1. Frank Cowell: Oviedo – Inequality & Poverty Deprivation, Complaints and Inequality March 2007 Inequality, Poverty and Income Distribution University of Oviedo Frank Cowell http://darp.lse.ac.uk/oviedo2007

2. Frank Cowell: Oviedo – Inequality & Poverty Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality Themes and methodology

3. Frank Cowell: Oviedo – Inequality & Poverty Purpose of lecture We will look at recent theoretical developments in distributional analysis We will look at recent theoretical developments in distributional analysis Consider some linked themes Consider some linked themes  alternative approaches to inequality  related welfare concepts Use ideas from sociology and philosophy Use ideas from sociology and philosophy Focus on the way modern methodology is applied Focus on the way modern methodology is applied

4. Frank Cowell: Oviedo – Inequality & Poverty Themes Cross-disciplinary concepts Cross-disciplinary concepts Income differences Income differences Reference incomes Reference incomes Formal methodology Formal methodology

5. Frank Cowell: Oviedo – Inequality & Poverty Methodology Exploit common structure Exploit common structure  poverty  deprivation  complaints and inequality  see Cowell (2007) Cowell (2007)Cowell (2007) Axiomatic method Axiomatic method  minimalist approach  characterise structure  introduce ethics

6. Frank Cowell: Oviedo – Inequality & Poverty Basic components Income distribution: x Income distribution: x  an n-vector  population of size n  person i has income x i Space of all income distributions: D  R n Space of all income distributions: D  R n  specification of this captures nature of income  include zeros? negatives? An evaluation function An evaluation function   : D → R Axioms of two broad types of axiom Axioms of two broad types of axiom  to impose standard structure  to give meaning to a particular economic problem

7. Frank Cowell: Oviedo – Inequality & Poverty “Structural” axioms Take some social evaluation function  Take some social evaluation function   welfare  inequality  poverty Axiom 1 (Continuity). Axiom 1 (Continuity).   is a continuous function D → R. Axiom 2 (Linear homogeneity). Axiom 2 (Linear homogeneity).  For all x  D and > 0:   ( x) =  (x) Axiom 3 (Translation independence). Axiom 3 (Translation independence).  For all x  D and such that  R such that x  1  D   (x  1) =  (x)

8. Frank Cowell: Oviedo – Inequality & Poverty Structural axioms: illustration x1x1 x3x3 x2x2 D for n=3 D for n=3 An income distribution An income distribution Perfect equality Perfect equality Contours of “Absolute” Gini Contours of “Absolute” Gini Continuity Continuity  Continuous approach to I = 0 Linear homogeneity Linear homogeneity  Proportionate increase in I Translation invariance Translation invariance  I constant D for n=3 D for n=3 An income distribution An income distribution Perfect equality Perfect equality Contours of “Absolute” Gini Contours of “Absolute” Gini Continuity Continuity  Continuous approach to I = 0 Linear homogeneity Linear homogeneity  Proportionate increase in I Translation invariance Translation invariance  I constant 0 1 x *

9. Frank Cowell: Oviedo – Inequality & Poverty Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality An alternative approach

10. Frank Cowell: Oviedo – Inequality & Poverty Poverty concepts (1) The poverty line z The poverty line z  a reference point  exogenously given Define the number of the poor: Define the number of the poor:   (x, z) := #{i: x i ≤ z} Proportional headcount Proportional headcount   (x, z)/n Poverty gap Poverty gap  fundamental income difference  g i (x, z) = max (0, z  x i )

11. Frank Cowell: Oviedo – Inequality & Poverty Poverty concepts (2) Foster et al (1984) poverty index Foster et al (1984) poverty index Foster et al (1984) Foster et al (1984)   ≥ 0 is a sensitivity parameter Cumulative poverty gap Cumulative poverty gap  counterpart to income cumulations

12. Frank Cowell: Oviedo – Inequality & Poverty TIP / Poverty profile i/n  (x,z)/n G(x,z) 0 Cumulative gaps versus population proportions Cumulative gaps versus population proportions Proportion of poor Proportion of poor TIP curve TIP curve Cumulative gaps versus population proportions Cumulative gaps versus population proportions Proportion of poor Proportion of poor TIP curve TIP curve  Shorrocks 1983)  TIP curves have same interpretation as GLC (Shorrocks 1983) Shorrocks 1983)Shorrocks 1983)   TIP dominance implies unambiguously greater poverty

13. Frank Cowell: Oviedo – Inequality & Poverty Poverty: Axiomatic approach Characterise an ordinal poverty index P(x, z) Characterise an ordinal poverty index P(x, z)  See Ebert and Moyes (2002) See Ebert and Moyes (2002) See Ebert and Moyes (2002) Use some of the standard axioms we introduced for analysing social welfare Use some of the standard axioms we introduced for analysing social welfare Apply them to n+1 incomes – those of the n individuals and the poverty line Apply them to n+1 incomes – those of the n individuals and the poverty line Show that Show that  given just these axioms…  …you are bound to get a certain type of poverty measure.

