Transcript No Slide Title

Frank Cowell: UB Public Economics
June 2005
Distributional Equity, Social
Welfare
Public Economics: University of Barcelona
Frank Cowell
http://darp.lse.ac.uk/ub
Frank Cowell:
Onwards from welfare
economics...
UB Public 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.
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
How to represent
problems in
distributional
analysis
Rankings
Welfare and
needs
Compensation and
responsibility
•Income distributions
•Comparisons
Frank Cowell:
Representing a distribution
Recall our two standard
approaches:
UB Public Economics

Irene and Janet
particularly appropriate in
approaches to the subject
based primarily upon
individualistic welfare criteria

The F-form
especially useful in cases
where it is appropriate to
adopt a parametric model
of income distribution
Frank Cowell:
Pen's parade
 Plot income against proportion of
population
 Parade in ascending order of
"income" / height
UB Public Economics
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Now for some
formalisation:
x0.2
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qthe population
proportion of
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Frank Cowell:
A distribution function
F(x)
UB Public Economics
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F(x0)
x
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x0
Frank Cowell:
The set of distributions
UB Public Economics
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We can imagine a typical distribution as belonging to
some class F F
How should members of F be described or compared?
Sets of distributions are, in principle complicated
entities
We need some fundamental principles
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
Methods and
criteria of
distributional
analysis
Rankings
Welfare and
needs
Compensation and
responsibility
•Income distributions
•Comparisons
Frank Cowell:
Comparing Income Distributions
UB Public Economics

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”.
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Which is easy. Isn’t it?
Frank Cowell:
Some comparisons self-evident...
UB Public Economics
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Frank Cowell:
A fundamental issue...
UB Public Economics
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Can distributional orderings be modelled using the twoperson paradigm?
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If so then comparing distributions in terms of inequality
or other concepts of equity will be almost trivial.
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Then the comparison of tax systems in terms of
distributive effect presents no problem
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But, consider a simple example with three persons and
fixed incomes
Frank Cowell:
The 3-Person problem:
two types of income difference
 Which do you think is “better”?
 Top Sensitivity
 Bottom Sensitivity
UB Public Economics
Monday
P
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High
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inequality
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Frank Cowell:
Distributional Orderings and
Rankings
 In an ordering we unambiguously arrange
distributions
UB Public Economics
 But a ranking may include distributions that
cannot be ordered
more welfare
Syldavia
Ruritania
Arcadia
Borduria
less welfare
{Syldavia, Arcadia, Borduria} is an
ordering.
 {Syldavia, Ruritania, Borduria} is also
an ordering.
 But the ranking
{Syldavia, Arcadia, Ruritania, Borduria}
is not an ordering.
Frank Cowell:
Comparing income distributions - 2
UB Public Economics
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Distributional comparisons are more complex when
more than two individuals are involved.
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To make progress we need an axiomatic approach.

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There are other logical bases.
Apply the approach to general ranking principles
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Make precise “one distribution is better than another”
Axioms could be rooted in welfare economics
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P-Q and Q-R gaps important
Lorenz comparisons
Social-welfare rankings
Also to specific indices

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Welfare functions
Inequality measures
Frank Cowell:
The Basics: Summary
UB Public Economics

Income distributions can be represented in two main
ways
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The F-form is characterised by Pen’s Parade
Distributions are complicated entities:
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Irene-Janet
F-form
compare them using tools with appropriate properties.
A useful class of tools can be found from Welfare
Functions with suitable properties…
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
How to
incorporate
fundamental
principles
Rankings
Welfare and
needs
Compensation and
responsibility
•Axiomatic structure
•Classes
•Values
Frank Cowell:
Social-welfare functions
UB Public Economics

Basic tool is a social welfare function (SWF)
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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)
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Properties will depend on economic principles
Simple example of a SWF:
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Total income in the economy W = Si xi
 Perhaps not very interesting
Consider principles on which SWF could be based
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Frank Cowell:
Another fundamental question

