Frank Cowell: TU Lisbon – Inequality & Poverty Inequality Measurement July 2006 Inequality measurement Measurement Technical University of Lisbon Frank

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Inequality Measurement

July 2006 July 2006

Inequality measurement Measurement Inequality measurement Measurement

Technical University of LisbonTechnical University of Lisbon

Frank CowellFrank Cowellhttp://darp.lse.ac.uk/lisbon2006http://darp.lse.ac.uk/lisbon2006

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Issues to be addressed

Builds on lecture 3Builds on lecture 3 ““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 measurementComposition of Inequality measurement 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 measures

Inequality axioms

Inequality decomposition

Inequality in practice

Inequality measurement

Relationship with welfare rankings

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Inequality rankings

Begin by using welfare analysis of previous lectureBegin by using welfare analysis of previous lecture 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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Inequality rankings

Using the welfare analysis above…Using the welfare analysis above… Seek an Seek an inequalityinequality ranking ranking Take as a basis the 2nd-order distributional rankingTake as a basis the 2nd-order distributional ranking

……but introduce a small modificationbut introduce a small modification Normalise by dividing by the meanNormalise by dividing by the mean Away of forcing an “iso-inequality” path as mean income Away of forcing an “iso-inequality” path as mean income

changeschanges The 2nd-order dominance concept was originally The 2nd-order dominance concept was originally

expressed in this more restrictive form…expressed in this 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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The Lorenz diagram

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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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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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Proportion of population

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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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(Equality Line)

“Final income” – Lorenz

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“Original income” – Lorenz

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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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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 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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Inequality measures

Inequality axioms




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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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 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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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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By contrast: Gini contours

Not additively separableNot additively separable

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Inequality measures

Inequality axioms




A fundamentalist approach

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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 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

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The 3-Person income distribution

0 Irene's income

Jan

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inco

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Karen's income

ix

kx

xj

ray of

equali

ty

Income DistributionsWith Given Total

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Inequality contours

0

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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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Themes Cross-disciplinary conceptsCross-disciplinary concepts Income differencesIncome differences Reference incomesReference incomes Formal methodologyFormal methodology

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Methodology Exploit common structureExploit common structure

povertypoverty deprivationdeprivation complaints and inequalitycomplaints and inequality see see Cowell (2005)Cowell (2005)

Axiomatic methodAxiomatic method minimalist approachminimalist approach characterise structurecharacterise structure introduce ethicsintroduce ethics

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“Structural” axioms

Take some social evaluation function Take some social evaluation function ContinuityContinuity

Linear homogeneityLinear homogeneity

Translation invarianceTranslation invariance

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Structural axioms: illustration

x1

x3

x2

DD for for nn=3=3 An income distributionAn income distribution Perfect equalityPerfect equality Contours of “Absolute” GiniContours of “Absolute” Gini ContinuityContinuity

Continuous approach to Continuous approach to I I = 0= 0 Linear homogeneityLinear homogeneity

Proportionate increase in Proportionate increase in II Translation invarianceTranslation invariance

II constant constant

DD for for nn=3=3 An income distributionAn income distribution Perfect equalityPerfect equality Contours of “Absolute” GiniContours of “Absolute” Gini ContinuityContinuity

Continuous approach to Continuous approach to I I = 0= 0 Linear homogeneityLinear homogeneity

Proportionate increase in Proportionate increase in II Translation invarianceTranslation invariance

II constant constant

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Amiel-Cowell (1999) approach

BC

Irene's income

Jan

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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

A “fair” tax?

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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Translation independence

Example 2.

Equal absolute additions or subtractions keep inequality constant

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Intermediate case

Example 3.

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

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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 measures

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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 the Normalised area under the Lorenz curveLorenz 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...

x*

N1 N2

0

x**N1

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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Inequality measures

Inequality axioms




Performance of inequality measures

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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

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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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Absolute vs Relative measures

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Inequality measures and US experience

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19

65

19

70

19

75

19

80

19

85

19

90

19

95

20

00

20

05

GiniGE0GE1A.25A.50A.75

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References AmielAmiel, Y. and Cowell, F. A. (1999), Y. and Cowell, F. A. (1999) Thinking about InequalityThinking about Inequality, Cambridge , Cambridge

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

Economic Theory, 2, 244-263 Atkinson, A. B. and Brandolini. A. (2004) “Global World Inequality:

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

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

Cowell, F. A. (2006)Cowell, F. A. (2006) “Theil, Inequality Indices and Decomposition,” “Theil, Inequality Indices and Decomposition,” Research Research on Economic Inequalityon Economic Inequality, , 1313, 345-360, 345-360

Piketty, T. and E. Saez (2003) “Income inequality in the United States, 1913-1998,” Quarterly Journal of Economics, 118, 1-39.

ShorrocksShorrocks, A. F. (1983), A. F. (1983) “Ranking Income Distributions,” “Ranking Income Distributions,” EconomicaEconomica, , 5050, 3-17, 3-17 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

Documents

Frank Cowell: TU Lisbon – Inequality & Poverty Inequality Measurement July 2006 Inequality measurement Measurement Technical University of Lisbon Frank