Presentation on theme: "Inequality and Poverty"— Presentation transcript:
1 Inequality and Poverty Public Economics: University of BarcelonaFrank CowellJune 2005
2 Issues to be addressed Builds on lecture 2 “Distributional Equity, Social Welfare”Extension of ranking criteriaParade diagramsGeneralised Lorenz curveExtend SWF analysis to inequalityExamine structure of inequalityLink with the analysis of poverty
3 Major Themes Contrast three main approaches to the subject intuitivevia SWFvia analysis of structureStructure of the populationComposition of inequality and povertyImplications for measuresThe use of axiomatisationCapture what is “reasonable”?Find a common set of axioms for related problems
4 Overview... Relationship with welfare rankings Inequality and Poverty Inequality rankingsInequality measurementRelationship with welfare rankingsInequality and decompositionPoverty measuresPoverty rankings
5 Inequality rankingsBegin by using welfare analysis of previous lectureSeek inequality rankingWe take as a basis the second-order distributional ranking…but introduce a small modificationThe 2nd-order dominance concept was originally expressed in a more restrictive form.
6 Yet another important relationship The share of the proportion q of distribution F is given by L(F;q) := C(F;q) / m(F)Yields Lorenz dominance, or the “shares” rankingG Lorenz-dominates F means:for every q, L(G;q) ³ L(F;q),for some q, L(G;q) > L(F;q)The Atkinson (1970) result:For given m, G Lorenz-dominates FÛW(G) > W(F) for all WÎW2
7 The Lorenz diagram L(.; q) q L(G;.) L(F;.) proportion of income 10.8L(.; q)0.6L(G;.)proportion of incomeLorenz curve for F0.4L(F;.)0.2practical example, UK0.20.40.60.81qproportion of population
8 Official concepts of income: UK original income+ cash benefitsgross income- direct taxesdisposable income- indirect taxespost-tax income+ non-cash benefitsfinal incomeWhat distributional ranking would we expect to apply to these 5 concepts?
9 Impact of Taxes and Benefits. UK 2000/1. Lorenz Curve
10 Assessment of example We might have guessed the outcome… In most countries:Income tax progressiveSo are public expendituresBut indirect tax is regressiveSo Lorenz-dominance is not surprising.But what happens if we look at the situation over time?
13 Inequality ranking: Summary Second-order (GL)-dominance is equivalent to ranking by cumulations.From the welfare lectureLorenz dominance equivalent to ranking by shares.Special case of GL-dominance normalised by means.Where Lorenz-curves intersect unambiguous inequality orderings are not possible.This makes inequality measures especially interesting.
14 Overview... Three ways of approaching an index Inequality and Poverty Inequality rankingsInequality measurementIntuitionSocial welfareDistanceThree ways of approaching an indexInequality and decompositionPoverty measuresPoverty rankings
15 An intuitive approachLorenz comparisons (second-order dominance) may be indecisiveBut we may want to “force a solution”The problem is essentially one of aggregation of informationWhy worry about aggregation?It may make sense to use a very simple approachGo for something that you can “see”Go back to the Lorenz diagram
16 The best-known inequality measure? 10.8proportion of income0.6Gini Coefficient0.50.40.20.220.127.116.11proportion of population
17 The Gini coefficient Equivalent ways of writing the Gini: Normalised area above Lorenz curveNormalised difference between income pairs.
