Inequality: Advanced Topics

1. Inequality: Advanced Topics Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell http://darp.lse.ac.uk/lisbon2006 July 2006

2. Overview... Inequality: Advanced Topics Introduction Themes and methodology Inequality & responsibility Deprivation Complaints

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

4. Overview... Inequality: Advanced Topics Introduction An alternative approach Inequality & responsibility Deprivation Complaints

5. Responsibility • Standard approach to case for redistribution • Use reference point of equality • How effective is tax/benefit system in moving actual distribution toward reference point? • Does not take account of individual responsibility • Role of individual actions • The responsibility “cut” • Dworkin (1981a, 1981b) • Distinguish between • things that are your fault • things for which you deserve compensation

6. Responsibility and redistribution • Should affect the evaluation of distributions • Both case for redistribution... • ... and effectiveness of taxation. • Need to differentiate between • characteristics for which people can be held responsible • characteristics for which people should not • Assume that these characteristics are known and agreed...

7. Basic structure • Each person i has a vector of attributes ai: • Attributes partitioned into two classes • R-attributes: for which the individual is responsible • C-attributes: for which the individual may be compensated • Situation before intervention: • Determined by income function f • f maps attributes into incomes f(ai) • Only person i’s attributes involved • Situation after intervention: • Determined by distribution rule F • We need to compare fairness of outcomes from f and F.

8. Distribution rule • The rule F: • depends on whole profile of attributes • maps the attributes into income of i. • Assume feasibility: • Also assume that the rule F is anonymous Profile of attributes • But what other principles should the rule F satisfy?

9. Responsibility: Principle EIER • Bossert and Fleurbaey (1996) • Equal Income for Equal Responsibility • Focus on distribution itself • Full compensation

10. Responsibility: Principle ETEC • Equal Transfers for Equal C-attributes • Focus on changes in distribution • Strict Compensation

11. A difficulty • For large populations... • EIER and ETEC are incompatible except for... • Additive separability: • Fleurbaey (1995a,b) • In this special case... • ...a natural redistribution mechanism Consider two compromise approaches

12. Compromise (1) • Insist on Full compensation (EIER) • Weaken ETEC • Egalitarian-equivalent mechanisms Reference profile • Every agent has a post-tax income equal to • the pre-tax income earned given reference compensation characteristics plus... • a uniform transfer

13. Compromise (2) • Insist on strict compensation (ETEC) • Weaken EIER • Conditionally egalitarian mechanisms Reference profile • Every agent k is guaranteed the average income of a hypothetical economy • In this economy all agents have characteristics equal to reference profile

14. Application • The responsibility approach gives a reference income distribution • Exact version depends on balance of compensation rules • And on income function f. • Redefine inequality measurement • not based on perfect equality as a norm • use the norm income distribution from the responsibility approach • Devooght (2005) bases this on Cowell (1985) • Cowell approach based on Theil’s conditional entropy • Instead of looking at information content in going from perfect equality to actual distribution... • Start from the reference distribution

15. Overview... Inequality: Advanced Topics Introduction An economic interpretation of a sociological concept Inequality & responsibility Deprivation Complaints

16. Themes • Cross-disciplinary concepts • Income differences • Reference incomes • Formal methodology

17. Methodology • Exploit common structure • poverty • deprivation • complaints and inequality • see Cowell (2005) • Axiomatic method • minimalist approach • characterise structure • introduce ethics

18. “Structural” axioms • Take some social evaluation function F... • Continuity • Linear homogeneity • Translation invariance

19. Common structure • These assumptions underlie several problems • Already seen this with poverty axiomatisation • Ebert and Moyes (2002) • Apply this to other issues in distributional analysis • Individual deprivation • Aggregate deprivation • Inequality and complaints • Need to endow each individual problem with • Ethical assumptions • Reference level of income

20. Individual deprivation • The Yitzhaki (1979) definition • Equivalent form • In present notation • Use the conditional mean

