1. Inequality and Poverty
Public Economics: University of Barcelona
Frank Cowell
June 2005

2. Issues to be addressed Builds on lecture 2
“Distributional Equity, Social Welfare”
Extension of ranking criteria
Parade diagrams
Generalised Lorenz curve
Extend SWF analysis to inequality
Examine structure of inequality
Link with the analysis of poverty

3. Major Themes Contrast three main approaches to the subject
intuitive
via SWF
via analysis of structure
Structure of the population
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

4. Overview... Relationship with welfare rankings Inequality and Poverty
Inequality rankings
Inequality measurement
Relationship with welfare rankings
Inequality and decomposition
Poverty measures
Poverty rankings

5. Inequality rankings
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.

6. Yet another important relationship
The share of the proportion q of distribution F is given 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) ³ L(F;q),
for some q, L(G;q) > L(F;q)
The Atkinson (1970) result:
For given m, G Lorenz-dominates F Û
W(G) > W(F) for all WÎW2

7. The Lorenz diagram L(.; q) q L(G;.) L(F;.) proportion of income1
0.8
L(.; q)
0.6
L(G;.)
proportion of income
Lorenz curve for F
0.4
L(F;.)
0.2
practical example, UK
0.2
0.4
0.6
0.8 1 q
proportion of population

8. Official concepts of income: UK
original income
+ cash benefits
gross income
- direct taxes
disposable income
- indirect taxes
post-tax income
+ non-cash benefits
final income
What distributional ranking would we expect to apply to these 5 concepts?

9. Impact of Taxes and Benefits. UK 2000/1. Lorenz Curve

10. Assessment of example 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?

11. “Final income” – Lorenz

12. “Original income” – Lorenz
0.5
0.6
0.7
0.8
0.9
1.0
Lorenz curves intersect
0.0
0.1
0.2
0.3
0.4
0.5
Is 1993 more equal?
Or ?

13. Inequality ranking: Summary
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.

14. Overview... Three ways of approaching an index Inequality and Poverty
Inequality rankings
Inequality measurement
Intuition
Social welfare
Distance
Three ways of approaching an index
Inequality and decomposition
Poverty measures
Poverty rankings

15. An intuitive approach
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

16. The best-known inequality measure?1
0.8
proportion of income
0.6
Gini Coefficient
0.5
0.4
0.2
0.2
0.4
0.6
0.8 1
proportion of population

17. The Gini coefficient Equivalent ways of writing the Gini:
Normalised area above Lorenz curve
Normalised difference between income pairs.

18. Intuitive approach: difficulties
Essentially arbitrary
Does not mean that Gini is a bad index
But what is the basis for it?
What is the relationship with social welfare?
The Gini index also has some “structural” problems
We will see this in the next section
Examine the welfare-inequality relationship directly

19. Overview... Three ways of approaching an index Inequality and Poverty
Inequality rankings
Inequality measurement
Intuition
Social welfare
Distance
Three ways of approaching an index
Inequality and decomposition
Poverty measures
Poverty rankings

20. SWF and inequality Issues to be addressed: Begin with the SWF W
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

21. Equally-Distributed Equivalent Income
The Irene &Janet diagram
A given distribution
Distributions with same mean xi xj
Contours of the SWF
Construct an equal distribution E such that W(E) = W(F)
EDE income
Social waste from inequality
Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality E F O
x(F)
m(F)

22. Welfare-based inequality
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) = ò u(x) dF(x)
Introduced an extra assumption: Iso-elastic welfare.
x 1 - e – 1
u(x) = ————, e ³ 0
1 – e

23. The Atkinson Index
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

24. Social utility and relative income4
 = 0 3
 = 1/2 2
 = 1 1
 = 2
 = 5 1 2 3 4 5
x / m -1 -2 -3

25. Relationship between welfare weight and income =1 U'  =2  =5 4 3 2  =0 1 
=1/2  =1
x / m 1 2 3 4 5

26. Overview... Three ways of approaching an index Inequality and Poverty
Inequality rankings
Inequality measurement
Intuition
Social welfare
Distance
Three ways of approaching an index
Inequality and decomposition
Poverty measures
Poverty rankings

27. A further look at inequality
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

28. The 3-Person income distributionx j
Income Distributions
With Given Total
ray of
Janet's income
equality k x
Karen's income
Irene's income i x

29. Inequality contours x x x j k i Set of distributions for given total
Set of distributions for a higher (given) total
Perfect equality
Inequality contours for original level
Inequality contours for higher level k x i x

30. A distance interpretation
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?

31. Another class of indices
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

32. Inequality and a distance concept
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

33. The Generalised Entropy Class
GE class is rich
Includes two indices from Henri Theil:
a = 1:  [ x / m(F)] log (x / m(F)) dF(x)
a = 0: –  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.

