Session 7: Social Choice and Welfare Economics
Subgroup Consistency and DecomposabilityInequalityPovertyWelfareIncome Standards
Review
InequalityFour basic axioms
SymmetryReplication invarianceScale invarianceTransfer principle
Unanimity is LorenzSD2 over mean normalized distributions
Additional axiomTransfer sensitivity
Unanimity more complete than LorenzSD3 over mean normalized distributions
Review
WelfareFour basic axioms
SymmetryReplication invarianceMonotonicityTransfer principle
Unanimity is generalized LorenzSD2
Fixing mean yields Atkinson’s Th
Additional axiomTransfer sensitivity
Unanimity more complete than generalized LorenzSD3 Fixing the mean yields analog of Atkinson for transfer
sensitive
Unanimity for first three is SD1
Review
PovertyFive basic axioms
SymmetryReplication invarianceFocusMonotonicityTransfer principle
Types of quasiorderingsVariable poverty lineVariable measure
Review
PovertyVariable Poverty Line unlimited range
Headcount ordering P0 is SD1
Poverty gap ordering P1 is SD2
FGT ordering P2 is SD3
Variable Poverty Line up to z*
Headcount ordering P*0 is SD*1
Poverty gap ordering P*1 is SD*2
FGT ordering P*2 is SD*3
Variable Measure for a given z* and continuous measures
Unanimity for (sym, rep. inv., foc., mon..) is is SD*1
Unanimity for (above and trans.) is is SD*2
Unanimity for (above and trans. sens.) is is SD*3
Variable Measure up to z* and for continuous measures
Same
Review
RecallWelfare conclusions identical if
Consider only additive welfare functions Replace mon., trans., trans. sens. with respective deriv.
cond’s.Entirely analogous to expected utilitySame quasiordering with or without additivity assumption
NoteSen’s broadening suggests generalizing expected
utilityNon-additive forms such as Sen measure S(x)Indeed decision theory took this step to address paradoxesAdditivity assumption important here
Restricts predicted behavior in meaningful ways
Preview
This SessionConsiders two axioms related to additive form
Context of: inequality, poverty, welfare, income standards
Subgroup consistencyConceptual axiom Requires coherence between overall and subgroup changes
DecomposabilityPractical axiom for empirical applicationsOverall index is weighted additive sum of subgroup values
Plus between group term in case of inequalityInequality is perhaps most restrictive application
Starting point
Inequality
DecomposabilityHelps answer questions like:
Is most of global inequality within countries or between countries?
How much of total inequality in wages is due to gender inequality?
How much of today’s inequality is due to purely demographic factors?
Idea (Theil, 1967)
Analysis of variance (ANOVA)Total variance V(.) is divided into:
part that is ‘explained’part that is ’unexplained’
ANOVA
An exampleA program is made available to a randomly selected
population (the treatment group). Outcomes are
A second randomly selected group does not have access to the program. Outcomes are
Q/ Did the program have an impact?
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x =(x1, ...,xnx)
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y=(y1, ..., yny)
ANOVA
Notation x y (x,y)Population nx ny n
Mean μx μy μ
Variances V(x) V(y) V(x,y)Smoothed
DecompositionV(x,y) =
(i1nx (xi x x )2 j1
ny (yj y y )2) / n
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Proof : V(x, y) =(i=1nx (x i −)
2 + j=1ny (yj −)
2 ) / n
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=nx
ni=1
nx (xi −x)2 / nx +
ny
nj=1
ny (yj −y)2 / ny + (i=1
nx (x −)2 + j=1
ny (y −)2 ) / n
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nx
nV(x) +
ny
nV(y) +V( x, y)
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x =(x, ...,x)
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y=(y, ...,y)
€
( x, y)
€
=nx
nV(x) +
ny
nV(y) +V( x, y)
ANOVA
IdeaV(x,y) = overall variance
= within group variance
= between group variance the part of the variance ‘explained’ by the treatment
= share of the variance ‘explained’ by treatment
Q/ What makes this analysis possible?A/ Additive decomposition of variance
nx
nV(x)
ny
nV(y)
V(x , y )
V(x , y )/ V(x,y)
Inequality Decompositions
Theil’s entropy measure
where sx = |x|/|(x,y)| is the income share of x
Theil’s second measure mean log deviation
where px = nx/n is the population share of x
NoteWeights > 0 and Depend on subgroup statistics: nx, ny, μx and μy
T(x,y) sxT(x) syT(y) T(x , y )
D(x,y) pxD(x) pyD(y) D(x , y )
Inequality Decompositions
Axiom Additive DecomposabilityFor any x, y we have
where weights wx and wy and nonnegative and
depend on nx, ny, μx and μy.