14. Frank Cowell: Oviedo – Inequality & Poverty Poverty: The key axioms Adapt standard axioms from social welfare Adapt standard axioms from social welfare  anonymity  independence  monotonicity Strengthen two other axioms Strengthen two other axioms  scale invariance  translation invariance Also need continuity Also need continuity Plus a focus axiom Plus a focus axiom  income changes only affect poverty…  …if they concern the incomes of those where i ≤ 

15. Frank Cowell: Oviedo – Inequality & Poverty A closer look at the axioms Let D denote the set of ordered income vectors Let D denote the set of ordered income vectors The monotonicity axiom is The monotonicity axiom is  for x  D,  > 0 and x i ≤ z:  P(x 1, x 2,…, x i +  …, z) < P(x 1, x 2,…, x i, …, z) The focus axiom is The focus axiom is  for x  D and x i > z, P is constant in x i Scale invariance now becomes Scale invariance now becomes  if P(x, z) = P(y, z) then P( x, z) = P( y, z ) Independence means: Independence means:  consider x,y  D such that P(x, z) = P(y, z) where, for some i ≤ , x i = y i ; then, for any xº such that x i─1 ≤ xº≤ x i+1 and y i─1 ≤ xº ≤ y i+1 P(x 1, x 2, …, x i─1, xº, x i+1,…,x n, z) = P(y 1, y 2, …, y i─1, xº, y i+1,…,y n, z)

16. Frank Cowell: Oviedo – Inequality & Poverty Ebert-Moyes (2002) Gives two types of FGT measures Gives two types of FGT measures  “relative” version  “absolute” version Additivity follows from the independence axiom Additivity follows from the independence axiom

17. Frank Cowell: Oviedo – Inequality & Poverty Poverty: lessons Poverty indexes can be constructed from scratch Poverty indexes can be constructed from scratch Exploit the poverty line as a reference point Exploit the poverty line as a reference point Use standard axioms Use standard axioms  applied to n+1 incomes Impose structure Impose structure  independence  scale invariance Axioms to give meaning Axioms to give meaning  monotonicity  focus Use the same method in other areas Use the same method in other areas

18. Frank Cowell: Oviedo – Inequality & Poverty Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality An economic interpretation of a sociological concept

19. Frank Cowell: Oviedo – Inequality & Poverty Individual deprivation The Yitzhaki (1979) definition The Yitzhaki (1979) definitionYitzhaki (1979)Yitzhaki (1979) Equivalent form Equivalent form In present notation In present notation Use the conditional mean Use the conditional mean

20. Frank Cowell: Oviedo – Inequality & Poverty Deprivation: Axiomatic approach 1 The Better-than set for i The Better-than set for i Focus Focus  works like the poverty concept

21. Frank Cowell: Oviedo – Inequality & Poverty Deprivation: Axiomatic approach 2 Normalisation Normalisation Additivity Additivity  works like the independence axiom

22. Frank Cowell: Oviedo – Inequality & Poverty Bossert-D’Ambrosio (2006) This is just the Yitzhaki individual deprivation index This is just the Yitzhaki individual deprivation index There is an alternative axiomatisation There is an alternative axiomatisation  Ebert and Moyes (2000). Ebert and Moyes (2000) Ebert and Moyes (2000)  Different structure of reference group

23. Frank Cowell: Oviedo – Inequality & Poverty Aggregate deprivation Simple approach: just sum individual deprivation Simple approach: just sum individual deprivation Could consider an ethically transformed variant Could consider an ethically transformed variant As with poverty consider relative as well as absolute indices As with poverty consider relative as well as absolute indices  Chakravarty and Chakraborty (1984) Chakravarty and Chakraborty (1984) Chakravarty and Chakraborty (1984)  Chakravarty and Mukherjee (1999a) (1999b) (1999a)(1999b)(1999a)(1999b)

