1. Distributional Equity, Social Welfare
Public Economics: University of Barcelona
Frank Cowell
June 2005

2. Onwards from welfare economics...
We’ve seen the welfare-economics basis for redistribution as a public-policy objective
How to assess the impact and effectiveness of such policy?
We need appropriate criteria for comparing distributions of income and personal welfare
This requires a treatment of issues in distributional analysis.

3. Overview... How to represent problems in distributional analysis
Equity and social welfare
Overview...
Welfare comparisons
Income distributions
Comparisons
SWFs
How to represent problems in distributional
analysis
Rankings
Welfare and needs
Compensation and responsibility

4. Representing a distribution
Recall our two standard approaches:
Irene and Janet
The F-form
particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria
especially useful in cases where it is appropriate to adopt a parametric model of income distribution

5. Now for some formalisation:
Pen's parade
Plot income against proportion of population
Parade in ascending order of "income" / height x
"income" (height)
x0.8
Now for some formalisation:
x0.2 q
0.2
0.8 1
proportion of the population

6. A distribution function
F(x) 1
F(x0) x x0

7. The set of distributions
We can imagine a typical distribution as belonging to some class F Î F
How should members of F be described or compared?
Sets of distributions are, in principle complicated entities
We need some fundamental principles

8. Overview... Methods and criteria of distributional analysis
Equity and social welfare
Overview...
Welfare comparisons
Income distributions
Comparisons
SWFs
Methods and criteria of distributional analysis
Rankings
Welfare and needs
Compensation and responsibility

9. Comparing Income Distributions
Consider the purpose of the comparison...
…in this case to get a handle on the redistributive impact of government activity - taxes and benefits.
This requires some concept of distributional “fairness” or “equity”.
The ethical basis rests on some aspects of the last lecture…
…and the practical implementation requires an comparison in terms of “inequality”.
Which is easy. Isn’t it?

10. Some comparisons self-evident...1 2 3 4 5 6 7 8 9 10 $ P R 1 2 3 4 5 6 7 8 9 10 $ P R 1 2 3 4 5 6 7 8 9 10 $ P R 1 2 3 4 5 6 7 8 9 10 $ R P

11. A fundamental issue...
Can distributional orderings be modelled using the two-person paradigm?
If so then comparing distributions in terms of inequality or other concepts of equity will be almost trivial.
Then the comparison of tax systems in terms of distributive effect presents no problem
But, consider a simple example with three persons and fixed incomes

12. The 3-Person problem: two types of income difference
Which do you think is “better”?
Top Sensitivity
Bottom Sensitivity 1 2 3 4 5 6 7 8 9 10 11 12 13 $ P Q R
Monday
Low inequality
High inequality
Low inequality
High inequality 1 2 3 4 5 6 7 8 9 10 11 12 13 $ P Q R
Tuesday

13. Distributional Orderings and Rankings
In an ordering we unambiguously arrange distributions
But a ranking may include distributions that cannot be ordered
more welfare
Syldavia
{Syldavia, Arcadia, Borduria} is an ordering.
{Syldavia, Ruritania, Borduria} is also an ordering.
But the ranking {Syldavia, Arcadia, Ruritania, Borduria} is not an ordering.
Arcadia
Ruritania
Borduria
less welfare

14. Comparing income distributions - 2
Distributional comparisons are more complex when more than two individuals are involved.
P-Q and Q-R gaps important
To make progress we need an axiomatic approach.
Make precise “one distribution is better than another”
Axioms could be rooted in welfare economics
There are other logical bases.
Apply the approach to general ranking principles
Lorenz comparisons
Social-welfare rankings
Also to specific indices
Welfare functions
Inequality measures

15. The Basics: Summary
Income distributions can be represented in two main ways
Irene-Janet
F-form
The F-form is characterised by Pen’s Parade
Distributions are complicated entities:
compare them using tools with appropriate properties.
A useful class of tools can be found from Welfare Functions with suitable properties…

16. Overview... How to incorporate fundamental principles
Equity and social welfare
Overview...
Welfare comparisons
SWFs
Axiomatic structure
Classes
Values
How to incorporate fundamental principles
Rankings
Welfare and needs
Compensation and responsibility

