Frank Cowell: UB Public Economics
June 2005
Inequality and Poverty
Public Economics: University of Barcelona
Frank Cowell
http://darp.lse.ac.uk/ub
Frank Cowell:
Issues to be addressed
UB Public Economics

Builds on lecture 2


Extension of ranking criteria





“Distributional Equity, Social Welfare”
Parade diagrams
Generalised Lorenz curve
Extend SWF analysis to inequality
Examine structure of inequality
Link with the analysis of poverty
Frank Cowell:
Major Themes

Contrast three main approaches to the subject
UB Public Economics




Structure of the population



intuitive
via SWF
via analysis of structure
Composition of inequality and poverty
Implications for measures
The use of axiomatisation


Capture what is “reasonable”?
Find a common set of axioms for related problems
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
Relationship with
welfare rankings
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
Frank Cowell:
Inequality rankings
UB Public Economics




Begin by using welfare analysis of previous
lecture
Seek inequality ranking
We take as a basis the second-order distributional
ranking
 …but introduce a small modification
The 2nd-order dominance concept was originally
expressed in a more restrictive form.
Frank Cowell:
Yet another important relationship
The
share of the proportion q of distribution F is given
UB Public Economics
by L(F;q) := C(F;q) / m(F)
Yields Lorenz dominance, or the “shares” ranking
G Lorenz-dominates F means:
 for every q, L(G;q)  L(F;q),
 for some q, L(G;q) > L(F;q)

The Atkinson (1970) result:
For given m, G Lorenz-dominates F

W(G) > W(F) for all WW2
Frank Cowell:
The Lorenz diagram
1
UB Public Economics
0.8
L(.; q)
0.6
L(G;.)
Lorenz
curve for F
0.4
L(F;.)
0.2
0
0
0.2
0.4
0.6
proportion of population
q
0.8
1
practical
example, UK
Frank Cowell:
Official concepts of income: UK
original income
UB Public Economics
+ cash benefits
What
distributional
ranking would
we expect to
apply to these 5
concepts?
gross income
- direct taxes
disposable income
- indirect taxes
post-tax income
+ non-cash benefits
final income
1.0
UB Public Economics
Original Income
Gross Income
Disposable Income
After Tax Income
Final Income
(Equality Line)
0.9
0.8
0.7
Proportion of Income
Frank Cowell:
Impact of Taxes and Benefits.
UK 2000/1. Lorenz Curve
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Proportion of population
0.8
0.9
1.0
Frank Cowell:
Assessment of example
UB Public Economics


We might have guessed the outcome…
In most countries:





Income tax progressive
So are public expenditures
But indirect tax is regressive
So Lorenz-dominance is not surprising.
But what happens if we look at the situation over time?
1.0
0.9
UB Public Economics
1993
2000-1
(Equality Line)
0.8
0.7
Proportion of Income
Frank Cowell:
“Final income” – Lorenz
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Proportion of population
0.8
0.9
1.0
Frank Cowell:
“Original income” – Lorenz
1.01.0
0.9
0.9
0.8
0.80.7
Proportion of Income
UB Public Economics
1993
2000-1
(Equality Line)
0.6
0.7
0.5
0.6
0.4
0.50.3
Lorenz
curves
intersect
Is 1993
more equal?
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Proportion of population
0.8
0.9
1.0
Or 2000-1?
Frank Cowell:
Inequality ranking: Summary

UB Public Economics



Second-order (GL)-dominance is equivalent to
ranking by cumulations.
 From the welfare lecture
Lorenz dominance equivalent to ranking by shares.
 Special case of GL-dominance normalised by
means.
Where Lorenz-curves intersect unambiguous
inequality orderings are not possible.
This makes inequality measures especially interesting.
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
Three ways of
approaching an
index
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
•Intuition
•Social welfare
•Distance
Frank Cowell:
An intuitive approach
UB Public Economics



Lorenz comparisons (second-order dominance) may be
indecisive
But we may want to “force a solution”
The problem is essentially one of aggregation of
information



Why worry about aggregation?
It may make sense to use a very simple approach
Go for something that you can “see”
 Go back to the Lorenz diagram
Frank Cowell:
The best-known inequality
measure?
1
UB Public Economics
0.8
0.6
Gini
Coefficient
0.5
0.4
0.2
0
0
0.2
0.4
0.6
proportion of population
0.8
1
Frank Cowell:
The Gini coefficient
UB Public Economics
1.
Equivalent ways of writing the Gini:
Normalised area above Lorenz curve
2.
Normalised difference between income pairs.

