frank cowell: oviedo – inequality & poverty inequality measurement march 2007 inequality,...

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viedo – Inequality & P

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Inequality Measurement

March 2007 March 2007

Inequality, Poverty and Income Distribution Inequality, Poverty and Income Distribution

University of OviedoUniversity of Oviedo

Frank CowellFrank Cowellhttp://darp.lse.ac.uk/oviedo2007http://darp.lse.ac.uk/oviedo2007

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Issues to be addressed

Builds on lecture 3Builds on lecture 3 ““Income Distribution and Welfare” Income Distribution and 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 measurementComposition of Inequality measurement Implications for measuresImplications for measures

The use of axiomatisationThe use of axiomatisation Capture what is “reasonable”?Capture what is “reasonable”? Use principles similar to welfare and povertyUse principles similar to welfare and poverty

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Overview...Inequality rankings

Inequality measures

Inequality axiomatics

Inequality in practice

Inequality measurement

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 an Seek an inequalityinequality ranking 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 Normalise by dividing by the meanNormalise by dividing by the mean

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 given is given by 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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All the above has been done in terms of All the above has been done in terms of FF-form notation.-form notation. Can do the almost same in Irene-Janet notation.Can do the almost same in Irene-Janet notation. Use the order statistics Use the order statistics xx[[ii]] where where

is the is the iith smallest member of… th smallest member of… ……the income vector (the income vector (xx11,,xx22,…,,…,xxnn))

Then, define Then, define ParadeParade

income cumulationsincome cumulations

GLCGLC

LCLC

For discrete distributions

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The Lorenz diagram

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

prop

orti

on o

f in

com

e

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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Application of ranking

The tax and -benefit system maps one distribution into The tax and -benefit system maps one distribution into another...another...

Use ranking tools to assess the impact of this in welfare Use ranking tools to assess the impact of this in welfare terms.terms.

Typically this uses one or other concept of Lorenz Typically this uses one or other concept of Lorenz dominance.dominance.

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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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0.0

0.1

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1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Proportion of population

Pro

po

rtio

n o

f In

co

me

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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0.0

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Pro

po

rtio

n o

f In

co

me

1993

2000-1

(Equality Line)

“Final income” – Lorenz

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0.0

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Pro

po

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me

1993

2000-1

(Equality Line)

“Original income” – Lorenz

0.0 0.1 0.2 0.3 0.4 0.5

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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 ranking Second-order (GL)-dominance is equivalent to ranking

by cumulations. 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 means.Special case of GL-dominance normalised by means.

Where Lorenz-curves intersect unambiguous inequality Where Lorenz-curves intersect unambiguous inequality orderings are not possible.orderings are not possible.

This makes inequality measures especially interesting.This makes inequality measures especially interesting.

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Inequality measures




Three ways of approaching an index

•Intuition•Social welfare•Distance

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Inequality measures

What is an inequality measure?What is an inequality measure? Formally very simpleFormally very simple

function (or functional) from set of distributions… function (or functional) from set of distributions… ……to the real lineto the real line contrast this with ranking principlescontrast this with ranking principles

Nature of the measure?Nature of the measure? Some simple regularity properties…Some simple regularity properties… ……such as continuitysuch as continuity Beyond that we need some theoryBeyond that we need some theory

Alternative approaches to the theory:Alternative approaches to the theory: intuitionintuition social welfaresocial welfare distancedistance

Begin with intuitionBegin with intuition

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Intuitive inequality measures Perhaps borrow from other disciplines…Perhaps borrow from other disciplines… A standard measure of spread…A standard measure of spread…

variancevariance

But maybe better to use a normalised versionBut maybe better to use a normalised version coefficient of variationcoefficient of variation

Comparison between these two is instructiveComparison between these two is instructive Same iso-inequality contours for a given Same iso-inequality contours for a given .. Different behaviour as Different behaviour as alters. alters.

