Income Standards, Inequality and Poverty James E. Foster

Income Standards, Inequality and Income Standards, Inequality and PovertyPoverty

James E. FosterJames E. Foster

IntroductionIntroduction

QuestionsQuestionsWhat is inequality?What is inequality?

What is poverty?What is poverty?

How communicate concepts?How communicate concepts?To policymakers, publicTo policymakers, public

Ex/ Miguel Székely (2006),“Números que Mueven al Mundo: La Medición de la Pobreza en México”

This PaperThis PaperUnifying frameworkUnifying framework

IntroductionIntroduction

Key ConceptKey ConceptIncome StandardIncome StandardSummarizes distribution into a single incomeSummarizes distribution into a single income

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

Related papersRelated papers“Inequality Measurement” in The Elgar Companion to

Development Studies (David Clark, Ed.), Cheltenham: Edward Elgar 2006

“Poverty Measurement” in Poverty, Inequality and Development: Essays in Honor of Erik Thorbecke (Alain de Janvry and Ravi Kanbur, eds.), Amsterdam: Kluwer Academic Publishers, 2005

Income StandardsIncome Standards

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



x1

x2same



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)



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

s(x) = s(x) = = 30/16= 30/16

1 1 1 1

1 2 2 2

1 2 3 3

1 2 3 4



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



Another view



F = cdf

income

freq

p



F = cdf

income

freq

pA



F = cdf

income

freq

pA

p

A



F = cdf

income

freq

pA

p

A

Generalized LorenzGeneralized Lorenz



Generalize Lorenz CurveGeneralize Lorenz Curve

cumulative pop share

cum

ula

tive

inco

me



Generalize Lorenz CurveGeneralize 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



x1

x2same same 00



x1

x2same 0

x.1(x)0(x)



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



2 + x22 +...+ xn

2)/n )1/2

x1

x2

same same 22



2 + x22 +...+ xn

2)/n )1/2

x1

x2

same 2

1(x) 2(x)



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

Hardy Littlewood Polya 1952; Kolm 1969; Atkinson 1970Hardy Littlewood Polya 1952; Kolm 1969; Atkinson 1970

= 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

InequalityInequality

QuestionQuestion

What is inequality?What is inequality?

Universal framework for inequalityUniversal framework for inequalityTwo dimensions for evaluationTwo dimensions for evaluation

Reduces to discussion of twin “income standards”Reduces to discussion of twin “income standards”

What is inequality?What is inequality?

Canonical caseCanonical caseTwo persons Two persons 1 and 21 and 2

Two incomes Two incomes xx11 and x and x22

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

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

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

CaveatsCaveatsRatio scale Ratio scale

Relative inequality Relative inequality

Inequality between GroupsInequality between Groups

Group Based InequalityGroup Based InequalityTwo groups Two groups 1 and 21 and 2Two income distributions Two income distributions xx11 and x and x22

Income standard Income standard s(x) “rs(x) “representative income”epresentative income”Lower income standard Lower income standard a = min(s(xa = min(s(x11), s(x), s(x22))))Upper income standard Upper income standard b = max(s(xb = 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 analysisRelevant if individual is unit of analysis


Group Based Inequality - ExamplesGroup Based Inequality - ExamplesSpatial disparities Spatial disparities geographically determinedgeographically determinedGender inequality Gender inequality male/femalemale/femaleGrowthGrowth two points in timetwo points in time

Example: Racial Health Disparities in USExample: Racial Health Disparities in USTwo groups Two groups Black and WhiteBlack and WhiteTwo distributions Two distributions xx11 and x and x2 2 each with 1 = alive, 0 = noteach with 1 = alive, 0 = notIncome standard Income standard s(x) = s(x) = “1 - mortality rate”“1 - mortality rate”Lower income standard Lower income standard a = min(s(xa = min(s(x11), s(x), s(x22))))Upper income standard Upper income standard b = max(s(xb = max(s(x11), s(x), s(x22))))

InequalityInequalityI = (b - a)/bI = (b - a)/b or some function of ratio a/b, or some function of ratio a/b, Next graph uses ratios of mortality rates in log terms

Inequality between Races in USInequality between Races in US

Black/White Age Adjusted Mortality

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Year

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

Log Mortality


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

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 measures (yawn)A wide array of measures (yawn)

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

measuresmeasures 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 characteristics of the standardsstandardsCan reverse process to Can reverse process to assembleassemble new measures of new measures of

inequality, well being, povertyinequality, well being, poverty (where poverty line (where poverty line plays role of one of the income standards).plays role of one of the income standards).