34. Inequality contours
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

35. GE contours: a = 2

36. GE contours: a < 2
a = 0.25
a = 0
a = −0.25
a = −1

37. GE contours: a limiting case
Total priority to the poorest

38. GE contours: another limiting case
Total priority to the richest

39. By contrast: Gini contours
Not additively separable

40. Distance: a generalisation
The responsibility approach gives a reference income distribution
Exact version depends on balance of compensation rules
And on income function.
Redefine inequality measurement
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

41. Overview... Structural issues Inequality and Poverty
Inequality rankings
Inequality measurement
Structural issues
Inequality and decomposition
Poverty measures
Poverty rankings

42. first, some terminology
Why decomposition?
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

43. A partition pj sj Ij population share (4) (3) (6) (5) (2) (1) income
The population
Attribute 1
Attribute 2
One subgroup
population
share
(1)
(2)
(3)
(4)
(5)
(6) pj
(ii)
(i)
(iii)
(iv)
income
share sj Ij
subgroup
inequality

44. What type of decomposition?
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

45. 1:Inequality accounting
This is the most restrictive form of decomposition:
accounting equation
weight function
adding-up property

46. 2:Additive Decomposability
As type 1, but no adding-up constraint:

47. 3:General Consistency The weakest version: population shares
increasing in each subgroup’s inequality
income shares

48. A class of decomposable indices
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

49. What type of decomposition?
Assume scale independence…
Inequality accounting:
Theil indices only (a = 0,1)
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

50. What type of partition? 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
Non-overlapping in incomes
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

51. The Gini coefficient Different (equivalent) ways of writing the Gini:
0.2
0.4
0.6
0.8 1
proportion of income
proportion of population
Gini Coefficient
Different (equivalent) ways of writing the Gini:
Normalised area under the Lorenz curve
Normalised pairwise differences
A ranking-weighted average
But ranking depends on reference distribution

52. Partitioning by income...
Non-overlapping income groups
Overlapping income groups
Consider a transfer:Case 1
Consider a transfer:Case 2 N1 N2 N1 x*
x** x x x x' x'
Case 1: effect on Gini is same in subgroup and population
Case 2: effect on Gini differs in subgroup and population

53. Non-overlapping decomposition
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?

54. Top income shares in US

55. Choosing an inequality measure
Do you want an index that accords with intuition?
If so, what’s the basis for the intuition?
Is decomposability essential?
If so, what type of decomposability?
Do you need a welfare interpretation?
If so, what welfare principles to apply?

56. Overview... …Identification and measurement Inequality and Poverty
Inequality rankings
Inequality measurement
…Identification and measurement
Inequality and decomposition
Poverty measures
Poverty rankings

57. Poverty analysis – overview
Basic ideas
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

58. Poverty measurement How to break down the basic issues.
Sen (1979): Two main types of issues
Identification problem
Aggregation problem
Jenkins and Lambert (1997): “3Is”
Identification
Intensity
Inequality
Present approach:
Fundamental partition
Individual identification
Aggregation of information
population
non-poor
poor

59. Poverty and partition Depends on definition of poverty line
Exogeneity of partition?
Asymmetric treatment of information

60. 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

61. The poverty line and poverty gaps
poverty evaluation gi gj x* x xi xj
income

62. Poverty evaluation g gj gi Non-Poor Poor x = 0 B A poverty evaluation
the “head-count”
the “poverty deficit”
sensitivity to inequality amongst the poor
Income equalisation amongst the poor
poverty evaluation
Non-Poor
Poor B
x = 0 A g gj gi
poverty gap

63. Brazil 1985: How Much Poverty?
A highly skewed distribution
A “conservative” x*
A “generous” x*
An “intermediate” x*
The censored income distribution $0
$20
$40
$60
$80
$100
$120
$140
$160
$180
$200
$220
$240
$260
$280
$300
Rural Belo Horizonte
poverty line
compromise
poverty line
Brasilia
poverty line

64. The distribution of poverty gaps$0
$20
$40
$60
gaps

65. ASP Additively Separable Poverty measures
ASP approach simplifies poverty evaluation
Depends on own income and the poverty line.
p(x, x*)
Assumes decomposability amongst the poor
Overall poverty is an additively separable function
P =  p(x, x*) dF(x)
Analogy with decomposable inequality measures

66. A class of poverty indices
ASP leads to several classes of measures
Make poverty evaluation depends on poverty gap.
Normalise by poverty line
Foster-Greer-Thorbecke class

67. Poverty evaluation functions
p(x,x*)
x*-x

68. Empirical robustness Does it matter which poverty criterion you use?
Look at two key measures from the ASP class
Head-count ratio
Poverty deficit (or average poverty gap)
Use two standard poverty lines
$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

69. Poverty rates by region 1981

70. Poverty rates by region 2001

71. Poverty: East Asia

72. Poverty: South Asia

73. Poverty: Latin America, Caribbean

74. Poverty: Middle East and N.Africa

75. Poverty: Sub-Saharan Africa

76. Poverty: Eastern Europe and Central Asia

77. Empirical robustness (2)
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?

78. Poverty in Spaion 1993—2000

79. Overview... Another look at ranking issues Inequality and Poverty
Inequality rankings
Inequality measurement
Another look at ranking issues
Inequality and decomposition
Poverty measures
Poverty rankings

80. An extension of poverty analysis
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?

81. Poverty rankings (1) 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*.

82. 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.)

83. Poverty concepts Given poverty line z Poverty gap
a reference point
Poverty gap
fundamental income difference
Foster et al (1984) poverty index again
Cumulative poverty gap

84. TIP / Poverty profile TIP curves have same interpretation as GLC
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
p(x,z)/n

85. Poverty: Axiomatic approach
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.

86. Poverty: The key axioms
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

87. A closer look at the axioms
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:

88. Ebert-Moyes (2002) Gives two types of FGT measures
“relative” version
“absolute” version
Additivity follows from the independence axiom

89. Brief conclusion
Framework of distributional analysis covers a number of related problems:
Social Welfare
Inequality
Poverty
Commonality of approach can yield important insights
Ranking principles provide basis for broad judgments
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