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Another intuitive approach

Alternative intuition based on Lorenz approach Alternative intuition based on Lorenz approach Lorenz comparisons (second-order dominance) may be Lorenz comparisons (second-order dominance) may be

indecisiveindecisive Use the diagram to “force a solution”Use the diagram to “force a solution” Problem is essentially one of aggregation of informationProblem is essentially one of aggregation of information

It may make sense to use a very simple approachIt may make sense to use a very simple approach Try something that you can “see”Try something that you can “see” Go back to the Lorenz diagramGo back to the Lorenz diagram

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.5

prop

ortion of in

come

proportion of population

Gini CoefficientGini Coefficient

The best-known inequality measure?

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Natural expression of measure…Natural expression of measure… Normalised area above Lorenz curveNormalised area above Lorenz curve

The Gini coefficient (1)

Can express this also in Irene-Janet termsCan express this also in Irene-Janet terms for discrete distributions.for discrete distributions.

But alternative representations more usefulBut alternative representations more useful each of these each of these equivalentequivalent to the above to the above expressible in expressible in F-F-form or Irene-Janet termsform or Irene-Janet terms

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Normalised difference between income pairs:Normalised difference between income pairs: In In FF-form:-form:

In Irene-Janet terms:In Irene-Janet terms:


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Finally, express Gini as a weighted sumFinally, express Gini as a weighted sum In In FF-form-form

Or, more illuminating, in Irene-Janet termsOr, more illuminating, in Irene-Janet terms

Note that the weights Note that the weights are very special are very special depend on depend on rankrank or position in distribution or position in distribution will change as other members added/removed from distributionwill change as other members added/removed from distribution perhaps in interesting ways perhaps in interesting ways


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Intuitive approach: difficulties

Essentially arbitraryEssentially arbitrary Does not mean that CV or Gini is a bad indexDoes not mean that CV or 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 later in the lectureWe will see this later in the lecture

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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Inequality measures






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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 incomeSocial 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 (1970) From the concept of social waste Atkinson (1970) suggested an inequality measure: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 statement of social valuesMore generally see it as a statement 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

1 2 3 4 5

-3

-2

-1

0

1

2

3

4

U

x /

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Relationship between welfare weight and income

0 1 2 3 4 50

1

2

3

4

U'

x /

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Inequality measures






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A further look at inequality

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

But do we need to use an approach via social welfare?But do we need to use an approach via social welfare? An indirect approachAn indirect approach Maybe introduces unnecessary assumptionsMaybe 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

0 Irene's income

Jan

et's

inco

me

Karen's income

ix

kx

xj

ray of

equali

ty

Income DistributionsWith Given Total

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Inequality contours

0

ix

kx

xj

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” to How are inequality contours on one level “hooked up” to

those on another?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 of Each family of contours related to a different concept of distancedistance

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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Inequality measures




A fundamentalist approach

•The approach•Inequality and income levels•Decomposition•Results

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Axiomatic approach Can be applied to any of the three version of Can be applied to any of the three version of

inequalityinequality Reminder – what makes a good axiom system?Reminder – what makes a good axiom system?

Can’t be “right” or “wrong”Can’t be “right” or “wrong” But could be appropriate/inappropriateBut could be appropriate/inappropriate Capture commonly held ideas?Capture commonly held ideas?

Exploit similarity of form across related Exploit similarity of form across related problemsproblems inequalityinequality welfarewelfare povertypoverty

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Axiom systems Already seen many standard axioms in terms of Already seen many standard axioms in terms of WW

anonymityanonymity population principlepopulation principle principle of transfersprinciple of transfers scale/translation invariancescale/translation invariance

Could use them to characterise inequality Could use them to characterise inequality Use Atkinson type approachUse Atkinson type approach

But why use an indirect approach?But why use an indirect approach? Some welfare issues don’t need to concern us…Some welfare issues don’t need to concern us… ……monotonicity of welfare?monotonicity of welfare?