18 Intuitive approach: difficulties Essentially arbitraryDoes not mean that Gini is a bad indexBut what is the basis for it?What is the relationship with social welfare?The Gini index also has some “structural” problemsWe will see this in the next sectionExamine the welfare-inequality relationship directly
19 Overview... Three ways of approaching an index Inequality and Poverty Inequality rankingsInequality measurementIntuitionSocial welfareDistanceThree ways of approaching an indexInequality and decompositionPoverty measuresPoverty rankings
20 SWF and inequality Issues to be addressed: Begin with the SWF W the derivation of an indexthe nature of inequality aversionthe structure of the SWFBegin with the SWF WExamine contours in Irene-Janet space
21 Equally-Distributed Equivalent Income The Irene &Janet diagramA given distributionDistributions with same meanxixjContours of the SWFConstruct an equal distribution E such that W(E) = W(F)EDE incomeSocial waste from inequalityCurvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equalityEFOx(F)m(F)
22 Welfare-based inequality From the concept of social waste Atkinson (1970) suggested an inequality measure:Ede incomex(F)I(F) = 1 – ——m(F)Mean incomeAtkinson assumed an additive social welfare function that satisfied the other basic axioms.W(F) = ò u(x) dF(x)Introduced an extra assumption: Iso-elastic welfare.x 1 - e – 1u(x) = ————, e ³ 01 – e
23 The Atkinson IndexGiven scale-invariance, additive separability of welfareInequality takes the form:Given the Harsanyi argument…index of inequality aversion e based on risk aversion.More generally see it as a stament of social valuesExamine the effect of different values of erelationship between u(x) and xrelationship between u′(x) and x
24 Social utility and relative income 4 = 03 = 1/22 = 11 = 2 = 512345x / m-1-2-3
25 Relationship between welfare weight and income =1U'=2=5432=01=1/2=1x / m12345
26 Overview... Three ways of approaching an index Inequality and Poverty Inequality rankingsInequality measurementIntuitionSocial welfareDistanceThree ways of approaching an indexInequality and decompositionPoverty measuresPoverty rankings
27 A further look at inequality The Atkinson SWF route provides a coherent approach to inequality.But do we need to approach via social welfareAn indirect approachMaybe introduces unnecessary assumptions,Alternative route: “distance” and inequalityConsider a generalisation of the Irene-Janet diagram
28 The 3-Person income distribution xjIncome DistributionsWith Given Totalray ofJanet's incomeequalitykxKaren's incomeIrene's incomeix
29 Inequality contours x x x j k i Set of distributions for given total Set of distributions for a higher (given) totalPerfect equalityInequality contours for original levelInequality contours for higher levelkxix
30 A distance interpretation Can see inequality as a deviation from the normThe norm in this case is perfect equalityTwo key questions……what distance concept to use?How are inequality contours on one level “hooked up” to those on another?
31 Another class of indices Consider the Generalised Entropy class of inequality measures:The parameter a is an indicator sensitivity of each member of the class.a large and positive gives a “top -sensitive” measurea negative gives a “bottom-sensitive” measureRelated to the Atkinson class
32 Inequality and a distance concept The Generalised Entropy class can also be written:Which can be written in terms of income shares sUsing the distance criterion s1−a/ [1−a] …Can be interpreted as weighted distance of each income shares from an equal share
33 The Generalised Entropy Class GE class is richIncludes two indices from Henri Theil:a = 1: [ x / m(F)] log (x / m(F)) dF(x)a = 0: – log (x / m(F)) dF(x)For a < 1 it is ordinally equivalent to Atkinson classa = 1 – e .For a = 2 it is ordinally equivalent to (normalised) variance.
34 Inequality contoursEach family of contours related to a different concept of distanceSome are very obvious……others a bit more subtleStart with an obvious onethe Euclidian case
37 GE contours: a limiting case Total priority to the poorest
38 GE contours: another limiting case Total priority to the richest
39 By contrast: Gini contours Not additively separable
40 Distance: a generalisation The responsibility approach gives a reference income distributionExact version depends on balance of compensation rulesAnd on income function.Redefine inequality measurementnot based on perfect equality as a normuse the norm income distribution from the responsibility approachDevooght (2004) bases this on Cowell (1985)Cowell approach based on Theil’s conditional entropyInstead of looking at distance going from perfect equality to actual distribution...Start from the reference distribution
