Review

1
Session 9

• Review
• Decomposability
• Characterizations
• Gini Breakdown
• Today
• Subgroup Consistency
• Income Standards
• Other Characterizations
• Unifying Framework

2
Decomposability

• Helps 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 todays inequality is due to purely
  demographic factors?
• Source
• Analysis of variance (ANOVA)
• Total variance can be divided into a term
  representing the part that is explained by a
  particular characteristic and a second part that
  is unexplained.

3

• Example
• A development program is made available to a
  randomly selected population (the treatment
  group). Outcomes are x
• A second group that is randomly selected does not
  have access to the program. Outcomes are y
• Q/ Did the program have an impact?

4

• Notation
• µx and nx mean and pop size of x
• µy and ny mean and pop size of y
• µ and n - mean outcome and population overall
• V(.) is the variance
• Decomposition
• V(x,y)

5

• Idea
• V(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/ Decomposition of variance

6
Inequality Decompositions

• Additive Decomposability
• Note
• Usually stated for any number of groups
• Between group contribution
• Explained inequality
• Ex Amount due to gender inequality, differences
  across countries
• Within group contribution
• Unexplained inequality

7

• Specific Decompositions
• Theils entropy measure
• where sx x/(x,y) is the income share of x
• Theils second measure mean log deviation
• where px nx/n is the population share of x

8

• Specific Decompositions
• Squared Coefficient of Variation CV/µ2
• Note
• Follows from variance decomposition
• using C(x) V(x/µ)
• Generalized entropy measures Ia

9
Ex Generalized Entropy with a -1 transfer sens.

• Income Distributions
• x (12,21,12) y (15,32,10) (x,y)
  (12,21,12,15,32,10)
• Populations and Means
• nx 3 ny 3 n 6
• µx 15 µy 19 µ 17
• Inequality Levels
• Ia(x) 0.036 Ia(y) 0.127 Ia(x,y) 0.084
• Weights
• wx 0.567 wy 0.447
• Within Group
• wxIa(x) wyIa(y) (0.020 0.057) 0.077
• Between Group
• Ia(x,y) Ia(15,15,15,19,19,19) 0.00702
• Note Adds to total inequality 0.084 Betw group
  contr. 8.3

10
Ex Generalized Entropy with a 0 Theils second

• Income Distributions
• x (12,21,12) y (15,32,10) (x,y)
  (12,21,12,15,32,10)
• Populations and Means
• nx 3 ny 3 n 6
• µx 15 µy 19 µ 17
• Inequality Levels
• Ia(x) 0.037 Ia(y) 0.119 Ia(x,y) 0.085
• Weights
• wx 0.500 wy 0.500
• Within Group
• wxIa(x) wyIa(y) (0.018 0.059) 0.078
• Between Group
• Ia(x,y) Ia(15,15,15,19,19,19) 0.00697
• Note Adds to total inequality 0.085 Betw group
  contr. 8.2

11
Ex Generalized Entropy with a 1/2

• Income Distributions
• x (12,21,12) y (15,32,10) (x,y)
  (12,21,12,15,32,10)
• Populations and Means
• nx 3 ny 3 n 6
• µx 15 µy 19 µ 17
• Inequality Levels
• Ia(x) 0.037 Ia(y) 0.118 Ia(x,y) 0.087
• Weights
• wx 0.470 wy 0.529
• Within Group
• wxIa(x) wyIa(y) (0.017 0.062) 0.080
• Between Group
• Ia(x,y) Ia(15,15,15,19,19,19) 0.00695
• Note Adds to total inequality 0.087 Betw group
  contr. 8.0

12
Ex Generalized Entropy with a 1 Theils entropy

• Income Distributions
• x (12,21,12) y (15,32,10) (x,y)
  (12,21,12,15,32,10)
• Populations and Means
• nx 3 ny 3 n 6
• µx 15 µy 19 µ 17
• Inequality Levels
• Ia(x) 0.038 Ia(y) 0.118 Ia(x,y) 0.090
• Weights
• wx 0.441 wy 0.559
• Within Group
• wxIa(x) wyIa(y) (0.017 0.066) 0.083
• Between Group
• Ia(x,y) Ia(15,15,15,19,19,19) 0.00694
• Note Adds to total inequality 0.090 Betw group
  contr. 7.8

