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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

- Extend SWF analysis to inequality
- Examine structure of inequality
- Link with the analysis of poverty

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 rankings

Inequality measurement

Relationship with welfare rankings

Inequality and decomposition

Poverty measures

Poverty 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

- 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

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

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

- 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?

“Final income” – Lorenz

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

- 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.

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

- 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?

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

- Equivalent ways of writing the Gini:
- Normalised area above Lorenz curve

- Normalised difference between income pairs.

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

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

- 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

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

- 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

- 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

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

=1

U'

=2

=5

4

3

2

=0

1

=1/2

x / m

=1

0

0

1

2

3

4

5

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

- 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

j

x

k

x

i

The 3-Person income distributionIncome Distributions

With Given Total

ray of

Janet's income

equality

Karen's income

0

Irene's income

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

- 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

- 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

- 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

- 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 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 = 2

By contrast: Gini contours

- Not additively separable

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

Overview...

Inequality rankings

Inequality measurement

Structural issues

Inequality and decomposition

Poverty measures

Poverty rankings

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

(3)

(6)

(5)

(2)

(1)

(i)

(ii)

(iii)

(iv)

- The population

- Attribute 1

- Attribute 2

- One subgroup

population

share

pj

income

share

sj

Ij

subgroup

inequality

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
- Each type induces a class of inequality measures
- The “stronger” the decomposition requirement…
- …the “narrower” the class of inequality measures

1:Inequality accounting

This is the most restrictive form of decomposition:

accounting equation

weight function

adding-up property

2:Additive Decomposability

As type 1, but no adding-up constraint:

3:General Consistency

The weakest version:

population shares

increasing in each subgroup’s inequality

income shares

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 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?

- 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

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...

- 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

- 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

- 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?

Overview...

Inequality rankings

Inequality measurement

…Identification and measurement

Inequality and decomposition

Poverty measures

Poverty rankings

Poverty analysis – overview

- Basic ideas
- Income – similar to inequality problem?
- Consumption, expenditure or income?
- Time period
- Risk

- Income receiver – as before
- Relation to decomposition

- Income – similar to inequality problem?
- Development of specific measures
- Relation to inequality
- What axiomatisation?

- Use of ranking techniques
- Relation to welfare rankings

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

- Depends on definition of poverty line
- Exogeneity of partition?
- Asymmetric treatment of information

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

- 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

$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

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

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

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)

- 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?

Overview...

Inequality rankings

Inequality measurement

Another look at ranking issues

Inequality and decomposition

Poverty measures

Poverty rankings

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)

- 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)

- 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

- 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

- 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

- 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

- 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

- 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)

- Gives two types of FGT measures
- “relative” version
- “absolute” version

- Additivity follows from the independence axiom

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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