption has a cost in terms of agricultural inputs. If we buy seeds or
fodder this is accounted for as expenditure and included into consumption. Including into
consumption the total market price of grains produced with the seeds bought and
consumed or estimatting the value of cattles produced with the fodder bought and
consumed would over-estimate total consumption. We add the value of agricultural inputs
to the value of the final product.
The household that produces goods and services should be considered as an enterprise
and treated economically and financially as such. This means that only the value added
which is consumed should be considered. The most sophisticated HBS have entire
sections dedicated to self-production and self-consumption.
25
Adjusting Consumption for Welfare Comparisons
26
Properties of a Welfare Measure
We have established that consumption may be the best measure of welfare we can use and we have
adjusted consumption with the imputation of rents, durable goods and self-consumption.
Is household total consumption sufficient to compare individuals, households or larger communities?
Not really.
A measure of welfare should have the following properties:
Horizontally equitable. All equal individuals should be treated equally. But individuals are not
equal, they have different needs. Ex: Food and non-food requirements are different for
adults and children. We need to ‘equalize’ the welfare measure.
Fixed over time and space. A measure of welfare should be comparable across time and space.
But prices change over time and space. Ex: Inflation over time and price differentials across
regions. We need to adjust the welfare measure to comparable prices.
=> The poverty line or the poverty measure need to be adjusted accordingly.
Adjust the poverty line or the welfare measure? It’s your choice. In this course, we adjust the welfare
measure.
27
Adjusting Consumption for Welfare Comparisons
Adjust for what?
Household size
Household composition
Household purchasing power
Adjust to what?
Economies of scale (ES)
Adult Equivalent Consumption (AEC)
Purchasing Power Parity (PPP)
Adjust how?
Equivalence Scales
Deflators
28
Adjust Consumption for HH Size and Composition
Consumption per capita. Consumption is estimated on households but
households have different sizes. We need per capita estimates.
Economies of scale. Household size has an impact on economies of scales.
The more people live under the same roof and share the same resources the
more the fixed costs are spread, the more the unit costs are small, the greater
is individual welfare.
Consumption capacity. Households are composed of different type of members
such as adults, children and pensioners with different needs, costs and
consumption capacity. A child eat less than an adult. If a child and an adult
have the same monetary consumption, the child is better off. These differences
in welfare should be taken into account.
From HH consumption to Per capita adult equivalent consumption
This is done with equivalence scales
29
Equivalence Scales
Two simple and popular equivalence scales are the Oxford and OECD
scales:
Oxford:
1 = First adult
0.7 = Other adults (adjust economies of scale)
0.5 = Children (adjust for calories needs)
OECD:
1 = First adult
0.5 = Other adults (adjust economies of scale)
0.3 = Children (adjust for calories needs)
30
Equivalence Scales
But there are also much more sophisticated scales. A mathematical
formula which may capture both adult equivalent consumption and
economies of scale is the following.
ES = (A+αC)^β
A= Number of adults
C=Number of children
α=Child adult equivalent parameter
β=Economies of scale parameter
By dividing consumption by ES, we obtain the per capita adult
equivalent consumption. With α=1 and β=1, we obtain per capita
consumption. However, the general assumption is that α<1 and β<1.
31
Equivalence Scales
Note that consumption and, by consequence, poverty measures are very
sensitive to the choice of equivalence scales.
Note also that equivalence scales are very arbitrary. In fact, there is no
scientific ground to argue that the same equivalence scale should be applied
universally as suggested by the Oxford and OECD scales for various reasons:
– A child in a poor country may cost little more than the food required. But a
child in a rich country may cost more than an adult.
– Countries have different population structures. In some countries over 50%
of the population is composed of children, in others less than 20%. A small
change in the child parameter of the equivalence scale can lead to a
substantial increase or decrease in the poverty gap observed between two
countries.
– Economies of scale may be very different across countries. In Africa,
people tend to live in extended and elastic families and fixed costs such as
heating, water and electricity are very small. In Europe, people live in small
and rather established households and fixed costs may be a substantial
part of consumption.
