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

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Inequality and Poverty

June 2005 June 2005

Public Economics: University of Barcelona Public Economics: University of Barcelona

Frank CowellFrank Cowell

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

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Issues to be addressed

Builds on lecture 2 Builds on lecture 2 ““Distributional Equity, Social Welfare” Distributional Equity, Social Welfare”

Extension of ranking criteriaExtension of ranking criteria Parade diagramsParade diagrams Generalised Lorenz curveGeneralised Lorenz curve

Extend SWF analysis to inequalityExtend SWF analysis to inequality Examine structure of inequalityExamine structure of inequality Link with the analysis of povertyLink with the analysis of poverty

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Major Themes

Contrast three main approaches to the subjectContrast three main approaches to the subject intuitiveintuitive via SWF via SWF via analysis of structurevia analysis of structure

Structure of the populationStructure of the population Composition of inequality and povertyComposition of inequality and poverty Implications for measuresImplications for measures

The use of axiomatisationThe use of axiomatisation Capture what is “reasonable”?Capture what is “reasonable”? Find a common set of axioms for related problemsFind a common set of axioms for related problems

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Overview...Inequality rankings

Inequality measurement

Inequality and decomposition

Poverty measures

Poverty rankings


Relationship with welfare rankings

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Inequality rankings

Begin by using welfare analysis of previous Begin by using welfare analysis of previous lecturelecture

Seek inequality rankingSeek inequality ranking We take as a basis the second-order distributional We take as a basis the second-order distributional

rankingranking ……but introduce a small modificationbut introduce a small modification

The 2nd-order dominance concept was originally The 2nd-order dominance concept was originally expressed in a more restrictive form.expressed in a more restrictive form.

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Yet another important relationship

The The shareshare of the proportion of the proportion qq of distribution of distribution FF is is given by given by L(F;q) := C(F;q) / (F)Yields Lorenz dominance, or the “shares” rankingYields Lorenz dominance, or the “shares” ranking

For given , G Lorenz-dominates FW(G) > W(F) for all WW2

The Atkinson (1970) result:The Atkinson (1970) result:

G Lorenz-dominates Fmeans: for every q, L(G;q) L(F;q), for some q, L(G;q) > L(F;q)

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The Lorenz diagram

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pro

por

tion

of

inco

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proportion of population

L(G;.)

L(F;.)

L(.; q)

q

Lorenz curve for FLorenz curve for F

practical example, UK

practical example, UK

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original income+ cash benefits

gross income- direct taxes

disposable income- indirect taxes

post-tax income+ non-cash benefits

final income

Official concepts of income: UK

What distributional ranking would we expect to apply to these 5 concepts?

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Proportion of population

Pro

po

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Original Income

Gross Income

Disposable Income

After Tax Income

Final Income

(Equality Line)

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

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Assessment of example

We might have guessed the outcome…We might have guessed the outcome… In most countries:In most countries:

Income tax progressiveIncome tax progressive So are public expendituresSo are public expenditures But indirect tax is regressiveBut indirect tax is regressive

So Lorenz-dominance is not surprising.So Lorenz-dominance is not surprising. But what happens if we look at the situation over time?But what happens if we look at the situation over time?

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2000-1

(Equality Line)

“Final income” – Lorenz

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(Equality Line)

“Original income” – Lorenz

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Lorenz curves intersect

Is 1993 more equal?

Or 2000-1?

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Inequality ranking: Summary Second-order (GL)-dominance is equivalent to Second-order (GL)-dominance is equivalent to

ranking by cumulations. ranking by cumulations. From the welfare lectureFrom the welfare lecture

Lorenz dominance equivalent to ranking by shares. Lorenz dominance equivalent to ranking by shares. Special case of GL-dominance normalised by Special case of GL-dominance normalised by

means.means. Where Lorenz-curves intersect unambiguous Where Lorenz-curves intersect unambiguous

inequality orderings are not possible.inequality orderings are not possible. This makes inequality measures especially interesting.This makes inequality measures especially interesting.

