Frank Cowell: Oviedo – Inequality & Poverty Deprivation, Complaints and Inequality March 2007 Inequality, Poverty and Income Distribution University of.

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Deprivation, Complaints and Inequality

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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Overview...

Introduction

Poverty

Deprivation

Complaints

Deprivation, complaints, inequality

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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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 (2007)Cowell (2007)

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

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Basic components Income distribution: Income distribution: xx

an an nn-vector-vector population of size population of size nn person person ii has income has income xxii

Space of all income distributions: Space of all income distributions: DD RRnn

specification of this captures nature of income specification of this captures nature of income include zeros? negatives? include zeros? negatives?

An evaluation function An evaluation function :: D D → → RR

Axioms of two broad types of axiomAxioms of two broad types of axiom to impose standard structureto impose standard structure to give meaning to a particular economic problemto give meaning to a particular economic problem

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“Structural” axioms

Take some social evaluation function Take some social evaluation function welfarewelfare inequalityinequality povertypoverty

Axiom 1 (Continuity)Axiom 1 (Continuity). . is a continuous function is a continuous function DD→→RR..

Axiom 2 (Linear homogeneity).Axiom 2 (Linear homogeneity). For all For all xxDD and and > 0: > 0: ((xx) = ) = ((xx))

Axiom 3 (Translation independencAxiom 3 (Translation independence).e). For all For all xxDD and such that and such that RR such that such that xx1 1 DD ((xx11) = ) = ((xx))

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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

0

1•

x*

•

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Overview...

Introduction

Poverty

Deprivation

Complaints


An alternative approach

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Poverty concepts (1)

The poverty line The poverty line zz a reference pointa reference point exogenously givenexogenously given

Define the number of the poor:Define the number of the poor: ((xx, z, z) := #{) := #{ii:: x xii ≤≤ z z}}

Proportional headcountProportional headcount ((xx, z, z)/)/nn

Poverty gapPoverty gap fundamental income differencefundamental income difference ggii((xx, z, z) = max (0, ) = max (0, z z x xii))

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Poverty concepts (2) Foster et al (1984)Foster et al (1984) poverty index poverty index

≥≥ 0 is a sensitivity parameter0 is a sensitivity parameter

Cumulative poverty gapCumulative poverty gap

counterpart to income cumulationscounterpart to income cumulations

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TIP / Poverty profile

i/n

(x,z)/n

G(x,z)

0

Cumulative gaps versus Cumulative gaps versus population proportionspopulation proportions

Proportion of poorProportion of poor TIP curveTIP curve

Cumulative gaps versus Cumulative gaps versus population proportionspopulation proportions

Proportion of poorProportion of poor TIP curveTIP curve

TIP curves have same interpretation as GLC (Shorrocks 1983)

TIP dominance implies unambiguously greater poverty

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Poverty: Axiomatic approach

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

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

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

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

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Poverty: The key axioms

Adapt standard axioms from social welfare Adapt standard axioms from social welfare anonymityanonymity independenceindependence monotonicitymonotonicity

Strengthen two other axiomsStrengthen two other axioms scale invariancescale invariance translation invariancetranslation invariance

Also need continuityAlso need continuity Plus a Plus a focusfocus axiom axiom

income changes only affect poverty…income changes only affect poverty… ……if they concern the incomes of those where if they concern the incomes of those where i i ≤≤

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A closer look at the axioms Let Let DD denote the set of ordered income vectors denote the set of ordered income vectors The The monotonicity axiommonotonicity axiom is is

for for xx DD, , > 0 and > 0 and xxii ≤≤ zz: : PP((xx11, , xx22,…, ,…, xxii + + …… , z , z) < ) < PP((xx11, , xx22,…, ,…, xxii , , …… , z , z) )

The The focus axiomfocus axiom is is for for xx DD and and xxii > > zz, , PP is constant in is constant in xxii

Scale invariance now becomesScale invariance now becomes if if PP((xx, , zz) = ) = PP((yy, , zz) then ) then PP((xx, , zz) = ) = PP((yy, , z z ))

Independence means:Independence means: consider consider x,yx,y DD such that such that PP((xx, , zz) = ) = PP((yy, , zz) where, for some ) where, for some i i ≤≤

,, xxii = = yyii; then, for any ; then, for any xxºº such that such that xxii─1─1≤ ≤ xxºº≤≤ xxii+1+1 and and yyii─1─1≤ ≤ xxº º ≤≤ yyii+1+1

PP((xx11, , xx22, …, , …, xxii─1─1, , xxºº, , xxii+1+1,…,,…,xxnn, , zz) = ) = PP((yy11, , yy22, …, , …, yyii─1─1, , xxºº, , yyii+1+1,…,,…,yynn, , zz))

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Ebert-Moyes (2002)

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

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

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Poverty: lessons

Poverty indexes can be constructed from scratch Poverty indexes can be constructed from scratch Exploit the poverty line as a reference pointExploit the poverty line as a reference point Use standard axiomsUse standard axioms

applied to applied to nn+1 incomes+1 incomes

Impose structureImpose structure independenceindependence scale invariancescale invariance

Axioms to give meaningAxioms to give meaning monotonicitymonotonicity focusfocus

Use the same method in other areasUse the same method in other areas

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Overview...