32
Poverty Lines
33
Poverty Lines
The monetary value of the minimum set of basic needs necessary to reach a
minimum standard of living
or, in economic terms:
The minimum consumption needed to achieve the minimum level of utility
We consider different types of poverty lines
» Absolute (APL)
» Relative (RPL)
» Subjective (SPL)
34
Absolute Poverty Lines
Different Approaches
Cost of basic needs (CBN)
– Least cost approach
– Expenditure based method
Food Energy Intake (FEI)
Direct Calories Intake Method (DCI)
All PL are composed of a food and a non food component.
However, it is the estimation of the food component which makes a
difference across methods.
35
APL-Cost of Basic Needs (CBN)
With a Cost of Basic Needs approach, the standard method is to start
with evaluating in monetary terms a minimum amount of food
necessary for having a healthy and active life that allows individuals to
fully participate in society; we call this the Minimum Food Basket (MFB).
The MFB is based on nutritional values. First, the minimum level of
energy intake is established.
Example. The FAO recommends a level of 2,100 calories/day for an
adult in working age.
There are at least two different approaches:
– Least cost approach
– Expenditure based method
36
APL-Cost of Basic Needs (CBN)
Least Cost Approach
With a least-cost approach, we select a number of food baskets that provide the same
calories intake and we then select the one that is less costly and use the value of this
basket as the poverty line.
First, we find a number of products which are traditionally part of the diet of the
population we are targeting.
Second, we compose an ideal basket of goods selected on the basis of their nutritional
values (MFB), in terms of a correct balance between carbohydrates, proteins, vitamins
and other nutritional composites to achieve the set number of calories.
Third, we evaluate in monetary terms using affordable prices. By multiplying quantities by
prices we obtain the value of the MFB. This value can be used as the Food Poverty Line
(FPL).
MFB=FPL= ∑ qi*pi
where qi= quantity of good i and pi=price of good I
The advantage of this approach is that we do not need to know detailed data on
household consumption. On the other hand, this approach will not provide a basket of
goods which will be necessarily consumed by any household.
37
APL-Cost of Basic Needs (CBN)
Expenditure Based Method
The expenditure based method looks first at consumption patterns in a
certain population. Usually, the sample of households that is used for
this evaluation is the sample of middle income or poorer households.
The food consumed by this population group is included into the basket
and the basket is weighted according to the share of different foods
consumed by the target population.
The basket is then transformed into calories and adjusted so as to
reach the minimum amount of calories required.
The method ensures that the food basket is relevant to the population
we are assessing. However, it does not guarantee the best possible
diet to reach the minimum amount of calories.
38
APL-Food Energy Intake
The Food Energy Intake method seeks in the data the consumption level at
which a person typically attains the minimum food energy intake.
First we estimate econometrically how calories intake change as consumption
changes:
Ln(x)=a+bC+u
Where X=Consumption on a food basked; C=Calories obtained from the food
basket; u=error term
Then the food poverty line can be estimated as:
Z=a*+b*R
Where Z=PL, a* and b* are the coefficients estimated from the first equation
and R is the recommended calories intake.
39
APL-Direct Calories Intake Method
With the DCI method we take first the quantities consumed by HH.
These quantities are transformed into HH calories.
HH calories are transformed into calories per capita.
Poor HH are those household with calories per capita below a minimum
threshold.
Problem: two households with the same calories intake may enjoy very
different standards of living
40
Relative Poverty Line
A relative poverty line is not related to absolute income or consumption
of the population but is established relatively to the particular
distribution of income or consumption observed in a given society.
Example. This is the approach followed by the European Union that
considers poor all people with per capita income or consumption below
50% of the median income.
The problem with a relative poverty line is that there is no fixed
benchmark. The PL moves together with the median of the distribution.
If all incomes in a distribution increase by 10%, the number of poor
below the poverty line does not change. Thus, it is more similar to a
measure of distribution than to a measure of poverty. It pinpoints those
who are worse off but does not tell us how poor these people really are.
41
Relative Poverty Line
Annual Consumption per Capita (Euro)
Data
10000
9200
13400
4500
7800
15000
9000
14000
12900
10500
6600
Sorted data
4500
6600
7800
9000
9200
10000
10500
12900
13400
14000
15000
Median = 10,000
Poverty Line = 10,000/2 = 5,000
42
Subjective Poverty Line
A subjective poverty line is established by asking people about poverty.