Note Usually stated for arbitrary number of subgroups
Ex Generalized entropy
Half Squared coefficient of variation
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I (x, y) =wxI (x) + wyI (y) + I ( x, y)
I (x,y) nx
n
x
I (x) ny
n
y
I (y) I (x , y )
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I 2 x, y nx
n
x
2
I 2 x ny
n
y
⎛ ⎝ ⎜ ⎞
⎠ ⎟2
I 2 y I 2 x , y
Generalized Entropy with = 2 half sq coef var
Income Distributionsx = (12,21,12) y = (15,32,10) (x,y) =
(12,21,12,15,32,10)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 15 μy = 19 μ = 17
Inequality LevelsIα(x) = 0.040 Iα(y) = 0.123 Iα(x,y) = 0.099
Weightswx = 0.389 wy = 0.625
Within GroupwxIα(x) + wyIα(y) = (0.016 + 0.077) = 0.092
Between GroupIα(x,y) =Iα(15,15,15,19,19,19) = 0.00692
Note Adds to total inequality = 0.099 Betw group contr. 7.1%
Example Decompositions
Generalized Entropy with = 1 Theil’s entropy
Income Distributionsx = (12,21,12) y = (15,32,10) (x,y) =
(12,21,12,15,32,10)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 15 μy = 19 μ = 17
Inequality LevelsIα(x) = 0.038 Iα(y) = 0.118 Iα(x,y) = 0.090
Weightswx = 0.441 wy = 0.559
Within GroupwxIα(x) + wyIα(y) = (0.017 + 0.066) = 0.083
Between GroupIα(x,y) =Iα(15,15,15,19,19,19) = 0.00694
Note Adds to total inequality = 0.090 Betw group contr. 7.8%
Example Decompositions
Generalized Entropy with = 1/2
Income Distributionsx = (12,21,12) y = (15,32,10) (x,y) =
(12,21,12,15,32,10)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 15 μy = 19 μ = 17
Inequality LevelsIα(x) = 0.037 Iα(y) = 0.118 Iα(x,y) = 0.087
Weightswx = 0.470 wy = 0.529
Within GroupwxIα(x) + wyIα(y) = (0.017 + 0.062) = 0.080
Between GroupIα(x,y) =Iα(15,15,15,19,19,19) = 0.00695
Note Adds to total inequality = 0.087 Betw group contr. 8.0%
Example Decompositions
Generalized Entropy with = 0 Theil’s second
Income Distributionsx = (12,21,12) y = (15,32,10) (x,y) =
(12,21,12,15,32,10)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 15 μy = 19 μ = 17
Inequality LevelsIα(x) = 0.037 Iα(y) = 0.119 Iα(x,y) = 0.085
Weightswx = 0.500 wy = 0.500
Within GroupwxIα(x) + wyIα(y) = (0.018 + 0.059) = 0.078
Between GroupIα(x,y) =Iα(15,15,15,19,19,19) = 0.00697
Note Adds to total inequality = 0.085 Betw group contr. 8.2%
Example Decompositions
Generalized Entropy with = -1 transfer sens.