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UB Public Economics

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

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Use a simple framework to list some of the basic axioms

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Need some standard of “reasonableness”
For example, how do people view income distribution
comparisons?
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)
Frank Cowell:
Basic Axioms:
UB Public Economics
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Anonymity
Population principle
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell:
Basic Axioms:
UB Public Economics

Anonymity

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Permute the individuals and social welfare does not
change
Population principle
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell:
Anonymity
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W(x′) = W(x)
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Frank Cowell:
Implication of anonymity
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End state principle: xy
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Frank Cowell:
Basic Axioms:
UB Public Economics

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Anonymity
Population principle

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Scale up the population and social welfare comparisons
remain unchanged
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell:
Population replication
UB Public Economics
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W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y)
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Frank Cowell:
A change of notation?
UB Public Economics
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Using the first two axioms
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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
Frank Cowell:
Basic Axioms:
UB Public Economics



Anonymity
Population principle
Monotonicity




Increase anyone’s income and social welfare increases
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell:
Monotonicity
UB Public Economics
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Frank Cowell:
Monotonicity
UB Public Economics
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Frank Cowell:
Monotonicity
UB Public Economics
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Frank Cowell:
Basic Axioms:
UB Public Economics

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Anonymity
Population principle
Monotonicity
Principle of Transfers



Poorer to richer transfer must lower social welfare
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell:
Transfer principle:
UB Public Economics

The Pigou (1912) approach:
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The Dalton (1920) extension
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Focused on a 2-person world
A transfer from poor P to rich R must lower social welfare
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…
Frank Cowell:
Which group seems to have the
more unequal distribution?
UB Public Economics
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Frank Cowell:
The issue viewed as two groups
UB Public Economics
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Frank Cowell:
Focus on just the affected persons
UB Public Economics
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Frank Cowell:
Basic Axioms:
UB Public Economics

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


Anonymity
Population principle
Monotonicity
Principle of Transfers
Scale Invariance


Rescaling incomes does not affect welfare comparisons
Strong independence / Decomposability
Frank Cowell:
Scale invariance (homotheticity)
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UB Public Economics
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W(x)  W(y)  W(lx)  W(ly)
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Frank Cowell:
Basic Axioms:
UB Public Economics





Anonymity
Population principle
Monotonicity
Principle of Transfers
Translation Invariance


Adding a constant to all incomes does not affect welfare
comparisons
Strong independence / Decomposability
Frank Cowell:
Translation invariance
x
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UB Public Economics
10
15
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Frank Cowell:
Basic Axioms:
UB Public Economics






Anonymity
Population principle
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability

merging with an “irrelevant” income distribution does not
affect welfare comparisons
Frank Cowell:
Decomposability / Independence
UB Public Economics
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Frank Cowell:
Using axioms
UB Public Economics

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Why the list of axioms?
We can use some, or all, of them to characterise
particular classes of SWF

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This then enables us to get fairly general results



More useful than picking individual functions W ad hoc
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.
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
Categorising
important types
Rankings
Welfare and
needs
Compensation and
responsibility
•Axiomatic structure
•Classes
•Values
Frank Cowell:
Classes of SWFs (1)
UB Public Economics

Anonymity and population principle imply we can
write SWF in either I-J form or F form

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Introduce decomposability and you get class of
Additive SWFs W :
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Most modern approaches use these assumptions
But you may need to standardise for needs etc
W(x)= Si u(xi)
or equivalently in F-form W(F) =  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
Frank Cowell:
Classes of SWFs (2)
UB Public Economics

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From W we get important subclasses
If we impose monotonicity we get
 W1  W : u(•) increasing
If we further impose the transfer principle we
get
 W2  W1: u(•) increasing and concave
We often need to use these special subclasses
Illustrate their behaviour with a simple
example…
Frank Cowell:
The density function
 Income growth at x0
f(x)
 Welfare increases if WW1
 A mean-preserving spread
 Welfare decreases if WW2
UB Public Economics
x
x1
x0
Frank Cowell:
An important family
UB Public Economics

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
1–e
Same as that in lecture 1:

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individual utility represented by x.
also same form as CRRA utility function
Parameter e captures society’s inequality aversion.