However, do need some additional axiomsHowever, do need some additional axioms How do inequality levels change with income…?How do inequality levels change with income…? ……not just inequality not just inequality rankingsrankings.. How does inequality overall relate to that in subpopulations?How does inequality overall relate to that in subpopulations?

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Inequality and income level

BC

Irene's income

Jan

et's

inco

me

xi

xj

0

ray o

f equal

ity

The Irene &Janet diagram A distribution

Possible distributions of a small increment

Does this direction keep inequality unchanged? Or this direction?

Consider the iso-inequality path.

Also gives what would be an inequality-preserving income reduction

See Amiel-Cowell (1999)

A

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xi

xj

Scale independence

Example 1.

Equal proportionate additions or subtractions keep inequality constant

Corresponds to regular Lorenz criterion

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xjx2

Translation independence

Example 2.

Equal absolute additions or subtractions keep inequality constant

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xi

xj

Intermediate case

Example 3.

Income additions or subtractions in the same “intermediate” direction keep inequality constant

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xi

xjx2

Dalton’s conjecture

Amiel-Cowell (1999) showed that individuals perceived inequality comparisons this way.

Pattern is based on a conjecture by Dalton (1920)

Note dependence of direction on income level

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Inequality and income level

Three different standard cases Three different standard cases scale independencescale independence translation independencetranslation independence intermediate (affine)intermediate (affine)

Consistent with different types of measureConsistent with different types of measure relative inequalityrelative inequality absoluteabsolute intermediateintermediate Blackorby and Donaldson, (Blackorby and Donaldson, (19781978, , 19801980))

A matter of judgment which version to useA matter of judgment which version to use

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Inequality decomposition

Decomposition enables us to relate inequality overall to Decomposition enables us to relate inequality overall to inequality in constituent parts of the populationinequality in constituent parts of the population

Distinguish three types, in increasing order of generality:Distinguish three types, in increasing order of generality: Inequality accountingInequality accounting Additive decomposabilityAdditive decomposability General consistencyGeneral consistency

Which type is a matter of judgmentWhich type is a matter of judgment 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

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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adding-up propertyadding-up property

weight functionweight function

Type 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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Type 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

Type 3: Subgroup 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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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

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

Distinction between them is crucial Distinction between them is crucial

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Partitioning by income...

x*

N1 N2

0

x**N1

Non-overlapping income groups

Overlapping income groups

x

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A class of decomposable indices

Given scale-independence and additive decomposability,Given scale-independence 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

11−−ssjj

Weights only sum to 1 if Weights only sum to 1 if = 0 or 1 (Theil indices) = 0 or 1 (Theil indices)

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Another class of decomposable indices Given translation-independence and additive decomposability,Given translation-independence and additive decomposability, Inequality takes the Inequality takes the KolmKolm form ( form (Kolm 1976))

Another class of additive measuresAnother class of additive measures But these are absolute indicesBut these are absolute indices There is a relationship to There is a relationship to Theil indices (Theil indices (Cowell 2006 Cowell 2006 ))

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Generalisation (1) Suppose we don’t insist on additive decomposability?Suppose we don’t insist on additive decomposability? Given subgroup consistency…Given subgroup consistency… ……with scale independence: with scale independence:

transforms of GE indices transforms of GE indices moments, Atkinson class ...moments, Atkinson class ...

……with translation independence:with translation independence: transforms of Kolmtransforms of Kolm

But we never get Gini indexBut we never get Gini index Gini is not decomposable!Gini is not decomposable! i.e., given general partition will not satisfy subgroup consistencyi.e., given general partition will not satisfy subgroup consistency to see why, recall definition of Gini in terms of positions:to see why, recall definition of Gini in terms of positions:

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x*

N1 N2

0

x**N1

x'x

Case 2: effect on Gini is proportional to [i-j]: differs in subgroup and population

x'x

Case 1: effect on Gini is proportional to [i-j]: same in subgroup and population

x

Partitioning by income... Overlapping income groups

Consider a transfer:Case 1

Consider a transfer:Case 2

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Generalisation (2) Relax decomposition furtherRelax decomposition further Given nonoverlapping decomposability…Given nonoverlapping decomposability… ……with scale independence: with scale independence:

transforms of GE indices transforms of GE indices moments, Atkinson classmoments, Atkinson class + Gini+ Gini