Income Distributionsx = (12,21,12) y = (15,32,10) (x,y) =
(12,21,12,15,32,10)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 15 μy = 19 μ = 17
Inequality LevelsIα(x) = 0.036 Iα(y) = 0.127 Iα(x,y) = 0.084
Weightswx = 0.567 wy = 0.447
Within GroupwxIα(x) + wyIα(y) = (0.020 + 0.057) = 0.077
Between GroupIα(x,y) =Iα(15,15,15,19,19,19) = 0.00702
Note Adds to total inequality = 0.084 Betw group contr. 8.3%
Example Decompositions
NoteOnly Theil measures have weights summing to 1 Between group term
smallerrose slightly as α fellcontribution increased as α fell
Within group termlargerdecreased as α fellcontribution decreased as α fell
Q/What about other additively decomposable measures?
A/ Explored by Bourguignon (1979), Cowell and Kuga (1979), Shorrocks (1980,
1984), Foster (1984), and others.
Example Decompositions
MethodsUse functional equations (Aczel, 1966, Aczel&Dhombres,
1989)Ex Cauchy equations Continuous solutions f(a+b) = f(a) + f(b) f(s) = ksf(a+b) = f(a) f(b) f(s) = eks or 0f(ab) = f(a) + f(b) f(s) = klnsf(ab) = f(a) f(b) f(s) = sk or 0
OutlineAssume I(x) satisfies various axioms (and reqularity condition)Derive a function f(s) from I(x)Show f must satisfy a particular functional equationDerive functional form for fUse f and axioms to reconstruct allowable forms for I
Econ to math to econ
Axiomatic Characterizations
Q/ Is there any other relative measure that has this decomposition?
Theorem Foster (1983)
I is a Lorenz consistent inequality measure satisfying Theil decomposability if and only if there is some positive constant k such that
I(x) = kT(x) for all x.Idea Only the Theil measure has its decompositionProof (Key step) Set f(s) = I(s,1-s) for 0 < s < 1. Note f(s) =
f(1-s) by sym. By TD and LC, f(s) + (1-s)f(t/(1-s)) = f(t) + (1-t)f(s/(1-t)) for all s,t with s+t < 1. This functional equation has the solution: f(t) = t ln t + (1-t)ln (1-t) which eventually yields T(x).
I (x,y) sxI (x) syI (y) I (x , y )
Characterization of Theil
Axiom (Theil Decomposability)For any x,y we have
Key papersBourguignon (1979), Shorrocks (1980)Characterize Theil measures and GE measuresHowever – Assumed that I is (twice) differentiable
Violated by Gini
Characterization of Generalized Entropy
Recall Additive decomposabilityFor any x,y we have
where weights wx and wy and nonnegative and depend on nx, ny, μx and μy.
Q/ Other measures satisfying this axiom?Theorem Shorrocks (1984)
I is a continuous, normalized, Lorenz consistent inequality measure satisfying additive decomposability if and only if there is some positive constant k and some α such that
I(x) = kIα(x) for all x.Idea Only the generalized entropy measures
Shorrocks proved more general result: aggregability
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I x, y wxI x wyI y I x , y
Characterization of Generalized Entropy
NoteGini coefficient is not additively decomposableOk for some subgroups not for others
ConsiderFormula
with weights
wx = (nx/n)2(μx/μ)
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G(x, y) =wxG(x) + wyG(y) + G( x, y)
Gini Breakdown
Example
Income Distributionsx = (10,12,12) y = (15,21,32) (x,y) =
(10,12,12,15,21,32)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 11.33μy = 22.67μ = 17
Inequality LevelsG(x) = 0.039 G(y) = 0.167 G(x,y) = 0.229
Weightswx = 0.167 wy = 0.333
Within GroupwxG(x) + wyG(y) = (0.007 + 0.056) = 0.062
Between GroupG(x,y) =G(11.3,11.3,11.3,22.7,22.7,22.7) = 0.167
Note Adds to total inequality = 0.229 Non-overlapping groups!
