Frank Cowell: Siena – Inequality Summer School Deprivation, Complaints and Inequality June 2007...

Frank Cow

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owell: Siena – Inequality Sum

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Siena – Inequality Summ

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Deprivation, Complaints and Inequality

June 2007 June 2007

Summer School on InequalitySummer School on InequalityUniversity of SienaUniversity of Siena

Frank CowellFrank Cowellhttp://darp.lse.ac.uk/siena2007http://darp.lse.ac.uk/siena2007

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Overview...

Introduction

Poverty and inequality

Deprivation

Complaints

Deprivation, complaints, inequality

Themes and methodology

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Purpose of lecture Look at recent theoretical developments in Look at recent theoretical developments in

distributional analysisdistributional 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 Use and reuse common concepts:Use and reuse common concepts:

Income differencesIncome differences Reference incomesReference incomes Formal methodologyFormal methodology

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Methodology Focus on the way modern methodology is appliedFocus on the way modern methodology is applied 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

http://darp.lse.ac.uk/papersdb/Cowell_(DARP84).pdf

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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

could be inequality, poverty, social welfarecould be inequality, poverty, social welfare apply standard axioms on structureapply standard axioms on structure

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))

Illustrate these using an exampleIllustrate these using an example the Absolute Gini coefficientthe Absolute Gini coefficient useful inequality measure: equals Gini useful inequality measure: equals Gini × mean income× mean income

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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

0 1•

x*

•These axioms repeatedly used in the following applications

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Overview...

Introduction


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

Plot Plot GGii against population proportions against population proportions Get Get TIP curveTIP curve (Jenkins and Lambert 1997) (Jenkins and Lambert 1997) TIP: “TIP: “TThree ‘hree ‘I’I’s of s of PPoverty”: (Incidence, Intensity, Inequality)overty”: (Incidence, Intensity, Inequality)

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

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“Three ‘I’s of Poverty”

i/n

(x,z)/n

Gi(x,z)

0

Population proportions versus Population proportions versus cumulative gapscumulative gaps

TIP curveTIP curve Proportion of poorProportion of poor Poverty deficitPoverty deficit

Incidence of poverty is the horizontal distance

•

•

G stays constant if i >

Intensity of poverty is the vertical distance

Inequality among the poor is the curvature of the TIP

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Poverty orderings TIP curves have same interpretation TIP curves have same interpretation

as Generalised Lorenz Curves as Generalised Lorenz Curves GLC-dominance implies welfare GLC-dominance implies welfare

dominancedominance for all monotonic, separable, SWFs for all monotonic, separable, SWFs

satisfying transfer principle satisfying transfer principle (Shorrocks 1983)(Shorrocks 1983)

TIP dominance implies TIP dominance implies unambiguously greater povertyunambiguously greater poverty

holds for given poverty lineholds for given poverty line ……and virtually all poverty and virtually all poverty

measures in usemeasures in use A simple link with inequality A simple link with inequality

orderings and welfareorderings and welfare For details see Zheng (2000)For details see Zheng (2000) 0 •

•

•

•

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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 the standard axioms we introduced earlierUse the standard axioms we introduced earlier

some of these slightly modifiedsome of these slightly modified supplement with axioms to give meaning to povertysupplement with axioms to give meaning to poverty

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.

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

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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

Similarity to certain families of inequality measuresSimilarity to certain families of inequality measures Additivity follows from the independence axiom Additivity follows from the independence axiom

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Poverty and inequality: 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 deprivationdeprivation new approaches to inequalitynew approaches to inequality

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Overview...

Introduction


Deprivation

Complaints


An economic interpretation of a sociological concept

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Relative deprivation Individual Individual deprivationdeprivation

concern with position relative to others in society (Runciman, 1966) negative welfare effects when friends and neighbours become better-off?

