frank cowell: tu lisbon – inequality & poverty inequality: advanced topics july 2006...

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Inequality: Advanced Topics

July 2006 July 2006

Inequality and Poverty Measurement Inequality and Poverty 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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Overview...

Introduction

Inequality & responsibility

Deprivation

Complaints


Themes and methodology

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Purpose of lecture We will look at recent theoretical developments We will look at recent theoretical developments

in distributional analysisin distributional analysis Consider some linked themes Consider some linked themes

alternative approaches to inequalityalternative approaches to inequality related welfare conceptsrelated welfare concepts

Use ideas from sociology and philosophyUse ideas from sociology and philosophy Focus on the way modern methodology is Focus on the way modern methodology is

appliedapplied

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Overview...

Introduction


Deprivation

Complaints


An alternative approach

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Responsibility

Standard approach to case for redistributionStandard approach to case for redistribution Use reference point of equalityUse reference point of equality How effective is tax/benefit system in moving actual How effective is tax/benefit system in moving actual

distribution toward reference point?distribution toward reference point?

Does not take account of individual responsibility Does not take account of individual responsibility Role of individual actionsRole of individual actions The responsibility “cut” The responsibility “cut” Dworkin (1981a, 1981b)Dworkin (1981a, 1981b)

Distinguish between Distinguish between things that are your fault things that are your fault things for which you deserve compensationthings for which you deserve compensation

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Responsibility and redistribution

Should affect the evaluation of distributionsShould affect the evaluation of distributions Both case for redistribution...Both case for redistribution... ... and effectiveness of taxation.... and effectiveness of taxation.

Need to differentiate between Need to differentiate between characteristics for which people can be held characteristics for which people can be held

responsibleresponsible characteristics for which people should notcharacteristics for which people should not

Assume that these characteristics are known Assume that these characteristics are known and agreed...and agreed...

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Each person Each person ii has a vector of attributes has a vector of attributes a aii:: Attributes partitioned into two classesAttributes partitioned into two classes RR-attributes: for which the individual is responsible-attributes: for which the individual is responsible CC-attributes: for which the individual may be compensated-attributes: for which the individual may be compensated

Situation before intervention:Situation before intervention: Determined by income function Determined by income function ff ff maps attributes into incomes maps attributes into incomes ff((aaii)) Only person Only person ii’s attributes involved’s attributes involved

Situation after intervention:Situation after intervention: Determined by distribution rule Determined by distribution rule FF We need to compare fairness of outcomes from We need to compare fairness of outcomes from ff and and FF..

Basic structure

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Distribution rule

Also assume that the rule Also assume that the rule F F is anonymousis anonymous

The rule The rule FF: : depends on whole profile of attributesdepends on whole profile of attributes maps the attributes into income of maps the attributes into income of ii..

Assume feasibility:Assume feasibility:Profile of attributes

But what other principles should the rule But what other principles should the rule FF satisfy? satisfy?

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Responsibility: Principle EIER Bossert and Fleurbaey (1996)Bossert and Fleurbaey (1996) EEqual qual IIncome for ncome for EEqual qual RResponsibilityesponsibility

Focus on distribution itselfFocus on distribution itself Full compensationFull compensation

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Responsibility: Principle ETEC

EEqual qual TTransfers for ransfers for EEqual qual CC-attributes-attributes Focus on Focus on changeschanges in distribution in distribution Strict CompensationStrict Compensation

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A difficulty

Fleurbaey (1995a,b) In this special case... ...a natural redistribution mechanism

For large populations...For large populations... EIER and ETEC are incompatible except for...EIER and ETEC are incompatible except for... Additive separability:Additive separability:

Consider two compromise approaches

Consider two compromise approaches

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Compromise (1)

Insist on Full compensation (EIER) Weaken ETEC Egalitarian-equivalent mechanisms

Every agent has a post-tax income equal to the pre-tax income earned given reference compensation

characteristics plus... a uniform transfer

Reference profile

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Compromise (2)

Insist on strict compensation (ETEC) Weaken EIER Conditionally egalitarian mechanismsConditionally egalitarian mechanisms

Every agent Every agent kk is guaranteed the average income of a is guaranteed the average income of a hypothetical economyhypothetical economy In this economy all agents have characteristics equal to In this economy all agents have characteristics equal to

reference profilereference profile

Reference profile

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Application

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

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 approachuse the norm income distribution from the responsibility approach

Devooght (2005) bases this on bases this on Cowell (1985) Cowell approach based on Theil’s conditional entropyCowell approach based on Theil’s conditional entropy Instead of looking at information content in going from perfect equality to Instead of looking at information content in going from perfect equality to

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

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Overview...