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Poverty measures

Poverty rankings


Three ways of approaching an index

•Intuition•Social welfare•Distance

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An intuitive approach

Lorenz comparisons (second-order dominance) may be Lorenz comparisons (second-order dominance) may be indecisiveindecisive

But we may want to “force a solution”But we may want to “force a solution” The problem is essentially one of aggregation of The problem is essentially one of aggregation of

informationinformation Why worry about aggregation?Why worry about aggregation?

It may make sense to use a very simple approachIt may make sense to use a very simple approach Go for something that you can “see”Go for something that you can “see”

Go back to the Lorenz diagramGo back to the Lorenz diagram

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Gini CoefficientGini Coefficient

The best-known inequality measure?

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Equivalent ways of writing the Gini:Equivalent ways of writing the Gini:1.1. Normalised area above Lorenz curveNormalised area above Lorenz curve

The Gini coefficient

2.2. Normalised difference between income pairs. Normalised difference between income pairs.

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Intuitive approach: difficulties

Essentially arbitraryEssentially arbitrary Does not mean that Gini is a bad indexDoes not mean that Gini is a bad index But what is the basis for it?But what is the basis for it?

What is the relationship with social welfare?What is the relationship with social welfare? The Gini index also has some “structural” problemsThe Gini index also has some “structural” problems

We will see this in the next sectionWe will see this in the next section

What is the relationship with social welfare?What is the relationship with social welfare? Examine the welfare-inequality relationship directlyExamine the welfare-inequality relationship directly

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Poverty measures

Poverty rankings




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SWF and inequality

Issues to be addressed:Issues to be addressed: the derivation of an indexthe derivation of an index the nature of inequality aversionthe nature of inequality aversion the structure of the SWFthe structure of the SWF

Begin with the SWF Begin with the SWF WW Examine contours in Irene-Janet spaceExamine contours in Irene-Janet space

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Equally-Distributed Equivalent Income

O xi

xj

The Irene &Janet diagram A given distribution Distributions with same mean

Contours of the SWF

•E

(F) (F)

•F

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

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Atkinson assumed an additive social welfare Atkinson assumed an additive social welfare function that satisfied the other basic axioms.function that satisfied the other basic axioms.

(F)I(F) = 1 – ——

(F)Mean incomeMean income

Ede incomeEde income

Welfare-based inequality

x1 - – 1 u(x) = ————, 1 –

Introduced an extra assumption: Iso-elastic Introduced an extra assumption: Iso-elastic welfare.welfare.

From the concept of social waste Atkinson From the concept of social waste Atkinson (1970) suggested an inequality measure:(1970) suggested an inequality measure:

W(F) = u(x) dF(x)

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The Atkinson Index Given scale-invariance, additive separability of welfareGiven scale-invariance, additive separability of welfare Inequality takes the form:Inequality takes the form:

Given the Harsanyi argument…Given the Harsanyi argument… index of inequality aversion index of inequality aversion based on risk aversion.based on risk aversion.

More generally see it as a stament of social valuesMore generally see it as a stament of social values Examine the effect of different values of Examine the effect of different values of

relationship between relationship between uu((xx) and ) and xx relationship between relationship between uu′′((xx) and ) and xx

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Social utility and relative income

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Relationship between welfare weight and income

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Poverty measures

Poverty rankings




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A further look at inequality

The Atkinson SWF route provides a coherent approach The Atkinson SWF route provides a coherent approach to inequality.to inequality.

But do we need to approach via social welfareBut do we need to approach via social welfare An indirect approachAn indirect approach Maybe introduces unnecessary assumptions,Maybe introduces unnecessary assumptions,

Alternative route: “distance” and inequalityAlternative route: “distance” and inequality Consider a generalisation of the Irene-Janet diagramConsider a generalisation of the Irene-Janet diagram

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The 3-Person income distribution

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Income DistributionsWith Given Total

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Inequality contours

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Set of distributions for given total

Set of distributions for a higher (given) total Perfect equality

Inequality contours for original levelInequality contours for higher level

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A distance interpretation

Can see inequality as a deviation from the normCan see inequality as a deviation from the norm The norm in this case is perfect equalityThe norm in this case is perfect equality Two key questions…Two key questions… ……what distance concept to use?what distance concept to use? How are inequality contours on one level “hooked up” How are inequality contours on one level “hooked up”

to those on another?to those on another?

