1. Inequality: Advanced Topics
Inequality and Poverty Measurement
Universitat Autònoma de Barcelona
Frank Cowell
July 2006

2. Overview... Themes and methodology Inequality & responsibility
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... An alternative approach Inequality & responsibility
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: Also assume that the rule F is anonymous
depends on whole profile of attributes
maps the attributes into income of i.
Assume feasibility:
Profile of attributes
Also assume that the rule F is anonymous
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. Consider two compromise approaches
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... An economic interpretation of a sociological concept
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 Axiomatic method 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... Reference groups and distributional judgments
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... A new approach to inequality Inequality & responsibility
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 If    then inequality just becomes the range, xn–x1 .
“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. Which is more unequal? A B 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 22 4 6 8 10 12 14 16 18 20 22 24 26 28 30 A 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 B

47. Focus on one type of BOP complaint2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 A 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 B

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

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?

51. The “sequence” (2) This is increasing in m if  > 0
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

52. Overview... A replacement for the Lorenz order?
Inequality: Advanced Topics
Introduction
A replacement for the Lorenz order?
Inequality & responsibility
Deprivation
Complaints
Model
Inequality results
Rankings and welfare

53. 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.

54. Cumulative complaints
Define cumulative complaints
Gives the CCC
cumulative-complaint contour
Just like TIP / Poverty profile
Use this to get a ranking principle
i/n
r(x) / n
K(x)

55. Complaint-ranking The class of BOP-complaint indices
Define complaint ranking
Like the generalised-Lorenz result

56. Social welfare again Linear approximation:
Temkin’s complaints approach to income distribution was to be viewed in terms of “better” or “worse”
Not just “less” or “more” inequality.
Can incorporate the complaint-inequality index in a welfare-economic framework:
Total income
Inequality
Linear approximation:

57. Welfare contours (φ=1)
Janet’s income
Irene’s income

58. Welfare contours (φ<1)
Janet’s income
Irene’s income

59. Welfare contours (φ>1)
Janet’s income
Meade’s “superegalitarianism”
Irene’s income

60. The ATBO Complaint Again, a natural definition of complaint:
Similar to fundamental difference for deprivation:
Use this complaint in the Temkin class
Get a form similar to Chakravarty deprivation

61. Summary: complaints
“Complaints” provide a useful basis for inequality analysis.
Intuitive links with poverty and deprivation as well as conventional inequality.
BOP extension provides an implementable inequality measure.
CCCs provide an implementable ranking principle

62. References (1)
Bossert, W. and C. D’Ambrosio (2006) “Reference groups and individual deprivation,” Economics Letters, 90,
Bossert, W. and M. Fleurbaey (1996) “Redistribution and compensation,” Social Choice and Welfare, 13,
Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative deprivation,” Economics Letters, 14,
Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and their meaning in terms of social satisfaction.” Theory and Decision 47,
Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions by deprivation orderings,” Social Indicators Research 46,
Cowell, F. A. (1985) “The measurement of distributional change: an axiomatic approach.” Review of Economic Studies, 52,
Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London, WC2A 2AE.
Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice and Welfare 23,
Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory and empirics,” Economics and Philosophy, 19, ,

63. References (2)
Devooght, K. (2005) “To each the same and to each his own. A proposal to measure responsibility-sensitive income inequality,” Working paper, University of Kortrijk.
Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and the measurement of poverty,” Review of Income and Wealth 48,
Dworkin, R. (1981a) “What is equality? Part I: Equality of welfare.” Philosophy and Public Affairs, 10,
Dworkin, R. (1981b) “What is equality? Part I: Equality of resources.” Philosophy and Public Affairs, 10,
Dutta, B. and D. Ray (1989) “A concept of egalitarianism under participation constraints” Econometrica, 57,
Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s index of individual deprivation. Economics Letters 68,
Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-Greer-Thorbecke poverty orderings,” Journal of Public Economic Theory 4,
Fleurbaey, M. (1995a) “Equal opportunity or equal social outcome?” Economics and Philosophy 11,

64. References (3)
Fleurbaey, M. (1995b) “Equality and responsibility,” European Economic Review, 39,
Fleurbaey, M. (1995c) “Three solutions to the compensation problem,” Journal of Economic Theory, 65,
Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable poverty measures,” Econometrica, 52,
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,
Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17
Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15,
Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press.
Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly Journal of Economics 93,