This can be done in several ways:
Ask people what they think is a minimum amount necessary for living.
Answers to this question vary from person to person but one could use
an average or median value to estimate a poverty line. The question
could be formulated as follows: What do you think is the minimum
income necessary to your family every month? And answers could be
structured as open ended or multiple choice answers.
Alternatively, we can ask people how they would rank themselves on a
poverty scale with several steps and estimate overall poverty by taking
the mean of the answers. One example is the following: On a scale
from 1 to 10 where one is “extremely poor” and 10 is “extremely rich”
how would you rank the income status of your household?
43
Poverty Lines. What is better?
Absolute, relative and poverty lines are not substitutes but complement each
other and provide different types of information.
Absolute poverty lines provide information on poverty as compared to a
recognized minimum threshold determined on the basis of normative and
positive criteria. Such lines can be used to compare people across space and
time. These lines are used to measure real welfare.
Relative poverty lines provide information on poverty based on the position of
individuals relative to the position of other individuals within the same
consumption distribution. Such lines cannot be used to evaluate changes in
real welfare, only changes in relative welfare. They can be seen as a measure
of distribution.
A subjective poverty line provides a picture of self-perceived poverty. This may
be very different from absolute or relative poverty but is nevertheless a useful
tool for policy. It can be seen as a measure of the feeling of individual
deprivation and can used for political purposes.
44
Poverty Indexes
45
Poverty Indexes
In this section we look at three popular measures of poverty:
» Headcount Index
» Poverty Gap
» Severity of Poverty
We will see why we need three measures and we will see that these three
measures belong to the same class of poverty measures.
We will also have a brief look at some other indexes:
»
»
»
»
Watts
Atkinson
Sen
Sen-Shorrocks-Thon
46
The Poverty Headcount Index
H
1
P 0  n

q
i1
i
or
H = q/n
yi= Individual consumption i
z = Poverty line
n = Population
q = Number of individuals below the poverty line
47
The Poverty Gap and the Severity of Poverty Index
PG
P 1  1n
SP
P 2  1n


zy i
q
z
i1
zy i
q
z
i1

2

yi= Individual consumption i
z = Poverty line
n = Population
q = Number of individuals below the poverty line
48
The Common Structure of the Three Poverty
Indexes
1
P 
y;z  n

zy i
q

z
i1


α=0 => P(0) = Headcount Index
α=1 => P(1) = Poverty Gap Index
α=2 => P(2) = Severity of Poverty Index
α is defined as the poverty aversion parameter. The larger is α the more
weight we give to the poorest people. This is a normative choice which
may reflect, for example, the weight that governments wish to attribute to
poverty in public policy.
49
A Graphical Illustration of the Three Indexes
1
P0
Poverty
Index
P1
P2
0
Consumption
z
If all people have zero consumption, all the three indexes are equal to 1. All people reach maximum poverty.
If all people consume the same amount as the poverty line z (or above), the poverty indexes are all equal to zero. There is
no poverty.
In between the two extremes:
P0 (headcount index) is constant. Each additional poor add an equal amount of poverty.
P1 (poverty gap) is linear and increasing in poverty. Each additional poor increases poverty proportionally to the level of
poverty. The poorest contribute to the index more than the less poor.
P2 (severity of poverty) is exponential and increasing in poverty. Each additional poor increases poverty more than
50
proportionally to the level of poverty. The poorest contribute to the index much more than the less poor.