13
Ex Generalized Entropy with a 2 alf sq coef var

• Income Distributions
• x (12,21,12) y (15,32,10) (x,y)
  (12,21,12,15,32,10)
• Populations and Means
• nx 3 ny 3 n 6
• µx 15 µy 19 µ 17
• Inequality Levels
• Ia(x) 0.040 Ia(y) 0.123 Ia(x,y) 0.099
• Weights
• wx 0.389 wy 0.625
• Within Group
• wxIa(x) wyIa(y) (0.016 0.077) 0.092
• Between Group
• Ia(x,y) Ia(15,15,15,19,19,19) 0.00692
• Note Adds to total inequality 0.099 Betw group
  contr. 7.1

14

• Note
• Only Theil measures have weights summing to 1
• Between group term
• smaller
• fell slightly as a rose
• contribution decreased with a
• Within group term
• larger
• increased as a rose
• contribution increased with a
• Recall Theil measure used by AnandSegal to
  evaluate global inequality

15

• Characterizations
• Q/What other inequality measures are
  decomposable?
• A/Explored by Bourguignon (1979), Shorrocks
  (1980), Foster (1984), and others
• Idea
• Axiomatic approach
• - Start with generic I(x)
• - Assume various axioms
• - They place certain mathematical restrictions
  on some function f related to I
• - Use f to construct I (or Is) satisfying
  axioms
• Econ to math to econ
• What form of math?
• Functional equations solve for functional forms

16

• Ex
• Suppose we love the decomposition of Theils
  entropy measure.
• Axiom (Theil Decomposability)
• For any x,y we have
• 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 decomposition

17

• Key initial papers
• Bourguignon (1979), Shorrocks (1980)
• Characterize Theil measures and GE measures
• However Assumed that I must be differentiable
• Violated by Gini
• G (µ S)/µ
• S(x) SiSjmin(xi,xj)/n2

18

• Shorrocks (1984)
• Assumed following
• Continuity
• Satisfied by Gini and all
• Normalization
• Four basic axioms or Lorenz consistency
• Axiom Aggregation
• There exists a function A such that for any x, y
  we have I(x,y) A(I(x), I(y), nx, ny, µx, µy)
• Note Can get n and µ from subgroup levels,
  generalizes decomp.
• Q/
• Are there other relative measures that are
  aggregative?

19

• Theorem Shorrocks (1984)
• I is a Lorenz consistent, continuous, normalized
  inequality measure satisfying aggregation if and
  only if there is some a and a continuous,
  strictly increasing function f with f(0)0 such
  that
• I(x) f(Ia(x)) for all x.
• Idea Only the GE measures and their monotonic
  transformations satisfy aggregation
• Idea If you want to be able to recover overall
  inequality from subgroup data, then essentially
  you can only use GE

20

• Gini Breakdown
• Q/ Does Gini violate decomposability?
• Could there be weights such that
• Try wx (nx/n)2(µx/µ)

21
Ex Gini

• Income Distributions
• x (10,12,12) y (15,21,32) (x,y)
  (10,12,12,15,21,32)
• Populations and Means
• nx 3 ny 3 n 6
• µx 11.33 µy 22.67 µ 17
• Inequality Levels
• G(x) 0.039 G(y) 0.167 G(x,y) 0.229
• Weights
• wx 0.167 wy 0.333
• Within Group
• wxG(x) wyG(y) (0.007 0.056) 0.062
• Between Group
• G(x,y) G(11.3,11.3,11.3,22.7,22.7,22.7)
  0.167
• Note Adds to total inequality 0.229
  Nonoverlapping groups!

