Frank Cowell: Oviedo – Inequality & Poverty
March 2007
Poverty Measurement
Inequality, Poverty and Income Distribution
University of Oviedo
Frank Cowell
http://darp.lse.ac.uk/oviedo2007
Frank Cowell: Oviedo – Inequality & Poverty
Issues to be addressed

Builds on Lectures 3 and 4


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Extension of ranking criteria


Generalised Lorenz curve again
Examine structure of poverty indices


“Income Distribution and Welfare”
“Inequality measurement”
Link with inequality analysis
Axiomatics of poverty
Frank Cowell: Oviedo – Inequality & Poverty
Overview...
Poverty
measurement
Poverty
concepts
…Identification
and representation
Poverty
measures
Empirical
robustness
Poverty
rankings
Conclusion
Frank Cowell: Oviedo – Inequality & Poverty
Poverty analysis – overview

Basic ideas

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Income – similar to inequality problem?

Consumption, expenditure or income?

Time period

Risk

Income receiver – as before

Relation to decomposition
Development of specific measures

Relation to inequality

What axiomatisation?
Use of ranking techniques

Relation to welfare rankings
Frank Cowell: Oviedo – Inequality & Poverty
Poverty measurement


How to break down the basic issues.
Sen (1979): Two main types of issues

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Jenkins and Lambert (1997): “3Is”

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Identification problem
Aggregation problem
Incidence
Intensity
Inequality
Present approach:



population
Fundamental partition
Individual identification
Aggregation of information
non-poor
poor
Frank Cowell: Oviedo – Inequality & Poverty
Poverty and partition

A link between this subject and inequality
decomposition.

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Asymmetric treatment of information
Exogeneity of partition?


Partitioning of population is crucial
Depends on definition of poverty line
Does it depend on the distribution of income?
Uniqueness of partition?

May need to deal with ambiguities in definition of
poverty line
Frank Cowell: Oviedo – Inequality & Poverty
Counting the poor


Use the concept of individual poverty evaluation
Simplest version is (0,1)



(non-poor, poor)
headcount
Perhaps make it depend on income

poverty deficit

Or on the whole distribution?

Convenient to work with poverty gaps
poverty evaluation
Frank Cowell: Oviedo – Inequality & Poverty
The poverty line and poverty gaps
gi
0
gj
xi
xj
z
x
income
 the “head-count”
 the “poverty deficit”
 sensitivity to inequality
amongst the poor
 Income equalisation
amongst the poor
poverty
evaluation
Poor
Non-Poor
x=0
Frank Cowell: Oviedo – Inequality & Poverty
Poverty evaluation
B
A
gj
0
g
gi
poverty
gap
Frank Cowell: Oviedo – Inequality & Poverty
Brazil 1985: How Much Poverty?
 A highly skewed distribution
 A “conservative” z
 A “generous” z
 An “intermediate” z
 The censored income
distribution
Rural Belo Horizonte
poverty line
compromise
poverty line Brasilia
poverty line
$0
$20
$40
$60
$80 $100 $120 $140 $160 $180 $200 $220 $240 $260 $280 $300
Frank Cowell: Oviedo – Inequality & Poverty
The distribution of poverty gaps
$0
$20
$40
$60
gaps
Frank Cowell: Oviedo – Inequality & Poverty
Overview...
Poverty
measurement
Poverty
concepts
Aggregation
information about
poverty
Poverty
measures
Empirical
robustness
Poverty
rankings
Conclusion
Frank Cowell: Oviedo – Inequality & Poverty
ASP



Additively Separable Poverty measures
ASP approach simplifies poverty evaluation
Depends on own income and the poverty line.



Assumes decomposability amongst the poor
Overall poverty is an additively separable function


p(x, z)
P =  p(x, z) dF(x)
Analogy with decomposable inequality measures
Frank Cowell: Oviedo – Inequality & Poverty
A class of poverty indices

ASP leads to several classes of measures
Make poverty evaluation depend on poverty gap
Normalise by poverty line
Foster-Greer-Thorbecke class

Important special case a = 0







poverty evaluation is simple: {0,1}
gives poverty rate
= poverty count / n
Important special case a = 1



poverty evaluation is simple: normalised poverty gap g/z
gives poverty deficit
measures resources needed to remove poverty
Frank Cowell: Oviedo – Inequality & Poverty
Poverty evaluation functions
1
p(x,z)
0.8
0.6
a=0
a=1
0.4
a=1.5
a=2
0.2
a=2.5
z-x
0
-0.2
0
0.2
0.4
0.6
0.8
1
Frank Cowell: Oviedo – Inequality & Poverty
Other ASP measures

Other ASP indices focus directly on incomes rather than gaps
Clark et al (1981)

where b < 1 is a sensitivity parameter
Watts



Both can give rise to empirical problems Cowell. and Victoria-Feser,
(1996)
Frank Cowell: Oviedo – Inequality & Poverty
Quasi ASP measures


Consider also quasi-ASP
This allows ranks or position in the evaluation function


Sen (1976) is the primary example



p(x, z, F(x) )
Based on an axiomatic approach
incorporates, poverty count, poverty deficit, Gini amongst poor
Poverty evaluation function:
Frank Cowell: Oviedo – Inequality & Poverty
Poverty measures: assessment

ASP class is fruitful



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But which members of it are appropriate?
Questionnaire experiments again?