24. Frank Cowell: Oviedo – Inequality & Poverty Aggregate deprivation (2) Alternative approach Alternative approach Based aggregate deprivation on the generalised- Gini Based aggregate deprivation on the generalised- Gini  where w i are positional weights  Duclos and Grégoire (2002) Duclos and Grégoire (2002) Duclos and Grégoire (2002)

25. Frank Cowell: Oviedo – Inequality & Poverty Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality Reference groups and distributional judgments Model Inequality results Rankings and welfare

26. Frank Cowell: Oviedo – Inequality & Poverty The Temkin approach Larry Temkin (1986, 1993) approach to inequality Larry Temkin (1986, 1993) approach to inequality  Unconventional  Not based on utilitarian welfare economics  But not a complete “outlier” Common ground with other distributional analysis Common ground with other distributional analysis  Poverty  deprivation Contains the following elements: Contains the following elements:  Concept of a complaint  The idea of a reference group  A method of aggregation

27. Frank Cowell: Oviedo – Inequality & Poverty What is a “complaint?” Individual’s relationship with the income distribution Individual’s relationship with the income distribution The complaint exists independently The complaint exists independently  does not depend on how people feel  does not invoke “utility” or (dis)satisfaction Requires a reference group Requires a reference group  effectively a reference income  a variety of specifications  see also Devooght (2003) Devooght (2003)Devooght (2003)

28. Frank Cowell: Oviedo – Inequality & Poverty Types of reference point BOP BOP  The Best-Off Person  Possible ambiguity if there is more than one  By extension could consider the best-off group AVE AVE  The AVErage income  Obvious tie-in with conventional inequality measures  A conceptual difficulty for those above the mean? ATBO ATBO  All Those Better Off  A “conditional” reference point

29. Frank Cowell: Oviedo – Inequality & Poverty Aggregation The complaint is an individual phenomenon. The complaint is an individual phenomenon. How to make the transition from this to society as a whole? How to make the transition from this to society as a whole? Temkin makes two suggestions: Temkin makes two suggestions: Simple sum Simple sum  Just add up the complaints Weighted sum Weighted sum  Introduce distributional weights  Then sum the weighted complaints

30. Frank Cowell: Oviedo – Inequality & Poverty The BOP Complaint Let r(x) be the first richest person you find in N. Let r(x) be the first richest person you find in N. Person r (and higher) has income x n. Person r (and higher) has income x n. For “lower” persons, there is a natural definition of complaint: For “lower” persons, there is a natural definition of complaint:  k i (x) := x n  x i Similar to fundamental difference for poverty: Similar to fundamental difference for poverty:  g i (x, z) = max (0, z  x i ) Other similarities: Other similarities:  replace “  ” with “r”  instead of the last poor person we now have the first rich person

31. Frank Cowell: Oviedo – Inequality & Poverty BOP-Complaint: Axiomatisation Use same structural axioms as before. Plus… Use same structural axioms as before. Plus… Monotonicity: income increments reduce complaint Monotonicity: income increments reduce complaint Independence Independence Normalisation Normalisation

32. Frank Cowell: Oviedo – Inequality & Poverty Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality A new approach to inequality Model Inequality results Rankings and welfare

33. Frank Cowell: Oviedo – Inequality & Poverty Implications for inequality Broadly two types of axioms with different roles. Broadly two types of axioms with different roles. Axioms on structure: Axioms on structure:  use these to determine the “shape” of the measures. Transfer principles and properties of measures: Transfer principles and properties of measures:  use these to characterise ethical nature of measures

34. Frank Cowell: Oviedo – Inequality & Poverty A BOP-complaint class The Cowell-Ebert (SCW 2004) result The Cowell-Ebert (SCW 2004) result Similarity of form to FGT Similarity of form to FGT Characterises a family of distributions … Characterises a family of distributions …

35. Frank Cowell: Oviedo – Inequality & Poverty The transfer principle Do BOP-complaint measures satisfy the transfer principle? Do BOP-complaint measures satisfy the transfer principle?  If transfer is from richest, yes  But if transfers are amongst hoi polloi, maybe not Cowell-Ebert (SCW 2004): Cowell-Ebert (SCW 2004): Look at some examples that satisfy this Look at some examples that satisfy this

36. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours To examine the properties of the derived indices… To examine the properties of the derived indices… …take the case n = 3 …take the case n = 3 Draw contours of T  –inequality Draw contours of T  –inequality Note that both the sensitivity parameter  and the weights w are of interest… Note that both the sensitivity parameter  and the weights w are of interest…

37. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours (  =2) w 1 =0.5 w 2 =0.5 Now change the weights…

38. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours (  =2) w 1 =0.75 w 2 =0.25

39. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours (  = 1) w 1 =0.75 w 2 =0.25

40. Frank Cowell: Oviedo – Inequality & Poverty By contrast: Gini contours

41. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours (  = 0) w 1 =0.5 w 2 =0.5 Again change the weights… Again change the weights…

42. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours (  = –1) w 1 =0.75 w 2 =0.25

43. Frank Cowell: Oviedo – Inequality & Poverty Inequality contours (  = –1) w 1 =0.5 w 2 =0.5

44. Frank Cowell: Oviedo – Inequality & Poverty Special cases If    then inequality just becomes the range, x n – x 1. If    then inequality just becomes the range, x n – x 1. If   –  then inequality just becomes the “upper- middle class” complaint: x n –x n-1. If   –  then inequality just becomes the “upper- middle class” complaint: x n –x n-1. If  = 1 then inequality becomes a generalised absolute Gini. If  = 1 then inequality becomes a generalised absolute Gini. “triangles” “Y-shapes” Hexagons

45. Frank Cowell: Oviedo – Inequality & Poverty Which is more unequal? 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B

46. Frank Cowell: Oviedo – Inequality & Poverty Focus on one type of BOP complaint 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B

47. Frank Cowell: Oviedo – Inequality & Poverty Orthodox approach 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B

48. Frank Cowell: Oviedo – Inequality & Poverty T  – inequality

49. Frank Cowell: Oviedo – Inequality & Poverty The “sequence” Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality. Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality. Take a simple model of a ladder with just two rungs. Take a simple model of a ladder with just two rungs. The rungs are fixed, but the numbers on them are not. The rungs are fixed, but the numbers on them are not. Initially everyone is on the upper rung. Initially everyone is on the upper rung. Then, one by one, people are transferred to the lower rung. Then, one by one, people are transferred to the lower rung.  Start with m = 0 on lower rung  Carry on until m = n on lower rung What happens to inequality? What happens to inequality?  Obviously zero at the two endpoints of the sequence  But in between?

50. Frank Cowell: Oviedo – Inequality & Poverty The “sequence” (2) For the case of T  –inequality we have For the case of T  –inequality we have This is increasing in m if  > 0 This is increasing in m if  > 0 For other cases there is a degenerate sequence in the same direction For other cases there is a degenerate sequence in the same direction

51. Frank Cowell: Oviedo – Inequality & Poverty Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality A replacement for the Lorenz order? Model Inequality results Rankings and welfare

52. Frank Cowell: Oviedo – Inequality & Poverty Rankings Move beyond simple inequality measures Move beyond simple inequality measures The notion of complaint can also be used to generate a ranking principle that can be applied quite generally. The notion of complaint can also be used to generate a ranking principle that can be applied quite generally. This is rather like the use of Lorenz curves to specify a Lorenz ordering that characterises inequality comparisons. This is rather like the use of Lorenz curves to specify a Lorenz ordering that characterises inequality comparisons. Also similar to poverty rankings with arbitrary poverty lines. Also similar to poverty rankings with arbitrary poverty lines.

53. Frank Cowell: Oviedo – Inequality & Poverty Cumulative complaints Define cumulative complaints Define cumulative complaints Gives the CCC Gives the CCC  cumulative-complaint contour  Just like TIP / Poverty profile Use this to get a ranking principle Use this to get a ranking principle i/n r(x) / n K(x)K(x)

54. Frank Cowell: Oviedo – Inequality & Poverty Complaint-ranking The class of BOP-complaint indices The class of BOP-complaint indices Define complaint ranking Define complaint ranking Like the generalised-Lorenz result Like the generalised-Lorenz result

55. Frank Cowell: Oviedo – Inequality & Poverty Social welfare again Temkin’s complaints approach to income distribution was to be viewed in terms of “better” or “worse” Temkin’s complaints approach to income distribution was to be viewed in terms of “better” or “worse” Not just “less” or “more” inequality. Not just “less” or “more” inequality. Can incorporate the complaint-inequality index in a welfare- economic framework: Can incorporate the complaint-inequality index in a welfare- economic framework:  W(x) =  (X, T)  X: total income  T: Temkin inequality Linear approximation: Linear approximation:  W(x) = X  φT  φ is the weight attached to inequality in welfare  gives three types of distinct pattern:

