inequality: advanced topics

Frank Cow

ell: Frank C

owell: TU

Lisbon – Inequality & Poverty

TU Lisbon – Inequality &

Poverty

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

Frank Cow

ell: Frank C

owell: TU



Poverty

Overview...

Introduction

Deprivation

Deprivation

Complaints

Inequality: Advanced Topics

Themes and methodology

Frank Cow

ell: Frank C

owell: TU



Poverty

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

Frank Cow

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owell: TU



Poverty

Overview...

Introduction

Deprivation

Deprivation

Complaints

Deprivation, complaints, inequality

An alternative approach

Frank Cow

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owell: TU



Poverty

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

Frank Cow

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Poverty

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...

Frank Cow

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Poverty

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 Has similar flavour to claims problemHas similar flavour to claims problem

Basic structure

Frank Cow

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Poverty

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?

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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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Poverty

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

Frank Cow

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Poverty

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

Frank Cow

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owell: TU



Poverty

Overview...

Introduction

Deprivation

Deprivation

Complaints


An economic interpretation of a sociological concept

Frank Cow

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owell: TU



Poverty

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

http://darp.lse.ac.uk/papersdb/Yitzhaki_(QJE_79).pdf


Frank Cow

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owell: TU



Poverty

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

Frank Cow

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Poverty

Deprivation: Axiomatic approach 2 NormalisationNormalisation

Additivity Additivity works like the independence axiomworks like the independence axiom

Frank Cow

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Poverty

Bossert-D’Ambrosio (2004)

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

Frank Cow

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owell: TU



Poverty

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…

http://darp.lse.ac.uk/papersdb/Chakravarty-Chakraborty_(EL84).pdf




http://darp.lse.ac.uk/papersdb/Chakravarty-Mukherjee_(SIR_1999).pdf




Frank Cow

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Poverty

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)

http://darp.lse.ac.uk/papersdb/Chakravarty-Mukherjee_(TD_1999).pdf




http://darp.lse.ac.uk/papersdb/Duclos-Gregoire_(RIW02).pdf



Frank Cow

ell: Frank C

owell: TU



Poverty

Overview...

Introduction

Deprivation

Deprivation

Complaints


Reference groups and distributional judgments

•Model•Inequality results•Rankings and welfare

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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)

http://darp.lse.ac.uk/papersdb/Devooght_(EconPhil03).pdf

http://darp.lse.ac.uk/papersdb/Devooght_(EconPhil03).pdf

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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””

Frank Cow

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Poverty

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

Frank Cow

ell: Frank C

owell: TU



Poverty

Overview...

Introduction

Deprivation

Deprivation

Complaints


A new approach to inequality


Frank Cow

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owell: TU



Poverty

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

Frank Cow

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Poverty

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 …

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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…

Frank Cow

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Poverty

Inequality contours (=2)

w1=0.5 w2=0.5

•Now change the weights…

Frank Cow

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Poverty

Inequality contours (=2)

w1=0.75 w2=0.25

Frank Cow

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Poverty

Inequality contours (= 1)

w1=0.75 w2=0.25

Frank Cow

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Poverty

By contrast: Gini contours

Frank Cow

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Poverty

Inequality contours (= 0)

w1=0.5 w2=0.5

Again change the weights…Again change the weights…

Frank Cow

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Poverty

Inequality contours (= –1)

w1=0.75 w2=0.25

Frank Cow

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Poverty

Inequality contours (= –1)

w1=0.5 w2=0.5

Frank Cow

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Poverty

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”

“Y-shapes”

Hexagons

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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

Frank Cow

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Poverty

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)

Frank Cow

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Poverty

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?

Frank Cow

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Poverty

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

Frank Cow

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Poverty

Overview...

Introduction

Deprivation

Deprivation

Complaints


A replacement for the Lorenz order?


Frank Cow

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owell: TU



Poverty

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.

Frank Cow

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Poverty

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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Poverty

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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Poverty

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 incomeInequality

Frank Cow

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Poverty

Welfare contours (φ=1)

A’s income

B’s

inco

me

Frank Cow

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Poverty

Welfare contours (φ<1)

A’s income

B’s

inco

me

Frank Cow

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Poverty

Welfare contours (φ>1)

A’s income

B’s

inco

me

Meade’s “superegalitarianism”

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Poverty

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

Frank Cow

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Poverty

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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Poverty

References (1) Bossert, W. and M. Fleurbaey (1996) “Redistribution and compensation,” Social Choice

and Welfare, 13, 343-355. Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”. Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”.

