session 9 review subgroup consistency income standards other characterizations unifying framework...

Session 9

ReviewSubgroup ConsistencyIncome StandardsOther CharacterizationsUnifying Framework

TodayPoverty - IntroductionSpaceIdentificationAggregation

Subgroup Consistency

Helps answer questions like:Are local inequality reductions going to

decrease overall inequality?If gender inequality stays the same and

inequality within the groups of men and women rises, must overall inequality rise?

SourceCowell “three bad measures”

Holding population sizes and means fixed, overall inequality should rise when when subgroup inequalities rise.

Subgroup Consistency Suppose that x’ and x share means and

population sizes, while y’ and y also share means and population sizes. If I(x’) > I(x) and I(y’) = I(y), then I(x’,y’) > I(x,y).

Ex (from book)x = (1,3,8,8) y = (2,2) (x,y) = (1,3,8,8,2,2) x’ = (2,2,7,8) y’ = (2,2) (x’,y’) = (2,2,7,8,2,2)G(x) = G(x’), G(y) = G(y’), G(x,y) > G(x’,y’)

Why? Residual R fellI2(x) = I2(x’), I2(y) = I2(y’), I2(x,y) > I2(x’,y’) Assignment: Find x, y that shows G violates SC

Note All decomposable indices are subgroup

consistentAll GE indicesWhy?

QAny others?

Theorem Shorrocks (1988)I is a Lorenz consistent, continuous, normalized inequality measure satisfying subgroup consistency if and only if there is some α and a continuous, strictly increasing function f with f(0)=0 such that

I(x) = f(Iα(x)) for all x.

A/ No!

Income StandardsIncome Standards

Key ConceptKey ConceptSummarizes distribution in a single Summarizes distribution in a single

incomeincome

Ex/ Ex/ Mean, median, income at 90th Mean, median, income at 90th percentile, mean of top 40%, Sen’s mean, percentile, mean of top 40%, Sen’s mean, Atkinson’s ede income…Atkinson’s ede income…

Measures ‘size’ of the distributionMeasures ‘size’ of the distributionCan have normative interpretationCan have normative interpretation

Related papersRelated papersFoster (2006) “Inequality MeasurementFoster and Shneyerov (1999, 2000)Foster and Szekely (2008)


NotationNotation

Income distributionIncome distribution x = (x1,…,xn)

xi > 0 income of the ith person

n population size

Dn = R++n set of all n-person income distributions

D = Dn set of all income distributions

s: D R income standard


PropertiesPropertiesSymmetrySymmetry If x is a permutation of y, then s(x) = s(y).

Replication InvarianceReplication Invariance If x is a replication of y, then s(x) = s(y).

Linear HomogeneityLinear Homogeneity If x = ky for some scalar k > 0, then s(x) = ks(y).

NormalizationNormalization If x is completely equal, then s(x) = x1.

ContinuityContinuity s is continuous on each Dn.

Weak MonotonicityWeak Monotonicity If x > y, then s(x) > s(y).

NoteNoteSatisfied by all examples given above and below.Satisfied by all examples given above and below.


ExamplesExamplesMeanMean s(x) =s(x) = (x) = (x1+...+xn)/n

F = cdf

income

freq


ExamplesExamplesMedian Median x = (3, 8, 9, 10, 20), s(x)s(x) = 9= 9

F = cdf

income

freq

0.5

median


ExamplesExamples1010thth percentile percentile

F = cdf

income

freq

0.1

s =s = Income at10th percentile


ExamplesExamplesMean of bottom fifth Mean of bottom fifth

x = (3, 5, 6, 6, 8, 9, 15, 17, 23, 25)

s(x) = 4s(x) = 4


ExamplesExamplesMean of top 40% Mean of top 40%

x = (3, 5, 6, 6, 8, 9, 15, 17, 23, 25)

s(x) = 20s(x) = 20


ExamplesExamplesSen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b)

Ex/ x = (1,2,3,4)

s(x) = s(x) = = 30/16= 30/16 < < (1,2,3,4) = 40/16(1,2,3,4) = 40/16

€

1 1 1 1

1 2 2 2

1 2 3 3

1 2 3 4

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟


ExamplesExamplesSen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b)