56. Frank Cowell: Oviedo – Inequality & Poverty Welfare contours (φ=1) Janet’s income Irene’s income 0 ray of equality

57. Frank Cowell: Oviedo – Inequality & Poverty Welfare contours (φ < 1) Janet’s income Irene’s income 0 ray of equality

58. Frank Cowell: Oviedo – Inequality & Poverty Welfare contours (φ > 1) Janet’s income Irene’s income 0 ray of equality Meade’s “superegalitarianism”

59. Frank Cowell: Oviedo – Inequality & Poverty The ATBO Complaint Again, a natural definition of complaint: Again, a natural definition of complaint: Similar to fundamental difference for deprivation: Similar to fundamental difference for deprivation: Use this complaint in the Temkin class Use this complaint in the Temkin class Get a form similar to Chakravarty deprivation Get a form similar to Chakravarty deprivation

60. Frank Cowell: Oviedo – Inequality & Poverty Summary: complaints “Complaints” provide a useful basis for inequality analysis. “Complaints” provide a useful basis for inequality analysis. Intuitive links with poverty and deprivation as well as conventional inequality. Intuitive links with poverty and deprivation as well as conventional inequality. BOP extension provides an implementable inequality measure. BOP extension provides an implementable inequality measure. CCCs provide an implementable ranking principle CCCs provide an implementable ranking principle

61. Frank Cowell: Oviedo – Inequality & Poverty References (1) Bossert, W. and C. D’Ambrosio (2006) “Reference groups and individual deprivation,” Economics Letters, 90, 421-426 Bossert, W. and C. D’Ambrosio (2006) Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative deprivation,” Economics Letters, 14, 283-287 Chakravarty, S. R. and A. B. Chakraborty (1984) Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” Theory and Decision 47, 89-100 Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” Theory and Decision 47, 89-100 Chakravarty, S. R. and D. Mukherjee (1999a) Chakravarty, S. R. and D. Mukherjee (1999a) Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” Social Indicators Research 46, 125-135.. Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” Social Indicators Research 46, 125-135.. Chakravarty, S. R. and D. Mukherjee (1999b) Chakravarty, S. R. and D. Mukherjee (1999b) Cowell, F. A. (2007) Cowell, F. A. (2007) in Betti, G. and Lemmi, A. (ed.) Advances in income inequality and concentration measures, Routledge, London. Chapter 3. Cowell, F. A. (2007) Cowell, F. A. (2007) Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Welfare 23, 71-89. Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Welfare 23, 71-89. Cowell, F. A. and U. Ebert (2004) Cowell, F. A. and U. Ebert (2004) Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” Economics and Philosophy, 19, 241 - 263, Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” Economics and Philosophy, 19, 241 - 263, Devooght, K. (2003) Devooght, K. (2003) Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the measurement of poverty,” Review of Income and Wealth 48, 471-492. Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the measurement of poverty,” Review of Income and Wealth 48, 471-492. Duclos, J.-Y. and P. Grégoire (2002) Duclos, J.-Y. and P. Grégoire (2002)

62. Frank Cowell: Oviedo – Inequality & Poverty References (2) Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters 68, 263-270. Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters 68, 263-270. Ebert, U. and P. Moyes (2000) Ebert, U. and P. Moyes (2000) Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer- Thorbecke poverty orderings,” Journal of Public Economic Theory 4, 455- 473. Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer- Thorbecke poverty orderings,” Journal of Public Economic Theory 4, 455- 473. Ebert, U. and P. Moyes (2002) Ebert, U. and P. Moyes (2002) Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, 761-776 Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52, 761-776 Foster, J. E., Greer, J. and Thorbecke, E. (1984) Foster, J. E., Greer, J. and Thorbecke, E. (1984) Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, 317-327. Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an analysis of UK poverty trends,” Oxford Economic Papers, 49, 317-327. Jenkins, S. P. and Lambert, P. J. (1997) Jenkins, S. P. and Lambert, P. J. (1997) Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Shorrocks, A. F. (1983) Shorrocks, A. F. (1983) Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, 99-121. Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, 99-121. Temkin, L. S. (1986) Temkin, L. S. (1986) Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press. Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press. Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Journal of Economics 93, 321.324. Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Journal of Economics 93, 321.324. Yitzhaki, S. (1979) Yitzhaki, S. (1979)