1. Frank Cowell: Siena – Inequality Summer School Deprivation, Complaints and Inequality June 2007 Summer School on Inequality University of Siena Frank Cowell http://darp.lse.ac.uk/siena2007

2. Frank Cowell: Siena – Inequality Summer School Overview... Introduction Poverty and inequality Deprivation Complaints Deprivation, complaints, inequality Themes and methodology

3. Frank Cowell: Siena – Inequality Summer School Purpose of lecture Look at recent theoretical developments in distributional analysis 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 Use and reuse common concepts: Use and reuse common concepts:  Income differences  Reference incomes  Formal methodology

4. Frank Cowell: Siena – Inequality Summer School Methodology Focus on the way modern methodology is applied Focus on the way modern methodology is applied 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

5. Frank Cowell: Siena – Inequality Summer School 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

6. Frank Cowell: Siena – Inequality Summer School “Structural” axioms Take some social evaluation function  Take some social evaluation function   could be inequality, poverty, social welfare  apply standard axioms on structure 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) Illustrate these using an example Illustrate these using an example  the Absolute Gini coefficient  useful inequality measure: equals Gini × mean income

7. Frank Cowell: Siena – Inequality Summer School 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 *   These axioms repeatedly used in the following applications

8. Frank Cowell: Siena – Inequality Summer School Overview... Introduction Poverty and inequality Deprivation Complaints Deprivation, complaints, inequality An alternative approach

9. Frank Cowell: Siena – Inequality Summer School 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 )

10. Frank Cowell: Siena – Inequality Summer School 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 Plot G i against population proportions Plot G i against population proportions  Get TIP curve (Jenkins and Lambert 1997) Jenkins and Lambert 1997Jenkins and Lambert 1997  TIP: “Three ‘I’s of Poverty”: (Incidence, Intensity, Inequality)

11. Frank Cowell: Siena – Inequality Summer School “Three ‘I’s of Poverty” i/n  (x,z)/n G i (x,z) 0 Population proportions versus cumulative gaps Population proportions versus cumulative gaps TIP curve TIP curve Proportion of poor Proportion of poor Poverty deficit Poverty deficit Population proportions versus cumulative gaps Population proportions versus cumulative gaps TIP curve TIP curve Proportion of poor Proportion of poor Poverty deficit Poverty deficit   Incidence of poverty is the horizontal distance G stays constant if i >    Intensity of poverty is the vertical distance   Inequality among the poor is the curvature of the TIP

12. Frank Cowell: Siena – Inequality Summer School Poverty orderings TIP curves have same interpretation as Generalised Lorenz Curves TIP curves have same interpretation as Generalised Lorenz Curves  GLC-dominance implies welfare dominance  for all monotonic, separable, SWFs satisfying transfer principle (Shorrocks 1983) Shorrocks 1983)Shorrocks 1983) TIP dominance implies unambiguously greater poverty TIP dominance implies unambiguously greater poverty  holds for given poverty line  …and virtually all poverty measures in use A simple link with inequality orderings and welfare A simple link with inequality orderings and welfare  For details see Zheng (2000) Zheng (2000)Zheng (2000) 0

13. Frank Cowell: Siena – Inequality Summer School 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 the standard axioms we introduced earlier Use the standard axioms we introduced earlier  some of these slightly modified  supplement with axioms to give meaning to poverty 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: Siena – Inequality Summer School 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: Siena – Inequality Summer School 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: Siena – Inequality Summer School Ebert-Moyes (2002) Gives two types of FGT measures Gives two types of FGT measures  “relative” version  “absolute” version Similarity to certain families of inequality measures Similarity to certain families of inequality measures Additivity follows from the independence axiom Additivity follows from the independence axiom

17. Frank Cowell: Siena – Inequality Summer School Poverty and inequality: 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  deprivation  new approaches to inequality

18. Frank Cowell: Siena – Inequality Summer School Overview... Introduction Poverty and inequality Deprivation Complaints Deprivation, complaints, inequality An economic interpretation of a sociological concept

19. Frank Cowell: Siena – Inequality Summer School Relative deprivation Individual deprivation Individual deprivation   concern with position relative to others in society (Runciman, 1966)   negative welfare effects when friends and neighbours become better-off? Related to income satisfaction?  D’Ambrosio and Frick (2007)  D’Ambrosio and Frick (2007) focus on E. and W. Germany, 1990-2004 D’Ambrosio and Frick (2007) D’Ambrosio and Frick (2007)   show happiness/satisfaction is a relative notion   derive perceived well-being from being richer not simply from being rich Do all people care about it? Do all people care about it?  Ravallion and Lokshin 2005) test for perceived welfare effects of relative deprivation in Malawi Ravallion and Lokshin 2005 Ravallion and Lokshin 2005  Relative deprivation is not a concern for most people, although it is for the comparatively well off. Aggregate deprivation Aggregate deprivation  a social concept  related to individual deprivation  relationship to inequality and poverty? Runciman Runciman, 1966 : “If people have no reason to expect or hope for more than they can achieve, they will be less discontent with what they have, or even grateful simply to be able to hold on to it. But if, on the other hand, they have been led to see as a possible goal the relative prosperity of some more fortunate community with which they can directly compare themselves, then they will remain discontent with their lot until they have succeeded in catching up”. Runciman, 1966 : “If people have no reason to expect or hope for more than they can achieve, they will be less discontent with what they have, or even grateful simply to be able to hold on to it. But if, on the other hand, they have been led to see as a possible goal the relative prosperity of some more fortunate community with which they can directly compare themselves, then they will remain discontent with their lot until they have succeeded in catching up”.