Exact Breakdown
Example overlapping groups
Income Distributionsx = (12,21,12) y = (15,32,10) (x,y) =
(12,21,12,15,32,10)
Populations and Meansnx = 3 ny = 3 n = 6
μx = 15 μy = 19 μ = 17
Inequality LevelsG(x) = 0.133 G(y) = 0.257 G(x,y) = 0.229
Weightswx = 0.221 wy = 0.279
Within GroupwxG(x) + wyG(y) = (0.029 + 0.072) = 0.101
Between GroupG(x,y) =G(15,15,15,19,19,19) = 0.059
Note Adds to 0.160 < 0.229 R = residual = 0.69 Why?
Breakdown with Residual
Answers
Use rank order definition of Gini (def 3)Impact of transfer on Gini depends on:
Size of transferNumber (or percent) of population between the two
Use expected difference definition of Gini (def 1)Construct difference matrix
Submatrices on diagonal relate to within groupSubmatrices off diagonal relate to between group
So long as nonoverlappingOtherwise sum of entries is too large - residual in proportion to number of overlapping entries
Try for example x = (2,6) y = (6,10) vs. x’ = (1,7), y’ = (5,11)
Breakdown with Residual
Now examine related axiomSubgroup consistency
Idea: Helps answer questions likeWill local inequality reductions decrease overall
inequality?If gender inequality stays the same and
inequality within the groups of men and women rises, must overall inequality rise?
SourcesCowell (1988) “Inequality decomposition: three bad
measures”
FGT (1984) similar concept called subgroup monotonicity
Shorrocks (1988) axiomatic result
Subgroup Consistency
Axiom (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’ = (2,2,7,9) 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’) ‘Almost’ a violation of SC
Idea Residual R fellModified Exx” = (2,2,6.9,9.1) y” = (2,2) (x”,y”) =
(2,2,6.9,9.1,2,2)G(x) < G(x”), G(y) = G(y”), G(x,y) > G(x”,y”) Gini violates. How about the coeff of var?I2(x) = I2(x’), I2(y) = I2(y’), I2(x,y) = I2(x’,y’) I2(x) < I2(x”), I2(y) = I2(y”), I2(x,y) < I2(x”,y”)
Subgroup Consistency
Result If I is additively decomposable then I is
subgroup consistent
Proof Stare at this
ImplicationAll generalized entropy measures are SCSame true for increasing transformation of
additively decomposable measureExample Atkinson’s measure (μ – eα)/μ
An alternative Atkinson’s measure (μ – eα)/eα
Others?
Subgroup Consistency
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I (x, y) =wxI (x) + wyI (y) + I ( x, y)
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.
ImplicationsOnly the GE measures or monotonic
transformationsLimited scope for satisfying SC
Subgroup Consistency
Discussion
Decomposability is usefulAllows analyses otherwise impossibleDisentangle demographic changes from economic
changes
Not always neededMany analyses concern overall inequality – not
broken down
Represents a strong restrictionOnly generalized entropyIs that all there is to inequality?
But can be generalizedTo restrict domain of applicability (say non-
overlapping)To allow other income standards as basis of
decompositionWider classes of measures
Subgroup Consistency and Additive Decomposability
Discussion
Subgroup consistency very compellingEspecially in policy context
Most policies rely on regional componentTry explaining away the inconsistencyRelativity run rampant
But inequality centrally relativeWhy can’t our determination of greater or less
inequality depend on all the incomes? (where do we stop)
Ex (from book)x = (1,3,8,8) y = (2,2) (x,y) = (1,3,8,8,2,2) x’ = (2,2,7,9) y’ = (2,2) (x’,y’) = (2,2,7,8,2,2)
Alt exx = (1,3,8,8) y = (8,8) (x,y) = (1,3,8,8,8,8) x’ = (2,2,7,9) y’ = (8,8) (x’,y’) = (2,2,7,9,8,8)Must they go in same direction?
Subgroup Consistency and Additive Decomposability
Subgroup Consistency and Additive Decomposability
Should comparison depend on whether a = 2 or a = 8?
Now change context - PovertyAxiom Decomposability
Decomposability: For any distributions x and y, we have P(x,y;z) = (nx/n) P(x;z) + (ny/n) P(y;z).
Q/ Why useful?
