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 today’s 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 μ x and n x – mean and pop size of x μ y and n y – 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 Theil’s entropy measure where s x = |x|/|(x,y)| is the income share of x Theil’s second measure mean log deviation where p x = n x /n is the population share of x

8. Specific Decompositions Squared Coefficient of Variation C=V/ μ 2 Note Follows from variance decomposition using C(x) = V(x/ μ ) Generalized entropy measures I α

9. Ex Generalized Entropy with  = -1 transfer sens. Income Distributions x = (12,21,12)y = (15,32,10)(x,y) = (12,21,12,15,32,10) Populations and Means n x = 3 n y = 3n = 6 μ x = 15 μ y = 19 μ = 17 Inequality Levels I α (x) = 0.036I α (y) = 0.127I α (x,y) = 0.084 Weights w x = 0.567w y = 0.447 Within Group w x I α (x) + w y I α (y) = (0.020 + 0.057) = 0.077 Between Group I α (x,y) =I α (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  = 0 Theil’s second Income Distributions x = (12,21,12)y = (15,32,10)(x,y) = (12,21,12,15,32,10) Populations and Means n x = 3 n y = 3n = 6 μ x = 15 μ y = 19 μ = 17 Inequality Levels I α (x) = 0.037I α (y) = 0.119I α (x,y) = 0.085 Weights w x = 0.500w y = 0.500 Within Group w x I α (x) + w y I α (y) = (0.018 + 0.059) = 0.078 Between Group I α (x,y) =I α (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  = 1/2 Income Distributions x = (12,21,12)y = (15,32,10)(x,y) = (12,21,12,15,32,10) Populations and Means n x = 3 n y = 3n = 6 μ x = 15 μ y = 19 μ = 17 Inequality Levels I α (x) = 0.037I α (y) = 0.118I α (x,y) = 0.087 Weights w x = 0.470w y = 0.529 Within Group w x I α (x) + w y I α (y) = (0.017 + 0.062) = 0.080 Between Group I α (x,y) =I α (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  = 1 Theil’s entropy Income Distributions x = (12,21,12)y = (15,32,10)(x,y) = (12,21,12,15,32,10) Populations and Means n x = 3 n y = 3n = 6 μ x = 15 μ y = 19 μ = 17 Inequality Levels I α (x) = 0.038I α (y) = 0.118I α (x,y) = 0.090 Weights w x = 0.441w y = 0.559 Within Group w x I α (x) + w y I α (y) = (0.017 + 0.066) = 0.083 Between Group I α (x,y) =I α (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  = 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 n x = 3 n y = 3n = 6 μ x = 15 μ y = 19 μ = 17 Inequality Levels I α (x) = 0.040I α (y) = 0.123I α (x,y) = 0.099 Weights w x = 0.389w y = 0.625 Within Group w x I α (x) + w y I α (y) = (0.016 + 0.077) = 0.092 Between Group I α (x,y) =I α (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 α rose contribution decreased with α Within group term larger increased as α rose contribution increased with α Recall Theil measure used by Anand&Segal 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 I’s) 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 Theil’s 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) = Σ i Σ j min(x i,x j )/n 2

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), n x, n y, μ x, μ y ) 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 α and a continuous, strictly increasing function f with f(0)=0 such that I(x) = f(I α (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 w x = (n x /n) 2 ( μ x / 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 n x = 3 n y = 3n = 6 μ x = 11.33 μ y = 22.67 μ = 17 Inequality Levels G(x) = 0.039G(y) = 0.167G(x,y) = 0.229 Weights w x = 0.167w y = 0.333 Within Group w x G(x) + w y G(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 n x = 3 n y = 3n = 6 μ x = 15 μ y = 19 μ = 17 Inequality Levels G(x) = 0.133G(y) = 0.257G(x,y) = 0.229 Weights w x = 0.221w y = 0.279 Within Group w x G(x) + w y G(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 < 0.229R = 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’) > 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 fell I 2 (x) = I 2 (x’), I 2 (y) = I 2 (y’), I 2 (x,y) > I 2 (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 α and a continuous, strictly increasing function f with f(0)=0 such that I(x) = f(I α (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%, Sen’s mean, Atkinson’s 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 Income distribution x = (x 1,…,x n ) x i > 0 income of the ith person n population size D n = R ++ n set of all n-person income distributions D =  D n set of all income distributions s: D  R income standard

30. Income Standards Properties Symmetry Symmetry If x is a permutation of y, then s(x) = s(y). Replication Invariance Replication Invariance If x is a replication of y, then s(x) = s(y). Linear Homogeneity Linear Homogeneity If x = ky for some scalar k > 0, then s(x) = ks(y). Normalization Normalization If x is completely equal, then s(x) = x 1. Continuity Continuity s is continuous on each D n. Weak Monotonicity Weak Monotonicity If x > y, then s(x) > s(y).Note Satisfied by all examples given above and below.