17. Social-welfare functions
Basic tool is a social welfare function (SWF)
Maps set of distributions into the real line
I.e. for each distribution we get one specific number
In Irene-Janet notation W = W(x)
Properties will depend on economic principles
Simple example of a SWF:
Total income in the economy W = Si xi
Perhaps not very interesting
Consider principles on which SWF could be based

18. Another fundamental question
What makes a “good” set of principles?
There is no such thing as a “right” or “wrong” axiom.
However axioms could be appropriate or inappropriate
Need some standard of “reasonableness”
For example, how do people view income distribution comparisons?
Use a simple framework to list some of the basic axioms
Assume a fixed population of size n.Assume that individual utility can be measured by x
Income normalised by equivalence scales
Rules out utility interdependence
Welfare is just a function of the vector x := (x1, x2,…,xn )
Follow the approach of Amiel-Cowell (1999)

19. Basic Axioms: Anonymity Population principle Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability

20. Basic Axioms: Anonymity Population principle Monotonicity
Permute the individuals and social welfare does not change
Population principle
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability

21. Anonymity x W(x′) = W(x) x' $ $ 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 41 2 3 4 5 6 7 8 9 10 11 12 13 $ x
W(x′) = W(x) 1 2 3 4 5 6 7 8 9 10 11 12 13 $ x'

22. Implication of anonymity1 2 3 4 5 6 7 8 9 10 11 12 13 $ x y
End state principle: xy is equivalent to x′y . 1 2 3 4 5 6 7 8 9 10 11 12 13 $ x' y'

23. Basic Axioms: Anonymity Population principle Monotonicity
Scale up the population and social welfare comparisons remain unchanged
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability

24. Population replication1 2 3 4 5 6 7 8 9 10 $
W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y) 1 2 3 4 5 6 7 8 9 10 $

25. A change of notation? Using the first two axioms
Anonymity
Population principle
We can write welfare using F –form
Just use information about distribution
Sometimes useful for descriptive purposes
Remaining axioms can be expressed in either form

26. Basic Axioms: Anonymity Population principle Monotonicity
Increase anyone’s income and social welfare increases
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability

27. W(x1+,x2,..., xn ) > W(x1,x2,..., xn )
Monotonicity x $ 2 4 6 8 10 12 14 16 18 20 x′ $ 2 4 6 8 10 12 14 16 18 20
W(x1+,x2,..., xn ) > W(x1,x2,..., xn )

28. W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)
Monotonicity x′ $ 2 4 6 8 10 12 14 16 18 20 x $ 2 4 6 8 10 12 14 16 18 20
W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)

29. W(x1,x2,..., xn+) > W(x1,x2,..., xn )
Monotonicity x′ $ 2 4 6 8 10 12 14 16 18 20
x′ $ 2 4 6 8 10 12 14 16 18 20
W(x1,x2,..., xn+) > W(x1,x2,..., xn )

30. Basic Axioms: Anonymity Population principle Monotonicity
Principle of Transfers
Poorer to richer transfer must lower social welfare
Scale / translation Invariance
Strong independence / Decomposability

31. Transfer principle: The Pigou (1912) approach:
Focused on a 2-person world
A transfer from poor P to rich R must lower social welfare
The Dalton (1920) extension
Extended to an n-person world
A transfer from (any) poorer i to (any) richer j must lower social welfare
Although convenient, the extension is really quite strong…

32. Which group seems to have the more unequal distribution?1 2 3 4 5 6 7 8 9 10 11 12 13 $

33. The issue viewed as two groups1 2 3 4 5 6 7 8 9 10 11 12 13 $

34. Focus on just the affected persons1 2 3 4 5 6 7 8 9 10 11 12 13 $

35. Basic Axioms: Anonymity Population principle Monotonicity
Principle of Transfers
Scale Invariance
Rescaling incomes does not affect welfare comparisons
Strong independence / Decomposability

36. Scale invariance (homotheticity)x y $ 5 10 15
W(x)  W(y)  W(lx)  W(ly) lx $
500
1000
1500 ly

37. Basic Axioms: Anonymity Population principle Monotonicity
Principle of Transfers
Translation Invariance
Adding a constant to all incomes does not affect welfare comparisons
Strong independence / Decomposability

