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9.1 Distributional Analysis and Methods9.1 Distributional Analysis and Methods

14 December 2011

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Overview...The basics

SWF and rankings

Inequality measures

Evidence

Distributional Analysis and Methods

How to represent problems in distributionalanalysis

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Distributional analysis Covers a broad class of economic problemsCovers a broad class of economic problems

inequalityinequality social welfaresocial welfare povertypoverty

Similar techniquesSimilar techniques rankingsrankings measuresmeasures

Four basic components need to be clarifiedFour basic components need to be clarified ““income” conceptincome” concept

usually equivalised, disposable incomeusually equivalised, disposable income but could be other income types, consumption, wealth…but could be other income types, consumption, wealth…

““income receiving unit” conceptincome receiving unit” concept usually the individualusually the individual but could also be family, householdbut could also be family, household

a distributiona distribution method of assessment or comparisonmethod of assessment or comparison

See See Cowell (2000, , 2008, 2011), Sen and Foster (1997), 2011), Sen and Foster (1997)

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1: “Irene and Janet” approach

0Irene's income

Jane

t's in

com

e

Karen's income

ix

kx

xj

ray of eq

uality

A representation with 3 incomes Income distributions with given total X

xi + xj + xk = X

Feasible income distributions given X Equal income distributions

particularly appropriate in particularly appropriate in approaches to the subject approaches to the subject based primarily upon based primarily upon individualistic welfare criteriaindividualistic welfare criteria

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2: The parade

x

0.2 0.8 10

x0.8

q

"inc

ome"

(hei

ght)

proportion of the population

x0.2

especially useful in cases especially useful in cases where it is appropriate to where it is appropriate to adopt a parametric model adopt a parametric model of income distributionof income distribution

0

1

x

F(x)

x0

F(x0)distribution function F(∙)

Plot income against proportion of population

Parade in ascending order of "income" / height Related to familiar statistical concept

Pen (1971)Pen (1971)

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Overview...The basics

SWF and rankings

Inequality measures

Evidence

Distributional Analysis and Methods

How to incorporate fundamental principles

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Social-welfare functions A standard approach to a method of assessmentA standard approach to a method of assessment Basic tool is a Basic tool is a social welfare functionsocial welfare function (SWF) (SWF)

Maps set of distributions into the real lineMaps set of distributions into the real line I.e. for each distribution we get one specific numberI.e. for each distribution we get one specific number In Irene-Janet notation In Irene-Janet notation W = WW = W((xx))

Properties will depend on economic principlesProperties will depend on economic principles Simple example of a SWF:Simple example of a SWF:

Total income in the economy Total income in the economy WW = = xxii

Perhaps not very interestingPerhaps not very interesting Consider principles on whichConsider principles on which SWF could be basedSWF could be based

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Another fundamental question What makes a “good” set of principles?What makes a “good” set of principles? There is no such thing as a “right” or “wrong” axiom.There is no such thing as a “right” or “wrong” axiom. However axioms could be appropriate or inappropriateHowever axioms could be appropriate or inappropriate

Need some standard of “reasonableness”Need some standard of “reasonableness” For example, how do people view income distribution comparisons?For example, how do people view income distribution comparisons?

Use a simple framework to list some of the basic axiomsUse a simple framework to list some of the basic axioms Assume a fixed population of size Assume a fixed population of size nn.. Assume that individual utility can be measured by Assume that individual utility can be measured by xx Income normalised by equivalence scalesIncome normalised by equivalence scales Rules out utility interdependenceRules out utility interdependence Welfare is just a function of the vector Welfare is just a function of the vector xx := ( := (xx11, , xx22,…,,…,xxnn ) )

Follow the approach of Follow the approach of Amiel-Cowell (1999) Appendix A Appendix A

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SWF axioms Anonymity. Suppose x′ is a permutation of x. Then:

W(x′) = W(x) Population principle.

