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Frank Cowell UAB Inequality PovertyInequality MeasurementInequality measurement MeasurementUniversitat Aut noma de BarcelonaFrank Cowell.

http darp lse ac uk uab2006December 2006 Issues to be addressedFrank Cowell UAB Inequality Poverty Builds on lecture 3.

Income Distribution and Welfare Extension of ranking criteria Parade diagrams Generalised Lorenz curve Extend SWF analysis to inequality.

Examine structure of inequality Link with the analysis of poverty Major ThemesFrank Cowell UAB Inequality Poverty Contrast three main approaches to the subject.

intuitive via SWF via analysis of structure Structure of the population Composition of Inequality measurement.

Implications for measures The use of axiomatisation Capture what is reasonable Use principles similar to welfare and poverty Inequality.

Frank Cowell UAB Inequality PovertymeasurementOverview InequalityInequality.

Relationship with measureswelfare rankingsInequalityaxiomaticsInequality in.

Frank Cowell UAB Inequality PovertyInequality rankings Begin by using welfare analysis of previous lecture Seek an inequality ranking We take as a basis the second order distributional.

but introduce a small modification Normalise by dividing by the mean The 2nd order dominance concept was originallyexpressed in a more restrictive form Yet another important relationship.

Frank Cowell UAB Inequality Poverty The share of the proportion q of distribution F is givenby L F q C F q F Yields Lorenz dominance or the shares rankingG Lorenz dominates F means .

for every q L G q L F q for some q L G q L F q The Atkinson 1970 result For given G Lorenz dominates FW G W F for all W W2.

Frank Cowell UAB Inequality PovertyFor discrete distributions All the above has been done in terms of F form notation Can do the almost same in Irene Janet notation Use the order statistics x i where.

is the ith smallest member of the income vector x1 x2 xn Then define Parade income cumulations.

The Lorenz diagramFrank Cowell UAB Inequality Povertyproportion of incomecurve for F0 0 2 0 4 0 6 0 8 1 practical.

proportion of population q practicalexample UK UK Application of rankingFrank Cowell UAB Inequality Poverty The tax and benefit system maps one distribution into.

another Use ranking tools to assess the impact of this in welfare Typically this uses one or other concept of Lorenzdominance Official concepts of income UK.

Frank Cowell UAB Inequality Povertyoriginal income cash benefitsgross income direct taxes.

distributionalranking would disposable incomewe expect to indirect taxesapply to these 5 post tax incomeconcepts non cash benefits.

final income Frank Cowell UAB Inequality Poverty Impact of Taxes and Benefits UK 2000 1 Lorenz CurveOriginal IncomeGross Income 0 9.

Disposable IncomeAfter Tax Income 0 8Final Income Equality Line Proportion of Income.

0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0Proportion of population Frank Cowell UAB Inequality PovertyAssessment of example We might have guessed the outcome .

In most countries Income tax progressive So are public expenditures But indirect tax is regressive So Lorenz dominance is not surprising .

But what happens if we look at the situation over time Final income LorenzFrank Cowell UAB Inequality Poverty2000 1 0 8 Equality Line .

Proportion of Income0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0Proportion of population Original income LorenzFrank Cowell UAB Inequality Poverty.

2000 1 0 8 Equality Line 0 80 7 LorenzProportion of Income0 6 curves.

Is 1993 more0 50 3 equal 0 0 0 1 0 2 0 3 0 4 0 5 Or 2000 1 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0.

Proportion of population Inequality ranking SummaryFrank Cowell UAB Inequality Poverty Second order GL dominance is equivalent to rankingby cumulations .

From the welfare lecture Lorenz dominance equivalent to ranking by shares Special case of GL dominance normalised by means Where Lorenz curves intersect unambiguous inequalityorderings are not possible .

This makes inequality measures especially interesting InequalityFrank Cowell UAB Inequality PovertymeasurementOverview .

Inequality IntuitionInequality Social welfareThree ways of measures Distanceapproaching an.

InequalityaxiomaticsInequality in Frank Cowell UAB Inequality PovertyInequality measures.

What is an inequality measure Formally very simple function or functional from set of distributions to the real line contrast this with ranking principles.

Nature of the measure Some simple regularity properties such as continuity Beyond that we need some theory Alternative approaches to the theory .

intuition social welfare distance Begin with intuition Frank Cowell UAB Inequality Poverty.

Intuitive inequality measures Perhaps borrow from other disciplines A standard measure of spread variance But maybe better to use a normalised version.

coefficient of variation Comparison between these two is instructive Same iso inequality contours for a given Different behaviour as alters Frank Cowell UAB Inequality Poverty.

Another intuitive approach Alternative intuition based on Lorenz approach Lorenz comparisons second order dominance may beindecisive Use the diagram to force a solution .

Problem is essentially one of aggregation of information It may make sense to use a very simple approach Try something that you can see Go back to the Lorenz diagram Frank Cowell UAB Inequality Poverty The best known inequality.

proportion of incomeCoefficient 0 50 0 2 0 4 0 6 0 8 1proportion of population Frank Cowell UAB Inequality Poverty.