38. Translation invariancex y $ 5 10 15
W(x)  W(y)  W(x+d1)  W(y+d1)
x+d1 $ 5 10 15 20
y+d1 $ 5 10 15 20

39. Basic Axioms: Anonymity Population principle Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
merging with an “irrelevant” income distribution does not affect welfare comparisons

40. Decomposability / Independence1 2 3 4 5 6 7 8 9 10 11 12 13 $
Before merger... x y
W(x)  W(y)  W(x')  W(y')
After merger... 1 2 3 4 5 6 7 8 9 10 11 12 13 $ x' y'

41. Using axioms Why the list of axioms?
We can use some, or all, of them to characterise particular classes of SWF
More useful than picking individual functions W ad hoc
This then enables us to get fairly general results
Depends on richness of the class
The more axioms we impose (perhaps) the less general the result
This technique will be applied to other types of tool
Inequality
Poverty
Deprivation.

42. Overview... Categorising important types Equity and social welfare
Welfare comparisons
SWFs
Axiomatic structure
Classes
Values
Categorising important types
Rankings
Welfare and needs
Compensation and responsibility

43. Classes of SWFs (1)
Anonymity and population principle imply we can write SWF in either I-J form or F form
Most modern approaches use these assumptions
But you may need to standardise for needs etc
Introduce decomposability and you get class of Additive SWFs W :
W(x)= Si u(xi)
or equivalently in F-form W(F) = ò u(x) dF(x)
The class W is of great importance
Already seen this in lecture 1.
But W excludes some well-known welfare criteria

44. Classes of SWFs (2) From W we get important subclasses
If we impose monotonicity we get
W1 Ì W : u(•) increasing
If we further impose the transfer principle we get
W2 Ì W1: u(•) increasing and concave
We often need to use these special subclasses
Illustrate their behaviour with a simple example…

45. The density function x x f(x) x 1 Income growth at x0
Welfare increases if WÎ W1
A mean-preserving spread
Welfare decreases if WÎ W2 x x x 1

46. An important family x 1 – e – 1
Take the W2 subclass and impose scale invariance.
Get the family of SWFs where u is iso-elastic:
x 1 – e – 1
u(x) = ————, e ³ 0
1 – e
Same as that in lecture 1:
individual utility represented by x.
also same form as CRRA utility function
Parameter e captures society’s inequality aversion.
Similar interpretation to individual risk aversion
See Atkinson (1970)

47. Another important family
Take the W2 subclass and impose translation invariance.
Get the family of SWFs where u is iso-elastic:
1 – exp–kx
u(x) = ——— k
Same form as CARA utility function
Parameter k captures society’s absolute inequality aversion.
Similar to individual absolute risk aversion

48. Overview... …Can we deduce how inequality-averse “society” is?
Equity and social welfare
Overview...
Welfare comparisons
SWFs
Axiomatic structure
Classes
Values
…Can we deduce how inequality-averse “society” is?
Rankings
Welfare and needs
Compensation and responsibility

49. Values: the issues
In previous lecture we saw the problem of adducing social values.
Here we will focus on two questions…
First: do people care about distribution?
Justify a motive for considering positive inequality aversion
Second: What is the shape of u?
What is the value of e?
Examine survey data and other sources

50. Happiness and welfare? Alesina et al (2004)
Use data on happiness from social survey
Construct a model of the determinants of happiness
Use this to see if income inequality makes a difference
Seems to be a difference in priorities between US and Europe
US Continental Europe
Share of government in GDP 30% 45%
Share of transfers in GDP 11% 18%
But does this reflect values?
Do people in Europe care more about inequality?

51. The Alesina et al model An ordered logit
“Happy” is categorical; built from three (0,1) variables:
not too happy
fairly happy
very happy
individual, state, time, group.
Macro variables include inflation, unemployment rate
Micro variables include personal characteristics
h, m are state, time dummies

52. The Alesina et al. results
People tend to declare lower happiness levels when inequality is high.
Strong negative effects of inequality on happiness of the European poor and leftists.
No effects of inequality on happiness of US poor and the left-wingers are not affected by inequality
Negative effect of inequality on happiness of US rich
No differences across the American right and the European right.
No differences between the American rich and the European rich

53. The shape of u: approaches
Direct estimates of inequality aversion
See Cowell-Gardiner (2000)
Carlsson et al (2005)
Direct estimates of risk aversion
Use as proxy for inequality aversion
Base this on Harsanyi arguments?
Indirect estimates of risk aversion
Indirect estimates of inequality aversion
From choices made by government

54. Direct evidence on risk aversion
Barsky et al (1997) estimated relative risk-aversion from survey evidence.
Note dependence on how well-off people are.