……with translation independence:with translation independence: transforms of Kolmtransforms of Kolm + absolute Gini+ absolute Gini

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Gini contours

Not additively separableNot additively separable

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Gini axioms: illustration

x1

x3

x2

•Distributions for n=3•An income distribution•Perfect equality•Contours of “Absolute” Gini•Continuity

•Continuous approach to I = 0

•Linear homogeneity•Proportionate increase in I

•Translation invariance•I constant

0

1•

x*

•

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Inequality measures




Performance of inequality measures

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Why decomposition?

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

Gives us a handle on axiomatising inequality Gives us a handle on axiomatising inequality measuresmeasures Decomposability imposes structureDecomposability imposes structure Like separability in demand analysisLike separability in demand analysis

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Non-overlapping decomposition

Can be particularly valuable in empirical Can be particularly valuable in empirical applicationsapplications

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

Frank C

owell:

Frank C

owell: O


overty O


overty

Choosing an inequality measure

Do you want an index that accords with intuition?Do you want an index that accords with 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?

What difference does it make?What difference does it make? Example 1: recent US experienceExample 1: recent US experience Example 2: relative measures and world inequalityExample 2: relative measures and world inequality Example 3: Absolute/Relative for worldExample 3: Absolute/Relative for world

Frank C

owell:

Frank C

owell: O


overty O


overty

Example 1: US

Re-examine the data from Lecture 1Re-examine the data from Lecture 1 DeNavas-Walt et al. (2005)

Recall the impression of rising inequalityRecall the impression of rising inequality ““fanning-out” of quantile ratios fanning-out” of quantile ratios increasing disparity of income sharesincreasing disparity of income shares

Is this borne out by inequality measures?Is this borne out by inequality measures? GiniGini Atkinson indices – does it matter what Atkinson indices – does it matter what ?? GE indices – does it matter what GE indices – does it matter what ??

Frank C

owell:

Frank C

owell: O


overty O


overty

Inequality measures and US experience

Source: Source: DeNavas-Walt et al. (2005)

0

0.1

0.2

0.3

0.4

0.5

0.6

GiniGE0GE1A.25A.50A.75

Frank C

owell:

Frank C

owell: O


overty O


overty

Example 2: International trends Recent debate on “convergence” / “divergence” Traditional approach takes each country as separate unit

shows divergence – increase in inequality but, in effect, weights countries equally it is obviously debatable that huge countries like China… …get the same weight as very small countries

New conventional view:New conventional view: within-country disparities have increased within-country disparities have increased not enough to offset reduction in cross-country disparities. not enough to offset reduction in cross-country disparities. (Sala-i-Martin 2006)(Sala-i-Martin 2006)

Components of change in distribution are importantComponents of change in distribution are important ““correctly” compute world income distributioncorrectly” compute world income distribution decomposition is then crucialdecomposition is then crucial what drives cross-country reductions in inequality?what drives cross-country reductions in inequality? Large growth rate of the incomes of the 1.2 billion ChineseLarge growth rate of the incomes of the 1.2 billion Chinese

Frank C

owell:

Frank C

owell: O


overty O


overty

Inequality measures and World experience

00.10.20.30.40.50.60.70.80.9

1

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

GiniGE0GE1A.50A1.0

Source: Sala-i-Martin (2006)Source: Sala-i-Martin (2006)

Frank C

owell:

Frank C

owell: O


overty O


overty

Inequality measures and World experience: breakdown

00.10.20.30.40.50.60.70.80.9

1

1970

1973

1976

1979

1982

1985

1988

1991

1994

1997

2000

GE0GE0 betwGE0 within GE1GE1 betwGE1 within

Source: Sala-i-Martin (2006)Source: Sala-i-Martin (2006)