21. Deprivation: Axiomatic approach 1 • The Better-than set for i • Focus • works like the poverty concept

22. Deprivation: Axiomatic approach 2 • Normalisation • Additivity • works like the independence axiom

23. Bossert-D’Ambrosio (2006) • This is just the Yitzhaki individual deprivation index • There is an alternative axiomatisation • Ebert-Moyes (Economics Letters 2000) • Different structure of reference group

24. Aggregate deprivation • Simple approach: just sum individual deprivation • Could consider an ethically weighted variant • Chakravarty and Chakraborty (1984) • Chakravarty and Mukherjee (1999b) • As with poverty consider relative as well as absolute indices…

25. Aggregate deprivation (2) • An ethically weighted relative index • Chakravarty and Mukherjee (1999a) • One based on the generalised-Gini • Duclos and Grégoire (2002)

26. Overview... Inequality: Advanced Topics Introduction Reference groups and distributional judgments Inequality & responsibility Deprivation Complaints • Model • Inequality results • Rankings and welfare

27. The Temkin approach • 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 • Poverty • deprivation • Contains the following elements: • Concept of a complaint • The idea of a reference group • A method of aggregation

28. What is a “complaint?” • Individual’s relationship with the income distribution • The complaint exists independently • does not depend on how people feel • does not invoke “utility” or (dis)satisfaction • Requires a reference group • effectively a reference income • a variety of specifications • see also Devooght (2003)

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

30. Aggregation • The complaint is an individual phenomenon. • How to make the transition from this to society as a whole? • Temkin makes two suggestions: • Simple sum • Just add up the complaints • Weighted sum • Introduce distributional weights • Then sum the weighted complaints

31. The BOP Complaint • Let r(x) be the first richest person you find in N. • Person r (and higher) has income xn. • For “lower” persons, natural definition of complaint: • Similar to fundamental difference for poverty: • Now we replace “p” with “r”

32. BOP-Complaint: Axiomatisation • Use same structural axioms as before. Plus… • Monotonicity: income increments reduce complaint • Independence • Normalisation

33. Overview... Inequality: Advanced Topics Introduction A new approach to inequality Inequality & responsibility Deprivation Complaints • Model • Inequality results • Rankings and welfare

34. Implications for inequality • Broadly two types of axioms with different roles. • Axioms on structure: • use these to determine the “shape” of the measures. • Transfer principles and properties of measures: • use these to characterise ethical nature of measures

35. A BOP-complaint class • The Cowell-Ebert (SCW 2004) result • Similarity of form to FGT • Characterises a family of distributions …

36. 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): • Look at some examples that satisfy this

37. Inequality contours • To examine the properties of the derived indices… • …take the case n = 3 • Draw contours of T–inequality • Note that both the sensitivity parameter  and the weights w are of interest…

38. Inequality contours (e=2) • Now change the weights… w1=0.5 w2=0.5

39. Inequality contours (e=2) w1=0.75 w2=0.25

40. Inequality contours (e = 1) w1=0.75 w2=0.25

41. By contrast: Gini contours

42. Inequality contours (e = 0) • Again change the weights… w1=0.5 w2=0.5

43. Inequality contours (e = –1) w1=0.75 w2=0.25

44. Inequality contours (e = –1) w1=0.5 w2=0.5

45. Special cases “triangles” • If    then inequality just becomes the range, xn–x1 . • If   – then inequality just becomes the “upper-middle class” complaint: xn–xn-1 . • If  = 1 then inequality becomes a generalised absolute Gini. “Y-shapes” Hexagons

46. A 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 B 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Which is more unequal?

47. A 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 B 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Focus on one type of BOP complaint

48. A 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Orthodox approach B 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26

49. Te– inequality

50. The “sequence” • Temkin’s seminal contributions offer an intuitive approach to considering changes in inequality. • Take a simple model of a ladder with just two rungs. • The rungs are fixed, but the numbers on them are not. • Initially everyone is on the upper 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? • Obviously zero at the two endpoints of the sequence • But in between?