Related to income satisfaction? D’Ambrosio and Frick (2007) D’Ambrosio and Frick (2007) focus on E. and W. Germany, 1990-2004 show happiness/satisfaction is a relative notion derive perceived well-being from being richer not simply from being rich

Do all people care about it?Do all people care about it? Ravallion and Lokshin 2005) Ravallion and Lokshin 2005) test for perceived welfare effects of relative test for perceived welfare effects of relative

deprivation in Malawideprivation in Malawi Relative deprivation is not a concern for most people, although it is for the Relative deprivation is not a concern for most people, although it is for the

comparatively well off.comparatively well off. AggregateAggregate deprivation deprivation

a social concepta social concept related to individual deprivationrelated to individual deprivation relationship to inequality and poverty?relationship to inequality and poverty?

Runciman Runciman, 1966 :“If people have no reason to expect or hope for more than they can achieve, they will be less discontent with what they have, or even grateful simply to be able to hold on to it. But if, on the other hand, they have been led to see as a possible goal the relative prosperity of some more fortunate community with which they can directly compare themselves, then they will remain discontent with their lot until they have succeeded in catching up” .

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Individual deprivation: model YitzhakiYitzhaki (1979) (1979) definition of individual deprivatoin: definition of individual deprivatoin:

Can write this in equivalent formCan write this in equivalent form

In discrete notationIn discrete notation

Use the conditional mean Use the conditional mean

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


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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

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

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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

ChakravartyChakravarty and and ChakrabortyChakraborty (1984) (1984) Chakravarty and Mukherjee Chakravarty and Mukherjee (1999a)(1999a) (1999b)(1999b)

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




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

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

where where wwii are positional weightsare positional weights

Duclos and Duclos and GrégoireGrégoire (2002) (2002)

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



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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 (Larry Temkin (1986 1986 , 1993) approach to inequality, 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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A “complaint?”

Involves the individual’s relationship Involves the individual’s relationship with the income distributionwith the income distribution

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

Complaint depends on position in Complaint depends on position in distributiondistribution

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)

Temkin(Temkin 1986, p. 102):“To say that the best-off have nothing to complain about is in no way to impugn their moral sensibilities. They may be just as concerned about the inequality in their world as anyone else. Nor is it to deny that, insofar as one is concerned about inequality, one might have a complaint about them being as well o. as they are. It is only to recognize that, since they are at least as well o. as every other member of their world, they have nothing to complain about. Similarly, to say that the worst-off have a complaint is not to claim that they will in fact complain (they may not). It is only to recognize that it is a bad thing (unjust or unfair) for them to be worse o. than the other members of their world through no fault of their own”

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


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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


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 The Cowell and Ebert (2004)Cowell and Ebert (2004) result 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 transfer principle?Do BOP-complaint measures satisfy transfer 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

From From Cowell and Ebert (2004) Cowell and Ebert (2004) ::

Look at some examples that do/do not satisfy this:Look at some examples that do/do not satisfy this: take the case take the case nn = 3 = 3 draw contours of draw contours of TT––inequality inequality both the sensitivity parameter 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––xx11

contour map becomes a set of trianglescontour map becomes a set of triangles 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 . . contour map becomes a set of “Y-shapes”contour map becomes a set of “Y-shapes”

If If = 1 then inequality is a generalised absolute Gini = 1 then inequality is a generalised absolute Gini contour map is a set of hexagonscontour map is a set of hexagons

Different values of Different values of may give very different rankings may give very different rankings not all concur with orthodox view…not all concur with orthodox view… ……. corresponding to Dalton transfer principle. corresponding to Dalton transfer principle

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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

Two points of view...

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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

B is more unequal?

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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

A is more unequal?

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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

in

equa

lity

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

A is more unequal for high values of

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The “sequence” Temkin also offers an intuitive approach to considering Temkin also offers an intuitive approach to considering

changes in inequalitychanges 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 notThe rungs are fixed, but the numbers on them are not Initially everyone is on the upper rungInitially everyone is on the upper rung

One by one, people are transferred to the lower rungOne 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 generallyranking 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 comparisonsLorenz ordering that characterises inequality comparisons Also similar to poverty rankings with given poverty linesAlso similar to poverty rankings with given poverty 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 TIPJust like TIP