Poverty in a PopulationPoverty in a Population

Poverty in a Population - DiscussionPoverty in a Population - DiscussionA wide array of measures (yawn)A wide array of measures (yawn)

Sen Sen

Thon/Sen/ShorrocksThon/Sen/Shorrocks

Clark, Hemming, Ulph/Chakravarty (CHU)Clark, Hemming, Ulph/Chakravarty (CHU)

Headcount RatioHeadcount Ratio

Per Capita Poverty GapPer Capita Poverty Gap

Watts IndexWatts Index

FGTFGT

What do they have in common? What do they have in common?

Poverty as Twin Income StandardsPoverty as Twin Income Standards

Framework for Population PovertyFramework for Population PovertyOne income distribution One income distribution xxTwo income standards:Two income standards:

Lower income standard Lower income standard a = sa = sLL(x) (x) (usually employs censored x)

Upper income standard Upper income standard b = z b = z (the absolute poverty line)

Note: Note: ssLL(x) (x) << z z for all xfor all x

PovertyPovertyP = (b - a)/bP = (b - a)/b or some function of ratio a/bor some function of ratio a/b

ObservationObservationFramework encompasses Watt’s, CHU, Sen, Thon, Framework encompasses Watt’s, CHU, Sen, Thon,

headcount, poverty gap. headcount, poverty gap.

Poverty as Twin Gap StandardsPoverty as Twin Gap Standards

Framework for Population PovertyFramework for Population PovertyOne gap distribution One gap distribution g g (positive entries are z - xi )

Two gap standards:Two gap standards:

Lower gap standard Lower gap standard a = sa = sLL(g)(g)

Upper gap standard Upper gap standard b = z b = z (the absolute poverty line)

Note: Note: ssLL(g) (g) << z z for all xfor all x

PovertyPovertyP = a/bP = a/b or some function of ratio a/bor some function of ratio a/b

ObservationObservationFramework encompasses FGT, Sen, Thon, headcount, Framework encompasses FGT, Sen, Thon, headcount,

poverty gap. poverty gap.

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!

Poverty and TimePoverty and TimeCalculate growth rate for respective standard.Calculate growth rate for respective standard.

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

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 (see Foster and Sen, 1997)

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 (2006)

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 (2006)

General Means and AtkinsonGeneral Means and Atkinson

Application: Atkinson’s FamilyApplication: Atkinson’s Family

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

Atkinson 1970Atkinson 1970

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

General Means and AtkinsonGeneral Means and Atkinson

InterpretationInterpretation

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

x1

x2

x.

General MeansGeneral Means

Application: Assembling Decomposable Application: Assembling Decomposable Inequality MeasuresInequality Measures

Define Define Icq(x) =

1

1

1

1

1

c(c - q)[(

(x)

(x)) - 1], c 0, q c

1

q n

x

(x)ln(

x

(x)), c 0, q c

(x) c 0, q c

2V (x), c 0, q = c

c

q

c

i

q

q

i

qi

n

q

L

q g xln(

( ))

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)

A Class of Distribution Sensitive HDI’sA Class of Distribution Sensitive HDI’s

H(D) = [(x), (y), (z)] for = 1- > 0

= 0 = 0 HH00 = = [(x), (y), (z)] usual HDI usual HDI

= 1 = 1 HH11 = = 0[0(x), 0(y), 0(z)] based on geometric meanbased on geometric mean

sensitive to inequalitysensitive to inequality

= 2= 2 HH22 = = [(x), (y), (z)] based on harmonic mean based on harmonic mean

even more sensitiveeven more sensitiveFoster, Lopez-Calva, and Szekely (2005, 2006)

SummarySummary

Income standards provide Income standards provide unifying frameworkunifying framework for measuring inequality, poverty and well for measuring inequality, poverty and well beingbeing

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

Plan to explore more thoroughly the Plan to explore more thoroughly the theoretical theoretical linklink between the properties of income standards between the properties of income standards and associated measuresand associated measures

Thank you!Thank you!

Documents

Income Standards, Inequality and Poverty James E. Foster