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Inequality and Poverty. Public Economics: University of Barcelona Frank Cowell http://darp.lse.ac.uk/ub. June 2005. Issues to be addressed. Builds on lecture 2 “Distributional Equity, Social Welfare” Extension of ranking criteria Parade diagrams Generalised Lorenz curve

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inequality and poverty

Inequality and Poverty

Public Economics: University of Barcelona

Frank Cowell

http://darp.lse.ac.uk/ub

June 2005

issues to be addressed
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
major themes
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
overview

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

Relationship with welfare rankings

Inequality and decomposition

Poverty measures

Poverty rankings

inequality rankings
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.
yet another important relationship
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 Fmeans:

  • for every q, L(G;q) ³L(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

the lorenz diagram
The Lorenz diagram

1

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

0

0.2

0.4

0.6

0.8

1

q

proportion of population

official concepts of income uk
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?

assessment of example
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?
original income lorenz

1.0

0.9

0.8

0.7

0.6

0.5

0.0

0.1

0.2

0.3

0.4

0.5

“Original income” – Lorenz
  • Lorenz curves intersect
  • Is 1993 more equal?
  • Or 2000-1?
inequality ranking summary
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.
overview1

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

  • Intuition
  • Social welfare
  • Distance

Three ways of approaching an index

Inequality and decomposition

Poverty measures

Poverty rankings

an intuitive approach
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
the best known inequality measure
The best-known inequality measure?

1

0.8

proportion of income

0.6

Gini Coefficient

0.5

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

proportion of population

the gini coefficient
The Gini coefficient
  • Equivalent ways of writing the Gini:
  • Normalised area above Lorenz curve
  • Normalised difference between income pairs.
intuitive approach difficulties
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
  • What is the relationship with social welfare?
    • Examine the welfare-inequality relationship directly
overview2

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

  • Intuition
  • Social welfare
  • Distance

Three ways of approaching an index

Inequality and decomposition

Poverty measures

Poverty rankings

swf and inequality
SWF and inequality
  • 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
equally distributed equivalent income

xj

xi

Equally-Distributed Equivalent Income
  • The Irene &Janet diagram
  • A given distribution
  • Distributions with same mean
  • 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

m(F)

x(F)

welfare based inequality
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) = òu(x) dF(x)

  • Introduced an extra assumption: Iso-elastic welfare.

x1 - e– 1

u(x) = ————, e ³ 0

1 – e

the atkinson index
The Atkinson Index
  • Given scale-invariance, additive separability of welfare
  • Inequality takes the form:
  • Given the Harsanyi argument…
    • index of inequality aversion ebased 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
social utility and relative income
Social utility and relative income

U

 = 0

4

3

 = 1/2

2

 = 1

1

 = 2

 = 5

0

1

2

3

4

5

x / m

-1

-2

-3

relationship between welfare weight and income
Relationship between welfare weight and income

=1

U\'

=2

=5

4

3

2

=0

1

=1/2

x / m

=1

0

0

1

2

3

4

5

overview3

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

  • Intuition
  • Social welfare
  • Distance

Three ways of approaching an index

Inequality and decomposition

Poverty measures

Poverty rankings

a further look at inequality
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
the 3 person income distribution

x

j

x

k

x

i

The 3-Person income distribution

Income Distributions

With Given Total

ray of

Janet\'s income

equality

Karen\'s income

0

Irene\'s income

inequality contours

x

j

x

k

x

i

Inequality contours
  • 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

0

a distance interpretation
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?
another class of indices
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
inequality and a distance concept
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
the generalised entropy class
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.
inequality contours1
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
ge contours a 21
GE contours: a < 2

a = 0.25

a = 0

a = −0.25

a = −1

ge contours a limiting case
GE contours: a limiting case

a = −∞

  • Total priority to the poorest
ge contours another limiting case
GE contours: another limiting case

a = +∞

  • Total priority to the richest
by contrast gini contours
By contrast: Gini contours
  • Not additively separable
distance a generalisation
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
overview4