20. Frank Cowell: Siena – Inequality Summer School Individual deprivation: model Yitzhaki (1979) definition of individual deprivatoin: Yitzhaki (1979) definition of individual deprivatoin: Yitzhaki (1979) Yitzhaki (1979) Can write this in equivalent form Can write this in equivalent form In discrete notation In discrete notation Use the conditional mean Use the conditional mean

21. Frank Cowell: Siena – Inequality Summer School Deprivation: Axiomatic approach 1 The Better-than set for i The Better-than set for i Focus Focus  works like the poverty concept

22. Frank Cowell: Siena – Inequality Summer School Deprivation: Axiomatic approach 2 Normalisation Normalisation Additivity Additivity  works like the independence axiom

23. Frank Cowell: Siena – Inequality Summer School 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

24. Frank Cowell: Siena – Inequality Summer School 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)

25. Frank Cowell: Siena – Inequality Summer School 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)

26. Frank Cowell: Siena – Inequality Summer School Overview... Introduction Poverty and inequality Deprivation Complaints Deprivation, complaints, inequality Reference groups and distributional judgments Model Inequality results Rankings and welfare

27. Frank Cowell: Siena – Inequality Summer School The Temkin approach Larry Temkin (1986, 1993) approach to inequality Larry Temkin (1986, 1993) approach to inequality1986  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

28. Frank Cowell: Siena – Inequality Summer School A “complaint?” Involves the individual’s relationship with the income distribution Involves the 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 Complaint depends on position in distribution Complaint depends on position in distribution Requires a reference group Requires a reference group  effectively a reference income  a variety of specifications  see also Devooght (2003) Devooght (2003)Devooght (2003) Temkin (Temkin 1986, p. 102): “To say that the best-off have nothing to complain about is in no way to impugn their moral sensibilities. They may be just as concerned about the inequality in their world as anyone else. Nor is it to deny that, insofar as one is concerned about inequality, one might have a complaint about them being as well o. as they are. It is only to recognize that, since they are at least as well o. as every other member of their world, they have nothing to complain about. Similarly, to say that the worst-off have a complaint is not to claim that they will in fact complain (they may not). It is only to recognize that it is a bad thing (unjust or unfair) for them to be worse o. than the other members of their world through no fault of their own” (Temkin 1986, p. 102): “To say that the best-off have nothing to complain about is in no way to impugn their moral sensibilities. They may be just as concerned about the inequality in their world as anyone else. Nor is it to deny that, insofar as one is concerned about inequality, one might have a complaint about them being as well o. as they are. It is only to recognize that, since they are at least as well o. as every other member of their world, they have nothing to complain about. Similarly, to say that the worst-off have a complaint is not to claim that they will in fact complain (they may not). It is only to recognize that it is a bad thing (unjust or unfair) for them to be worse o. than the other members of their world through no fault of their own”

29. Frank Cowell: Siena – Inequality Summer School 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

30. Frank Cowell: Siena – Inequality Summer School 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

31. Frank Cowell: Siena – Inequality Summer School 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

32. Frank Cowell: Siena – Inequality Summer School 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

33. Frank Cowell: Siena – Inequality Summer School Overview... Introduction Poverty and inequality Deprivation Complaints Deprivation, complaints, inequality A new approach to inequality Model Inequality results Rankings and welfare

34. Frank Cowell: Siena – Inequality Summer School 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

35. Frank Cowell: Siena – Inequality Summer School A BOP-complaint class The Cowell and Ebert (2004) result The Cowell and Ebert (2004) resultCowell and Ebert (2004)Cowell and Ebert (2004) Similarity of form to FGT Similarity of form to FGT Characterises a family of distributions … Characterises a family of distributions …

36. Frank Cowell: Siena – Inequality Summer School The transfer principle Do BOP-complaint measures satisfy transfer principle? Do BOP-complaint measures satisfy transfer principle?  If transfer is from richest, yes  But if transfers are amongst hoi polloi, maybe not From Cowell and Ebert (2004) : From Cowell and Ebert (2004) :Cowell and Ebert (2004)Cowell and Ebert (2004) Look at some examples that do/do not satisfy this: Look at some examples that do/do not satisfy this:  take the case n = 3  draw contours of T  –inequality  both the sensitivity parameter  and the weights w are of interest…