Headcount Index
Person
Consump.
y
1
1
1
4
Number of Poor=1 Non- Number of
obs.
poor=0
poor
n
Poor
q
4
1
3
4
1
3
4
1
3
4
0
3
Poverty
Headcount Index
P0
H=q/n
0.75
0.75
0.75
0.75
A
1
2
3
4
Poverty line
z
3
3
3
3
B
1
2
3
4
3
3
3
3
1
2
3
4
4
4
4
4
1
1
1
0
3
3
3
3
0.75
0.75
0.75
0.75
C
1
2
3
4
3
3
3
3
2
2
2
4
4
4
4
4
1
1
1
0
3
3
3
3
0.75
0.75
0.75
0.75
51
Poverty Gap Index
Sum of the
Relative
relative
Distance
distance
distances
from the
from the
from the Poverty Gap Index
poverty line poverty line poverty line
P1
z-y
(z-y)/z
sum(z-y)/z
sum((z-y)/z)n
2
0.67
2.00
0.5
2
0.67
2.00
0.5
2
0.67
2.00
0.5
0
0.00
0.00
0
A
1
2
3
4
Poverty line
z
3
3
3
3
B
1
2
3
4
3
3
3
3
1
2
3
4
2
1
0
0
0.67
0.33
0.00
0.00
1.00
1.00
1.00
0.00
0.25
0.25
0.25
0
C
1
2
3
4
3
3
3
3
2
2
2
4
1
1
1
0
0.33
0.33
0.33
0.00
1.00
1.00
1.00
0.00
0.25
0.25
0.25
0
Consump.
y
1
1
1
4
52
Severity of Poverty Index
Consump.
y
1
1
1
4
Square of
Relative
the relative
distance
distance
from the
from the
poverty line poverty line
(z-y)/z
((z-y)/z)^2
0.67
0.44
0.67
0.44
0.67
0.44
0.00
0.00
Sum of the
squares of
the relative
distance
from the
Severity of
poverty line Poverty Index P2
sum((z-y)/z)^2 sum(((z-y)/z)^2)/n
1.33
0.33
1.33
0.33
1.33
0.33
0.00
0.00
A
1
2
3
4
Poverty line
z
3
3
3
3
B
1
2
3
4
3
3
3
3
1
2
3
4
0.67
0.33
0.00
0.00
0.44
0.11
0.00
0.00
0.56
0.56
0.56
0.00
0.14
0.14
0.14
0.00
C
1
2
3
4
3
3
3
3
2
2
2
4
0.33
0.33
0.33
0.00
0.11
0.11
0.11
0.00
0.33
0.33
0.33
0.00
0.08
0.08
0.08
0.00
53
The Three Indexes Compared (Kazakhstan)
P(0)
P(1)
P(2)
(1)
(2)
(3)
Mean
Aver. income
income of
gap of the
the poor P(0)
poor P(0)
(4)
(5)
2001
Akmola
Aktubinsk
Almaty
Atirau
West-Kazakhstan
Jambul
Karaganda
Kostanay
Kizilorda
Magnistau
South-Kazakhstan
Pavlodar
North-Kazakhstan
East-Kazakhstan
Astana city
Almaty city
Kazakhstan
0.11
0.18
0.20
0.24
0.16
0.29
0.12
0.21
0.28
0.39
0.29
0.06
0.05
0.13
0.02
0.03
0.17
0.02
0.05
0.04
0.07
0.03
0.07
0.02
0.05
0.05
0.10
0.06
0.02
0.01
0.03
0.00
0.01
0.04
0.01
0.02
0.01
0.03
0.01
0.02
0.01
0.02
0.01
0.03
0.02
0.01
0.00
0.01
0.00
0.00
0.01
30000
27200
30500
27300
31900
28900
30000
28100
31000
28200
29500
27600
30300
29300
31300
31500
29538
7890
10700
7380
10600
5940
8990
7900
9740
6890
9680
8360
10300
7620
8560
6620
6420
8349
54
Interpretation of the three Indexes
The headcount index (P0) is easily interpreted. It is the percentage of poor people in the
population.
The poverty gap (P1) index can be interpreted as the cost necessary to eliminate poverty.
That is because P1 is the sum of the consumption gap between each poor and the
poverty line.
More difficult is to interpret the severity of poverty index (P2) and, more in general, all
indexes where α>1.
Note first, that the greater is α the smaller is the size of the index. Comparing the three
indexes between each other, only provides information on the size of poverty aversion
parameter, i.e. on the normative judgment made about the weight that we wish to give to
poverty.