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

neat and elegant
interesting axiomatisation – see next lecture
Amiel-Cowell (1999)
Many of Sen (1976) axioms rejected
In particular transfer principle rejected
which also rules out FGT measures for a > 1
Leading poverty measures are still


Poverty count or ratio
Poverty deficit
Frank Cowell: Oviedo – Inequality & Poverty
Overview...
Poverty
measurement
Poverty
concepts
Definitions and
consequences
Poverty
measures
Empirical
robustness
Poverty
rankings
Conclusion
Frank Cowell: Oviedo – Inequality & Poverty
Empirical robustness


Does it matter which poverty criterion you use?
Look at two key measures from the ASP class



Use two standard poverty lines





Head-count ratio
Poverty deficit (or average poverty gap)
$1.08 per day at 1993 PPP
$2.15 per day at 1993 PPP
How do different regions of the world compare?
What’s been happening over time?
Use World-Bank analysis

Chen-Ravallion “How have the world’s poorest fared since the early
1980s?” World Bank Policy Research Working Paper Series 3341
Frank Cowell: Oviedo – Inequality & Poverty
Poverty rates by region 1981
China
East Asia
India
South Asia
Sub-Saharan Africa
All regions
Latin America and Caribbean
Middle East and North Africa
Eastern Europe and Central Asia
Headcount
$1.08
$2.15
63.80
1
88.10
57.70
2
84.80
54.40
3
89.60
51.50
4
89.10
41.60
5
73.30
40.40
6
66.70
9.70
7
26.90
5.10
8
28.90
0.70
9
4.70
3
4
1
2
5
6
8
7
9
Poverty gap
$1.08
$2.15
23.41
1
50.82
20.58
2
47.20
17.27
3
47.22
16.06
5
45.78
17.03
4
38.54
13.92
6
35.02
2.75
7
10.66
1.00
8
8.81
0.17
9
1.39
1
3
2
4
5
6
7
8
9
Frank Cowell: Oviedo – Inequality & Poverty
Poverty rates by region 2001
Sub-Saharan Africa
India
South Asia
All regions
China
East Asia
Latin America and Caribbean
Eastern Europe and Central Asia
Middle East and North Africa
Headcount
$1.08
$2.15
46.90
1
76.60
34.70
2
79.90
31.30
3
77.20
21.10
4
52.90
16.60
5
46.70
14.90
6
47.40
9.50
7
24.50
3.70
8
19.70
2.40
9
23.20
3
1
2
4
6
5
7
9
8
Poverty gap
$1.08
$2.15
20.29
1
41.42
7.08
2
34.43
6.37
3
32.35
5.96
4
21.21
3.94
5
18.44
3.35
7
17.78
3.36
6
10.20
0.79
8
5.94
0.45
9
6.14
1
2
3
4
5
6
7
9
8
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: East Asia
Headcount at $1.08 per day
Headcount at $2.15 per day
Poverty gap at $1.08 per day
Poverty gap at $2.15 per day
100
90
80
70
60
50
40
30
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
20
10
0
1981
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: South Asia
Headcount at $1.08 per day
Headcount at $2.15 per day
Poverty gap at $1.08 per day
Poverty gap at $2.15 per day
35
30
25
20
15
10
5
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
0
1981
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: Latin America, Caribbean
Headcount at $1.08 per day
Headcount at $2.15 per day
Poverty gap at $1.08 per day
Poverty gap at $2.15 per day
35
30
25
20
15
10
5
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
0
1981
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: Middle East and N.Africa
Headcount at $1.08 per day
Headcount at $2.15 per day
Poverty gap at $1.08 per day
Poverty gap at $2.15 per day
90
80
70
60
50
40
30
20
10
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
0
1981
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: Sub-Saharan Africa
Headcount at $1.08 per day
Headcount at $2.15 per day
Poverty gap at $1.08 per day
Poverty gap at $2.15 per day
25
20
15
10
5
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
0
1981
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: Eastern Europe and
Central Asia
Frank Cowell: Oviedo – Inequality & Poverty
Empirical robustness (2)
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Does it matter which poverty criterion you use?
An example from Spain
 Bárcena and Cowell (2006)
Data are from ECHP
OECD equivalence scale
Poverty line is 60% of 1993 median income
Does it matter which FGT index you use?
Frank Cowell: Oviedo – Inequality & Poverty
Poverty in Spain 1993—2000
Frank Cowell: Oviedo – Inequality & Poverty
Overview...
Poverty
measurement
Poverty
concepts
Another look at
ranking issues
Poverty
measures
Empirical
robustness
Poverty
rankings
Conclusion
Frank Cowell: Oviedo – Inequality & Poverty
Extension of poverty analysis