22
Ex Gini (overlapping groups)

• Income Distributions
• x (12,21,12) y (15,32,10) (x,y)
  (12,21,12,15,32,10)
• Populations and Means
• nx 3 ny 3 n 6
• µx 15 µy 19 µ 17
• Inequality Levels
• G(x) 0.133 G(y) 0.257 G(x,y) 0.229
• Weights
• wx 0.221 wy 0.279
• Within Group
• wxG(x) wyG(y) (0.029 0.072) 0.101
• Between Group
• G(x,y) G(15,15,15,19,19,19) 0.059
• Note Adds to 0.160 lt 0.229 R residual 0.69
  Why? Assignment

23
Session 9

• Review
• Decomposability
• Characterizations
• Gini Breakdown
• Today
• Subgroup Consistency
• Income Standards
• Other Characterizations
• Unifying Framework

24
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?
• Source
• Cowell three bad measures
• Holding population sizes and means fixed,
  overall inequality should rise when when subgroup
  inequalities rise.

25

• 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) gt I(x) and I(y)
  I(y), then I(x,y) gt 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) gt G(x,y)
• Why? Residual R fell
• I2(x) I2(x), I2(y) I2(y), I2(x,y) gt
  I2(x,y)
• Assignment Find x, y that shows G violates SC

26

• Note
• All decomposable indices are subgroup consistent
• All GE indices
• Why?
• Q
• Any others?
• Theorem Shorrocks (1988)
• I is a Lorenz consistent, continuous, normalized
  inequality measure satisfying subgroup
  consistency if and only if there is some a and a
  continuous, strictly increasing function f with
  f(0)0 such that
• I(x) f(Ia(x)) for all x.
• A/ No!

27
Session 9

• Review
• Decomposability
• Characterizations
• Gini Breakdown
• Today
• Subgroup Consistency
• Income Standards
• Other Characterizations
• Unifying Framework

28
Income Standards

• Key Concept
• Summarizes distribution in a single income
• Ex/ Mean, median, income at 90th percentile, mean
  of top 40, Sens mean, Atkinsons ede income
• Measures size of the distribution
• Can have normative interpretation
• Related papers
• Foster (2006) Inequality Measurement
• Foster and Shneyerov (1999, 2000)
• Foster and Szekely (2008)

29
Income Standards

• Notation
• Income distribution x (x1,,xn)
• xi gt 0 income of the ith person
• n population size
• Dn Rn set of all n-person income
  distributions
• D ? Dn set of all income distributions
• s D ? R income standard

30
Income Standards

• Properties
• Symmetry If x is a permutation of y, then s(x)
  s(y).
• Replication Invariance If x is a replication of
  y, then s(x) s(y).
• Linear Homogeneity If x ky for some scalar k gt
  0, then s(x) ks(y).
• Normalization If x is completely equal, then
  s(x) x1.
• Continuity s is continuous on each Dn.
• Weak Monotonicity If x gt y, then s(x) gt s(y).
• Note
• Satisfied by all examples given above and below.

31
Income Standards

• Examples
• Mean s(x) ?(x) (x1...xn)/n

32
Income Standards

• Examples
• Mean s(x) ?(x) (x1...xn)/n

x2
same ?
x1
33
Income Standards

• Examples
• Mean s(x) ?(x) (x1...xn)/n

freq
F cdf
?
income
34
Income Standards

• Examples
• Median x (3, 8, 9, 10, 20), s(x) 9

freq
F cdf
0.5
income
median
35
Income Standards

• Examples
• 10th percentile

freq
F cdf
0.1
income
s Income at10th percentile
36
Income Standards

• Examples
• Mean of bottom fifth
• x (3, 5, 6, 6, 8, 9, 15, 17, 23, 25)
• s(x) 4

37
Income Standards

• Examples
• Mean of top 40
• x (3, 5, 6, 6, 8, 9, 15, 17, 23, 25)
• s(x) 20

38
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

39
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)
• Ex/ x (1,2,3,4)

40
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)
• Ex/ x (1,2,3,4)
• s(x) ? 30/16

41
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)
• Ex/ x (1,2,3,4)
• s(x) ? 30/16 lt
  ?(1,2,3,4) 40/16

42
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)
• Another view

43
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

freq
F cdf
p
income
44
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

freq
F cdf
p
A
income
45
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

freq
F cdf
p
p
A
income
A
?
46
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

Generalized Lorenz
freq
F cdf
p
p
A
income
A
?
47
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

Generalized Lorenz Curve
cumulative income
cumulative pop share
48
Income Standards

• Examples
• Sen Mean or Welfare Function S(x) E min(a,b)