W(x) W(y) W(x,x,…,x) W(y,y,…,y) Decomposability. Suppose x' is formed by joining x with z and

y' is formed by joining y with z. Then :W(x) W(y) W(x') W(y')

Monotonicity. W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn) Transfer principle. (Dalton 1920) Suppose xi< xj then, for

small : W(x1,x2..., xi+ ,..., xj ,..., xn) > W(x1,x2,..., xi,..., xn)

Scale invariance. W(x) W(y) W(x) W(y)

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Classes of SWFs Anonymity and population principle:

can write SWF in either Irene-Janet form or F form may need to standardise for needs etc

Introduce decomposability get class of Additive SWFs W : WW((xx) = ) = u(xxii) or equivalently W(F) = u(x) dF(x)

If we impose monotonicity we get W1 W : u(•) increasing

If we further impose the transfer principle we get W2 W1: u(•) increasing and concave

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An important family Take the Take the W2 subclass and impose subclass and impose scale invariance. scale invariance. Get the family of SWFs where Get the family of SWFs where uu is iso-elastic: is iso-elastic:

xx1 – – 1 u(xx) = ————— , 1 – has same form as CRRA utility functionhas same form as CRRA utility function

Parameter Parameter captures society’s inequality aversion. captures society’s inequality aversion. Similar interpretation to individual risk aversionSimilar interpretation to individual risk aversion See See Atkinson (1970)Atkinson (1970)

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Ranking and dominance Introduce two simple conceptsIntroduce two simple concepts

first illustrate using the Irene-Janet representationfirst illustrate using the Irene-Janet representation take income vectors take income vectors xx and and yy for a given for a given nn

First-order dominance:First-order dominance: yy[1] [1] > > xx[1][1], y, y[2] [2] > > xx[2][2], y, y[3] [3] > > xx[3][3] Each ordered income in Each ordered income in yy larger than that in larger than that in xx

Second-order dominance:Second-order dominance: yy[1] [1] > > xx[1][1], y, y[1][1]+y+y[2] [2] > > xx[1][1]+x+x[2][2], y, y[1][1]+y+y[2] [2] +…+ y+…+ y[[nn] ] > > xx[1][1]+x+x[2] [2] …+ x…+ x[[nn]] Each cumulated income sum in Each cumulated income sum in yy larger than that in larger than that in xx

Need to generalise this a littleNeed to generalise this a little represent distributions in represent distributions in FF-form (anonymity, population principle)-form (anonymity, population principle) qq: population proportion (0 : population proportion (0 ≤≤ qq ≤≤ 1) 1) FF((xx): proportion of population with incomes ): proportion of population with incomes ≤≤ xx ((FF): mean of distribution ): mean of distribution FF

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1st-Order approach Basic tool isBasic tool is the quantile, expressed as

Q(F; q) := inf {x | F(x) q} = xq “smallest income such that cumulative frequency is at least as great as q”

Use this to derive a number of intuitive concepts interquartile range, decile-ratios, semi-decile ratios graph of Q is Pen’s Parade

Also to characterise the idea of 1st-order (quantile) dominance: “G quantile-dominates F”means:

for every q, Q(G;q) Q(F;q), for some q, Q(G;q) > Q(F;q)

A fundamental result: G quantile-dominates F iff W(G) > W(F) for all WW1

Illustrate using Parade:

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Parade and 1st-order dominance

F

G

Q(.; q)

10 q

Plot quantiles against proportion of population

Parade for distribution F again

Parade for distribution G

In this case In this case GG clearly clearly quantile-dominates quantile-dominates FFBut (as often happens) But (as often happens) what if it doesn’t?what if it doesn’t?Try second-order methodTry second-order method

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2nd-Order approach Basic tool is the income cumulant, expressed as

C(F; q) := ∫ Q(F; q) x dF(x) “The sum of incomes in the Parade, up to and including position q”

Use this to derive a number of intuitive concepts the “shares” ranking, Gini coefficient graph of C is the generalised Lorenz curve

Also to characterise the idea of 2nd-order (cumulant) dominance: “G cumulant-dominates F”means:

for every q, C(G;q) C(F;q), for some q, C(G;q) > C(F;q)