Generalized Lorenz CurveGeneralized Lorenz Curve

cumulative pop share

cum

ula

tive

inco

me

s =s = S = 2 x Areabelow curve


ExamplesExamplesGeometric MeanGeometric Mean s(x) =s(x) = 0(x) = (x1x2...xn)1/n

ThusThus s(x) = s(x) = 0

- emphasizes lower incomes

- is lower than the usual mean Unless distribution is completely equalUnless distribution is completely equal

x1

x2same 0

x.1(x)0(x)


ExamplesExamplesEuclidean MeanEuclidean Mean s(x) =s(x) = 2(x) = [(x1

2 + x22 +...+ xn

2)/n )1/2

ThusThus s(x) = s(x) = 2 - emphasizes higher incomes- is higher than the usual mean Unless distribution is completely equalUnless distribution is completely equal

x1

x2

same 2

1(x) 2(x)


Examples Examples General MeansGeneral Means

[(x1 + … + xn

)/n] 1/ for all 0

(x) = (x1

…xn)1/n for = 0

= 1 = 1 arithmetic meanarithmetic mean

= 0 = 0 geometric meangeometric mean = 2= 2 Euclidean meanEuclidean mean = -1= -1 harmonic meanharmonic mean

For For < 1: Distribution sensitive < 1: Distribution sensitiveLowerLower implies greatergreater emphasis on lowerlower incomes

Other CharacterizationsOther Characterizations

Idea Idea Use income standard s in decompositionUse income standard s in decompositions(x) replaces (x) in

- between group term ‘smoothed dist’- within group term ‘weights’

Ex: x = (2,8) y = (4,4) (x) = 6 (y) = 4 smoothed (6,6,4,4)Alt/ s is geometric mean g(x) = 4 g(y) = 4 smoothed (4,4,4,4)

Q/ What happens?

Additional CharacterizationsAdditional Characterizations

Theorem Theorem A measure has such a ‘weak additive decomposition’ if and only if it

takes the following form (or a positive multiple):

cf. gen. ent.

cf. Theil ent.

Icq(x) =

cf. Theil sec.

Var. Logs

Note All are functions of ratios of 2 gen. means or the limit of such functions. Not all are Lorenz consistent. Gen. ent. obtains when q = 1.

Example: Levels

0

500

1000

1500

2000

PP

P A

djus

ted

199

1 U

S D

olla

rs

M(-3) M(-2) M(-1) M(+1) M(+2) M(+3)

General Means

Comparison of Living Standards in the USA, UK and Sweden

United States

UK

Sweden

InequalityInequality

Q/ SummaryQ/ Summary

How does it all fit together?How does it all fit together?

What What isis inequality? inequality?

How to explain to policymakers?How to explain to policymakers?

A/A/

Provide unifying framework for inequalityProvide unifying framework for inequalityAcross groups or individuals

All use two dimensions for evaluation

Inequality as a comparison of twin “income standards”

What is inequality?What is inequality?

Canonical caseCanonical caseTwo persons Two persons 1 and 21 and 2

Two incomes xTwo incomes x11 and x and x22

Min income a = min(xMin income a = min(x11, x, x22))

Max income b = max(xMax income b = max(x11, x, x22))

InequalityInequalityI = (b - a)/bI = (b - a)/b or some function of ratio a/b or some function of ratio a/b

CaveatsCaveatsCardinal variableCardinal variable

Relative inequality Relative inequality

Inequality between GroupsInequality between Groups

Group Based InequalityGroup Based InequalityTwo groups 1 and 2Two groups 1 and 2Two income distributions xTwo income distributions x11 and x and x22

Income standard s(x) “representative income”Income standard s(x) “representative income”Lower income standard a = min(s(xLower income standard a = min(s(x11), s(x), s(x22))))Upper income standard b = max(s(xUpper income standard b = max(s(x11), s(x), s(x22))))


CaveatsCaveatsWhat about group size?What about group size?

Not relevant if group is unit of analysisNot relevant if group is unit of analysisRelevant if individual is unit of analysis Relevant if individual is unit of analysis –– Use smoothed dist. Use smoothed dist.

