Session 9

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 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 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

8. Specific Decompositions Squared Coefficient of Variation C=V/μ2 Note Follows from variance decomposition using C(x) = V(x/μ) Generalized entropy measures Iα

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 Iα(x) = 0.036 Iα(y) = 0.127 Iα(x,y) = 0.084 Weights wx = 0.567 wy = 0.447 Within Group wxIα(x) + wyIα(y) = (0.020 + 0.057) = 0.077 Between Group Iα(x,y) =Iα(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 Theil’s 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 Iα(x) = 0.037 Iα(y) = 0.119 Iα(x,y) = 0.085 Weights wx = 0.500 wy = 0.500 Within Group wxIα(x) + wyIα(y) = (0.018 + 0.059) = 0.078 Between Group Iα(x,y) =Iα(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 Iα(x) = 0.037 Iα(y) = 0.118 Iα(x,y) = 0.087 Weights wx = 0.470 wy = 0.529 Within Group wxIα(x) + wyIα(y) = (0.017 + 0.062) = 0.080 Between Group Iα(x,y) =Iα(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 Theil’s 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 Iα(x) = 0.038 Iα(y) = 0.118 Iα(x,y) = 0.090 Weights wx = 0.441 wy = 0.559 Within Group wxIα(x) + wyIα(y) = (0.017 + 0.066) = 0.083 Between Group Iα(x,y) =Iα(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 = 2alf 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 Iα(x) = 0.040 Iα(y) = 0.123 Iα(x,y) = 0.099 Weights wx = 0.389 wy = 0.625 Within Group wxIα(x) + wyIα(y) = (0.016 + 0.077) = 0.092 Between Group Iα(x,y) =Iα(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. IdeaOnly 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) = ΣiΣjmin(xi,xj)/n2

18. Shorrocks (1984) Assumed following Continuity Satisfied by Gini and all Normalization Four basic axioms or Lorenz consistency AxiomAggregation 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 α and a continuous, strictly increasing function f with f(0)=0 such that I(x) = f(Iα(x)) for all x. IdeaOnly the GE measures and their monotonic transformations satisfy aggregation IdeaIf 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 < 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’) > 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 I2(x) = I2(x’), I2(y) = I2(y’), I2(x,y) > 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 α and a continuous, strictly increasing function f with f(0)=0 such that I(x) = f(Iα(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 distributionx = (x1,…,xn) xi > 0income of the ith person npopulation size Dn = R++nset of all n-person income distributions D =  Dnset of all income distributions s: D  Rincome standard

30. Income Standards Properties SymmetryIf x is a permutation of y, then s(x) = s(y). Replication InvarianceIf x is a replication of y, then s(x) = s(y). Linear HomogeneityIf x = ky for some scalar k > 0, then s(x) = ks(y). NormalizationIf x is completely equal, then s(x) = x1. Continuitys is continuous on each Dn. Weak MonotonicityIf x > y, then s(x) > s(y). Note Satisfied by all examples given above and below.

31. Income Standards Examples Means(x) =(x) = (x1+...+xn)/n

32. Income Standards Examples Means(x) =(x) = (x1+...+xn)/n x2 same  x1

33. Income Standards Examples Means(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< (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 cumulativepop 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 cumulativepop share

49. Income Standards Examples Geometric Means(x) =0(x) = (x1x2...xn)1/n

50. Income Standards Examples Geometric Means(x) =0(x) = (x1x2...xn)1/n x2 same 0 x1