Now consider some further generalisations
What if we do not know the poverty line?
Can we find a counterpart to second order dominance in
welfare analysis?
What if we try to construct poverty indices from first
principles?
Frank Cowell: Oviedo – Inequality & Poverty
Poverty rankings (1)


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Atkinson (1987) connects poverty and welfare.
Based results on the portfolio literature concerning “belowtarget returns”
Theorem




Given a bounded range of poverty lines (zmin, zmax)
and poverty measures of the ASP form
a necessary and sufficient condition for poverty to be lower in
distribution F than in distribution G is that the poverty deficit
be no greater in F than in G for all z ≤ zmax.
Equivalent to requiring that the second-order dominance
condition hold for all z.
Frank Cowell: Oviedo – Inequality & Poverty
Poverty rankings (2)



Foster and Shorrocks (1988a, 1988b) have a similar
approach to orderings by P,
But concentrate on the FGT index’s particular functional
form:
Theorem: Poverty rankings are equivalent to



first-order welfare dominance for a = 0
second-degree welfare dominance for a = 1
(third-order welfare dominance for a = 2.)
Frank Cowell: Oviedo – Inequality & Poverty
Poverty concepts – more

Given poverty line z


Poverty gap


fundamental income difference
Define the number of the poor as:


a reference point
p(x, z) := #{i: xi ≤ z}
Cumulative poverty gap
Frank Cowell: Oviedo – Inequality & Poverty
TIP / Poverty profile
•Cumulative gaps versus
population proportions
•Proportion of poor
•TIP curve
G(x,z)
 TIP curves have
same
interpretation as
GLC
TIP dominance
implies
unambiguously
greater poverty
i/n
0
p(x,z)/n
Frank Cowell: Oviedo – Inequality & Poverty
Overview...
Poverty
measurement
Poverty
concepts
Building from first
principles?
Poverty
measures
Empirical
robustness
Poverty
rankings
Conclusion
Frank Cowell: Oviedo – Inequality & Poverty
Brief conclusion

Framework of distributional analysis covers a number of
related problems:





Commonality of approach can yield important insights
Ranking principles provide basis for broad judgments



Social Welfare
Inequality
Poverty
May be indecisive
specific indices could be used
Poverty trends will often be robust to choice of poverty
index
Frank Cowell: Oviedo – Inequality & Poverty
Poverty: a way forward

Introduce a formal axiomatisation of ASP class?



Use standard axioms introduced earlier



for analysing social welfare
for inequality
Show how this is related to



In particular FGT measures
See Ebert and Moyes (2002)
deprivation
inequality
See next lecture
Frank Cowell: Oviedo – Inequality & Poverty
References (1)
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Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality,
Cambridge University Press
Atkinson, A. B. (1987) “On the measurement of poverty,”
Econometrica, 55, 749-764
Bárcena, E. and Cowell, F.A. (2006) “Static and Dynamic Poverty in
Spain, 1993-2000,” Hacienda Pública Española 179
Chen, S. and Ravallion, M. (2004) “How have the world’s poorest
fared since the early 1980s?” World Bank Policy Research Working
Paper Series, 3341
Clark, S.,Hemming, R. and Ulph, D. (1981) “On indices for the
measurement of poverty, The Economic Journal, 91, 515-526
Cowell, F. A. and Victoria-Feser, M.-P. (1996) “Poverty Measurement
with Contaminated Data: A Robust Approach,” European Economic
Review, 40, 1761-1771
Ebert, U. and P. Moyes (2002) “A simple axiomatization of the FosterGreer-Thorbecke poverty orderings,” Journal of Public Economic
Theory 4, 455-473.
Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of
decomposable poverty measures,” Econometrica, 52, 761-776
Frank Cowell: Oviedo – Inequality & Poverty
References (2)
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Foster , J. E. and Shorrocks, A. F. (1988a) “Poverty orderings,”
Econometrica, 56, 173-177
Foster , J. E. and Shorrocks, A. F. (1988b) “Poverty orderings and
welfare dominance,” Social Choice and Welfare, 5,179-198
Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves,
with an analysis of UK poverty trends,” Oxford Economic Papers, 49,
317-327.
Sen, A. K. (1976) “Poverty: An ordinal approach to measurement,”
Econometrica, 44, 219-231
Sen, A. K. (1979) “Issues in the measurement of poverty,”
Scandinavian Journal of Economics, 91, 285-307
Watts, H. W. (1968) “An economic definition of poverty,” in
Moynihan, D. P. (ed) Understanding Poverty, Basic Books, New York,
Chapter, 11, 316-329
Zheng, B. (1993) “An axiomatic characterization of the Watts index,”
Economics Letters, 42, 81-86
Zheng, B. (2000) “Minimum Distribution-Sensitivity, Poverty
Aversion, and Poverty Orderings,” Journal of Economic Theory, 95,
116-137