55. Indirect evidence on risk aversion
Blundell et al (1994) inferred relative risk-aversion from estimated parameter of savings using expenditure data.
Use two models: second version includes variables to capture anticipated income growth.
Again note dependence on how well-off people are.

56. Indirect evidence on social values
Assume constant absolute sacrifice
Assume isoelastic social utility
Then estimate e from
Results for UK:

57. SWFs: Summary A small number of key axioms
Generate an important class of SWFs with useful subclasses.
Need to make a decision on the form of the SWF
Decomposable?
Scale invariant?
Translation invariant?
If we use the isoelastic model perhaps a value of around 1.5 – 2 is reasonable.

58. Overview... ...rankings, orderings and practical tools
Equity and social welfare
Overview...
Welfare comparisons
SWFs
...rankings, orderings and practical tools
Rankings
Welfare and needs
Compensation and responsibility

59. Ranking and dominance
We pick up on the problem of comparing distributions
Two simple concepts based on elementary axioms
Anonymity
Population principle
Monotonicity
Transfer principle
Illustrate these tools with a simple example
Use the Irene-Janet representation of the distribution
Fixed population (so we don’t need pop principle)

60. First-order Dominancex $ 2 4 6 8 10 12 14 16 18 20
y[1] > x[1], y[2] > x[2], y[3] > x[3] y $ 2 4 6 8 10 12 14 16 18 20
Each ordered income in y larger than that in x.

61. Second-order Dominancex $ 2 4 6 8 10 12 14 16 18 20
y[1] > x[1], y[1]+y[2] > x[1]+x[2], y[1]+y[2] +y[3] > x[1]+x[2] +x[3] y $ 2 4 6 8 10 12 14 16 18 20
Each cumulated income sum in y larger than that in x.
Weaker than first-order dominance

62. Social-welfare criteria and dominance
Why are these concepts useful?
First these concepts and classes of SWF
Recall the class of additive SWFs
W : W(F) = ò u(x) dF(x)
… and its important subclasses
W1 Ì W : u(•) increasing
W2 Ì W1: u(•) increasing and concave
Now for the special relationship.
We need to move on from the example by introducing formal tools of distributional analysis.

63. 1st-Order approach
The basic tool is the quantile. This can be expressed in general as the functional
Use this to derive a number of intuitive concepts
Interquartile range
Decile-ratios
Semi-decile ratios
The graph of Q is Pen’s Parade
Extend it to characterise the idea of dominance…

64. An important relationship
The idea of quantile (1st-order) dominance:
G quantile-dominates F means:
for every q, Q(G;q) ³ Q(F;q),
for some q, Q(G;q) > Q(F;q)
A fundamental result:
G quantile-dominates F Û
W(G) > W(F) for all WÎW1
To illustrate, use Pen's parade

65. First-order dominance
Q(.; q) G F q 1

66. 2nd-Order approach
The basic tool is the income cumulant. This can be expressed as the functional
Use this to derive three intuitive concepts
The (relative) Lorenz curve
The shares ranking
Gini coefficient
The graph of C is the generalised Lorenz curve
Again use it to characterise dominance…

67. Another important relationship
The idea of cumulant (2nd-order) dominance:
G cumulant-dominates F means:
for every q, C (G;q) ³ C (F;q),
for some q, C (G;q) > C (F;q)
A fundamental result:
G cumulant-dominates F Û
W(G) > W(F) for all WÎW2
To illustrate, draw the GLC

68. Second order dominance
C(.; q)
cumulative income
m(G)
C(G; . )
m(F)
practical example, UK
C(F; . ) q 1

69. Application of ranking
The tax and -benefit system maps one distribution into another...
Use ranking tools to assess the impact of this in welfare terms.
Typically this uses one or other concept of Lorenz dominance.

70. UK “Final income” – GLC

71. “Original income” – GLC

72. Ranking Distributions: Summary
First-order (Parade) dominance is equivalent to ranking by quantiles.
A strong result.
Where Parades cross, second-order methods may be appropriate.
Second-order (GL)-dominance is equivalent to ranking by cumulations.
Another strong result.

