Table Of ContentLecture Notes in Economics and Mathematical Systems 685
Antonio Villar
Lectures on
Inequality,
Poverty and
Welfare
Lecture Notes in Economics
and Mathematical Systems
685
FoundingEditors
M.Beckmann
H.P.K€unzi
ManagingEditors
Prof.Dr.G.Fandel
FachbereichWirtschaftswissenschaften
Fernuniversita¨tHagen
Hagen,Germany
Prof.Dr.W.Trockel
MuratSertelInstituteforAdvancedEconomicResearch
IstanbulBilgiUniversity
Istanbul,Turkey
and
Institutf€urMathematischeWirtschaftsforschung(IMW)
Universita¨tBielefeld
Bielefeld,Germany
EditorialBoard
H.Dawid,D.Dimitrov,A.Gerber,C-J.Haake,C.Hofmann,T.Pfeiffer,
R.Slowin´ski,W.H.M.Zijm
More information about this series at http://www.springer.com/series/300
Antonio Villar
Lectures on Inequality,
Poverty and Welfare
AntonioVillar
DepartmentofEconomics
UniversidadPablodeOlavide
Seville,Spain
ISSN0075-8442 ISSN2196-9957 (electronic)
LectureNotesinEconomicsandMathematicalSystems
ISBN978-3-319-45561-7 ISBN978-3-319-45562-4 (eBook)
DOI10.1007/978-3-319-45562-4
LibraryofCongressControlNumber:2016959767
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To Flavia
Preface
ThismonographoriginatesinashortcourseItaughtattheEconomicsDepartment
oftheEuropeanUniversityInstitute(Florence,Italy)inthefirstsemesterof2015.
The course consisted of a series of five 2-hour lessons aimed at graduate students
pursuing their PhD in Economics. Its contents, though, can also be of interest to
graduates of other disciplines with a minimal mathematical training who are
interestedinthosetopics.
The length of the course imposed severe restrictions on the material to be
covered but also constituted a challenge on the selection of topics. I chose to
focusthoseLectures onunderstandingthelogicsandtheconstructionofthemain
indicatorsthatpermitonehavingsensiblemeasuresofinequalityandpoverty,with
special attentiontotheirsocialwelfareunderpinnings.Noattemptatcoveringthe
whole spectrum of indices or measurement problems is made, and hence, many
complementarytopicshavebeenputaside,particularlythoseregardingtheempir-
ical implementation. The reader will find suitable references along the Lectures.
The emphasis is on theory and applicability rather than on specific applications,
which were left as part of the students’ course activities. Nevertheless, I include
here some empirical illustrations that help understanding the possible uses of
inequalityandpovertymeasures.
The first three lectures are largely based on a former monograph, written in
Spanish with F. Goerlich.1 The remaining ones benefit from some former contri-
butionsoftheauthor.EachLectureprovidesitsownlistofreferences,tofacilitate
thereaderfurtherinsights,andtriestobeself-contained.
I have finished the writing of these Lectures while I was visiting the OECD in
ParisasaThomasJ.AlexanderFellow.IwouldliketothanktheFellowshipforthe
support provided. Thanks are also due to the funding by the Spanish Ministry of
EconomicsandInnovation,underprojectECO2015-65408-R(MINECO/FEDER).
1See Goerlich, F., & Villar, A. (2009). Desigualdad y Bienestar: De la Teoría a la Pra(cid:2)ctica.
Fundacio´nBBVA.
vii
viii Preface
Finally, I would like to thank Ricardo Mart´ınez for his critical reading and his
numerous suggestions. Needless to say, all remaining errors and imprecisions are
mysoleresponsibility.
Paris AntonioVillarPh.D.
June2016
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 EqualityandWelfare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 WeHaveaProblem.... . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 MeasuringInequalityandPoverty. . . . . . . . . . . . . . . . . . 4
1.1.3 Purpose. . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . 5
1.2 EqualityofWhat?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 IncomeDistributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Wealth,IncomeandExpenditures. . . . . . . . . . . . . . . . . . 7
1.2.3 TheUnitsofReference:EqualityAmongWhom?. . . . . . . 9
1.2.4 NeedsandDeserts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.5 Multidimensionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 PlanoftheBook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
PartI Inequality
2 InequalityIndices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 PropertiesofInequalityIndices. . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 InequalityandDispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 TheVariance(σ2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 TheCoefficientofVariation(CV). . . . . . . . . . . . . . . . . . 29
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 PositiveInequalityIndices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 TheLorenzCurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 TheGiniIndex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 FromLorenztoGini. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 TheGeneralisedGiniIndex. . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Theil’sInequalityIndices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ix
x Contents
3.3.1 TheFirstIndexofTheil:FromInformationTheoryto
InequalityMeasurement. . . . . . . . . . . . . . . . . . . .. . . . . . 39
3.3.2 TheSecondIndexofTheil(MeanLogarithmic
Deviation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.3 TheGeneralisedIndicesofTheil(Tθ). . . . . . . . . . . . . . . . 44
3.4 QuantileMeasures:ThePalmaRatio. . . . . . . . . . . . . . . . . . . . . . 48
Appendix:DecomposabilityoftheGeneralisedIndicesofTheil. . . . . . 53
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 NormativeInequalityIndices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 TheArrowianApproachtoNormativeInequalityIndices. . . 57
4.1.2 TheSocialEvaluationFunctionApproach. . . . . . . . . . . . 60
4.2 Dalton’sApproachtoInequalityMeasurement. . . . . . . . . . . . . . . 60
4.3 AtkinsonInequalityMeasures. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.2 TheAtkinsonFamilyofInequalityIndices. . . . . . . . . . . . 64
4.3.3 TheAtkinsonandtheEntropyFamiliesofInequality
Indices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 InequalityofOpportunity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 MeasuringInequalityofOpportunitybyTheil’sInequality
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.2 TheModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.3 TheSecondIndexofTheil. . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 EqualityofOpportunityWithCategoricalData. . . . . . . . . . . . . . 81
5.3.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.2 TheModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.3 EmpiricalIllustration:InequalityofOpportunityin
CompulsoryEducationintheOECD,AccordingtoPISA
(2012). . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . 84
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 InequalityandWelfare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 SocialEvaluationFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 MultidimensionalInequalityandWelfare. . . . . . . . . . . . . . . . . . 101
6.3.1 MultidimensionalWelfareIndicators. . . . . . . . . . . . . . . . 103
6.3.2 MultidimensionalInequality. . . .. . . . .. . . .. . . . .. . . .. 105
6.3.3 WeightsandUnits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4 TheHumanDevelopmentIndex. . . . . . . . . . . . . . . . . . . . . . . . . 109
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