37. Frank Cowell: Siena – Inequality Summer School Inequality contours (  =2) w 1 =0.5 w 2 =0.5 Now change the weights…

38. Frank Cowell: Siena – Inequality Summer School Inequality contours (  =2) w 1 =0.75 w 2 =0.25

39. Frank Cowell: Siena – Inequality Summer School Inequality contours (  = 1) w 1 =0.75 w 2 =0.25

40. Frank Cowell: Siena – Inequality Summer School By contrast: Gini contours

41. Frank Cowell: Siena – Inequality Summer School Inequality contours (  = 0) w 1 =0.5 w 2 =0.5 Again change the weights… Again change the weights…

42. Frank Cowell: Siena – Inequality Summer School Inequality contours (  = –1) w 1 =0.75 w 2 =0.25

43. Frank Cowell: Siena – Inequality Summer School Inequality contours (  = –1) w 1 =0.5 w 2 =0.5

44. Frank Cowell: Siena – Inequality Summer School Special cases If    then inequality just becomes the range, x n –x 1 If    then inequality just becomes the range, x n –x 1  contour map becomes a set of triangles 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.  contour map becomes a set of “Y-shapes” If  = 1 then inequality is a generalised absolute Gini If  = 1 then inequality is a generalised absolute Gini  contour map is a set of hexagons Different values of  may give very different rankings Different values of  may give very different rankings  not all concur with orthodox view…  …. corresponding to Dalton transfer principle

45. Frank Cowell: Siena – Inequality Summer School Which is more unequal? 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B   Two points of view...

46. Frank Cowell: Siena – Inequality Summer School Focus on one type of BOP complaint 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B   B is more unequal?

47. Frank Cowell: Siena – Inequality Summer School Orthodox approach 0 2468101214161820222426 2830 A 02468101214161820222426 2830 B   A is more unequal?

48. Frank Cowell: Siena – Inequality Summer School T  – inequality   A is more unequal for high values of 

49. Frank Cowell: Siena – Inequality Summer School The “sequence” Temkin also offers an intuitive approach to considering changes in inequality Temkin also offers 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  Initially everyone is on the upper rung One by one, people are transferred to the lower rung 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: Siena – Inequality Summer School 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: Siena – Inequality Summer School Overview... Introduction Poverty and inequality Deprivation Complaints Deprivation, complaints, inequality A replacement for the Lorenz order? Model Inequality results Rankings and welfare

52. Frank Cowell: Siena – Inequality Summer School 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 given poverty lines Also similar to poverty rankings with given poverty lines

53. Frank Cowell: Siena – Inequality Summer School Cumulative complaints Define cumulative complaints Define cumulative complaints Gives the CCC Gives the CCC  cumulative-complaint contour  Just like TIP 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: Siena – Inequality Summer School 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: Siena – Inequality Summer School Social welfare 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: Siena – Inequality Summer School Welfare contours (φ = 1) Janet’s income Irene’s income 0 ray of equality Two person case Two person case Diagram is symmetric Diagram is symmetric x-values giving constant W x-values giving constant W Two person case Two person case Diagram is symmetric Diagram is symmetric x-values giving constant W x-values giving constant W   Similar to “max-min” welfare function

57. Frank Cowell: Siena – Inequality Summer School Welfare contours (φ < 1) Janet’s income 0 Irene’s income Two person case Two person case Diagram is symmetric Diagram is symmetric x-values giving constant W x-values giving constant W Two person case Two person case Diagram is symmetric Diagram is symmetric x-values giving constant W x-values giving constant W   Similar to Gini case

58. Frank Cowell: Siena – Inequality Summer School Welfare contours (φ > 1) Janet’s income 0 Irene’s income Two person case Two person case Diagram is symmetric Diagram is symmetric x-values giving constant W x-values giving constant W Two person case Two person case Diagram is symmetric Diagram is symmetric x-values giving constant W x-values giving constant W   Captures “superegalitarianism” (Meade 1976)

59. Frank Cowell: Siena – Inequality Summer School 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: Siena – Inequality Summer School 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: Siena – Inequality Summer School 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) “Reference groups and individual deprivation,” Economics Letters, 90, 421-426 Bossert, W. and C. D’Ambrosio (2006) 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) “On indices of relative deprivation,” Economics Letters, 14, 283-287 Chakravarty, S. R. and A. B. Chakraborty (1984) 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) “Gini, Deprivation and Complaints.” in Betti, G. and Lemmi, A. (ed.) Advances in income inequality and concentration measures, Routledge, London. Chapter 3. Cowell, F. A. (2007) “Gini, Deprivation and Complaints.” 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) D’Ambrosio, C. and J. R. Frick (2007) “Income satisfaction and relative deprivation: an empirical link,” Social Indicators Research 81, 497–519 D’Ambrosio, C. and J. R. Frick (2007) “Income satisfaction and relative deprivation: an empirical link,” Social Indicators Research 81, 497–519 D’Ambrosio, C. and J. R. Frick (2007) D’Ambrosio, C. and J. R. Frick (2007) 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) 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)

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