73. Overview... Extensions of the ranking approach
Equity and social welfare
Overview...
Welfare comparisons
SWFs
Extensions of the ranking approach
Rankings
Welfare and needs
Compensation and responsibility

74. Difficulties with needs
Why equivalence scales?
Need a way of making welfare comparisons
Should be coherent
Take account of differing family size
Take account of needs
But there are irreconcilable difficulties:
Logic
Source information
Estimation problems
Perhaps a more general approach
“Needs” seems an obvious place for explicit welfare analysis

75. Income and needs reconsidered
Standard approach uses "equivalised income"
The approach assumes:
Given, known welfare-relevant attributes a
A known relationship n = n(a)
Equivalised income given by x = y / n
n is the "exchange-rate" between income types x, y
Set aside the assumption that we have a single n(•).
Get a general result on joint distribution of (y, a)
To do this need to recall results on ranking criteria

76. Social-welfare criteria
Recall the standard classes of SWF
Additive SWFs
W : W(F) = ò u(x) dF(x)
With principal subclasses
W1 Ì W : u(•) increasing
W2 Ì W1: u(•) increasing and concave
Recall the second-order result
G cumulant-dominates F Û
W(G) > W(F) for all WÎW2
Make progress by further restricting subclasses

77. Alternative approach to needs
Sort individuals into needs groups N1, N2 ,…
Suppose a proportion pj are in group Nj .
Then social welfare can be written:
To make this operational…
The utility people get from income depends on their needs:

78. A needs-related class of SWFs
Suppose we want j=1,2,… to reflect decreasing order of need.
Consider need and the marginal utility of income:
“Need” reflected in high MU of income?
If need falls with j then the above should be positive.
Let W3 Ì W2 be the subclass of welfare functions for which the above is positive and decreasing in y

79. Atkinson-Bourguignon result
Let F( j) denote distribution for all needs groups up to and including j.
Distinguish this from the marginal distribution
Theorem:
A UK example

80. Household types in Economic Trends
2+ads,3+chn/3+ads,chn
2 adults with 2 children
1 adult with children
2 adults with 1 child
2+ adults 0 children
1 adult, 0 children

81. Impact of Taxes and Benefits. UK 1991. Sequential GLCs (1)

82. Impact of Taxes and Benefits. UK 1991. Sequential GLCs (2)

83. Needs: summary Doing without equivalence scales seems attractive
Removes a level of arbitrariness
Simplifies computation?
But the sequential dominance principle is problematic
Demands one-dimensional needs categorisation
It is often indecisive
May get even more complicated for comparisons over time.
Can the approach be rescued?
Perhaps one is trying to do too much
May make sense to put upper and lower bounds on equivalence scales

84. Overview... What should be equalised? Equity and social welfare
Welfare comparisons
SWFs
What should be equalised?
Rankings
Welfare and needs
Compensation and responsibility

85. Responsibility (1) 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
The Responsibility “cut” of Dworkin (1981a, 1981b)
Distinguish between things that are your fault and things for which you deserve compensation
May need to revise our concept of “equality” or “equal treatment”

86. Responsibility (2)
Responsibility should affect the evaluation of redistribution
Both case for redistribution...
... and effectiveness of taxation.
Differentiate between
characteristics for which people can be held responsible
characteristics for which people should not
Assume that these characteristics are known and agreed
Follow the approach of Fleurbaey (1995a), (1995b), (1995c)

87. Basic structure Anonymity 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
The income function f maps attributes into incomes f(ai)
A distribution rule F:
Profile of attributes
Anonymity

88. Responsibility: Rules
Bossert and Fleurbaey (1996)
Equal Income for Equal Responsibility
Focus on distribution itself
Full compensation
Equal Transfers for Equal C-attributes
Focus on changes in distribution
Strict Compensation

89. 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

90. 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

91. 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

92. Conclusion
Axiomatisation of welfare can be accomplished using just a few basic principles
Ranking criteria can be used to provide broad judgments
These may be indecisive, so specific SWFs could be used
What shape should they have?
How do we specify them empirically?
The same basic framework of distributional analysis can be extended to a number of related problems:
Move on to consider inequality and poverty…
…in the next lecture component.