1. Income Standards, Inequality and Poverty James E. Foster

2. Introduction Questions What is inequality? What is poverty? How communicate concepts? To policymakers, public Ex/ Miguel Székely (2006),“Números que Mueven al Mundo: La Medición de la Pobreza en México” This Paper Unifying framework

3. Introduction Key Concept Income Standard Summarizes distribution into a single income Ex/ Mean, median, income at 90th percentile, mean of top 40%, Sen’s mean, Atkinson’s ede income... Related papers “Inequality Measurement” in The Elgar Companion to Development Studies (David Clark, Ed.), Cheltenham: Edward Elgar 2006 “Poverty Measurement” in Poverty, Inequality and Development: Essays in Honor of Erik Thorbecke (Alain de Janvry and Ravi Kanbur, eds.), Amsterdam: Kluwer Academic Publishers, 2005

4. 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

5. 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.

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

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

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

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

10. Income Standards Examples 10 th percentile F = cdf income freq 0.1 s = s = Income at10 th percentile

11. Income Standards Examples Mean of bottom fifth x = (3, 5, 6, 6, 8, 9, 15, 17, 23, 25) s(x) = 4

12. Income Standards Examples Mean of top 40% x = (3, 5, 6, 6, 8, 9, 15, 17, 23, 25) s(x) = 20

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

14. 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)

15. 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

16. 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

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

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

19. 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

20. 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 

21. 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

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

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

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

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

26. 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)

27. 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)

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

29. 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

30. 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)

31. 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)

32. 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 Hardy Littlewood Polya 1952; Kolm 1969; Atkinson 1970  = 1 arithmetic mean  = 0 geometric mean  = 2Euclidean mean  = -1harmonic mean For  < 1: Distribution sensitive Lowergreaterlower Lower  implies greater emphasis on lower incomes

33. Inequality Question What is inequality? Universal framework for inequality Two dimensions for evaluation Reduces to discussion of twin “income standards”

34. 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 Ratio scale Relative inequality

35. 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

36. 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

37. Inequality between Races in US Black/White Age Adjusted Mortality Year Source:CDC and Levine, Foster, et al Public Health Reports (2001) Log Mortality

38. 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 Question: How to measure “overall” inequality in a population? Answer: Analogous exercise

39. Inequality in a Population Population Inequality - Discussion A wide array of measures (yawn) Gini Coefficient Coefficient of Variation Mean Log Deviation Variance of logarithms Generalized Entropy Family 90/10 ratio Decile Ratio Atkinson Family

40. 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?

41. 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

42. Inequality as Twin Standards Population Inequality - Discussion Income Standardss 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

43. 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, well being, poverty (where poverty line plays role of one of the income standards).

44. Poverty in a Population Poverty in a Population - Discussion A wide array of measures (yawn) SenThon/Sen/Shorrocks Clark, Hemming, Ulph/Chakravarty (CHU) Headcount Ratio Per Capita Poverty Gap Watts Index FGT What do they have in common?

45. Poverty as Twin Income Standards Framework for Population Poverty One income distribution x Two income standards: Lower income standard a = s L (x) Lower income standard a = s L (x) (usually employs censored x) Upper income standard b = z Upper income standard b = z (the absolute poverty line) Note: s L (x) < z for all x Poverty P = (b - a)/b or some function of ratio a/b Observation Framework encompasses Watt’s, CHU, Sen, Thon, headcount, poverty gap.

46. Poverty as Twin Gap Standards Framework for Population Poverty One gap distribution g One gap distribution g (positive entries are z - x i ) Two gap standards: Lower gap standard a = s L (g) Upper gap standard b = z Upper gap standard b = z (the absolute poverty line) Note: s L (g) < z for all x Poverty P = a/b or some function of ratio a/b Observation Framework encompasses FGT, Sen, Thon, headcount, poverty gap.

47. 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! Poverty and Time Calculate growth rate for respective standard. Robustness Calculate growth rates for several standards at once

48. 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 Subgroup consistency (see Foster and Sen, 1997) 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 (2006)

49. 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 (2006)

50. General Means and Atkinson Application: Atkinson’s Family I = (  -   ) /   < 1 Atkinson 1970 Welfare interpretation of general mean and hence inequality measure Percentage welfare loss due to inequality

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

52. 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)

53. A Class of Distribution Sensitive HDI’s H  (D) =   [   (x),   (y),   (z)] for  = 1-  > 0  = 0 H 0 = usual HDI  = 0 H 0 =  [  (x),  (y),  (z)] usual HDI  = 1 H 1 =  = 1 H 1 =  0 [  0 (x),  0 (y),  0 (z)] based on geometric mean sensitive to inequality  = 2H 2 =  = 2H 2 =   [   (x),   (y),   (z)] based on harmonic mean even more sensitive Foster, Lopez-Calva, and Szekely (2005, 2006)

54. Summary Income standards provide unifying framework for measuring inequality, poverty and well being Income standards should receive more direct empirical attention Plan to explore more thoroughly the theoretical link between the properties of income standards and associated measures Thank you!