41 Overview... Structural issues Inequality and Poverty Inequality rankingsInequality measurementStructural issuesInequality and decompositionPoverty measuresPoverty rankings
42 first, some terminology Why decomposition?Resolve questions in decomposition and population heterogeneity:Incomplete informationInternational comparisonsInequality accountingGives us a handle on axiomatising inequality measuresDecomposability imposes structure.Like separability in demand analysisfirst, some terminology
43 A partition pj sj Ij population share (4) (3) (6) (5) (2) (1) income The populationAttribute 1Attribute 2One subgrouppopulationshare(1)(2)(3)(4)(5)(6)pj(ii)(i)(iii)(iv)incomesharesjIjsubgroupinequality
44 What type of decomposition? Distinguish three types of decomposition by subgroupIn increasing order of generality these are:Inequality accountingAdditive decomposabilityGeneral consistencyWhich type is a matter of judgmentMore on this belowEach type induces a class of inequality measuresThe “stronger” the decomposition requirement……the “narrower” the class of inequality measures
45 1:Inequality accounting This is the most restrictive form of decomposition:accounting equationweight functionadding-up property
46 2:Additive Decomposability As type 1, but no adding-up constraint:
47 3:General Consistency The weakest version: population shares increasing in each subgroup’s inequalityincome shares
48 A class of decomposable indices Given scale-invariance and additive decomposability,Inequality takes the Generalised Entropy form:Just as we had earlier in the lecture.Now we have a formal argument for this family.The weight wj on inequality in group j is wj = pjasj1−a
49 What type of decomposition? Assume scale independence…Inequality accounting:Theil indices only (a = 0,1)Here wj = pj or wj = sjAdditive decomposability:Generalised Entropy IndicesGeneral consistency:moments,Atkinson, ...But is there something missing here?We pursue this later
50 What type of partition? General Non-overlapping in incomes The approach considered so farAny characteristic used as basis of partitionAge, gender, region, incomeInduces specific class of inequality measures... but excludes one very important measureNon-overlapping in incomesA weaker versionPartition just on the basis of incomeAllows one to include the "missing" inequality measureDistinction between them is crucial for one special inequality measure
51 The Gini coefficient Different (equivalent) ways of writing the Gini: 0.20.40.60.81proportion of incomeproportion of populationGini CoefficientDifferent (equivalent) ways of writing the Gini:Normalised area under the Lorenz curveNormalised pairwise differencesA ranking-weighted averageBut ranking depends on reference distribution
52 Partitioning by income... Non-overlapping income groupsOverlapping income groupsConsider a transfer:Case 1Consider a transfer:Case 2N1N2N1x*x**xxxx'x'Case 1: effect on Gini is same in subgroup and populationCase 2: effect on Gini differs in subgroup and population
53 Non-overlapping decomposition Can be particularly valuable in empirical applicationsUseful for rich/middle/poor breakdownsEspecially where data problems in tailsMisrecorded dataIncomplete dataVolatile data componentsExample: Piketty-Saez on US (QJE 2003)Look at behaviour of Capital gains in top income shareShould this affect conclusions about trend in inequality?
55 Choosing an inequality measure Do you want an index that accords with intuition?If so, what’s the basis for the intuition?Is decomposability essential?If so, what type of decomposability?Do you need a welfare interpretation?If so, what welfare principles to apply?
56 Overview... …Identification and measurement Inequality and Poverty Inequality rankingsInequality measurement…Identification and measurementInequality and decompositionPoverty measuresPoverty rankings
57 Poverty analysis – overview Basic ideasIncome – similar to inequality problem?Consumption, expenditure or income?Time periodRiskIncome receiver – as beforeRelation to decompositionDevelopment of specific measuresRelation to inequalityWhat axiomatisation?Use of ranking techniquesRelation to welfare rankings
58 Poverty measurement How to break down the basic issues. Sen (1979): Two main types of issuesIdentification problemAggregation problemJenkins and Lambert (1997): “3Is”IdentificationIntensityInequalityPresent approach:Fundamental partitionIndividual identificationAggregation of informationpopulationnon-poorpoor
59 Poverty and partition Depends on definition of poverty line Exogeneity of partition?Asymmetric treatment of information
60 Counting the poor Use the concept of individual poverty evaluation Simplest version is (0,1)(non-poor, poor)headcountPerhaps make it depend on incomepoverty deficitOr on the whole distribution?Convenient to work with poverty gaps