The three indexes cannot be compared between each other, they are only relative to
themselves and they are only useful when we observe changes over time or space. They
mean little in absolute values.
Therefore, the severity of poverty index can only be interpreted as a measure of intensity
of poverty. If the index is larger, the intensity of poverty is larger. If it the index is smaller
the intensity of poverty is smaller.
55
Example
Small farmers
Large farmers
Unskilled workers
Herders/Fishermen
Retirees/disabled
P0
82
77
63
51
50
Rank
1
2
3
4
5
P1
41
35
26
28
24
Rank
1
2
4
3
5
P2
25
19
14
16
14
Rank
1
2
5
3
4
56
Watts and Atkinson Poverty Indexes
Watts (1968)
q
W   log( z / y i ) / n
i 1
y = Consumption of individual i
z = Poverty line
n = Population
q = Number of poor
ε= Poverty aversion parameter
Atkinson (1987)
 1 n  y 1 
Ag  1     i  
 n i 1  y  
1
1 
i
57
Sen and Sen-Shorrocks-Thon Indexes
Sen (1976)
Sen-Shorrocks-Thon (SST)
PS  P0 G  P1 (1  G )
q
q
PSST  P0 P (1  G
q
1
q
z q
)
P0=Poverty Headcount
P1=Poverty gap
G^q=Gini index among the poor
Gz-q^q= Gini index calculated on the poverty gaps
58
Poverty Decompositions
59
Poverty Decomposition into Sub-groups
K
P   vk Pk
k 1
pk
vk 
p
K = Number of sub-groups
k = Sub-groups
P = Poverty index
Pk = Poverty index for group k
p = Population
pk = Population of sub-group k
60
Poverty Shares by Sub-groups
v k Pk
Sk 
P
pk
vk 
p
K
S
k 1
k
1
K = Number of sub-groups
k = Sub-groups
P = Poverty index
Pk = Poverty index for group k
p = Population
pk = Population of sub-group k
61
Poverty Risk
Pk S k
Rk 

P vk
v k Pk
Sk 
P
pk
vk 
p
K = Number of sub-groups
k = Sub-groups
P = Poverty index
Pk = Poverty index for group k
p = Population
pk = Population of sub-group k
62
Poverty Indexes, Shares and Risk: Example
(Kosovo)
Poverty Headcount
(complete pov. line)
Poverty Headcount
(food pov. line)
Index
Share
Risk
Index
Share
Risk
Male
36.6
50.3
98.8
14.6
48.9
96.1
Female
37.5
49.7
101.2
15.9
51.1
104
Age<=25
38.4
56.2
103.6
16.5
58.8
108.3
Age 26-65
34.5
37.5
93.1
13.4
35.4
87.9
Age > 65
42.2
6.2
113.9
16.2
5.8
106.1
Gender
Age
63
Inequality
64
From Units to Population
Thus, in order of size we have:
– Population units (Individuals)
– Population units (Households)
– Sample units (Individuals)
– Sample units (Households)
– Quantiles (Equal groups of sample units)
– Sample
– Population
65
Units and Quantiles
For the measurement of inequality it is essential to understand the distinction between
units of measurement and quantiles.
Units of measurement can be individuals, households or larger communities.
Quantiles are groups of observations of equal size, i.e. containing the same number of
units. If we order the surveyed sample in ascending order of one variable such as income
and then we split the sample in groups of equal size we obtain ‘quantiles’. If we order
individuals in ascending order of income and then we split the sample into ten groups we
talk of deciles. We talk of quartiles if the groups are four and quintiles if the groups are
five and so on. Evidently the maximum number of quantiles we can obtain from the
sample is equal to the number of units in that sample.
Dividing the sample into quantiles is useful because we can then calculate statistics for
each quantile and compare these statistics across quantiles. For example, we can
calculate the percentage of people with access to basic health services and compare this
percentage across income quintiles to see if the poorest groups (quintiles one and two)
have more or less access than the richest groups (quintiles four and five) to these basic
services.
66
Measures of Inequality
There is a very wide range of measures of inequality. We focus here on a few
basic measures to get acquainted with the practice of inequality measurement.