The Gini coefficient 1 Natural expression of measure Normalised area above Lorenz curve Can express this also in Irene Janet terms for discrete distributions .

But alternative representations more useful each of these equivalent to the above expressible in F form or Irene Janet terms Frank Cowell UAB Inequality PovertyThe Gini coefficient 2 .

Normalised difference between income pairs In F form In Irene Janet terms Frank Cowell UAB Inequality PovertyThe Gini coefficient 3 .

Finally express Gini as a weighted sum In F form Or more illuminating in Irene Janet terms Note that the weights are very special depend on rank or position in distribution.

will change as other members added removed from distribution perhaps in interesting ways Frank Cowell UAB Inequality PovertyIntuitive approach difficulties Essentially arbitrary.

Does not mean that CV or Gini is a bad index But what is the basis for it What is the relationship with social welfare The Gini index also has some structural problems We will see this later in the lecture.

What is the relationship with social welfare Examine the welfare inequality relationship directly InequalityFrank Cowell UAB Inequality Povertymeasurement.

Overview Inequality IntuitionInequality Social welfareThree ways of measures Distance.

approaching anInequalityaxiomaticsInequality in Frank Cowell UAB Inequality Poverty.

SWF and inequality Issues to be addressed the derivation of an index the nature of inequality aversion the structure of the SWF.

Begin with the SWF W Examine contours in Irene Janet space Equally Distributed EquivalentFrank Cowell UAB Inequality PovertyIncome The Irene Janet diagram.

A given distribution Distributions with same meanxj Contours of the SWF Construct an equal distributionE such that W E W F .

EDE income Social waste from inequality Curvature ofcontour indicatessociety s willingness.

to tolerate efficiency loss in E pursuit of greater F equality F F xi.

Frank Cowell UAB Inequality PovertyWelfare based inequality From the concept of social waste Atkinson 1970 suggested an inequality measure Ede incomeI F 1 Mean income.

Atkinson assumed an additive social welfarefunction that satisfied the other basic axioms W F u x dF x Introduced an extra assumption Iso elasticx 1 1.

u x 1 The Atkinson IndexFrank Cowell UAB Inequality Poverty Given scale invariance additive separability of welfare.

Inequality takes the form Given the Harsanyi argument index of inequality aversion based on risk aversion More generally see it as a statement of social values Examine the effect of different values of .

relationship between u x and xrelationship between u x and x Frank Cowell UAB Inequality PovertySocial utility and relative income4 .

1 2 3 4 5 x Relationship between welfare weight.

Frank Cowell UAB Inequality Povertyand income 1 .

x 0 1 2 3 4 5 InequalityFrank Cowell UAB Inequality Poverty.

measurementOverview Inequality IntuitionInequality Social welfare.

Three ways of measures Distanceapproaching anInequalityaxiomaticsInequality in.

A further look at inequalityFrank Cowell UAB Inequality Poverty The Atkinson SWF route provides a coherent approach toinequality But do we need to use an approach via social welfare .

An indirect approach Maybe introduces unnecessary assumptions Alternative route distance and inequality Consider a generalisation of the Irene Janet diagram Frank Cowell UAB Inequality Poverty.

The 3 Person income distributionJanet s incomeIncome DistributionsWith Given Total y ofn s i nc xk.

Frank Cowell UAB Inequality PovertyInequality contours Set of distributions forgiven total Set of distributions for axj higher given total.

Perfect equality Inequality contours for original Inequality contours for higher Frank Cowell UAB Inequality PovertyA distance interpretation.

Can see inequality as a deviation from the norm The norm in this case is perfect equality Two key questions what distance concept to use How are inequality contours on one level hooked up to.

those on another Frank Cowell UAB Inequality PovertyAnother class of indices Consider the Generalised Entropy class of inequality The parameter is an indicator sensitivity of each.

member of the class large and positive gives a top sensitive measure negative gives a bottom sensitive measure Related to the Atkinson class Frank Cowell UAB Inequality Poverty.

Inequality and a distance concept The Generalised Entropy class can also be written Which can be written in terms of income shares s Using the distance criterion s1 1 Can be interpreted as weighted distance of each income shares from an equal share.

Frank Cowell UAB Inequality PovertyThe Generalised Entropy Class GE class is rich Includes two indices from Henri Theil 1 x F log x F dF x .

0 log x F dF x For 1 it is ordinally equivalent to Atkinson class 1 For 2 it is ordinally equivalent to normalised Frank Cowell UAB Inequality Poverty.

Inequality contours Each family of contours related to a different concept of Some are very obvious others a bit more subtle Start with an obvious one.

the Euclidian case GE contours 2Frank Cowell UAB Inequality Poverty Frank Cowell UAB Inequality PovertyGE contours 2.

2Title: Extensive Form Author: Frank Cowell Last modified by: Cowell Created Date: 4/11/2003 2:24:41 PM Document presentation format: On-screen Show