Introduction


Deprivation

Complaints


An economic interpretation of a sociological concept

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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 Cowell (2005) see 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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Common structure

These assumptions underlie several problemsThese assumptions underlie several problems Already seen this with poverty axiomatisationAlready seen this with poverty axiomatisation Ebert and Moyes (2002)Ebert and Moyes (2002)

Apply this to other issues in distributional analysis Apply this to other issues in distributional analysis Individual deprivationIndividual deprivation Aggregate deprivationAggregate deprivation Inequality and complaintsInequality and complaints

Need to endow each individual problem withNeed to endow each individual problem with Ethical assumptionsEthical assumptions Reference level of incomeReference level of income

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Individual deprivation The The YitzhakiYitzhaki (1979) (1979) definition definition

Equivalent formEquivalent form

In present notationIn present notation

Use the conditional mean Use the conditional mean

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Deprivation: Axiomatic approach 1

The Better-than set for The Better-than set for ii

Focus Focus works like the poverty conceptworks like the poverty concept

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Deprivation: Axiomatic approach 2 NormalisationNormalisation

Additivity Additivity works like the independence axiomworks like the independence axiom

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Bossert-D’Ambrosio (2006)

This is just the Yitzhaki individual deprivation This is just the Yitzhaki individual deprivation index index

There is an alternative axiomatisation There is an alternative axiomatisation Ebert-Moyes (Economics Letters 2000)Ebert-Moyes (Economics Letters 2000) Different structure of reference groupDifferent structure of reference group

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Aggregate deprivation Simple approach: just sum individual deprivationSimple approach: just sum individual deprivation

Could consider an ethically weighted variantCould consider an ethically weighted variant Chakravarty and Chakraborty (1984) ChakravartyChakravarty and and MukherjeeMukherjee (1999b) (1999b)

As with poverty consider relative as well as As with poverty consider relative as well as absolute indices…absolute indices…

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Aggregate deprivation (2) An ethically weighted relative indexAn ethically weighted relative index

ChakravartyChakravarty and and MukherjeeMukherjee (1999a) (1999a)

One based on the generalised-GiniOne based on the generalised-Gini Duclos and Duclos and GrégoireGrégoire (2002) (2002)

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Overview...

Introduction


Deprivation

Complaints


Reference groups and distributional judgments

•Model•Inequality results•Rankings and welfare

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The Temkin approach

Larry Temkin (1986, 1993) approach to inequalityLarry Temkin (1986, 1993) approach to inequality UnconventionalUnconventional Not based on utilitarian welfare economicsNot based on utilitarian welfare economics But not a complete “outlier” But not a complete “outlier”

Common ground with other distributional analysisCommon ground with other distributional analysis PovertyPoverty deprivationdeprivation

Contains the following elements:Contains the following elements: Concept of a complaintConcept of a complaint The idea of a reference groupThe idea of a reference group A method of aggregationA method of aggregation

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What is a “complaint?”

Individual’s relationship with the income Individual’s relationship with the income distributiondistribution

The complaint exists independentlyThe complaint exists independently does not depend on how people feeldoes not depend on how people feel does not invoke “utility” or (dis)satisfaction does not invoke “utility” or (dis)satisfaction

Requires a reference groupRequires a reference group effectively a reference incomeeffectively a reference income a variety of specifications a variety of specifications see also see also DevooghtDevooght (2003) (2003)

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Types of reference point

BOPBOP The Best-Off PersonThe Best-Off Person Possible ambiguity if there is more than onePossible ambiguity if there is more than one By extension could consider the best-off groupBy extension could consider the best-off group

AVEAVE The AVErage incomeThe AVErage income Obvious tie-in with conventional inequality measuresObvious tie-in with conventional inequality measures A conceptual difficulty for those above the mean?A conceptual difficulty for those above the mean?

ATBOATBO All Those Better OffAll Those Better Off A “conditional” reference pointA “conditional” reference point

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Aggregation

The complaint is an individual phenomenon.The complaint is an individual phenomenon. How to make the transition from this to society as How to make the transition from this to society as

a whole?a whole? Temkin makes two suggestions:Temkin makes two suggestions: Simple sumSimple sum

Just add up the complaintsJust add up the complaints Weighted sumWeighted sum

Introduce distributional weights Introduce distributional weights Then sum the weighted complaintsThen sum the weighted complaints

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The BOP Complaint Let Let rr((xx) be the first richest person you find in ) be the first richest person you find in NN.. Person Person rr (and higher) has income (and higher) has income xxnn..