Frank Cowell:
Intuitive approach: difficulties
UB Public Economics

Essentially arbitrary




What is the relationship with social welfare?
The Gini index also has some “structural” problems


Does not mean that Gini is a bad index
But what is the basis for it?
We will see this in the next section
What is the relationship with social welfare?

Examine the welfare-inequality relationship directly
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
Three ways of
approaching an
index
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
•Intuition
•Social welfare
•Distance
Frank Cowell:
SWF and inequality
UB Public Economics

Issues to be addressed:





the derivation of an index
the nature of inequality aversion
the structure of the SWF
Begin with the SWF W
Examine contours in Irene-Janet space
Frank Cowell:
Equally-Distributed Equivalent
Income
 The Irene &Janet diagram
 A given distribution
 Distributions with same mean
xj
 Contours of the SWF
UB Public Economics
 Construct an equal distribution
E such that W(E) = W(F)
 EDE income
Social waste from inequality
•E
O
x(F)
m(F)
 Curvature of
contour indicates
society’s willingness
to tolerate
“efficiency loss” in
pursuit of greater
equality
•F
xi
Frank Cowell:
Welfare-based inequality

UB Public Economics
From the concept of social waste Atkinson (1970)
suggested an inequality measure:
Ede income
x(F)
I(F) = 1 – ——
m(F)

Mean income
Atkinson assumed an additive social welfare
function that satisfied the other basic axioms.
W(F) =  u(x) dF(x)

Introduced an extra assumption: Iso-elastic
welfare.
x 1-e – 1
u(x) = ————, e
1–e
Frank Cowell:
The Atkinson Index
UB Public Economics

Given scale-invariance, additive separability of welfare
Inequality takes the form:

Given the Harsanyi argument…




index of inequality aversion e based on risk aversion.
More generally see it as a stament of social values
Examine the effect of different values of e


relationship between u(x) and x
relationship between u′(x) and x
Frank Cowell:
Social utility and relative income
U
UB Public Economics
e =
4
3
e =1/2
2
e =1
1
e =2
e =5
0
1
-1
-2
2
3
4
5
x/m
Frank Cowell:
Relationship between welfare
weight
and
income
e
=1
U'
UB Public Economics
e =2
e =5
4
3
2
e =0
1
e =1/2
e =1
0
0
1
2
3
4
5
x/m
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
Three ways of
approaching an
index
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
•Intuition
•Social welfare
•Distance
Frank Cowell:
A further look at inequality
UB Public Economics


The Atkinson SWF route provides a coherent approach
to inequality.
But do we need to approach via social welfare




An indirect approach
Maybe introduces unnecessary assumptions,
Alternative route: “distance” and inequality
Consider a generalisation of the Irene-Janet diagram
xj
UB Public Economics
Janet's income
Frank Cowell:
The 3-Person income distribution
Income Distributions
With Given Total
0
xi
Frank Cowell:
Inequality contours
 Set of distributions for
given total
 Set of distributions for a higher
(given) total
 Perfect equality
xj
UB Public Economics
 Inequality contours for original
level
Inequality contours for higher
level
0
xi
Frank Cowell:
A distance interpretation
UB Public Economics





Can see inequality as a deviation from the norm
The norm in this case is perfect equality
Two key questions…
…what distance concept to use?
How are inequality contours on one level “hooked up”
to those on another?
Frank Cowell:
Another class of indices
UB Public Economics

