Session 7: Social Choice and Welfare Economics Subgroup Consistency and Decomposability Inequality Poverty Welfare Income Standards

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?

€

x =(x1, ...,xnx)

€

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

€

Proof : V(x, y) =(i=1nx (x i −)

2 + j=1ny (yj −)

2 ) / n

€

=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

€

nx

nV(x) +

ny

nV(y) +V( x, y)

€

x =(x, ...,x)

€

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

€

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 )

€

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


(12,21,12,15,32,10)


μx = 15 μy = 19 μ = 17


Weightswx = 0.441 wy = 0.559





Generalized Entropy with = 1/2


(12,21,12,15,32,10)


μx = 15 μy = 19 μ = 17


Weightswx = 0.470 wy = 0.529





Generalized Entropy with = 0 Theil’s second


(12,21,12,15,32,10)


μx = 15 μy = 19 μ = 17


Weightswx = 0.500 wy = 0.500





Generalized Entropy with = -1 transfer sens.


(12,21,12,15,32,10)


μx = 15 μy = 19 μ = 17


Weightswx = 0.567 wy = 0.447





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.


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

€

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)

€

G(x, y) =wxG(x) + wyG(y) + G( x, y)

Gini Breakdown

Example


(10,12,12,15,21,32)


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


(12,21,12,15,32,10)


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”)


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?


€

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


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?



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


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.


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))))


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


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

€

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


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.


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


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.


Application: Growth and PovertyTo see how poverty changes over time

Calculate growth rate for respective standard

Works with relative poverty line too


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

Documents

Session 7: Social Choice and Welfare Economics Subgroup Consistency and Decomposability Inequality Poverty Welfare Income Standards