For “lower” persons, natural definition of complaint:For “lower” persons, natural definition of complaint:

Similar to fundamental difference for poverty:Similar to fundamental difference for poverty:

Now we replace “Now we replace “pp” with “” with “rr””

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BOP-Complaint: Axiomatisation

Use same structural axioms as before. Plus…Use same structural axioms as before. Plus… Monotonicity: income increments reduce complaintMonotonicity: income increments reduce complaint

IndependenceIndependence

NormalisationNormalisation

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Overview...

Introduction


Deprivation

Complaints


A new approach to inequality


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Implications for inequality

Broadly two types of axioms with different roles.Broadly two types of axioms with different roles. Axioms on structure: Axioms on structure:

use these to determine the “shape” of the measures. use these to determine the “shape” of the measures. Transfer principles and properties of measures: Transfer principles and properties of measures:

use these to characterise ethical nature of measures use these to characterise ethical nature of measures

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A BOP-complaint class The Cowell-Ebert (SCW 2004) resultThe Cowell-Ebert (SCW 2004) result

Similarity of form to FGTSimilarity of form to FGT Characterises a family of distributions …Characterises a family of distributions …

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The transfer principle Do BOP-complaint measures satisfy the transfer Do BOP-complaint measures satisfy the transfer

principle?principle? If transfer is from richest, yesIf transfer is from richest, yes But if transfers are amongst hoi polloi, maybe not But if transfers are amongst hoi polloi, maybe not

Cowell-Ebert (SCW 2004):Cowell-Ebert (SCW 2004):

Look at some examples that satisfy thisLook at some examples that satisfy this

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Inequality contours

To examine the properties of the derived indices…To examine the properties of the derived indices… ……take the case take the case nn = 3 = 3 Draw contours of Draw contours of TT––inequality inequality

Note that both the sensitivity parameter Note that both the sensitivity parameter and the weights and the weights ww are of interest… are of interest…

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Inequality contours (=2)

w1=0.5 w2=0.5

•Now change the weights…

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Inequality contours (=2)

w1=0.75 w2=0.25

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Inequality contours (= 1)

w1=0.75 w2=0.25

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By contrast: Gini contours

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Inequality contours (= 0)

w1=0.5 w2=0.5

Again change the weights…Again change the weights…

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Inequality contours (= –1)

w1=0.75 w2=0.25

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Inequality contours (= –1)

w1=0.5 w2=0.5

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Special cases

If If then inequality just becomes the range, then inequality just becomes the range, xxnn––xx1 1

.. If If –– then inequality just becomes the “upper- then inequality just becomes the “upper-

middle class” complaint: middle class” complaint: xxnn––xxn-n-1 1 . .

If If = 1 then inequality becomes a generalised = 1 then inequality becomes a generalised absolute Gini.absolute Gini.

“triangles”“triangles”

“Y-shapes”“Y-shapes”

HexagonsHexagons

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Which is more unequal?

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

A

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

B

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Focus on one type of BOP complaint

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

A

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

B

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Orthodox approach

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

A

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

B

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T – inequality

16

17

18

19

20

21

22

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

ineq

ualit

y

A: (2,5,9,20,30)B: (2,6,9,19,30)

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The “sequence”

Temkin’s seminal contributions offer an intuitive approach Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality.to considering changes in inequality.

Take a simple model of a ladder with just two rungs. Take a simple model of a ladder with just two rungs. The rungs are fixed, but the numbers on them are not.The rungs are fixed, but the numbers on them are not. Initially everyone is on the upper rung. Initially everyone is on the upper rung. Then, one by one, people are transferred to the lower rung.Then, one by one, people are transferred to the lower rung.

Start with Start with mm = 0 on lower rung = 0 on lower rung Carry on until Carry on until mm = = nn on lower rung on lower rung

What happens to inequality? What happens to inequality? Obviously zero at the two endpoints of the sequenceObviously zero at the two endpoints of the sequence But in between?But in between?

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The “sequence” (2) For the case of For the case of TT––inequality we haveinequality we have

This is increasing in This is increasing in mm if if > 0 > 0 For other cases there is a degenerate sequence in the For other cases there is a degenerate sequence in the

same directionsame direction

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Overview...

Introduction


Deprivation

Complaints


A replacement for the Lorenz order?


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Rankings

Move beyond simple inequality measuresMove beyond simple inequality measures The notion of complaint can also be used to generate a The notion of complaint can also be used to generate a

ranking principle that can be applied quite generally.ranking principle that can be applied quite generally. This is rather like the use of Lorenz curves to specify a This is rather like the use of Lorenz curves to specify a

Lorenz ordering that characterises inequality comparisons.Lorenz ordering that characterises inequality comparisons. Also similar to poverty rankings with arbitrary poverty Also similar to poverty rankings with arbitrary poverty

lines.lines.

