Frank Cowell: EC513 Public Economics EC513 PhD Public Economics 2005/6 Deprivation, Complaints and...

Frank Cow

ell: Frank C

owell: EC

513 EC

513 Public Economics

Public Economics

EC513 PhD Public Economics 2005/6

http://darp.lse.ac.uk/EC513.htm

Deprivation, Complaints and Inequality Deprivation, Complaints and Inequality

7 March 2006

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Overview...

Introduction

Poverty

Deprivation

Complaints

Deprivation, complaints, inequality

Themes and methodology

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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

ell: Frank C

owell: EC

513 EC


Public Economics

Themes Cross-disciplinary conceptsCross-disciplinary concepts Income differencesIncome differences Reference incomesReference incomes Formal methodologyFormal methodology

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Methodology Exploit common structureExploit common structure

povertypoverty deprivationdeprivation complaints and inequalitycomplaints and inequality see Cowell (2005) see Cowell (2005)

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

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

“Structural” axioms Take some social evaluation function Take some social evaluation function ContinuityContinuity

Linear homogeneityLinear homogeneity

Translation invarianceTranslation invariance

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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*

•

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Overview...

Introduction

Poverty

Deprivation

Complaints


An alternative approach

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Poverty concepts Given poverty line Given poverty line zz

a reference pointa reference point HeadcountHeadcount

p(x,z)/n Poverty gapPoverty gap

fundamental income differencefundamental income difference

Foster et al (1984) poverty index Foster et al (1984) poverty index Cumulative poverty gapCumulative poverty gap

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

TIP / Poverty profile

i/n

p(x,z)/n

G(x,z)

0

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 Shorrocks 1983)1983)

TIP dominance implies unambiguously greater poverty

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Poverty: Axiomatic approach Characterise an ordinal poverty index Characterise an ordinal poverty index PP((xx , ,zz))

See Ebert and See Ebert and MoyesMoyes (2002) (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.

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



Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Poverty: The key axioms Adapt standard axioms from social welfare Adapt standard axioms from social welfare

anonymityanonymity independenceindependence monotonicitymonotonicity

income increments reduce povertyincome increments reduce poverty 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

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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

Scale invariance now becomesScale invariance now becomes

Independence means:Independence means:

Define the number of the poor asDefine the number of the poor as

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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

Frank Cow

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owell: EC

513 EC


Public Economics

Brief conclusion

Poverty indexes can be constructed from Poverty indexes can be constructed from scratch scratch

Use standard axiomsUse standard axioms Exploit the poverty line as a reference pointExploit the poverty line as a reference point Impose structureImpose structure

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Overview...

Introduction

Poverty

Deprivation

Complaints


An economic interpretation of a sociological concept

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Individual deprivation The The Yitzhaki (1979) Yitzhaki (1979) definition definition

Equivalent formEquivalent form

In present notationIn present notation

Use the conditional mean Use the conditional mean

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

Deprivation: Axiomatic approach 2 NormalisationNormalisation

Additivity Additivity works like the independence axiomworks like the independence axiom

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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: EC

513 EC


Public Economics

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) Chakravarty and Mukherjee (1999b)Chakravarty and Mukherjee (1999b)

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

Frank Cow

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owell: EC

513 EC


Public Economics

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

Chakravarty and Mukherjee (1999a) Chakravarty and Mukherjee (1999a)

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

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Overview...

Introduction

Poverty

Deprivation

Complaints


Reference groups and distributional judgments

•Model•Inequality results•Rankings and welfare

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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 Devooght (2003) Devooght (2003)

Frank Cow

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owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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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513 EC


Public Economics

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

ell: Frank C

owell: EC

513 EC


Public Economics

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: EC

513 EC


Public Economics

Overview...

Introduction

Poverty

Deprivation

Complaints


A new approach to inequality


Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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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513 EC


Public Economics

Inequality contours (=2)

w1=0.5 w2=0.5

•Now change the weights…

Frank Cow

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513 EC


Public Economics

Inequality contours (=2)

w1=0.75 w2=0.25

Frank Cow

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513 EC


Public Economics

Inequality contours (= 1)

w1=0.75 w2=0.25

Frank Cow

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owell: EC

513 EC


Public Economics

By contrast: Gini contours

Frank Cow

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owell: EC

513 EC


Public Economics

Inequality contours (= 0)

w1=0.5 w2=0.5

Again change the weights…Again change the weights…

Frank Cow

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owell: EC

513 EC


Public Economics

Inequality contours (= –1)

w1=0.75 w2=0.25

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

Inequality contours (= –1)

w1=0.5 w2=0.5

Frank Cow

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owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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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513 EC


Public Economics

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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Public Economics

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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513 EC


Public Economics

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

ell: Frank C

owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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

ell: Frank C

owell: EC

513 EC


Public Economics

Overview...

Introduction

Poverty

Deprivation

Complaints


A replacement for the Lorenz order?


Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

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)

Frank Cow

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owell: EC

513 EC


Public Economics

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

Frank Cow

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owell: EC

513 EC


Public Economics

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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owell: EC

513 EC


Public Economics

Welfare contours (φ=1)

A’s income

B’s

inco

me

Frank Cow

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owell: EC

513 EC


Public Economics

Welfare contours (φ<1)

A’s income

B’s

inco

me

Frank Cow

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owell: EC

513 EC


Public Economics

Welfare contours (φ>1)

A’s income

B’s

inco

me

Meade’s “superegalitarianism”

Frank Cow

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owell: EC

513 EC


Public Economics

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

ell: Frank C

owell: EC

513 EC


Public Economics

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

Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

References (1) BossertBossert, W. and C. , W. and C. D’AmbrosioD’Ambrosio (2004) (2004) “Reference groups and individual “Reference groups and individual

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

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

Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” by deprivation orderings,” Social Indicators ResearchSocial Indicators Research 4646, 125-135.., 125-135..

Cowell, F. A. (2005)Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” “Gini, Deprivation and Complaints,” Distributional Distributional Analysis Discussion PaperAnalysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, , 84, STICERD, LSE, Houghton St., London, WC2A 2AE. WC2A 2AE.

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

DevooghtDevooght, K. (2003), K. (2003) “Measuring inequality by counting ‘complaints:’ theory “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. Duclos, J.-Y. and P. GrégoireGrégoire (2002) (2002) “Absolute and relative deprivation and “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.

http://darp.lse.ac.uk/papersdb/Bossert-d'Ambrosio_(Montreal2004-10).pdf




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




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

http://darp.lse.ac.uk/papersdb/Cowell-Ebert_(SCW04).pdf

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

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

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



Frank Cow

ell: Frank C

owell: EC

513 EC


Public Economics

References (2) 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) “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)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. 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.

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

Frank Cowell: EC513 Public Economics EC513 PhD Public Economics 2005/6 Deprivation, Complaints and...

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