Consider the Generalised Entropy class of inequality
measures:

The parameter a is an indicator sensitivity of each
member of the class.



a large and positive gives a “top -sensitive” measure
a negative gives a “bottom-sensitive” measure
Related to the Atkinson class
Frank Cowell:
Inequality and a distance concept
UB Public Economics

The Generalised Entropy class can also be written:

Which can be written in terms of income shares s

Using the distance criterion s1−a/ [1−a] …
Can be interpreted as weighted distance of each income
shares from an equal share

Frank Cowell:
The Generalised Entropy Class
UB Public Economics




GE class is rich
Includes two indices from Henri Theil:

a= 1:

a= 0:
 [ x / m(F)] log (x / m(F)) dF(x)
–  log (x / m(F)) dF(x)
For a < 1 it is ordinally equivalent to Atkinson class
 a= 1 – e.
For a = 2 it is ordinally equivalent to (normalised)
variance.
Frank Cowell:
Inequality contours
UB Public Economics




Each family of contours related to a different concept
of distance
Some are very obvious…
…others a bit more subtle
Start with an obvious one
 the Euclidian case
Frank Cowell:
GE contours: a=2
UB Public Economics
Frank Cowell:
GE contours: a<2
UB Public Economics
a=.25
a=
a=−.25
a=−1
Frank Cowell:
GE contours: a limiting case
UB Public Economics
a=−∞

Total priority to the poorest
Frank Cowell:
GE contours: another limiting case
UB Public Economics
a=+∞

Total priority to the richest
Frank Cowell:
By contrast: Gini contours
UB Public Economics

Not additively separable
Frank Cowell:
Distance: a generalisation
UB Public Economics

The responsibility approach gives a reference income
distribution



Redefine inequality measurement



Exact version depends on balance of compensation rules
And on income function.
not based on perfect equality as a norm
use the norm income distribution from the responsibility
approach
Devooght (2004) bases this on Cowell (1985)



Cowell approach based on Theil’s conditional entropy
Instead of looking at distance going from perfect equality to
actual distribution...
Start from the reference distribution
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
Structural issues
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
Frank Cowell:
Why decomposition?
UB Public Economics

Resolve questions in decomposition and population
heterogeneity:




Incomplete information
International comparisons
Inequality accounting
Gives us a handle on axiomatising inequality measures


Decomposability imposes structure.
Like separability in demand analysis
first, some
terminology
Frank Cowell:
• The population
• Attribute 1
• Attribute 2
• One subgroup
A partition
population
share
UB Public Economics
pj
income
share
(3)
(1)
(i)
sj
(ii)
Ij
subgroup
inequality
(iii)
(iv)
(2)
(4)
(5)
(6)
Frank Cowell:
What type of decomposition?
UB Public Economics


Distinguish three types of decomposition by subgroup
In increasing order of generality these are:




Inequality accounting
Additive decomposability
General consistency
Which type is a matter of judgment




More on this below
Each type induces a class of inequality measures
The “stronger” the decomposition requirement…
…the “narrower” the class of inequality measures
Frank Cowell:
1:Inequality accounting
UB Public Economics
This is the most restrictive form
accounting equation
of decomposition:
weight function
adding-up
property
Frank Cowell:
2:Additive Decomposability
UB Public Economics
As type 1, but no adding-up
constraint:
Frank Cowell:
3:General Consistency
UB Public Economics
The weakest version:
increasing in each
subgroup’s inequality
population shares
income
shares
Frank Cowell:
A class of decomposable indices
UB Public Economics





Given scale-invariance and additive decomposability,
Inequality takes the Generalised Entropy form:
Just as we had earlier in the lecture.
Now we have a formal argument for this family.
The weight wj on inequality in group j is wj = pjasj1−a
Frank Cowell:
What type of decomposition?