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

Structural issues

Inequality and decomposition

Poverty measures

Poverty rankings

why decomposition
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

a partition

(4)

(3)

(6)

(5)

(2)

(1)

(i)

(ii)

(iii)

(iv)

  • The population
  • Attribute 1
A partition
  • Attribute 2
  • One subgroup

population

share

pj

income

share

sj

Ij

subgroup

inequality

what type of decomposition
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
1 inequality accounting
1:Inequality accounting

This is the most restrictive form of decomposition:

accounting equation

weight function

adding-up property

2 additive decomposability
2:Additive Decomposability

As type 1, but no adding-up constraint:

3 general consistency
3:General Consistency

The weakest version:

population shares

increasing in each subgroup’s inequality

income shares

a class of decomposable indices
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
what type of decomposition1
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
what type of partition
What type of partition?
  • General
    • 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
the gini coefficient1

1

0.8

0.6

0.4

proportion of income

0.2

proportion of population

0

0

0.2

0.4

0.6

0.8

1

The Gini coefficient
  • Different (equivalent) ways of writing the Gini:
  • Normalised area under the Lorenz curve

Gini Coefficient

  • Normalised pairwise differences
  • A ranking-weighted average
  • But ranking depends on reference distribution
partitioning by income
Partitioning by income...
  • Non-overlapping income groups
  • Overlapping income groups
  • Consider a transfer:Case 1
  • Consider a transfer:Case 2

N1

N1

N2

x**

x*

x

x

x\'

x

x\'

0

  • Case 1: effect on Gini is same in subgroup and population
  • Case 2: effect on Gini differs in subgroup and population
non overlapping decomposition
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?
choosing an inequality measure
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?
overview5

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

…Identification and measurement

Inequality and decomposition

Poverty measures

Poverty rankings

poverty analysis overview
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
poverty measurement

population

non-poor

poor

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
poverty and partition
Poverty and partition
  • Depends on definition of poverty line
  • Exogeneity of partition?
  • Asymmetric treatment of information
counting the poor
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
the poverty line and poverty gaps
The poverty line and poverty gaps

poverty evaluation

gi

gj

x*

0

x

xi

xj

income

poverty evaluation
Poverty evaluation
  • the “head-count”
  • the “poverty deficit”
  • sensitivity to inequality amongst the poor
  • Income equalisation amongst the poor

poverty evaluation

Poor

Non-Poor

x = 0

B

A

g

gj

gi

poverty gap

0

brazil 1985 how much poverty

$0

$20

$40

$60

$80

$100

$120

$140

$160

$180

$200

$220

$240

$260

$280

$300

Brazil 1985: How Much Poverty?
  • A highly skewed distribution
  • A “conservative” x*
  • A “generous” x*
  • An “intermediate” x*
  • The censored income distribution

Rural Belo Horizonte

poverty line

compromise

poverty line

Brasilia

poverty line

slide65
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
a class of poverty indices
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
empirical robustness
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
empirical robustness 2
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?
overview6

Inequality and Poverty

Overview...

Inequality rankings

Inequality measurement

Another look at ranking issues

Inequality and decomposition

Poverty measures

Poverty rankings

an extension of poverty analysis
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?
poverty rankings 1
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*.
poverty rankings 2
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.)
poverty concepts
Poverty concepts
  • Given poverty line z
    • a reference point
  • Poverty gap
    • fundamental income difference
  • Foster et al (1984) poverty index again
  • Cumulative poverty gap
tip poverty profile
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

poverty axiomatic approach
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.
poverty the key axioms
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
a closer look at the axioms
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:
ebert moyes 2002
Ebert-Moyes (2002)
  • Gives two types of FGT measures
    • “relative” version
    • “absolute” version
  • Additivity follows from the independence axiom
brief conclusion
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
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