Session 9 Review Subgroup Consistency Income Standards Other Characterizations Unifying Framework...
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Transcript of 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)
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
Income StandardsIncome Standards
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.
Income StandardsIncome Standards
ExamplesExamplesMeanMean s(x) =s(x) = (x) = (x1+...+xn)/n
F = cdf
income
freq
Income StandardsIncome Standards
ExamplesExamplesMedian Median x = (3, 8, 9, 10, 20), s(x)s(x) = 9= 9
F = cdf
income
freq
0.5
median
Income StandardsIncome Standards
ExamplesExamples1010thth percentile percentile
F = cdf
income
freq
0.1
s =s = Income at10th percentile
Income StandardsIncome Standards
ExamplesExamplesMean of bottom fifth Mean of bottom fifth
x = (3, 5, 6, 6, 8, 9, 15, 17, 23, 25)
s(x) = 4s(x) = 4
Income StandardsIncome Standards
ExamplesExamplesMean of top 40% Mean of top 40%
x = (3, 5, 6, 6, 8, 9, 15, 17, 23, 25)
s(x) = 20s(x) = 20
Income StandardsIncome Standards
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
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
Income StandardsIncome Standards
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
Income StandardsIncome Standards
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)
Income StandardsIncome Standards
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)
Income StandardsIncome Standards
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))))
InequalityInequalityI = (b - a)/bI = (b - a)/b or some function of ratio a/b or some function of ratio a/b
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
InequalityInequalityI = (b - a)/bI = (b - a)/b or some function of ratio a/b or some function of ratio a/b
ObservationObservationFramework encompasses all common inequality Framework encompasses all common inequality
measures measures Theil, variance of logs Theil, variance of logs in limitin limit
Inequality as Twin StandardsInequality as Twin Standards
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
Inequality as Twin StandardsInequality as Twin Standards
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)