31. Income Standards Examples Means(x) = Mean s(x) =  (x) = (x 1 +...+x n )/n

32. Income Standards Examples Means(x) = Mean s(x) =  (x) = (x 1 +...+x n )/n x1x1 x2x2 same 

33. Income Standards Examples Means(x) = Mean s(x) =  (x) = (x 1 +...+x n )/n  F = cdf income freq

34. Income Standards Examples Median s(x)= 9 Median x = (3, 8, 9, 10, 20), s(x) = 9 F = cdf income freq 0.5 median

35. Income Standards Examples 10 th percentile F = cdf income freq 0.1 s = s = Income at10 th 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 Sen Mean or Welfare Function S(x) = E min(a,b)

39. Income Standards Examples Sen Mean or Welfare Function 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 Sen Mean or Welfare Function S(x) = E min(a,b) Ex/ x = (1,2,3,4) s(x) = = 30/16 s(x) =  = 30/16

41. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) Ex/ x = (1,2,3,4) s(x) = = 30/16<  (1,2,3,4) = 40/16 s(x) =  = 30/16 <  (1,2,3,4) = 40/16

42. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) Another view

43. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) F = cdf income freq p

44. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) F = cdf income freq p A

45. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) F = cdf income freq p A p A 

46. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) F = cdf income freq p A p A  Generalized Lorenz

47. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) Generalized Lorenz Curve cumulative pop share cumulative income

48. Income Standards Examples Sen Mean or Welfare Function Sen Mean or Welfare Function S(x) = E min(a,b) Generalized Lorenz Curve cumulative pop share cumulative income s = s = S = 2 x Area below curve

49. Income Standards Examples Geometric Means(x) = Geometric Mean s(x) =  0 (x) = (x 1 x 2...x n ) 1/n

50. Income Standards Examples Geometric Means(x) = Geometric Mean s(x) =  0 (x) = (x 1 x 2...x n ) 1/n x1x1 x2x2 same  0

51. Income Standards Examples Geometric Means(x) = Geometric Mean s(x) =  0 (x) = (x 1 x 2...x n ) 1/n x1x1 x2x2 same  0 x.  1 (x)  0 (x)

52. Income Standards Examples Geometric Means(x) = Geometric Mean s(x) =  0 (x) = (x 1 x 2...x n ) 1/n Thuss(x) = Thus s(x) =  0 - emphasizes lower incomes - is lower than the usual mean Unless distribution is completely equal x1x1 x2x2 same  0 x.  1 (x)  0 (x)

53. Income Standards Examples Euclidean Means(x) = Euclidean Mean s(x) =  2 (x) = [(x 1 2 + x 2 2 +...+ x n 2 )/n ) 1/2

54. Income Standards Examples Euclidean Means(x) = Euclidean Mean s(x) =  2 (x) = [(x 1 2 + x 2 2 +...+ x n 2 )/n ) 1/2 x1x1 x2x2 same  2

55. Income Standards Examples Euclidean Means(x) = Euclidean Mean s(x) =  2 (x) = [(x 1 2 + x 2 2 +...+ x n 2 )/n ) 1/2 x1x1 x2x2 same  2  1 (x)  2 (x)

56. Income Standards Examples Euclidean Means(x) = Euclidean Mean s(x) =  2 (x) = [(x 1 2 + x 2 2 +...+ x n 2 )/n ) 1/2 Thuss(x) = Thus s(x) =  2 - emphasizes higher incomes - is higher than the usual mean Unless distribution is completely equal x1x1 x2x2 same  2  1 (x)  2 (x)

57. Income Standards Income Standards Examples General Means [(x 1  + … + x n  )/n] 1/  for all   0   (x) = (x 1 … x n ) 1/n for  = 0  = 1 arithmetic mean  = 0 geometric mean  = 2Euclidean mean  = -1harmonic mean For  < 1: Distribution sensitive Lowergreaterlower Lower  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 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. I cq (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.

62. Example: Levels 0 500 1000 1500 2000 PPP Adjusted 1991 US Dollars 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

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 x 1 and x 2 Min income a = min(x 1, x 2 ) Max income b = max(x 1, x 2 ) 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 x 1 and x 2 Income standard s(x) “representative income” Lower income standard a = min(s(x 1 ), s(x 2 )) Upper income standard b = max(s(x 1 ), s(x 2 )) 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 x 1 and x 2 each with 1 = alive, 0 = not Income standard s(x) =  “1 - mortality rate” Lower income standard a = min(s(x 1 ), s(x 2 )) Upper income standard b = max(s(x 1 ), s(x 2 )) 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 Year Source:CDC and Levine, Foster, et al Public Health Reports (2001) Log Mortality

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 Gradient 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 Axiomatic Basis - Lorenz consistent, subgroup consistent, decomposable, decomposable by ordered subgroups Understandable Understandable. - Welfare basis, intuitive graph Data Availability Data Availability - Historical studies Easy to Use 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 = s L (x) Upper income standard b = s U (x) Note: s L (x) < s U (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 s L s U Gini Coefficient Gini Coefficient Sen mean Coefficient of Variation Coefficient of Variation mean euclidean mean Mean Log Deviation Mean Log Deviation geometric mean mean Generalized Entropy Family Generalized Entropy Family general mean mean ormean general mean 90/10 ratio 90/10 ratio income at 10 th pc income at 90 th pc Decile Ratio Decile Ratio mean mean of upper 10% Atkinson Family Atkinson Family general mean mean

74. Inequality as Twin Standards Population Inequality -Summary 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 s L Calculate growth rate for s U 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 Ex: Evolution of General Means in Taiwan 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 General Mean Income Relative to 1976 Value 19761978198019821984198619881990199219941996 Year    

77. 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 % Change in income standard   Costa Rica 1985-1995 Mexico 1984-1996         Foster and Szekely (2008)

78. General Means are Unique Q/ Why general means? A/ Satisfy Properties for an Income Standard and Symmetry, replication invariance, linear homogeneity, normalization, continuity and 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). Idea Idea Otherwise decentralized policy is impossible. general mean 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: Atkinson’s Family I = (  -   ) /   < 1 Welfare interpretation of general mean and hence inequality measure Percentage welfare loss due to inequality

80. General Means and Atkinson Interpretation I = (  -   ) /   < 1 x1x1 x2x2 x.  

81. General Means Application: Assembling Decomposable Inequality Measures Define I cq (x) = Foster Shneyerov 1999 I cq 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