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Another class of indices Consider the Consider the Generalised EntropyGeneralised Entropy class of inequality class of inequality

measures:measures:

The parameter The parameter is an indicator sensitivity of each is an indicator sensitivity of each member of the class.member of the class. large and positive gives a “top -sensitive” measurelarge and positive gives a “top -sensitive” measure negative gives a “bottom-sensitive” measurenegative gives a “bottom-sensitive” measure

Related to the Atkinson classRelated to the Atkinson class

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Inequality and a distance concept The Generalised Entropy class can also be written:The Generalised Entropy class can also be written:

Which can be written in terms of income shares Which can be written in terms of income shares ss

Using the distance criterion Using the distance criterion ss11−−/ [1/ [1−−] …] … Can be interpreted as weighted distance of each income shares from an equal shareCan be interpreted as weighted distance of each income shares from an equal share

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The Generalised Entropy Class

GE class is richGE class is rich Includes two indices from Henri Theil:Includes two indices from Henri Theil:

= 1: = 1: [ [ xx / / ((FF)] log ()] log (xx / / ((FF)) d)) dFF((xx))

= 0: = 0: – – log ( log (xx / / ((FF)) d)) dFF((xx)) For For < 1 it is ordinally equivalent to Atkinson class < 1 it is ordinally equivalent to Atkinson class

= 1 = 1 – – .. For For = 2 it is ordinally equivalent to (normalised) = 2 it is ordinally equivalent to (normalised)

variance.variance.

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Inequality contours

Each family of contours related to a different concept Each family of contours related to a different concept of distanceof distance

Some are very obvious…Some are very obvious… ……others a bit more subtleothers a bit more subtle Start with an obvious oneStart with an obvious one

the Euclidian casethe Euclidian case

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GE contours: 2

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GE contours: 225

− −

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GE contours: a limiting case

−∞

Total priority to the poorestTotal priority to the poorest

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GE contours: another limiting case

Total priority to the richestTotal priority to the richest

+∞

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By contrast: Gini contours

Not additively separableNot additively separable

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Distance: a generalisation

The responsibility approach gives a reference income The responsibility approach gives a reference income distributiondistribution Exact version depends on balance of compensation rulesExact version depends on balance of compensation rules And on income function.And on income function.

Redefine inequality measurementRedefine inequality measurement not based on perfect equality as a normnot based on perfect equality as a norm use the norm income distribution from the responsibility use the norm income distribution from the responsibility

approachapproach

Devooght (2004) bases this on Cowell (1985)Devooght (2004) bases this on Cowell (1985) Cowell approach based on Theil’s conditional entropyCowell approach based on Theil’s conditional entropy Instead of looking at distance going from perfect equality to Instead of looking at distance going from perfect equality to

actual distribution...actual distribution... Start from the reference distributionStart from the reference distribution

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Poverty measures

Poverty rankings


Structural issues

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Why decomposition?

Resolve questions in decomposition and population Resolve questions in decomposition and population heterogeneity:heterogeneity: Incomplete informationIncomplete information International comparisonsInternational comparisons Inequality accountingInequality accounting

Gives us a handle on axiomatising inequality measuresGives us a handle on axiomatising inequality measures Decomposability imposes structure.Decomposability imposes structure. Like separability in demand analysisLike separability in demand analysis

first, some terminology

first, some terminology

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A partition

population share

subgroupinequality

income share

j

sj

Ij

(ii)

(i)

(iii)

(iv)

• The populationThe population• Attribute 1Attribute 1

• One subgroupOne subgroup

• Attribute 2Attribute 2

(1)(2)

(3) (4)(5) (6)

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What type of decomposition?