61 The poverty line and poverty gaps poverty evaluationgigjx*xxixjincome
62 Poverty evaluation g gj gi Non-Poor Poor x = 0 B A poverty evaluation the “head-count”the “poverty deficit”sensitivity to inequality amongst the poorIncome equalisation amongst the poorpoverty evaluationNon-PoorPoorBx = 0Aggjgipoverty gap
63 Brazil 1985: How Much Poverty? A highly skewed distributionA “conservative” x*A “generous” x*An “intermediate” x*The censored income distribution$0$20$40$60$80$100$120$140$160$180$200$220$240$260$280$300Rural Belo Horizontepoverty linecompromisepoverty lineBrasiliapoverty line
64 The distribution of poverty gaps $0$20$40$60gaps
65 ASP Additively Separable Poverty measures ASP approach simplifies poverty evaluationDepends on own income and the poverty line.p(x, x*)Assumes decomposability amongst the poorOverall poverty is an additively separable functionP = p(x, x*) dF(x)Analogy with decomposable inequality measures
66 A class of poverty indices ASP leads to several classes of measuresMake poverty evaluation depends on poverty gap.Normalise by poverty lineFoster-Greer-Thorbecke class
68 Empirical robustness Does it matter which poverty criterion you use? Look at two key measures from the ASP classHead-count ratioPoverty deficit (or average poverty gap)Use two standard poverty lines$1.08 per day at 1993 PPP$2.15 per day at 1993 PPPHow do different regions of the world compare?What’s been happening over time?Use World-Bank analysisChen-Ravallion “How have the world’s poorest fared since the early 1980s?” World Bank Policy Research Working Paper Series 3341
77 Empirical robustness (2) Does it matter which poverty criterion you use?An example from SpainData are from ECHPOECD equivalence scalePoverty line is 60% of 1993 median incomeDoes it matter which FGT index you use?
79 Overview... Another look at ranking issues Inequality and Poverty Inequality rankingsInequality measurementAnother look at ranking issuesInequality and decompositionPoverty measuresPoverty rankings
80 An extension of poverty analysis Finally consider some generalisationsWhat if we do not know the poverty line?Can we find a counterpart to second order dominance in welfare analysis?What if we try to construct poverty indices from first principles?
81 Poverty rankings (1) Atkinson (1987) connects poverty and welfare. Based results on the portfolio literature concerning “below-target returns”TheoremGiven a bounded range of poverty lines (x*min, x*max)and poverty measures of the ASP forma necessary and sufficient condition for poverty to be lower in distribution F than in distribution G is that the poverty deficit be no greater in F than in G for all x* ≤ x*max.Equivalent to requiring that the second-order dominance condition hold for all x*.
82 Poverty rankings (2)Foster and Shorrocks (1988a, 1988b) have a similar approach to orderings by P,But concentrate on the FGT index’s particular functional form:Theorem: Poverty rankings are equivalent tofirst-order welfare dominance for a = 0second-degree welfare dominance for a = 1(third-order welfare dominance for a = 2.)
83 Poverty concepts Given poverty line z Poverty gap a reference pointPoverty gapfundamental income differenceFoster et al (1984) poverty index againCumulative poverty gap
84 TIP / Poverty profile TIP curves have same interpretation as GLC Cumulative gaps versus population proportionsProportion of poorTIP curveG(x,z)TIP curves have same interpretation as GLCTIP dominance implies unambiguously greater povertyi/np(x,z)/n
85 Poverty: Axiomatic approach Characterise an ordinal poverty index P(x ,z)See Ebert and Moyes (JPET 2002)Use some of the standard axioms we introduced for analysing social welfareApply them to n+1 incomes – those of the n individuals and the poverty lineShow thatgiven just these axioms……you are bound to get a certain type of poverty measure.
86 Poverty: The key axioms Standard ones from lecture 2anonymityindependencemonotonicityincome increments reduce povertyStrengthen two other axiomsscale invariancetranslation invarianceAlso need continuityPlus a focus axiom
87 A closer look at the axioms Let D denote the set of ordered income vectorsThe focus axiom isScale invariance now becomesDefine the number of the poor asIndependence means:
88 Ebert-Moyes (2002) Gives two types of FGT measures “relative” version“absolute” versionAdditivity follows from the independence axiom
89 Brief conclusionFramework of distributional analysis covers a number of related problems:Social WelfareInequalityPovertyCommonality of approach can yield important insightsRanking principles provide basis for broad judgmentsMay be indecisivespecific indices could be usedPoverty trends will often be robust to choice of poverty indexPoverty indexes can be constructed from scratch using standard axioms
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