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 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)Ja

net’

s inc

ome

Irene’s income0

ray o

f equali

ty

Two person caseTwo person case Diagram is symmetricDiagram is symmetric xx-values giving constant -values giving constant WW

Similar to “max-min” welfare function

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Welfare contours (φ < 1)Ja

net’

s inc

ome

0 Irene’s income


Similar to Gini case

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Welfare contours (φ > 1)Ja

net’

s inc

ome

0 Irene’s income


Captures “superegalitarianism” (Meade 1976)

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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) BossertBossert, W. and C. , W. and C. D’AmbrosioD’Ambrosio (2006) (2006) “Reference groups and individual deprivation,” “Reference groups and individual deprivation,”

Economics LettersEconomics Letters, , 9090, 421-426, 421-426 ChakravartyChakravarty, S. R. and A. B. , S. R. and A. B. ChakrabortyChakraborty (1984) (1984) “On indices of relative deprivation,” “On indices of relative deprivation,”

Economics Letters,Economics Letters, 1414, 283-287, 283-287 ChakravartyChakravarty, S. R. and D. , S. R. and D. MukherjeeMukherjee (1999a) (1999a) “Measures of deprivation and their “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 ChakravartyChakravarty, S. R. and D. , S. R. and D. MukherjeeMukherjee (1999b) (1999b) “Ranking income distributions by “Ranking income distributions by

deprivation orderings,” deprivation orderings,” Social Indicators ResearchSocial Indicators Research 4646, 125-135.., 125-135.. Cowell, F. A. (2007)Cowell, F. A. (2007) “Gini, Deprivation and Complaints.” “Gini, Deprivation and Complaints.” inin Betti, G. and Lemmi, A. Betti, G. and Lemmi, A.

(ed.) (ed.) Advances in income inequality and concentration measuresAdvances in income inequality and concentration measures , Routledge, London. , Routledge, London. Chapter 3. Chapter 3.

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

D’Ambrosio, C. and J. R. Frick (2007) “Income satisfaction and relative deprivation: an D’Ambrosio, C. and J. R. Frick (2007) “Income satisfaction and relative deprivation: an empirical link,” empirical link,” Social Indicators Research Social Indicators Research 8181, 497–519, 497–519

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

Duclos, J.-Y. and P. Duclos, J.-Y. and P. GrégoireGrégoire (2002) (2002) “Absolute and relative deprivation and the “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.

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.

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References (2) Ebert, U. and P. Moyes (2002)Ebert, U. and P. Moyes (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, 455-473., 455-473. Foster, J. E., Greer, J. and Thorbecke, E. (1984)Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty “A class of decomposable poverty

measures,” measures,” EconometricaEconometrica, , 5252, 761-776, 761-776 Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis

of UK poverty trends,” of UK poverty trends,” Oxford Economic PapersOxford Economic Papers, , 4949, 317-327., 317-327. Meade, J.E. (1976) Meade, J.E. (1976) The Just EconomyThe Just Economy, Allen and Unwin, London, Allen and Unwin, London Ravallion, M. and M. Lokshin (2005) “Who Cares About Relative Deprivation?” Ravallion, M. and M. Lokshin (2005) “Who Cares About Relative Deprivation?”

World Bank Policy Research,Working Paper, 3782World Bank Policy Research,Working Paper, 3782 Runciman, W.G. (1966) Runciman, W.G. (1966) Relative Deprivation and Social JusticeRelative Deprivation and Social Justice , Routledge, London., Routledge, London. 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) Inequality, Inequality, Oxford University Press, Oxford.Oxford University Press, Oxford. YitzhakiYitzhaki, S. (1979), S. (1979) “Relative deprivation and the Gini coefficient,” “Relative deprivation and the Gini coefficient,” Quarterly Journal Quarterly Journal

of Economicsof Economics 9393, 321-324. , 321-324. Zheng, B. (2000) “Poverty orderings,” Zheng, B. (2000) “Poverty orderings,” Journal of Economic SurveysJournal of Economic Surveys, , 1414, 427-466, 427-466

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Frank Cowell: Siena – Inequality Summer School Deprivation, Complaints and Inequality June 2007...

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