Distinguish three types of decomposition by subgroupDistinguish three types of decomposition by subgroup In increasing order of generality these are:In increasing order of generality these are:

Inequality accountingInequality accounting Additive decomposabilityAdditive decomposability General consistencyGeneral consistency

Which type is a matter of judgmentWhich type is a matter of judgment More on this belowMore on this below Each type induces a class of inequality measuresEach type induces a class of inequality measures The “stronger” the decomposition requirement…The “stronger” the decomposition requirement… ……the “narrower” the class of inequality measuresthe “narrower” the class of inequality measures

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adding-up propertyadding-up property

weight functionweight function

1:Inequality accounting

This is the most restrictive form This is the most restrictive form of decomposition:of decomposition: accounting equationaccounting equation

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2:Additive Decomposability

As type 1, but no adding-up As type 1, but no adding-up constraint:constraint:

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population sharespopulation shares

3:General Consistency

The weakest version:The weakest version:

income sharesincome shares

increasing in each subgroup’s inequalityincreasing in each subgroup’s inequality

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A class of decomposable indices

Given scale-invariance and additive decomposability,Given scale-invariance and additive decomposability, Inequality takes the Inequality takes the Generalised EntropyGeneralised Entropy form: form:

Just as we had earlier in the lecture.Just as we had earlier in the lecture. Now we have a formal argument for this family.Now we have a formal argument for this family. The weight The weight jj on inequality in group on inequality in group jj is is jj = = jj

ssjj11−−

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What type of decomposition? Assume scale independence…Assume scale independence… Inequality accounting:Inequality accounting:

Theil indices only (Theil indices only ( Here Here jj = = jj or or jj = = ssjj

Additive decomposability:Additive decomposability: Generalised Entropy IndicesGeneralised Entropy Indices

General consistency:General consistency: moments, moments, Atkinson, ...Atkinson, ...

But is there something missing here?But is there something missing here? We pursue this laterWe pursue this later

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What type of partition?

GeneralGeneral The approach considered so farThe approach considered so far Any characteristic used as basis of partitionAny characteristic used as basis of partition Age, gender, region, incomeAge, gender, region, income Induces specific class of inequality measuresInduces specific class of inequality measures ... but excludes one very important measure... but excludes one very important measure

Non-overlapping in incomesNon-overlapping in incomes A weaker versionA weaker version Partition just on the basis of incomePartition just on the basis of income Allows one to include the "missing" inequality measureAllows one to include the "missing" inequality measure

Distinction between them is crucial for one special Distinction between them is crucial for one special inequality measureinequality measure

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Different (equivalent) Different (equivalent) ways of writing the Gini:ways of writing the Gini:

Normalised area under Normalised area under the Lorenz curvethe Lorenz curve

The Gini coefficient

Normalised pairwise differencesNormalised pairwise differences

A ranking-weighted averageA ranking-weighted average

But ranking depends on reference distributionBut ranking depends on reference distribution

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Gini CoefficientGini Coefficient

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Partitioning by income...

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N1 N2

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x'x

Case 2: effect on Gini differs in subgroup and population

x'x

Case 1: effect on Gini is same in subgroup and population

Non-overlapping income groups

Overlapping income groups

Consider a transfer:Case 1

Consider a transfer:Case 2

x

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Non-overlapping decomposition

Can be particularly valuable in empirical applicationsCan be particularly valuable in empirical applications Useful for rich/middle/poor breakdownsUseful for rich/middle/poor breakdowns Especially where data problems in tailsEspecially where data problems in tails

Misrecorded dataMisrecorded data Incomplete dataIncomplete data Volatile data componentsVolatile data components

Example: Example: Piketty-SaezPiketty-Saez on US (QJE 2003) on US (QJE 2003) Look at behaviour of Capital gains in top income shareLook at behaviour of Capital gains in top income share Should this affect conclusions about trend in inequality?Should this affect conclusions about trend in inequality?