Introduction

Poverty

Deprivation

Complaints


An economic interpretation of a sociological concept

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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 and Moyes (2000)Ebert and Moyes (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 transformed variantCould consider an ethically transformed variant

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

Chakravarty and Chakraborty (1984)Chakravarty and Chakraborty (1984) Chakravarty and Mukherjee (1999a) Chakravarty and Mukherjee (1999a) (1999b)(1999b)

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Aggregate deprivation (2) Alternative approachAlternative approach Based aggregate deprivation on the generalised-Based aggregate deprivation on the generalised-

GiniGini

where where wwii are positional weightsare positional weights

Duclos and Duclos and GrégoireGrégoire (2002) (2002)

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Overview...

Introduction

Poverty

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, there is a natural definition of For “lower” persons, there is a natural definition of complaint:complaint: kkii((xx) := ) := xxnn x xii

Similar to fundamental difference for poverty:Similar to fundamental difference for poverty: ggii((xx, z, z) = max (0, ) = max (0, z z x xii))

Other similarities:Other similarities: replace “replace “” with “” with “rr” ” instead of the last poor person we now have the first rich personinstead of the last poor person we now have the first rich person

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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

Poverty

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

Poverty

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 distribution was to Temkin’s complaints approach to income distribution was to

be viewed in terms of “better” or “worse”be viewed in terms of “better” or “worse” Not just “less” or “more” inequality. Not just “less” or “more” inequality. Can incorporate the complaint-inequality index in a welfare-Can incorporate the complaint-inequality index in a welfare-

economic framework: economic framework: WW((xx) = ) = ((XX, , TT)) XX: total income: total income TT: Temkin inequality: Temkin inequality

Linear approximation:Linear approximation: WW((xx) = ) = XX φφTT φφ is the weight attached to inequality in welfare is the weight attached to inequality in welfare gives three types of distinct pattern:gives three types of distinct pattern:

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Welfare contours (φ=1)

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Welfare contours (φ < 1)

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Welfare contours (φ > 1)

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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 Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative

deprivation,” Economics Letters, 14, 283-287 ChakravartyChakravarty, S. R. and D. , S. R. and D. MukherjeeMukherjee (1999a) (1999a) “Measures of deprivation and “Measures of deprivation and

their meaning in terms of social satisfaction.” their meaning in terms of social satisfaction.” Theory and DecisionTheory and Decision 47, 89-100 47, 89-100 ChakravartyChakravarty, S. R. and D. , S. R. and D. MukherjeeMukherjee (1999b) (1999b) “Ranking income distributions “Ranking income distributions

by deprivation orderings,” by deprivation orderings,” Social Indicators ResearchSocial Indicators Research 4646, 125-135.., 125-135.. Cowell, F. A. (2007)Cowell, F. A. (2007) in Betti, G. and Lemmi, A. (ed.) Advances in income

inequality and concentration measures, Routledge, London. Chapter 3. Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice Social Choice

and Welfareand Welfare 2323, 71-89. , 71-89. Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory

and empirics,” and empirics,” Economics and PhilosophyEconomics and Philosophy, , 1919, 241 - 263,, 241 - 263, Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and

the measurement of poverty,” the measurement of poverty,” Review of Income and WealthReview of Income and Wealth 4848, 471-492. , 471-492.

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References (2) Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s Ebert, U. and P. Moyes (2000). 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. Moyes (2002) “A simple axiomatization of the Foster-Greer-Ebert, U. and P. Moyes (2002) “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) “A class of decomposable Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” poverty measures,” EconometricaEconometrica, , 5252, 761-776, 761-776

Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an Jenkins, S. P. and Lambert, P. J. (1997) “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.

Shorrocks, A. F. (1983) “Ranking Income Distributions,” Shorrocks, A. F. (1983) “Ranking Income Distributions,” EconomicaEconomica, , 5050, 3-17, 3-17 Temkin, L. S. (1986) “Inequality.” Temkin, L. S. (1986) “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. Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Quarterly

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

Frank Cowell: Oviedo – Inequality & Poverty Deprivation, Complaints and Inequality March 2007 Inequality, Poverty and Income Distribution University of.

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