# Inequality and Poverty

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Inequality and Poverty
Public Economics: University of Barcelona Frank Cowell 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... Relationship with welfare rankings Inequality and Poverty
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 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

The Lorenz diagram L(.; q) q L(G;.) L(F;.) proportion of income
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 0.2 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?

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

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

“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 ?

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... 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

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 0.2 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 Examine the welfare-inequality relationship directly

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

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

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)

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

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

Social utility and relative income
4  = 0 3  = 1/2 2  = 1 1  = 2  = 5 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 =1 x / m 1 2 3 4 5

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

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 Income Distributions With Given Total ray of Janet's income equality k x Karen's income Irene's income i x

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

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

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

GE contours: a limiting case
Total priority to the poorest

GE contours: another limiting case
Total priority to the richest

By contrast: Gini contours

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

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

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

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

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 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

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

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

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?

Top income shares in US

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... …Identification and measurement Inequality and Poverty
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 Development of specific measures Relation to inequality What axiomatisation? Use of ranking techniques Relation to welfare rankings

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

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

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

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

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

The distribution of poverty gaps
\$0 \$20 \$40 \$60 gaps

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

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

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

Poverty rates by region 1981

Poverty rates by region 2001

Poverty: East Asia

Poverty: South Asia

Poverty: Latin America, Caribbean

Poverty: Middle East and N.Africa

Poverty: Sub-Saharan Africa

Poverty: Eastern Europe and Central Asia

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?

Poverty in Spaion 1993—2000

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

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 Poverty gap
a reference point Poverty gap fundamental income difference Foster et al (1984) poverty index again Cumulative poverty gap

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

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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