A/ Poverty ProfilesSudhir Anand (1983) Inequality and poverty in Malaysia :
measurement and decomposition, Oxford University Press
Idea Who are the poor? Where are the poor?See the following guide at the World Bank’s site
http://info.worldbank.org/etools/docs/library/103073/ch7.pdf
Poverty
Decomposability
Sudhir Anand, (1977) Rev Inc Wealth
Decomposability and Poverty Profiles
Source: http://info.worldbank.org/etools/docs/library/103073/ch7.pdf
Q/ Which measures are decomposable?A/ FGT, CHU, and an infinity of othersNote Not like inequality gen. entropy unique class
ConsiderPf (x;z) = (1/n)Σif(xi;z) for any f
Note Pf satisfies decomposability for any fPrf (nx/n) Pf(x;z) + (ny/n) Pf(y;z)
= (1/n)Σif(xi;z) + (1/n)Σif(yi;z)
= Pf(x,y;z)
Note Foster and Shorrocks (1991) show that Pf can be axiomatically characterized by additive decomposability
Q/Other properties?
Decomposable Poverty Measures
A/ Every Pf satisfies symmetry, replication invariance, subgroup
consistency
Note Other properties of Pf depend on ffocus: f(s;z) is a constant function in s for z ≤ s
normalization: f(s;z) = 0 for z ≤ s
continuity: f(s;z) is continuous in sQ/ what is f for H?
monotonicity: f(s;z) is a decreasing function of s for s ≤ z
transfer: f(s;z) is a strictly convex function of s for s ≤ z
transfer sensitivity: the marginal impact of an extra dollar on individual poverty, or ∂f(s;z)/∂s, is a strictly concave function of s for s ≤ z
scale invariance: f(s;z) = r(s/z) for some function r
Decomposable Poverty Measures
Axiom Subgroup Consistency Let x, x’, y, and y’ be distributions satisfying nx =
nx’ and ny = ny’. If P(x;z) > P(x';z) and P(y;z) = P(y';z) then P(x,y;z) > P(x',y';z).
Recall Shorrocks Result for inequality measures: Only subgroup
consistent inequality measures are gen entropy and mon. transf.
Note Here All Pf at least!
Theorem Foster and Shorrocks (1991)If P is a continuous, symmetric, replication
invariant, monotonic poverty measure satisfying subgroup consistency, then P is a continuous increasing transformation of Pf for some continuous f that is decreasing in income.
Note Decomposable measure or some transformation
Subgroup Consistent Poverty Measures
Discussion
Additive Decomposability
Sudhir Anand, (1977) Rev Inc Wealth
Profiles are immensely important in empirical studiesNeed decomposability to construct them
Used in targeting – basis of Mexico’s Progresa programNot all evaluations require subgroup analysis
Property is needlessly restrictive
Discussion
Absolutely no hope of policy discussion if violatedShould hold for all partitions – since all can be relevantExisting critique based on differential links across
individuals is actually a critique of symmetry, transfer, and the construction and relevance of the income variable itself – not SC per se
If links are known, should be incorporated into variable Analogous to equivalence scales
This axiom is contingent and should only be applied in certain cases
Subgroup Consistency
Note Much overlap with povertyAxiom Decomposability
Decomposability: For any distributions x and y, we have W(x,y) = (nx/n) W(x) + (ny/n) W(y).
Note Per capita form – otherwise straight sumQ/ Which functions are decomposable?
ConsiderWu (x;z) = (1/n)Σiu(xi) for any increasing u
Clearly Wu satisfies decomposabilityNote Can adapt standard arguments as in Foster and
Shorrocks (1991) to axiomatically characterize Wu by additive decomposability (or see Hamada discus. p. 39)
Q/Other properties?
Welfare
A/ Every Wu satisfies symmetry, replication invariance, monotonicity,
subgroup consistency
Note Other properties of Wu depend on ucontinuity: u(s) is continuous in s
Q/ what is f for H?
transfer: u(s) is a strictly concave function of s
transfer sensitivity: the marginal utility is a strictly convex function of s
Decomposable Welfare Functions
Axiom Subgroup Consistency Let x, x’, y, and y’ be distributions satisfying nx =
nx’ and ny = ny’. If W(x) > W(x’) and W(y) = W(y’) then W(x,y) > W(x',y’).