Generalized Lorenz Curve
s S 2 x Area below curve
cumulative income
cumulative pop share
49
Income Standards

• Examples
• Geometric Mean s(x) ?0(x) (x1x2...xn)1/n

50
Income Standards

• Examples
• Geometric Mean s(x) ?0(x) (x1x2...xn)1/n

x2
same ?0
x1
51
Income Standards

• Examples
• Geometric Mean s(x) ?0(x) (x1x2...xn)1/n

x2
same ?0
.
x
x1
?1(x)
?0(x)
52
Income Standards

• Examples
• Geometric Mean s(x) ?0(x) (x1x2...xn)1/n
• Thus s(x) ?0
• - emphasizes lower incomes
• - is lower than the usual mean
• Unless distribution is completely equal

x2
same ?0
.
x
x1
?1(x)
?0(x)
53
Income Standards

• Examples
• Euclidean Mean s(x) ?2(x) (x12 x22 ...
  xn2)/n )1/2

54
Income Standards

• Examples
• Euclidean Mean s(x) ?2(x) (x12 x22 ...
  xn2)/n )1/2

x2
same ?2
x1
55
Income Standards

• Examples
• Euclidean Mean s(x) ?2(x) (x12 x22 ...
  xn2)/n )1/2

x2
same ?2
x1
?1(x) ?2(x)
56
Income Standards

• Examples
• Euclidean Mean s(x) ?2(x) (x12 x22 ...
  xn2)/n )1/2
• Thus s(x) ?2
• - emphasizes higher incomes
• - is higher than the usual mean
• Unless distribution is completely equal

x2
same ?2
x1
?1(x) ?2(x)
57
Income Standards

• Examples General Means
• (x1? xn?)/n 1/? for all ? ? 0
• ??(x)
• (x1xn)1/n
  for ? 0
• a 1 arithmetic mean
• a 0 geometric mean
• a 2 Euclidean mean
• a -1 harmonic mean
• For a lt 1 Distribution sensitive
• Lower a implies greater emphasis on lower incomes

58
Session 9

• Review
• Decomposability
• Characterizations
• Gini Breakdown
• Today
• Subgroup Consistency
• Income Standards
• Other Characterizations
• Unifying Framework

59
Other Characterizations

• Idea Use income standard s in decomposition
• s(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?

60
Additional Characterizations

• 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.

61
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62
Example Levels
Comparison of Living Standards in the
USA, UK and Sweden
2000
United States
UK
1500
1000
Sweden
PPP Adjusted 1991 US Dollars
500
0
M(-3)
M(-2)
M(-1)
M(1)
M(2)
M(3)
General Means
63
Session 9

• Review
• Decomposability
• Characterizations
• Gini Breakdown
• Today
• Subgroup Consistency
• Income Standards
• Other Characterizations
• Unifying Framework

64
Inequality

• Q/ Summary
• How does it all fit together?
• What is inequality?
• How to explain to policymakers?
• A/
• Provide unifying framework for inequality
• Across groups or individuals
• All use two dimensions for evaluation
• Inequality as a comparison of twin income
  standards

65
What is inequality?

• Canonical case
• Two persons 1 and 2
• Two incomes x1 and x2
• Min income a min(x1, x2)
• Max income b max(x1, x2)
• Inequality
• I (b - a)/b or some function of ratio a/b
• Caveats
• Cardinal variable
• Relative inequality

66
Inequality between Groups

• Group Based Inequality
• Two groups 1 and 2
• Two income distributions x1 and x2
• Income standard s(x) representative income
• Lower income standard a min(s(x1), s(x2))
• Upper income standard b max(s(x1), s(x2))
• Inequality
• I (b - a)/b or some function of ratio a/b
• Caveats
• What about group size?
• Not relevant if group is unit of analysis
• Relevant if individual is unit of analysis Use
  smoothed dist.