Working PaperWorking Paper 2004-10, Département de sciences économiques, Université de 2004-10, Département de sciences économiques, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada.Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada.

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 Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” meaning in terms of social satisfaction.” Theory and DecisionTheory and Decision 47, 89-100 47, 89-100

Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” deprivation orderings,” Social Indicators ResearchSocial Indicators Research 4646, 125-135.., 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,” Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Distributional Analysis Discussion PaperDiscussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. , 84, STICERD, LSE, Houghton St., London, WC2A 2AE.

Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Social Choice and WelfareWelfare 2323, 71-89. , 71-89.

Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” empirics,” Economics and PhilosophyEconomics and Philosophy, , 1919, 241 - 263,, 241 - 263,

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.

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Poverty

References (2) Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative

deprivation and the measurement of poverty,” deprivation and the measurement of poverty,” Review of Income and Review of Income and WealthWealth 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.

Fleurbaey, M. (1995a) “Equal opportunity or equal social outcome?” Economics and Philosophy 11, 25-55.

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.

http://darp.lse.ac.uk/papersDB/Dutta-Ray_(1989).pdf

http://darp.lse.ac.uk/papersDB/Dutta-Ray_(1989).pdf

http://darp.lse.ac.uk/papersDB/Fleurbaey_(EER1995).pdf

http://darp.lse.ac.uk/papersDB/Fleurbaey_(EER1995).pdf

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Poverty

References (3) Ebert, U. and P. Ebert, U. and P. MoyesMoyes (2000) (2000). An axiomatic characterization of Yitzhaki’s . An axiomatic characterization of Yitzhaki’s

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

Thorbecke poverty orderings,” Thorbecke poverty orderings,” Journal of Public Economic TheoryJournal of Public Economic Theory 44, 455-, 455-473.473.

Foster, J. E., Greer, J. and Thorbecke, E. (1984)Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable “A class of decomposable poverty measures,” poverty measures,” EconometricaEconometrica, , 5252, 761-776, 761-776

Jenkins, S. P. and Lambert, P. J. (1997)Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” analysis of UK poverty trends,” Oxford Economic PapersOxford Economic Papers, , 4949, 317-327., 317-327.

ShorrocksShorrocks, A. F. (1983), A. F. (1983) “Ranking Income Distributions,” “Ranking Income Distributions,” EconomicaEconomica, , 5050, 3-17, 3-17 TemkinTemkin, L. S. (1986), L. S. (1986) “Inequality.” “Inequality.” Philosophy and Public AffairsPhilosophy and Public Affairs 15, 99-121. 15, 99-121. Temkin, L. S. (1993) Temkin, L. S. (1993) InequalityInequality. Oxford: Oxford University Press.. Oxford: Oxford University Press. YitzhakiYitzhaki, S. (1979), S. (1979) “Relative deprivation and the Gini coefficient,” “Relative deprivation and the Gini coefficient,” Quarterly Quarterly

Journal of EconomicsJournal of Economics 9393, 321.324. , 321.324.

http://darp.lse.ac.uk/papersdb/Ebert-Moyes_(EL2000).pdf



http://darp.lse.ac.uk/papersDB/Ebert-Moyes_(JPET02).pdf



http://darp.lse.ac.uk/papersdb/Foster_et_al_(Econometrica_84).pdf

http://darp.lse.ac.uk/papersdb/Jenkins-Lambert_(OEP97).pdf

http://darp.lse.ac.uk/papersdb/Shorrocks_(Ecca_83).pdf

http://darp.lse.ac.uk/papersdb/Shorrocks_(Ecca_83).pdf

http://darp.lse.ac.uk/papersdb/Temkin_(PPA86).pdf

http://darp.lse.ac.uk/papersdb/Temkin_(PPA86).pdf



inequality: advanced topics

Documents

equal income

income of i

pretax income

average income

characteristics equal

attributes involvedsituation

posttax income equal

anonymousthe rule f