Recall Shorrocks result for inequality measures: Only subgroup
consistent inequality measures are gen entropy and mon. transf.
Note All Wu at least! Others?
TheoremIf W is a continuous, symmetric, replication
invariant, monotonic, welfare function satisfying subgroup consistency, then W is a continuous increasing transformation of Wu for some continuous and increasing u.
Note Decomposable measure or some transformation
Subgroup Consistent Welfare Functions
Discussion
Similar to above discussions.
Subgroup Consistency and Additive Decomposability
One final context: Income standard s(x)One final context: Income standard s(x)IdeaIdea
Income standard summarizes entire distribution x in Income standard summarizes entire distribution x in a single ‘representative’ income s(x)a single ‘representative’ income s(x)
ExEx Mean, median, income at 90th percentile, mean of Mean, median, income at 90th percentile, mean of
top 40%, Sen’s mean, Atkinson’s ede income…top 40%, Sen’s mean, Atkinson’s ede income…Measures ‘size’ of the distributionMeasures ‘size’ of the distributionCan have normative interpretation Can have normative interpretation Atkinson’s edeAtkinson’s ede
Related papersRelated papersFoster (2006) “Inequality MeasurementFoster and Shneyerov (1999, 2000)Foster and Szekely (2008)
Income Standard
Properties of s(x)Properties of s(x)SymmetrySymmetryReplication invarianceReplication invarianceLinear homogeneityLinear homogeneityNormalization Normalization s(s(μμ,,μμ,…,,…, μ μ) = ) = μμContinuityContinuityWeak monotonicityWeak monotonicity
ExamplesExamplesMean Mean , median, 10, median, 10thth percentile income, 90 percentile income, 90thth pc pc
income, mean of the bottom fifth, mean of top 40%, income, mean of the bottom fifth, mean of top 40%, Sen welfare function S(x), geometric mean Sen welfare function S(x), geometric mean 0, , Euclidean mean Euclidean mean 2, all the other general means, all the other general means
(x) = [(x1 + … + xn
)/n] 1/ for for ≠ 0
(where lower (where lower implies greater emphasis on lower incomes) implies greater emphasis on lower incomes)Income standards provide unifying framework for inequality, Income standards provide unifying framework for inequality,
poverty poverty
Income Standard
What is inequality?What is inequality?Canonical caseCanonical case
Two persons 1 and 2Two persons 1 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)/b or some function of ratio a/bI = (b - a)/b or some function of ratio a/b
CaveatsCaveatsCardinal variableCardinal variableRelative inequality Relative inequality
Inequality
Group Based InequalityGroup Based InequalityTwo groups 1 and 2Two groups 1 and 2
Two 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)/b or some function of ratio a/bI = (b - 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 analysis
Inequality between Groups
DiscussionDiscussionGroups can often be orderedGroups can often be ordered
Women/men wages, Men/women health, poor region/rich region, Women/men wages, Men/women health, poor region/rich region, Malay/Chinese incomes in MalaysiaMalay/Chinese incomes in Malaysia
Reflecting persistent inequalities of special concern Reflecting persistent inequalities of special concern or some underlying model or some underlying model Health of poor/health of nonpoorHealth of poor/health of nonpoorHealth of adjacent SES classes - GradientHealth of adjacent SES classes - Gradient
Note: Relevance depends on salience of groups.Note: Relevance depends on salience of groups.See discussion of subgroup consistency - Foster and Sen 1997See discussion of subgroup consistency - Foster and Sen 1997Can be more important than “overall” inequalityCan be more important than “overall” inequalityRecently interpreted as “inequality of opportunity”Recently interpreted as “inequality of opportunity”
Question: How to measure “overall” inequality in a Question: How to measure “overall” inequality in a population?population?