Inequality between Races in USInequality between Races in US

Black/White Age Adjusted Mortality

Year

Source:CDC and Levine, Foster, et al Public Health Reports (2001)

Log Mortality

Inequality between GroupsInequality between Groups

Group Based Inequality - DiscussionGroup Based Inequality - DiscussionNote: Groups can often be orderedNote: Groups can often be ordered

Women/men wages, Men/women health, poor region/rich region, Malay/Chinese incomes in Malaysia

Reflecting persistent inequalities of special concern or some Reflecting persistent inequalities of special concern or some underlying model underlying model

Health of poor/health of nonpoor

Health of adjacent SES classes - GradientGradient

Note: Relevance depends on salience of groups.Note: Relevance depends on salience of groups.

See discussion of subgroup consistency - Foster and Sen 1997

Can be more important than “overall” inequality

Recently interpreted as “inequality of opportunity”

Question: How to measure “overall” inequality in a population?Question: How to measure “overall” inequality in a population?

Answer: Answer: Analogous exerciseAnalogous exercise

Inequality in a PopulationInequality in a Population

Population Inequality - DiscussionPopulation Inequality - DiscussionA wide array of measuresA wide array of measures

Gini Coefficient Gini Coefficient

Coefficient of VariationCoefficient of Variation

Mean Log DeviationMean Log Deviation

Variance of logarithmsVariance of logarithms

Generalized Entropy FamilyGeneralized Entropy Family

90/10 ratio90/10 ratio

Decile RatioDecile Ratio

Atkinson FamilyAtkinson Family

Inequality in a PopulationInequality in a Population

Population Inequality - DiscussionPopulation Inequality - Discussion

Criteria for selectionCriteria for selection

Axiomatic BasisAxiomatic Basis - Lorenz consistent, subgroup consistent, decomposable, decomposable by ordered subgroupsUnderstandableUnderstandable. - Welfare basis, intuitive graphData AvailabilityData Availability - Historical studiesEasy to UseEasy to Use. - Is it in your software package?

What do the measures have in common?What do the measures have in common?

Inequality as Twin StandardsInequality as Twin Standards

Framework for Population InequalityFramework for Population InequalityOne income distribution One income distribution xxTwo income standards:Two income standards:

Lower income standard Lower income standard a = sa = sLL(x)(x)

Upper income standard Upper income standard b = sb = sUU(x)(x)

Note: Note: ssLL(x) (x) << s sUU(x) (x) for all xfor all x


ObservationObservationFramework encompasses all common inequality Framework encompasses all common inequality

measures measures Theil, variance of logs Theil, variance of logs in limitin limit


Population Inequality - DiscussionPopulation Inequality - DiscussionIncome StandardsIncome Standards ssLL ssUU

Gini CoefficientGini Coefficient Sen mean

Coefficient of VariationCoefficient of Variation mean euclidean mean

Mean Log DeviationMean Log Deviation geometric mean mean

Generalized Entropy FamilyGeneralized Entropy Family general mean mean

or mean general mean

90/10 ratio90/10 ratio income at 10th pc income at 90th pc

Decile RatioDecile Ratio mean mean of upper 10%

Atkinson Family Atkinson Family general mean mean


Population Inequality -Population Inequality - SummarySummaryInequality measures create Inequality measures create twin dimensionstwin dimensions of income of income

standardsstandardsCharacteristics of inequality measure depend on Characteristics of inequality measure depend on

characteristics of the standardscharacteristics of the standardsCan reverse process to assemble new measures of Can reverse process to assemble new measures of

inequalityinequality

Application of the MethodologiesApplication of the Methodologies

Growth and InequalityGrowth and InequalityTo see how inequality changes over timeTo see how inequality changes over time

Calculate growth rate for sCalculate growth rate for sLL

Calculate growth rate for sCalculate growth rate for sUU

Note: One of these is usually the meanNote: One of these is usually the meanCompare!Compare!