We also focus on discrete measures (with income measured in finite quantities
- we use formulas) as opposed to continuous measures (with income measured
in infinite quantities comprised in an interval - where integrals are used).
We consider the following inequality measures:
» The variance
» The coefficient of variation
» The quintile ratio
» The Lorenz curve
» The Gini coefficient
67
The Variance
The variance is the relative square of the distances of income from the mean of
income and is measured as follows:
1 n
V   ( yi  y) 2
n i 1
where:
yi=income of individual or family i
y = mean income
n=number of individual or family i
2
The variance varies from 0 (where all incomes are the same) to y (n  1)
This measure is not very useful because if, for example, we double all incomes, the
variance would quadruple while the shape of the distribution would stay the same. In other
words, the variance is affected by size while it is the distribution of income we are more
concerned with.
68
Coefficient of Variation
The coefficient of variation is the square root of the variance divided by mean income
as follows:
c
V
y
The coefficient of variation varies between 0 (where all incomes are the same) to
n  1 . This measure standardizes the variance making it relative to the mean income which
makes it a better measure of inequality because we can compare numbers of different
magnitude and with different means. However, the fact that it varies between 0 and n  1
makes it a difficult measure when we want to compare samples of different sizes.
69
Top/Bottom Quantile Ratio
One simple way to calculate an inequality index is to divide the population income in
quantiles and take the ratio of the total income of the top quantile(s) to the total income of the
bottom quantile(s). Suppose we have the following quintiles in a particular income
distribution.
Income
% of income
Cum % income
Population
% Population
Cum % Population
1st
10
3.3
3.3
220
20
20
2nd
30
10
13.3
220
20
40
3rd
50
16.7
30
220
20
60
4th
70
23.3
53.3
220
20
80
5th
140
46.7
100
220
20
100
Total
300
100
1100
100
The top/bottom quintile ratio would be 140/10=14. In other words, the top 20% of
earners in our income distribution earns 14 times the bottom 20% which is a rough but
indicative measure of inequality.
70
The Lorenz Curve
The Lorenz curve uses the concept of quantiles to construct a curve determined by
the cumulative income of a population and by its cumulative population, both measured
normalised to 1 or to 100. To see how a Lorenz curve is constructed we will use the same
data used for the quintile ratio above.
Cumulative
Income
1
0.8
0.6
A
0.4
B
0.2
0
0.2
0.4
0.6
0.8
1
Cumulative Population
71
The Gini Coefficient
The Gini coefficient is the average difference between all possible pairs of incomes in the
population expressed as a proportion of total income. It can be constructed starting from the
Lorenz curve and it represents the ratio between area A and area A+B:
G=A/(A+B)
It is evident that the larger is A (which is the area that represents the distance from the
diagonal) the smaller is B and the larger is the Gini coefficient. For simplicity, the area A+B is
normalised (put equal to) to 1 so that the Gini coefficient varies between 0 and 1. A Gini coefficient
of zero points to perfect equality. A Gini coefficient of 1 points to perfect inequality.
There many different ways of calculating the Gini coefficient. One possibility is to do it
geometrically as follows:
n
G  1   ( xi  xi 1 )( y i  y i 1 )
11
where x is the cumulated percentage population and y is the cumulated percentage income
(normalised to 1 in our example).
72
Properties of an Inequality Measure
Mean independence. This means that if all incomes were doubled, the measure would
not change.
Population size independence. If the population were to change, the measure of
inequality should not change, ceteris paribus. This is the problem we had with the
variance which changes in size if incomes change in size.
Symmetry. If you and I swap incomes, there should be no change in the measure of
inequality. This principle says that you can swap income and ranks between different
people and that this would leave the inequality measure unchanged. Inequality measures
are concerned with incomes, not people.
Pigou-Dalton Transfer sensitivity. Under this criterion, the transfer of income from richer
to poorer reduces measured inequality. If I take X income from the rich and I give it to the
poor the inequality measure should decrease.
Decomposability. This means that inequality may be broken down by population subgroups and that the inequality index may be reconstructed by summing-up the sub-group
indexes (additivity principle).
Statistical testability. One should be able to test for the significance of changes in the
index over time.
73