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Top income shares in US

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1925

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1937

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1961

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1969

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1997

P90–100

P99–100CG excl

P99–100CG Incl

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Choosing an inequality measure

Do you want an index that accords with Do you want an index that accords with intuition?intuition? If so, what’s the basis for the intuition?If so, what’s the basis for the intuition?

Is decomposability essential?Is decomposability essential? If so, what type of decomposability?If so, what type of decomposability?

Do you need a welfare interpretation?Do you need a welfare interpretation? If so, what welfare principles to apply?If so, what welfare principles to apply?

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Poverty measures

Poverty rankings


…Identification and measurement

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Poverty analysis – overview

Basic ideasBasic ideas Income – similar to inequality problem?Income – similar to inequality problem?

Consumption, expenditure or income?Consumption, expenditure or income? Time periodTime period RiskRisk

Income receiver – as beforeIncome receiver – as before Relation to decompositionRelation to decomposition

Development of specific measuresDevelopment of specific measures Relation to inequalityRelation to inequality What axiomatisation?What axiomatisation?

Use of ranking techniquesUse of ranking techniques Relation to welfare rankingsRelation to welfare rankings

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Poverty measurement

How to break down the basic issues.How to break down the basic issues. Sen (1979): Two main types of issues Sen (1979): Two main types of issues

Identification problemIdentification problem Aggregation problemAggregation problem

Jenkins and Lambert (1997)Jenkins and Lambert (1997): “3Is”: “3Is” IdentificationIdentification IntensityIntensity InequalityInequality

Present approach:Present approach: Fundamental partitionFundamental partition Individual identificationIndividual identification Aggregation of informationAggregation of information

population

non-poor

poor

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Poverty and partition

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

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Counting the poor

Use the concept of individual poverty evaluationUse the concept of individual poverty evaluation Simplest version is (0,1)Simplest version is (0,1)

(non-poor, poor)(non-poor, poor) headcountheadcount

Perhaps make it depend on incomePerhaps make it depend on income poverty deficitpoverty deficit

Or on the whole distribution?Or on the whole distribution?

Convenient to work with Convenient to work with poverty gapspoverty gaps

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The poverty line and poverty gaps

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pov

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luat

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incomexi xj

gi

gj

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Poverty evaluation

g

0

poverty evalu

ation

poverty gap

x = 0

Non-PoorNon-Poor PoorPoor

gi

A

gj

B

the “head-count”

the “poverty deficit” sensitivity to inequality amongst the poor

Income equalisation amongst the poor

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Brazil 1985: How Much Poverty?

Rural Belo Horizonte poverty lineRural Belo Horizonte poverty line

Brasilia poverty lineBrasilia poverty line

compromise poverty linecompromise poverty line

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

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The distribution of poverty gaps

$0 $20 $40 $60 gaps

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ASP

Additively Separable Poverty measures Additively Separable Poverty measures ASP approach simplifies ASP approach simplifies poverty evaluation poverty evaluation Depends on own income and the poverty line.Depends on own income and the poverty line.

pp((xx, , x*x*)) Assumes decomposability amongst the poorAssumes decomposability amongst the poor Overall poverty is an additively separable functionOverall poverty is an additively separable function

P = P = pp((xx, , x*x*) d) dFF((xx))

Analogy with decomposable inequality measuresAnalogy with decomposable inequality measures

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A class of poverty indices

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

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0

0.2

0.4

0.6

0.8

1

-0.2 0 0.2 0.4 0.6 0.8 1

a=0

a=1

a=1.5

a=2

a=2.5

Poverty evaluation functions

p(x,x*)

x*-x

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Empirical robustness

Does it matter which poverty criterion you use?Does it matter which poverty criterion you use? Look at two key measures from the ASP classLook at two key measures from the ASP class

Head-count ratioHead-count ratio Poverty deficit (or average poverty gap)Poverty deficit (or average poverty gap)

Use two standard poverty linesUse two standard poverty lines $1.08 per day at 1993 PPP$1.08 per day at 1993 PPP $2.15 per day at 1993 PPP$2.15 per day at 1993 PPP