RobustnessRobustnessCalculate growth rates for several standards at onceCalculate growth rates for several standards at once

Ex: Evolution of General Means in TaiwanEx: Evolution of General Means in Taiwan

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

Gen

eral

Mea

n In

com

e R

elat

ive

to 1

976

Val

ue

1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996Year

Application: Growth and Inequality over Time Application: Growth and Inequality over Time Growth in for Mexico vs. Costa Rica

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

140

160

180

200

% C

hang

e in

inco

me

stan

dar

d

Costa Rica

1985-1995

Mexico1984-1996

Foster and Szekely (2008)

General Means are UniqueGeneral Means are Unique

Q/ Why general means?Q/ Why general means?A/ Satisfy Properties for an Income StandardA/ Satisfy Properties for an Income Standard

Symmetry, replication invariance, linear homogeneity, normalization, continuity andand

Subgroup consistency Subgroup consistency Suppose that s(x') > s(x) and s(y') = s(y), where x' has the same

population size as x, and y' has the same population size as y.

Then s(x', y') > s(x, y).

IdeaIdea Otherwise decentralized policy is impossible.

Th An income standard satisfies all the above properties if and only if it is a general meangeneral mean

Foster and Székely (2008)

General Means and AtkinsonGeneral Means and Atkinson

Application: Atkinson’s FamilyApplication: Atkinson’s Family

I = (I = ( - - ) / ) / < 1 < 1

Welfare interpretation of general mean and hence Welfare interpretation of general mean and hence inequality measureinequality measurePercentage welfare loss due to inequality

General MeansGeneral Means

Application: Assembling Decomposable Application: Assembling Decomposable Inequality MeasuresInequality Measures

Define Define Icq(x) =

Foster Shneyerov 1999Foster Shneyerov 1999

IIcqcq is a function of a ratio of two general means, or the limit of such functions is a function of a ratio of two general means, or the limit of such functions Atkinson, Theil, coeff of variation, generalized entropy, var of logs (not Gini)

SummarySummary

Income standards provide Income standards provide unifying frameworkunifying framework for measuring inequality and well beingfor measuring inequality and well being

Income standards should receive more direct Income standards should receive more direct empiricalempirical attention attention

Session 9

ReviewSubgroup ConsistencyIncome StandardsOther CharacterizationsUnifying Framework

TodayPoverty - IntroductionSpaceIdentificationAggregation

Poverty – Introduction

Recall3 aspects of distributionsize, spread, poverty

NoteOnly poverty – official measure Q/ Why?Q/ Why the concern with poverty?

Sen (1976) Two steps1. Identification2. Aggregation0. Space

SpaceQ/ Which one?

Cumulative Distribution Function

Income s

Cum

ula

tive p

opu

lati

on

F(s)

H

Income s

Cum

ula

tive p

opu

lati

on

1

.5

Exx = (2, 8, 4, 1)

Fx(s)

2 4 6 8

Q/ Poverty of what?Here – income, consumption, or a single dimensional

achievementLater – Sen contends we should examined inequality in a different

space

Q/ Which income?Among whom?Family size?Over what period of time?What about durable goods?In kind income?Rich uncles?Gvt. transfersBribes and black market income?Price differences?Inflation?Taxes? Etc. See Citro and Michael

1. IdentificationBooth in LondonRowntree in YorkOrshansky in USCitro-Michael in US

Types of Poverty lines See Foster 1998

Absolute za

Relative zr

Subjective zs

Hybrid zh

Examples

Citro and Michael (National Academy)Proposed new method for USCorrected biggest problems

UpdatingSen “Poor Relatively Speaking”

Impact on policy? NothingWhy?

2. Aggregation

Find P(x;z)

Income s

Cum

ula

tive p

opu

lati

on

1

.5

μ=3.75

Exx = (2, 8, 4, 1)

Fx(s)

2 4 6 8z

2. Aggregation

Number of poor Q(x;z)Headcount ratio H(x;z)Aggregate poverty gap A(x;z)Income gap ratio I(x;z)Per capita poverty gap P1(x;z)

Q/ What about inequality among poor?Sen measure S(x;z) uses Gini among poor

FGT measure P2(x;z) uses sq Coeff of var among poor

FGT class Pα(x;z)

session 9 review subgroup consistency income standards other characterizations unifying framework...

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x share

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xn1nthus sx

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permutation of y

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