Frank C

owell:

Frank C

owell: O


overty O


overty

Example 3: World inequality again

All the previous used relative inequality measuresAll the previous used relative inequality measures GiniGini Atkinson indicesAtkinson indices GE indicesGE indices

What would happen if we switchewd to absolute What would happen if we switchewd to absolute measures?measures? absolute Giniabsolute Gini Kolm indicesKolm indices

Important role for changes in mean incomeImportant role for changes in mean income

Frank C

owell:

Frank C

owell: O


overty O


overty

Atkinson and Brandolini. (2004)

Absolute vs Relative measures

Frank C

owell:

Frank C

owell: O


overty O


overty

References (1) Amiel, Y. and Cowell, F. A. (1999) Amiel, Y. and Cowell, F. A. (1999) Thinking about InequalityThinking about Inequality, ,

Cambridge University Press, Cambridge, Chapter 7.Cambridge University Press, Cambridge, Chapter 7. Atkinson, A. B. (1970) “Atkinson, A. B. (1970) “On the Measurement of Inequality,” On the Measurement of Inequality,” Journal Journal

of Economic Theoryof Economic Theory, , 22, 244-263, 244-263 Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality: Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality:

Absolute, Relative or Intermediate?” Paper presented at the 28th Absolute, Relative or Intermediate?” Paper presented at the 28th General Conference of the International Association for Research on General Conference of the International Association for Research on Income and Wealth. August 22. Cork, Ireland.Income and Wealth. August 22. Cork, Ireland.

Blackorby, C. and Donaldson, D. (1978) “Measures of relative Blackorby, C. and Donaldson, D. (1978) “Measures of relative equality and their meaning in terms of social welfare,” equality and their meaning in terms of social welfare,” Journal of Journal of Economic TheoryEconomic Theory, , 1818, 59-80, 59-80

Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of Blackorby, C. and Donaldson, D. (1980) “A theoretical treatment of indices of absolute inequality,” indices of absolute inequality,” International Economic ReviewInternational Economic Review, , 2121, , 107-136107-136

Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) and Bourguignon, F. (eds) Handbook of Income DistributionHandbook of Income Distribution, North , North Holland, Amsterdam, Chapter 2, 87-166Holland, Amsterdam, Chapter 2, 87-166

Frank C

owell:

Frank C

owell: O


overty O


overty

References (2) Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition,” Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition,”

Research on Economic InequalityResearch on Economic Inequality, , 1313, 345-360, 345-360 Dalton, H. (1920) “Dalton, H. (1920) “Measurement of the inequality of incomes,” Measurement of the inequality of incomes,” The The

Economic JournalEconomic Journal, , 3030, 348-361, 348-361 DeNavas-Walt, C., Proctor, B. D. and Lee, C. H. (2005) “Income, DeNavas-Walt, C., Proctor, B. D. and Lee, C. H. (2005) “Income,

poverty, and health insurance coverage in the United States: 2004.” poverty, and health insurance coverage in the United States: 2004.” Current Population Reports P60-229, U.S. Census Bureau, U.S. Current Population Reports P60-229, U.S. Census Bureau, U.S. Government Printing Office, Washington, DC.Government Printing Office, Washington, DC.

Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Kolm, S.-Ch. (1976) “Unequal Inequalities I,” Journal of Economic Journal of Economic TheoryTheory, , 1212, 416-442, 416-442

Sala-i-Martin, X. (2006) “The world distribution of income: Falling Sala-i-Martin, X. (2006) “The world distribution of income: Falling poverty and ... convergence, period”, poverty and ... convergence, period”, Quarterly Journal of EconomicsQuarterly Journal of Economics, , 121121

Theil, H. (1967) Theil, H. (1967) Economics and Information TheoryEconomics and Information Theory, North Holland, , North Holland, Amsterdam, chapter 4, 91-134Amsterdam, chapter 4, 91-134

frank cowell: oviedo – inequality & poverty inequality measurement march 2007 inequality,...

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