How do different regions of the world compare?How do different regions of the world compare? What’s been happening over time?What’s been happening over time? Use World-Bank analysisUse World-Bank analysis

Chen-Ravallion “How have the world’s poorest fared since the early Chen-Ravallion “How have the world’s poorest fared since the early 1980s?” 1980s?” World Bank Policy Research Working Paper Series 3341World Bank Policy Research Working Paper Series 3341

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Poverty rates by region 1981

China 63.80 1 88.10 3 23.41 1 50.82 1

East Asia 57.70 2 84.80 4 20.58 2 47.20 3

India 54.40 3 89.60 1 17.27 3 47.22 2

South Asia 51.50 4 89.10 2 16.06 5 45.78 4

Sub-Saharan Africa 41.60 5 73.30 5 17.03 4 38.54 5

All regions 40.40 6 66.70 6 13.92 6 35.02 6

Latin America and Caribbean 9.70 7 26.90 8 2.75 7 10.66 7

Middle East and North Africa 5.10 8 28.90 7 1.00 8 8.81 8

Eastern Europe and Central Asia 0.70 9 4.70 9 0.17 9 1.39 9

Headcount Poverty gap$1.08 $2.15 $1.08 $2.15

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Poverty rates by region 2001

Sub-Saharan Africa 46.90 1 76.60 3 20.29 1 41.42 1India 34.70 2 79.90 1 7.08 2 34.43 2South Asia 31.30 3 77.20 2 6.37 3 32.35 3All regions 21.10 4 52.90 4 5.96 4 21.21 4

China 16.60 5 46.70 6 3.94 5 18.44 5East Asia 14.90 6 47.40 5 3.35 7 17.78 6Latin America and Caribbean 9.50 7 24.50 7 3.36 6 10.20 7Eastern Europe and Central Asia 3.70 8 19.70 9 0.79 8 5.94 9Middle East and North Africa 2.40 9 23.20 8 0.45 9 6.14 8

Headcount Poverty gap$1.08 $2.15 $1.08 $2.15

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Poverty: East Asia

0

10

20

30

40

50

60

70

80

90

Headcount at $1.08 per day


Poverty gap at $1.08 per day


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Poverty: South Asia

0

10

20

30

40

50

60

70

80

90

100

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001





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Poverty: Latin America, Caribbean

0

5

10

15

20

25

30

35

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001





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Poverty: Middle East and N.Africa

0

5

10

15

20

25

30

35

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001





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Poverty: Sub-Saharan Africa

0

10

20

30

40

50

60

70

80

90

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001





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Poverty: Eastern Europe and Central Asia

0

5

10

15

20

25

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001





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Empirical robustness (2) Does it matter which poverty criterion you use?Does it matter which poverty criterion you use? An example from SpainAn example from Spain Data are from ECHPData are from ECHP OECD equivalence scale OECD equivalence scale Poverty line is 60% of 1993 median incomePoverty line is 60% of 1993 median income Does it matter which FGT index you use?Does it matter which FGT index you use?

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Poverty in Spaion 1993—2000

40

50

60

70

80

90

100

110

120

1993 1994 1995 1996 1997 1998 1999 2000

FGT(1) FGT(2) FGT(3) FGT(4) FGT(5)

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Poverty measures

Poverty rankings


Another look at ranking issues

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An extension of poverty analysis

Finally consider some generalisationsFinally consider some generalisations What if we do not know the poverty line?What if we do not know the poverty line? Can we find a counterpart to second order dominance Can we find a counterpart to second order dominance

in welfare analysis?in welfare analysis? What if we try to construct poverty indices from first What if we try to construct poverty indices from first

principles?principles?

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Poverty rankings (1)

Atkinson (1987)Atkinson (1987) connects poverty and welfare. connects poverty and welfare. Based results on the portfolio literature concerning Based results on the portfolio literature concerning

“below-target returns” “below-target returns” Theorem Theorem

Given a bounded range of poverty lines (Given a bounded range of poverty lines (xx**minmin, x, x**

maxmax)) and poverty measures of the ASP form and poverty measures of the ASP form a necessary and sufficient condition for poverty to be a necessary and sufficient condition for poverty to be

lower in distribution lower in distribution FF than in distribution than in distribution GG is that the is that the poverty deficit be no greater in poverty deficit be no greater in FF than in than in GG for all for all xx** ≤ ≤ xx**

maxmax..

