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

2. Frank Cowell: EC513 Public Economics Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality Themes and methodology

3. Frank Cowell: EC513 Public Economics 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: EC513 Public Economics Themes Cross-disciplinary concepts Cross-disciplinary concepts Income differences Income differences Reference incomes Reference incomes Formal methodology Formal methodology

5. Frank Cowell: EC513 Public Economics Methodology Exploit common structure Exploit common structure  poverty  deprivation  complaints and inequality  see Cowell (2005) Cowell (2005)Cowell (2005) Axiomatic method Axiomatic method  minimalist approach  characterise structure  introduce ethics

6. Frank Cowell: EC513 Public Economics “Structural” axioms Take some social evaluation function  Take some social evaluation function  Continuity Continuity Linear homogeneity Linear homogeneity Translation invariance Translation invariance

7. Frank Cowell: EC513 Public Economics 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 *

8. Frank Cowell: EC513 Public Economics Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality An alternative approach

9. Frank Cowell: EC513 Public Economics Poverty concepts Given poverty line z Given poverty line z  a reference point Headcount Headcount   p(x,z)/n Poverty gap Poverty gap  fundamental income difference Foster et al (1984) poverty index Foster et al (1984) poverty index Foster et al (1984) Foster et al (1984) Cumulative poverty gap Cumulative poverty gap

10. Frank Cowell: EC513 Public Economics TIP / Poverty profile i/n p(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

11. Frank Cowell: EC513 Public Economics 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.

12. Frank Cowell: EC513 Public Economics Poverty: The key axioms Adapt standard axioms from social welfare Adapt standard axioms from social welfare  anonymity  independence  monotonicity  income increments reduce poverty 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

13. Frank Cowell: EC513 Public Economics 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 focus axiom is The focus axiom is Scale invariance now becomes Scale invariance now becomes Independence means: Independence means: Define the number of the poor as Define the number of the poor as

14. Frank Cowell: EC513 Public Economics 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

15. Frank Cowell: EC513 Public Economics Brief conclusion Poverty indexes can be constructed from scratch Poverty indexes can be constructed from scratch Use standard axioms Use standard axioms Exploit the poverty line as a reference point Exploit the poverty line as a reference point Impose structure Impose structure

16. Frank Cowell: EC513 Public Economics Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality An economic interpretation of a sociological concept

17. Frank Cowell: EC513 Public Economics 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

18. Frank Cowell: EC513 Public Economics Deprivation: Axiomatic approach 1 The Better-than set for i The Better-than set for i Focus Focus  works like the poverty concept

19. Frank Cowell: EC513 Public Economics Deprivation: Axiomatic approach 2 Normalisation Normalisation Additivity Additivity  works like the independence axiom

20. Frank Cowell: EC513 Public Economics Bossert-D’Ambrosio (2004) 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-Moyes (Economics Letters 2000)  Different structure of reference group

21. Frank Cowell: EC513 Public Economics Aggregate deprivation Simple approach: just sum individual deprivation Simple approach: just sum individual deprivation Could consider an ethically weighted variant Could consider an ethically weighted variant   Chakravarty and Chakraborty (1984) Chakravarty and Chakraborty (1984)  Chakravarty and Mukherjee (1999b) Chakravarty and Mukherjee (1999b) Chakravarty and Mukherjee (1999b) As with poverty consider relative as well as absolute indices… As with poverty consider relative as well as absolute indices…

22. Frank Cowell: EC513 Public Economics Aggregate deprivation (2) An ethically weighted relative index An ethically weighted relative index  Chakravarty and Mukherjee (1999a) Chakravarty and Mukherjee (1999a) Chakravarty and Mukherjee (1999a) One based on the generalised-Gini One based on the generalised-Gini  Duclos and Grégoire (2002) Duclos and Grégoire (2002) Duclos and Grégoire (2002)

23. Frank Cowell: EC513 Public Economics Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality Reference groups and distributional judgments Model Inequality results Rankings and welfare

24. Frank Cowell: EC513 Public Economics 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

25. Frank Cowell: EC513 Public Economics 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)

26. Frank Cowell: EC513 Public Economics 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

27. Frank Cowell: EC513 Public Economics 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

28. Frank Cowell: EC513 Public Economics 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, natural definition of complaint: For “lower” persons, natural definition of complaint: Similar to fundamental difference for poverty: Similar to fundamental difference for poverty: Now we replace “p” with “r” Now we replace “p” with “r”

29. Frank Cowell: EC513 Public Economics 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

30. Frank Cowell: EC513 Public Economics Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality A new approach to inequality Model Inequality results Rankings and welfare

31. Frank Cowell: EC513 Public Economics 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

32. Frank Cowell: EC513 Public Economics 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 …

33. Frank Cowell: EC513 Public Economics 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

34. Frank Cowell: EC513 Public Economics 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…

35. Frank Cowell: EC513 Public Economics Inequality contours (  =2) w 1 =0.5 w 2 =0.5 Now change the weights…

36. Frank Cowell: EC513 Public Economics Inequality contours (  =2) w 1 =0.75 w 2 =0.25

37. Frank Cowell: EC513 Public Economics Inequality contours (  = 1) w 1 =0.75 w 2 =0.25

38. Frank Cowell: EC513 Public Economics By contrast: Gini contours

39. Frank Cowell: EC513 Public Economics Inequality contours (  = 0) w 1 =0.5 w 2 =0.5 Again change the weights… Again change the weights…

40. Frank Cowell: EC513 Public Economics Inequality contours (  = –1) w 1 =0.75 w 2 =0.25

41. Frank Cowell: EC513 Public Economics Inequality contours (  = –1) w 1 =0.5 w 2 =0.5

42. Frank Cowell: EC513 Public Economics 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

43. Frank Cowell: EC513 Public Economics Which is more unequal? 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B

44. Frank Cowell: EC513 Public Economics Focus on one type of BOP complaint 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B

45. Frank Cowell: EC513 Public Economics Orthodox approach 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B

46. Frank Cowell: EC513 Public Economics T  – inequality

47. Frank Cowell: EC513 Public Economics 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?

48. Frank Cowell: EC513 Public Economics 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

49. Frank Cowell: EC513 Public Economics Overview... Introduction Poverty Deprivation Complaints Deprivation, complaints, inequality A replacement for the Lorenz order? Model Inequality results Rankings and welfare

50. Frank Cowell: EC513 Public Economics 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.

51. Frank Cowell: EC513 Public Economics 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)

52. Frank Cowell: EC513 Public Economics 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

53. Frank Cowell: EC513 Public Economics 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: Linear approximation: Linear approximation: Total income Inequality

54. Frank Cowell: EC513 Public Economics Welfare contours (φ=1) A’s income B’s income

55. Frank Cowell: EC513 Public Economics Welfare contours (φ<1) A’s income B’s income

56. Frank Cowell: EC513 Public Economics Welfare contours (φ>1) A’s income B’s income Meade’s “superegalitarianism”

57. Frank Cowell: EC513 Public Economics 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

58. Frank Cowell: EC513 Public Economics 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

59. Frank Cowell: EC513 Public Economics References (1) Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”. Working Paper 2004-10, Département de sciences économiques, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada. Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual deprivation”. Working Paper 2004-10, Département de sciences économiques, Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec) H3C 3J7, Canada. Bossert, W. and C. D’Ambrosio (2004) Bossert, W. and C. D’Ambrosio (2004) 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. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE. Cowell, F. A. (2005) Cowell, F. A. (2005) 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)

60. Frank Cowell: EC513 Public Economics 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)