A fundamental result (Shorrocks 1983): G cumulant-dominates F iff W(G) > W(F) for all WW2

Illustrate using GLC:

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GLC and 2nd-order dominance

10

0

C(G; . )

C(F; . )

C(.; q)

(F)

(G)

q

cum

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ive

inco

me

Plot cumulations against proportion of population

GLC for distribution F

GLC for distribution G

Intercept on vertical axis Intercept on vertical axis is at mean incomeis at mean income

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2nd-Order approach (continued) A useful tool: the A useful tool: the shareshare of the proportion of the proportion qq of distribution of distribution FF is is

L(F;q) := C(F;q) / (F) “income cumulation at q divided by total income”

Yields Lorenz dominance, or the “shares” ranking:Yields Lorenz dominance, or the “shares” ranking: “G Lorenz-dominates F”means:

for every q, L(G;q) L(F;q), for some q, L(G;q) > L(F;q)

Another fundamental result (Another fundamental result (Atkinson 1970):): For given , G Lorenz-dominates F iff W(G) > W(F) for

all W W2

Illustrate using Lorenz curve:

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Lorenz curve and ranking

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

prop

orti o

n of

i nc o

me

proportion of population

L(G;.)

L(F;.)

L(.; q)

q

Plot shares against proportion of population

Lorenz curve for distribution F

Lorenz curve for distribution G

In this case In this case GG clearly clearly Lorenz-dominates Lorenz-dominates FFSo So FF displays more displays more inequality than inequality than GGBut (as often happens) But (as often happens) what if it doesn’t?what if it doesn’t?No clear statement about No clear statement about inequality (or welfare) is inequality (or welfare) is possible without further possible without further informationinformation

Perfect equality

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Overview...The basics

SWF and rankings

Inequality measures

Evidence

Distributional Analysis and Methods

Three ways of approaching an index

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1: Intuitive inequality measures Perhaps borrow from other disciplines…Perhaps borrow from other disciplines… A standard measure of spread…A standard measure of spread…

variancevariance

But maybe better to use a normalised versionBut maybe better to use a normalised version coefficient of variationcoefficient of variation

Comparison between these two is instructiveComparison between these two is instructive Same iso-inequality contours for a given Same iso-inequality contours for a given .. Different behaviour as Different behaviour as alters alters

Alternative intuition based on Lorenz approach Alternative intuition based on Lorenz approach Lorenz comparisons (2Lorenz comparisons (2ndnd-order) may be indecisive-order) may be indecisive

problem is essentially one of aggregation of informationproblem is essentially one of aggregation of information so use the diagram to “force a solution”so use the diagram to “force a solution”

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0 1

1

L(.; q)

q

Gini coefficient Redraw Lorenz diagramRedraw Lorenz diagram A “natural” inequality measure…?A “natural” inequality measure…?

normalised area above Lorenz curvenormalised area above Lorenz curve can express this also in I-J termscan express this also in I-J terms

Also (equivalently) represented as Also (equivalently) represented as normalised difference between income pairs:normalised difference between income pairs:

In In FF-form:-form:

In Irene-Janet terms:In Irene-Janet terms:

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2: SWF and inequality

O xi

xj The Irene &Janet diagram A given distribution Distributions with same mean Contours of the SWF

•E

(F) (F)

•F

Construct an equal distribution E such that W(E) = W(F) Equally-Distributed Equivalent incomeSocial waste from inequality

Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality

contour: x values such that W(x) = const

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Welfare-based inequality From the concept of social waste From the concept of social waste Atkinson (1970)

suggested an inequality measure:suggested an inequality measure: ((FF))

II((FF)) = 1 = 1 –– —— ——

((FF)) Atkinson further assumed Atkinson further assumed

additive SWF, additive SWF, WW((FF)) = = uu((xx) d) dFF((xx)) isoelastic isoelastic uu

So inequality takes the formSo inequality takes the form

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3: “Distance” and inequality SWF route provides a coherent approach to inequalitySWF route provides a coherent approach to inequality But do we need to use an approach via social welfare?But do we need to use an approach via social welfare?

it’s indirectit’s indirect maybe introduces unnecessary assumptionsmaybe introduces unnecessary assumptions

Alternative route: “distance” and inequalityAlternative route: “distance” and inequality Can see inequality as a deviation from the normCan see inequality as a deviation from the norm

norm in this case is perfect equalitynorm in this case is perfect equality ……but what distance concept to use?but what distance concept to use?