67
Inequality between Groups

• Group Based Inequality - Examples
• Spatial disparities geographically determined
• Gender inequality male/female
• Growth two points in time
• Example Racial Health Disparities in US
• Two groups Black and White
• Two distributions x1 and x2 each with 1 alive,
  0 not
• Income standard s(x) m /1000 1 - mortality
  rate
• Lower income standard a min(s(x1), s(x2))
• Upper income standard b max(s(x1), s(x2))
• Inequality
• I (b - a)/b or some function of ratio a/b,
• Next graph uses ratios of mortality rates in log
  terms

68
Inequality between Races in US
Black/White Age Adjusted Mortality
Log Mortality
Year
SourceCDC and Levine, Foster, et al Public
Health Reports (2001)
69
Inequality between Groups

• Group Based Inequality - Discussion
• Note 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 underlying model
• Health of poor/health of nonpoor
• Health of adjacent SES classes - Gradient
• 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?
• Answer Analogous exercise

70
Inequality in a Population

• Population Inequality - Discussion
• A wide array of measures
• Gini Coefficient
• Coefficient of Variation
• Mean Log Deviation
• Variance of logarithms
• Generalized Entropy Family
• 90/10 ratio
• Decile Ratio
• Atkinson Family

71
Inequality in a Population

• Population Inequality - Discussion
• Criteria for selection
• Axiomatic Basis - Lorenz consistent, subgroup
  consistent, decomposable, decomposable by ordered
  subgroups
• Understandable. - Welfare basis, intuitive
  graph
• Data Availability - Historical studies
• Easy to Use. - Is it in your software package?
• What do the measures have in common?

72
Inequality as Twin Standards

• Framework for Population Inequality
• One income distribution x
• Two income standards
• Lower income standard a sL(x)
• Upper income standard b sU(x)
• Note sL(x) lt sU(x) for all x
• Inequality
• I (b - a)/b or some function of ratio a/b
• Observation
• Framework encompasses all common inequality
  measures
• Theil, variance of logs in limit

73
Inequality as Twin Standards

• Population Inequality - Discussion
• Income Standards sL sU
• Gini Coefficient Sen mean
• Coefficient of Variation mean
  euclidean mean
• Mean Log Deviation geometric mean mean
• Generalized Entropy Family general mean
  mean
• or mean general mean
• 90/10 ratio income at 10th pc income
  at 90th pc
• Decile Ratio mean mean of upper 10
• Atkinson Family general mean mean

74
Inequality as Twin Standards

• Population Inequality - Summary
• Inequality measures create twin dimensions of
  income standards
• Characteristics of inequality measure depend on
  characteristics of the standards
• Can reverse process to assemble new measures of
  inequality

75
Application of the Methodologies

• Growth and Inequality
• To see how inequality changes over time
• Calculate growth rate for sL
• Calculate growth rate for sU
• Note One of these is usually the mean
• Compare!
• Robustness
• Calculate growth rates for several standards at
  once

76
Ex Evolution of General Means in Taiwan
5.00
???
??
4.50
4.00
??
3.50
3.00
General Mean Income Relative to 1976 Value
2.50
2.00
1.50
1.00
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
Year
77
Application Growth and Inequality over Time
Growth in ?? for Mexico vs. Costa Rica
200
Costa Rica
Mexico
180
160
1985-1995
1984-1996
140
120
Change in income standard
100
80
60
40
20
0
-20
-40
-60
-80
-100
???
???
???
??
??
??
??
Foster and Szekely (2008)
78
General Means are Unique

• Q/ Why general means?
• A/ Satisfy Properties for an Income Standard
• Symmetry, replication invariance, linear
  homogeneity, normalization, continuity and
• Subgroup consistency
• Suppose that s(x') gt 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') gt
  s(x, y).
• Idea Otherwise decentralized policy is
  impossible.
• Th An income standard satisfies all the above
  properties if and only if it is a general mean
• Foster and Székely (2008)

79
General Means and Atkinson

• Application Atkinsons Family
• I (? - ?a) / ? a lt 1
• Welfare interpretation of general mean and hence
  inequality measure
• Percentage welfare loss due to inequality

80
General Means and Atkinson

• Interpretation
• I (? - ?a) / ? a lt 1

x2
.
x
x1
?
??
81
General Means

• Application Assembling Decomposable Inequality
  Measures
• Define

Icq(x)
Foster Shneyerov 1999 Icq 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)
82
Summary

• Income standards provide unifying framework for
  measuring inequality and well being
• Income standards should receive more direct
  empirical attention