Answer: Analogous exerciseAnswer: Analogous exercise
Inequality between Groups
Framework for Population InequalityFramework for Population InequalityOne income distribution One income distribution xxTwo income standards:Two income standards:
Lower income standard a = sLower income standard a = sLL(x) (x)
Upper income standard b = sUpper income standard b = sUU(x)(x)
Note: sNote: sLL(x) ≤ s(x) ≤ sUU(x) for all x(x) for all x
InequalityInequalityI = (b - a)/b or some function of ratio a/bI = (b - 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 in limitTheil, variance of logs in limit
Inequality within a Population
Inequality as twin income standards
MeasureMeasure StandardsStandards
ssLL s sUU
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 10th pc income 90th pc income
Decile RatioDecile Ratio mean upper 10% mean
Atkinson Family Atkinson Family general mean mean
Inequality within a Population
Application: Growth and Inequality
To see how inequality changes over timeCalculate growth rate for sL
Calculate growth rate for sU
Note: One of these is usually the meanCompare!
Inequality as Twin Income Standards
-100
-80
-60
-40
-20
0
20
40
60
80
100
120
140
160
180
200
% C
hang
e in
inco
me
stan
dard
Costa Rica1985-1995
Mexico1984-1996
Foster and Szekely (2008)
Application: Growth and Inequality over Time Application: Growth and Inequality over Time Growth in for Mexico vs. Costa Rica
Inequality as Twin Income Standards
Recall decomposable CHU
Monotonic transformation yields
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Cβ(x;z) =
1
βn1− xi
*
z
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
β ⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
β ≤1,β ≠0i=1
n
∑
1
n(lnz −lnxi
*) β =0i=1
n
∑
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
€
Cβ(x;z) =
1− 1
n
xi*
z
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
β
i=1
n
∑ ⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
1/ β
β ≤1,β ≠0
1− 1
n
xi*
z
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟i=1
n∏ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1/ n
β =0
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
Poverty and Income Standards
or
NoteSubgroup consistentIntuitive Analogous to Atkinson inequality measure
Poverty line in place of mean
IdeaPoverty is function of two standards
Poverty lineGeneral mean of censored income
€
Cβ(x;z) =z −β(x*)
zfor β ≤1
Poverty and Income Standards
Framework for Population PovertyOne income distribution xTwo income standards:Lower income standard a = sL(x) (usually
employs censored x) Upper income standard b = z (the absolute
poverty line) Note: sL(x) < z for all x
PovertyP = (b - a)/b or some function of ratio a/b
ObservationFramework encompasses Watt’s, CHU, Sen,
Thon, headcount, poverty gap.
Poverty and Income Standards
Recall decomposable FGT for α > 0
Monotonic transformation yields
or
€
P(x;z) =1
n
gi
z
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
i=1
n
∑ ⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
1/
€
P(x;z) =1
n
z −x i*
z
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
i=1
n
∑ =1
n
gi
z
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
i=1
n
∑
Poverty and Income Gap Standards
or
NoteSubgroup consistentIntuitive
IdeaPoverty is function of two standards
Poverty lineGeneral mean of gaps
€
P(x;z) =(g)
z
Poverty and Income Gap Standards
Framework for Population PovertyOne gap distribution g (positive entries are z - xi
)
Two gap standards:Lower gap standard a = sL(g) Upper gap standard b = z (the absolute poverty
line) Note: sL(g) < z for all x
PovertyP = a/b or some function of ratio a/b
ObservationFramework encompasses FGT, Sen, Thon,
headcount, poverty gap.
Poverty and Income Gap Standards
Application: Growth and PovertyTo see how poverty changes over time
Calculate growth rate for respective standard
Works with relative poverty line too
Poverty and Income Gap Standards
Q/ What if restrict to income standards that are subgroup consistent?
Theorem Foster and Szekely 2008s(x) is an income standard satisfying subgroup
consistency if and only if it is a positive multiple of μα(x)
Note Back to this important single parameter class
In contrast Any additive welfare function is subgroup consistent. For income standards only a class.
Q/ Why the difference?A/ linear homogeneity of standards
Subgroup Consistent Income Standards