EC513 PhD Public Economics 2005/6 darp.lse.ac.uk/EC513.htm

1. 7 March 2006 EC513 PhD Public Economics 2005/6http://darp.lse.ac.uk/EC513.htm Deprivation, Complaints and Inequality

2. Overview... Deprivation, complaints, inequality Introduction Themes and methodology Poverty 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. Themes • Cross-disciplinary concepts • Income differences • Reference incomes • Formal methodology

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

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

7. Structural axioms: illustration • D for n=3 • An income distribution • Perfect equality • Contours of “Absolute” Gini • Continuity • Continuous approach to I = 0 • Linear homogeneity • Proportionate increase in I • Translation invariance • I constant x2 x* • 1 • x3 0 x1

8. Overview... Deprivation, complaints, inequality Introduction An alternative approach Poverty Deprivation Complaints

9. Poverty concepts • Given poverty line z • a reference point • Headcount • p(x,z)/n • Poverty gap • fundamental income difference • Foster et al (1984) poverty index • Cumulative poverty gap

10. TIP / Poverty profile • Cumulative gaps versus population proportions • Proportion of poor • TIP curve G(x,z) • TIP curves have same interpretation as GLC (Shorrocks 1983) • TIP dominance implies unambiguously greater poverty i/n 0 p(x,z)/n

11. Poverty: Axiomatic approach • Characterise an ordinal poverty index P(x ,z) • See Ebert and Moyes (2002) • 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 • Show that • given just these axioms… • …you are bound to get a certain type of poverty measure.

12. Poverty: The key axioms • Adapt standard axioms from social welfare • anonymity • independence • monotonicity • income increments reduce poverty • Strengthen two other axioms • scale invariance • translation invariance • Also need continuity • Plus a focus axiom

13. A closer look at the axioms • Let D denote the set of ordered income vectors • The focus axiom is • Scale invariance now becomes • Define the number of the poor as • Independence means:

14. Ebert-Moyes (2002) • Gives two types of FGT measures • “relative” version • “absolute” version • Additivity follows from the independence axiom

15. Brief conclusion • Poverty indexes can be constructed from scratch • Use standard axioms • Exploit the poverty line as a reference point • Impose structure

16. Overview... Deprivation, complaints, inequality Introduction An economic interpretation of a sociological concept Poverty Deprivation Complaints

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

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

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

20. Bossert-D’Ambrosio (2004) • This is just the Yitzhaki individual deprivation index • There is an alternative axiomatisation • Ebert-Moyes (Economics Letters 2000) • Different structure of reference group

21. 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…

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

23. Overview... Deprivation, complaints, inequality Introduction Reference groups and distributional judgments Poverty Deprivation Complaints • Model • Inequality results • Rankings and welfare

24. 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

25. 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)

26. 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

27. 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

28. 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”

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

30. Overview... Deprivation, complaints, inequality Introduction A new approach to inequality Poverty Deprivation Complaints • Model • Inequality results • Rankings and welfare

31. 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

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

33. 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

34. 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…

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

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

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

38. By contrast: Gini contours

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

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

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

42. 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

43. 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?

44. 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

45. 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

46. Te– inequality

47. 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?

48. The “sequence” (2) • For the case of T–inequality we have • This is increasing in m if  > 0 • For other cases there is a degenerate sequence in the same direction

49. Overview... Deprivation, complaints, inequality Introduction A replacement for the Lorenz order? Poverty Deprivation Complaints • Model • Inequality results • Rankings and welfare

50. Rankings • Move beyond simple inequality measures • 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. • Also similar to poverty rankings with arbitrary poverty lines.