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Generalised Entropy measures Defines a Defines a classclass of inequality measures, given parameter of inequality measures, given parameter : :

GE class is rich. Some important special casesGE class is rich. Some important special cases for for < 1 it is ordinally equivalent to Atkinson ( < 1 it is ordinally equivalent to Atkinson (= 1 = 1 – – )) = 0: = 0: – – log ( log (xx / / ((FF)) d)) dFF((xx) (mean logarithmic deviation)) (mean logarithmic deviation) = 1: = 1: [ [ xx / / ((FF)] log ()] log (xx / / ((FF)) d)) dFF((xx) (the Theil index)) (the Theil index) or or = 2 it is ordinally equivalent to (normalised) variance. = 2 it is ordinally equivalent to (normalised) variance.

Parameter Parameter can be assigned any positive or negative value can be assigned any positive or negative value indicates sensitivity of each member of the classindicates sensitivity of each member of the class large and positive gives a “top-sensitive” measurelarge and positive gives a “top-sensitive” measure negative gives a “bottom-sensitive” measurenegative gives a “bottom-sensitive” measure each each gives a specific distance conceptgives a specific distance concept

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Inequality contours Each Each defines a set contours in the I-J diagram defines a set contours in the I-J diagram

each related to a different concept of distanceeach related to a different concept of distance For exampleFor example

the Euclidian casethe Euclidian case other typesother types

25

− −

2

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Overview...The basics

SWF and rankings

Inequality measures

Evidence

Distributional Analysis and Methods

Attitudes and perceptions

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Views on distributions Do people make distributional comparisons in the same way as Do people make distributional comparisons in the same way as

economists?economists? Summarised from Summarised from Amiel-Cowell (1999)

examine proportion of responses in conformity with standard axiomsexamine proportion of responses in conformity with standard axioms both directly in terms of inequality and in terms of social welfareboth directly in terms of inequality and in terms of social welfare

InequalityInequality SWFSWF NumNum VerbalVerbal NumNum VerbalVerbalAnonymityAnonymity 83%83% 72%72% 66%66% 54%54%PopulationPopulation 58%58% 66%66% 66%66% 53%53%DecomposabilityDecomposability 57%57% 40%40% 58%58% 37%37%MonotonicityMonotonicity -- -- 54%54% 55%55%TransfersTransfers 35%35% 31%31% 47%47% 33%33%Scale indep.Scale indep. 51%51% 47%47% -- --

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Inequality aversion Are people averse to inequality?Are people averse to inequality?

evidence of both inequality and risk aversion (evidence of both inequality and risk aversion (CarlssonCarlsson et al et al 20052005) risk-aversion may be used as proxy for inequality aversion? (risk-aversion may be used as proxy for inequality aversion? (

Cowell and Gardiner 2000Cowell and Gardiner 2000)) What value for What value for ??

affected by way the question is put? (Pirttilä and Uusitalo 2010) evidence on risk aversion is mixedevidence on risk aversion is mixed

high values from survey evidence (high values from survey evidence (Barsky et al 1997)) much lower from savings analysis (much lower from savings analysis (Blundell et al 1994))

from happiness studies 1.0 to 1.5 (from happiness studies 1.0 to 1.5 (Layard et al 2008)) related to the extent of inequality in the country? (related to the extent of inequality in the country? (Lambert et al 2003)) perhaps a value of around 0.7 – 2 is reasonableperhaps a value of around 0.7 – 2 is reasonable see also see also HM Treasury (2003) page 94 page 94

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Conclusion Axiomatisation of welfare can be accomplished using Axiomatisation of welfare can be accomplished using

just a few basic principlesjust a few basic principles Ranking criteria can provide broad judgmentsRanking criteria can provide broad judgments But may be indecisive, so specific SWFs could be usedBut may be indecisive, so specific SWFs could be used

What shape should they have?What shape should they have? How do we specify them empirically?How do we specify them empirically?