UB Public Economics



Assume scale independence…
Inequality accounting:
 Theil indices only (a=,1)
 Here wj = pj or wj = sj
Additive decomposability:
 Generalised Entropy Indices
General consistency:
 moments,
 Atkinson, ...
But is there something missing here?
 We pursue this later
Frank Cowell:
What type of partition?
UB Public Economics

General






Non-overlapping in incomes




The approach considered so far
Any characteristic used as basis of partition
Age, gender, region, income
Induces specific class of inequality measures
... but excludes one very important measure
A weaker version
Partition just on the basis of income
Allows one to include the "missing" inequality measure
Distinction between them is crucial for one special
inequality measure
Frank Cowell:
The Gini coefficient
1
0.8
UB Public Economics


Different (equivalent)
ways of writing the Gini:
Normalised area under
the Lorenz curve
Gini Coefficient
0.6
0.4
0.2
proportion of population
0

Normalised pairwise differences

A ranking-weighted average

0.2
0.4
0.6
But ranking depends on reference distribution
0.8
1
0
Frank Cowell:
Partitioning by income...
 Non-overlapping income groups
 Overlapping income groups
 Consider a transfer:Case 1
UB Public Economics
 Consider a transfer:Case 2
N1
0
x
x*
x
N2
x**
x'
N1
x
x'
 Case 1: effect on Gini is same in subgroup and population
 Case 2: effect on Gini differs in subgroup and population
Frank Cowell:
Non-overlapping decomposition
UB Public Economics



Can be particularly valuable in empirical applications
Useful for rich/middle/poor breakdowns
Especially where data problems in tails




Misrecorded data
Incomplete data
Volatile data components
Example: Piketty-Saez on US (QJE 2003)


Look at behaviour of Capital gains in top income share
Should this affect conclusions about trend in inequality?
UB Public Economics
1997
1993
1989
1985
1981
1977
1973
1969
1965
1961
1957
1953
1949
1945
1941
1937
1933
1929
1925
1921
1917
1913
Frank Cowell:
Top income shares in US
50
45
40
P90–100
35
30
25
P99–100
CG excl
20
15
10
P99–100
CG Incl
5
0
Frank Cowell:
Choosing an inequality measure

UB Public Economics
Do you want an index that accords with
intuition?


Is decomposability essential?


If so, what’s the basis for the intuition?
If so, what type of decomposability?
Do you need a welfare interpretation?

If so, what welfare principles to apply?
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
…Identification
and measurement
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
Frank Cowell:
Poverty analysis – overview

Basic ideas
UB Public Economics



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:
Poverty measurement
UB Public Economics


How to break down the basic issues.
Sen (1979): Two main types of issues



Jenkins and Lambert (1997): “3Is”




Identification problem
Aggregation problem
Identification
Intensity
Inequality
population
Present approach:



Fundamental partition
Individual identification
Aggregation of information
non-poor
poor
Frank Cowell:
Poverty and partition
UB Public Economics
Depends on definition of poverty line
 Exogeneity of partition?
 Asymmetric treatment of information

Frank Cowell:
Counting the poor
UB Public Economics


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
UB Public Economics
poverty evaluation
Frank Cowell:
The poverty line and poverty gaps
gi
0
gj
xi
xj
x*
x
income
Frank Cowell:
Poverty evaluation
 the “head-count”
poverty
evaluation
 Income equalisation amongst
the poor
Poor
Non-Poor
x=0
UB Public Economics
 the “poverty deficit”
 sensitivity to inequality amongst
the poor
B
A
gj
0
g
gi
poverty
gap
Frank Cowell:
Brazil 1985: How Much Poverty?
 A highly skewed distribution
 A “conservative” x*
UB Public Economics
 A “generous” x*
 An “intermediate” x*
 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:
The distribution of poverty gaps
UB Public Economics
$0
$20
$40
$60
gaps
Frank Cowell:
ASP
UB Public Economics



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, x*)
P =  p(x, x*) dF(x)
Analogy with decomposable inequality measures
Frank Cowell:
A class of poverty indices
UB Public Economics