Equivalent to requiring that the second-order Equivalent to requiring that the second-order dominance condition hold for all dominance condition hold for all xx**. .

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Poverty rankings (2)

Foster and Shorrocks (Foster and Shorrocks (1988a1988a, 1988b) have a similar , 1988b) have a similar approach to orderings by approach to orderings by PP, ,

But concentrate on the FGT index’s particular But concentrate on the FGT index’s particular functional form:functional form:

Theorem: Poverty rankings are equivalent to Theorem: Poverty rankings are equivalent to first-order welfare dominance for first-order welfare dominance for aa = 0 = 0 second-degree welfare dominance for second-degree welfare dominance for aa = 1 = 1 (third-order welfare dominance for (third-order welfare dominance for aa = 2.) = 2.)

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Poverty concepts Given poverty line Given poverty line zz

a reference pointa reference point

Poverty gapPoverty gap fundamental income differencefundamental income difference

Foster et al (1984) poverty index againFoster et al (1984) poverty index again

Cumulative poverty gapCumulative poverty gap

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TIP / Poverty profile

i/n

p(x,z)/n

G(x,z)

0

Cumulative gaps versus Cumulative gaps versus population proportionspopulation proportions

Proportion of poorProportion of poor TIP curveTIP curve

Cumulative gaps versus Cumulative gaps versus population proportionspopulation proportions

Proportion of poorProportion of poor TIP curveTIP curve

TIP curves have same interpretation as GLC

TIP dominance implies unambiguously greater poverty

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Poverty: Axiomatic approach

Characterise an ordinal poverty index Characterise an ordinal poverty index PP((xx , ,zz)) See Ebert and See Ebert and MoyesMoyes (JPET 2002) (JPET 2002)

Use some of the standard axioms we introduced for Use some of the standard axioms we introduced for analysing social welfareanalysing social welfare

Apply them to Apply them to nn+1 incomes – those of the +1 incomes – those of the nn individuals individuals and the poverty lineand the poverty line

Show that Show that given just these axioms…given just these axioms… ……you are bound to get a certain type of poverty measure.you are bound to get a certain type of poverty measure.

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Poverty: The key axioms

Standard ones from lecture 2Standard ones from lecture 2 anonymityanonymity independenceindependence monotonicitymonotonicity

income increments reduce povertyincome increments reduce poverty Strengthen two other axiomsStrengthen two other axioms

scale invariancescale invariance translation invariancetranslation invariance

Also need continuityAlso need continuity Plus a focus axiomPlus a focus axiom

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A closer look at the axioms

Let Let DD denote the set of ordered income vectors denote the set of ordered income vectors The focus axiom is The focus axiom is

Scale invariance now becomesScale invariance now becomes

Independence means:Independence means:

Define the number of the poor asDefine the number of the poor as

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Ebert-Moyes (2002)

Gives two types of FGT measuresGives two types of FGT measures ““relative” versionrelative” version ““absolute” versionabsolute” version

Additivity follows from the independence axiom Additivity follows from the independence axiom

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Brief conclusion

Framework of distributional analysis covers a number Framework of distributional analysis covers a number of related problems:of related problems: Social WelfareSocial Welfare InequalityInequality PovertyPoverty

Commonality of approach can yield important insightsCommonality of approach can yield important insights Ranking principles provide basis for broad judgments Ranking principles provide basis for broad judgments

May be indecisiveMay be indecisive specific indices could be usedspecific indices could be used

Poverty trends will often be robust to choice of poverty Poverty trends will often be robust to choice of poverty indexindex

Poverty indexes can be constructed from scratch using Poverty indexes can be constructed from scratch using standard axiomsstandard axioms

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

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