Several axioms survive scrutiny in experimentSeveral axioms survive scrutiny in experiment but Transfer Principle often rejectedbut Transfer Principle often rejected

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References (1) Amiel, Y. and Cowell, F.A. (1999) Thinking about InequalityThinking about Inequality, Cambridge University , Cambridge University

PressPress Atkinson, A. B. (1970) “ “On the Measurement of Inequality,” On the Measurement of Inequality,” Journal of Economic Journal of Economic

TheoryTheory, , 22, 244-263, 244-263 Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997) “Preference “Preference

parameters and behavioral heterogeneity: An Experimental Approach in the Health and parameters and behavioral heterogeneity: An Experimental Approach in the Health and Retirement Survey,” Retirement Survey,” Quarterly Journal of EconomicsQuarterly Journal of Economics, , 112112, 537-579, 537-579

Blundell, R., Browning, M. and Meghir, C. (1994) “Consumer Demand and the Life-“Consumer Demand and the Life-Cycle Allocation of Household Expenditures,” Cycle Allocation of Household Expenditures,” Review of Economic StudieReview of Economic Studies, s, 6161, 57-80, 57-80

Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) ““Are people inequality Are people inequality averse or just risk averse?” averse or just risk averse?” EconomicaEconomica, , 7272,,

Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) F. (eds) Handbook of Income DistributionHandbook of Income Distribution, North Holland, Amsterdam, Chapter 2, 87-, North Holland, Amsterdam, Chapter 2, 87-166166

Cowell, F.A. (2008) “Inequality: measurement,” Cowell, F.A. (2008) “Inequality: measurement,” The New PalgraveThe New Palgrave, second edition, second edition Cowell, F.A. (2011) Cowell, F.A. (2011) Measuring InequalityMeasuring Inequality, Oxford University Press, Oxford University Press Cowell, F.A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research Cowell, F.A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research

Paper 202, Office of Fair Training, Salisbury Square, LondonPaper 202, Office of Fair Training, Salisbury Square, London

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References (2)

Dalton, H. (1920) “Dalton, H. (1920) “Measurement of the inequality of incomes,” Measurement of the inequality of incomes,” The Economic The Economic JournalJournal, , 3030, 348-361, 348-361

HM Treasury (2003) HM Treasury (2003) The Green Book: Appraisal and Evaluation in Central The Green Book: Appraisal and Evaluation in Central GovernmentGovernment (and Technical Annex), TSO, London (and Technical Annex), TSO, London

Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) “Inequality aversion and Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) “Inequality aversion and the natural rate of subjective inequality,” the natural rate of subjective inequality,” Journal of Public EcoJournal of Public Economics,nomics, 8787, , 1061-1090.1061-1090.

Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of income,” income,” Journal of Public EconomicsJournal of Public Economics, 9, 922, 1846-1857., 1846-1857.

Pen, J. (1971) Pen, J. (1971) Income DistributionIncome Distribution, Allen Lane, The Penguin Press, London, Allen Lane, The Penguin Press, London Pirttilä, J. and Uusitalo, R. (2010) “A ‘Leaky Bucket’ in the Real World:

Estimating Inequality Aversion using Survey Data,” Economica, 77, 60–76 Sen, A. K. and Foster, J. E. (1997) Sen, A. K. and Foster, J. E. (1997) On Economic Inequality On Economic Inequality (Second ed.). (Second ed.).

Oxford: Clarendon Press.Oxford: Clarendon Press. Shorrocks, A. F. (1983) “Ranking Income Distributions,” Shorrocks, A. F. (1983) “Ranking Income Distributions,” EconomicaEconomica, , 5050, 3-17, 3-17