ASP leads to several classes of measures
Make poverty evaluation depends on poverty gap.
Normalise by poverty line
Foster-Greer-Thorbecke class
Frank Cowell:
Poverty evaluation functions
UB Public Economics
1
p(x,x*)
0.8
0.6
a=0
a=1
0.4
a=1.5
a=2
0.2
a=2.5
x*-x
0
-0.2
0
0.2
0.4
0.6
0.8
1
Frank Cowell:
Empirical robustness
UB Public Economics


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:
Poverty rates by region 1981
UB Public Economics
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:
Poverty rates by region 2001
UB Public Economics
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:
Poverty: East Asia
UB Public Economics
Headcount at $1.08 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
20
10
0
1983
UB Public Economics
Headcount at $2.15 per day
1981
Frank Cowell:
Poverty: South Asia
Headcount at $1.08 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
0
1983
UB Public Economics
Headcount at $2.15 per day
1981
Frank Cowell:
Poverty: Latin America, Caribbean
Headcount at $1.08 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
0
1983
UB Public Economics
Headcount at $2.15 per day
1981
Frank Cowell:
Poverty: Middle East and N.Africa
Headcount at $1.08 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
0
1983
UB Public Economics
Headcount at $2.15 per day
1981
Frank Cowell:
Poverty: Sub-Saharan Africa
Headcount at $1.08 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
0
1983
UB Public Economics
Headcount at $2.15 per day
1981
Frank Cowell:
Poverty: Eastern Europe and
Central Asia
Frank Cowell:
Empirical robustness (2)

UB Public Economics





Does it matter which poverty criterion you use?
An example from Spain
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:
Poverty in Spaion 1993—2000
UB Public Economics
Frank Cowell:
Overview...
Inequality and
Poverty
Inequality
rankings
UB Public Economics
Another look at
ranking issues
Inequality
measurement
Inequality and
decomposition
Poverty
measures
Poverty
rankings
Frank Cowell:
An extension of poverty analysis
UB Public Economics




Finally consider some 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:
Poverty rankings (1)
UB Public Economics



Atkinson (1987) connects poverty and welfare.
Based results on the portfolio literature concerning
“below-target returns”
Theorem




Given a bounded range of poverty lines (x*min, x*max)
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 x* ≤
x*max.
Equivalent to requiring that the second-order
dominance condition hold for all x*.
Frank Cowell:
Poverty rankings (2)
UB Public Economics



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:
Poverty concepts

Given poverty line z

UB Public Economics

a reference point
Poverty gap

fundamental income difference

Foster et al (1984) poverty index again

Cumulative poverty gap
Frank Cowell:
TIP / Poverty profile

UB Public Economics


G(x,z)
Cumulative gaps versus
population proportions
Proportion of poor
TIP curve
 TIP curves have
same
interpretation as
GLC
TIP dominance
implies
unambiguously
greater poverty
i/n
0
p(x,z)/n
Frank Cowell:
Poverty: Axiomatic approach
UB Public Economics

Characterise an ordinal poverty index P(x ,z)




See Ebert and Moyes (JPET 2002)
Use some of the standard axioms we introduced for
analysing social welfare
Apply them to n+1 incomes – those of the n individuals
and the poverty line
Show that


given just these axioms…
…you are bound to get a certain type of poverty measure.
Frank Cowell:
Poverty: The key axioms
UB Public Economics




Standard ones from lecture 2
 anonymity
 independence
 monotonicity
 income increments reduce poverty
Strengthen two other axioms
 scale invariance
 translation invariance
Also need continuity
Plus a focus axiom
Frank Cowell:
A closer look at the axioms
UB Public Economics

Let D denote the set of ordered income vectors
The focus axiom is

Scale invariance now becomes

Define the number of the poor as

Independence means:

Frank Cowell:
Ebert-Moyes (2002)
UB Public Economics

Gives two types of FGT measures



“relative” version
“absolute” version
Additivity follows from the independence axiom
Frank Cowell:
Brief conclusion

UB Public Economics
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
Poverty indexes can be constructed from scratch using
standard axioms