frank cowell: oviedo – inequality & poverty deprivation, complaints and inequality march 2007...
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Frank C
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Frank C
owell: O
viedo – Inequality & P
overty O
viedo – Inequality & P
overty
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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viedo – Inequality & P
overty O
viedo – Inequality & P
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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
Deprivation, complaints, inequality
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
Deprivation, complaints, inequality
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
Deprivation, complaints, inequality
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
Deprivation, complaints, inequality
A new approach to inequality
•Model•Inequality results•Rankings and welfare
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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
Deprivation, complaints, inequality
A replacement for the Lorenz order?
•Model•Inequality results•Rankings and welfare
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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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owell: O
viedo – Inequality & P
overty O
viedo – Inequality & P
overty
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