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Cumulative complaints Define cumulative complaintsDefine cumulative complaints

Gives the CCC Gives the CCC cumulative-complaint contourcumulative-complaint contour Just like TIP / Poverty profileJust like TIP / Poverty profile

Use this to get a ranking Use this to get a ranking principleprinciple

i/n

r(x) / n

K(x)

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Complaint-ranking The class of BOP-complaint indicesThe class of BOP-complaint indices

Define complaint rankingDefine complaint ranking

Like the generalised-Lorenz resultLike the generalised-Lorenz result

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Social welfare again Temkin’s complaints approach to income Temkin’s complaints approach to income

distribution was to be viewed in terms of “better” distribution was to be viewed in terms of “better” or “worse”or “worse”

Not just “less” or “more” inequality. Not just “less” or “more” inequality. Can incorporate the complaint-inequality index in a Can incorporate the complaint-inequality index in a

welfare-economic framework:welfare-economic framework:

Linear approximation:Linear approximation:

Total incomeTotal incomeInequalityInequality

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Welfare contours (φ=1)

Irene’s income

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Welfare contours (φ<1)

Irene’s income

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Welfare contours (φ>1)

Irene’s income

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Meade’s “superegalitarianism”

Meade’s “superegalitarianism”

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The ATBO Complaint Again, a natural definition of complaint:Again, a natural definition of complaint:

Similar to fundamental difference for deprivation:Similar to fundamental difference for deprivation:

Use this complaint in the Temkin classUse this complaint in the Temkin class

Get a form similar to Chakravarty deprivationGet a form similar to Chakravarty deprivation

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Summary: complaints ““Complaints” provide a useful basis for inequality Complaints” provide a useful basis for inequality

analysis.analysis. Intuitive links with poverty and deprivation as Intuitive links with poverty and deprivation as

well as conventional inequality. well as conventional inequality. BOP extension provides an implementable BOP extension provides an implementable

inequality measure.inequality measure. CCCs provide an implementable ranking principleCCCs provide an implementable ranking principle

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References (1) Bossert, W. and C. D’Ambrosio (2006) “Reference groups and individual deprivation,”

Economics Letters, 90, 421-426 Bossert, W. and M. Fleurbaey (1996) “Redistribution and compensation,” Social Choice

and Welfare, 13, 343-355. Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative deprivation,”

Economics Letters, 14, 283-287 Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their

meaning in terms of social satisfaction.” Theory and Decision 47, 89-100 Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by

deprivation orderings,” Social Indicators Research 46, 125-135.. Cowell, F. A. (1985) “The measurement of distributional change: an axiomatic

approach.” Review of Economic Studies, 52, 135.151. Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis

Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and

Welfare 23, 71-89. Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and

empirics,” Economics and Philosophy, 19, 241 - 263,

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References (2) Devooght, K. (2005) “To each the same and to each his own. A proposal to measure

responsibility-sensitive income inequality,” Working paper, University of Kortrijk. Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the

measurement of poverty,” measurement of poverty,” Review of Income and WealthReview of Income and Wealth 4848, 471-492. , 471-492. Dworkin, R. (1981a) “What is equality? Part I: Equality of welfare.” Philosophy and

Public Affairs, 10, 185- 246. Dworkin, R. (1981b) “What is equality? Part I: Equality of resources.” Philosophy and

Public Affairs, 10, 283-345. Dutta, B. and D. Ray (1989) “A concept of egalitarianism under participation

constraints” Econometrica, 57, 615.635. Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of

individual deprivation. individual deprivation. Economics LettersEconomics Letters 6868, 263-270., 263-270. Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer-Thorbecke Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer-Thorbecke

poverty orderings,” poverty orderings,” Journal of Public Economic TheoryJournal of Public Economic Theory 44, 455-473., 455-473. Fleurbaey, M. (1995a) “Equal opportunity or equal social outcome?” Economics and

Philosophy 11, 25-55.

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References (3) Fleurbaey, M. (1995b) “Equality and responsibility,” European Economic

Review, 39, 683-689. Fleurbaey, M. (1995c) “Three solutions to the compensation problem,”

Journal of Economic Theory, 65, 505-521. Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable

poverty measures,” Econometrica, 52, 761-776 Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an

analysis of UK poverty trends,” Oxford Economic Papers, 49, 317-327. Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, 99-121. Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press. Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly

Journal of Economics 93, 321.324.

frank cowell: tu lisbon – inequality & poverty inequality: advanced topics july 2006...

Documents

distribution rule f

methodology slide

vector of attributes

profile of attributes

uklisbon2006 slide

compensation slide

alternative approach

responsible cattributes