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Similar interpretation to individual risk aversion
See Atkinson (1970)
Frank Cowell:
Another important family
UB Public Economics

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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
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
…Can we deduce
how inequalityaverse “society”
is?
Rankings
Welfare and
needs
Compensation and
responsibility
•Axiomatic structure
•Classes
•Values
Frank Cowell:
Values: the issues
UB Public Economics

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
In previous lecture we saw the problem of adducing
social values.
Here we will focus on two questions…
First: do people care about distribution?
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Second: What is the shape of u?
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Justify a motive for considering positive inequality aversion
What is the value of e?
Examine survey data and other sources
Frank Cowell:
Happiness and welfare?
UB Public Economics

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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
Share of government in GDP
Share of transfers in GDP

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US
30%
11%
Continental Europe
45%
18%
But does this reflect values?
Do people in Europe care more about inequality?
Frank Cowell:
The Alesina et al model
UB Public Economics
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An ordered logit
“Happy” is categorical; built from three (0,1) variables:
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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
Frank Cowell:
The Alesina et al. results
UB Public Economics
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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
Frank Cowell:
The shape of u: approaches
UB Public Economics

Direct estimates of inequality aversion

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Direct estimates of risk aversion

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See Cowell-Gardiner (2000)
Carlsson et al (2005)
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
Frank Cowell:
Direct evidence on risk aversion
UB Public Economics

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Barsky et al (1997) estimated relative risk-aversion from
survey evidence.
Note dependence on how well-off people are.
Frank Cowell:
Indirect evidence on risk aversion
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
UB Public Economics

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.
Frank Cowell:
Indirect evidence on social values

Assume constant absolute sacrifice
UB Public Economics
Assume isoelastic social utility
 Then estimate e from
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Results for UK:
Frank Cowell:
SWFs: Summary
UB Public Economics
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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
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Decomposable?
Scale invariant?
Translation invariant?
If we use the isoelastic model perhaps a value of
around 1.5 – 2 is reasonable.
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
...rankings,
orderings and
practical tools
Rankings
Welfare and
needs
Compensation and
responsibility
Frank Cowell:
Ranking and dominance
UB Public Economics


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)
Frank Cowell:
First-order Dominance
UB Public Economics
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Each ordered income in y larger than that in x.
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Frank Cowell:
Second-order Dominance
UB Public Economics
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Each cumulated income sum in y larger than that in x.
Weaker than first-order dominance
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Frank Cowell:
Social-welfare criteria and
dominance
UB Public Economics

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
Why are these concepts useful?
First these concepts and classes of SWF
Recall the class of additive SWFs

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… and its important subclasses




W : W(F) =  u(x) dF(x)
W1  W : u(•) increasing
W2  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.
Frank Cowell:
st
1 -Order
approach
The
UB Public Economics
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…
Frank Cowell:
An important relationship
The
idea of quantile (1st-order) dominance:
UB Public Economics
G quantile-dominates F means:
 for every q, Q(G;q)  Q(F;q),
 for some q, Q(G;q) > Q(F;q)
 A fundamental
result:
G quantile-dominates F

W(G) > W(F) for all WW1

To illustrate, use Pen's parade
Frank Cowell:
First-order dominance
Q(.; q)
UB Public Economics
G
F
0
q
1
Frank Cowell:
nd
2 -Order
approach
The
basic tool is the income cumulant. This can
be expressed as the functional
UB Public Economics

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…
Frank Cowell:
Another important relationship
The
idea of cumulant (2nd-order) dominance:
UB Public Economics
G cumulant-dominates F means:
 for every q, C (G;q)  C (F;q),
 for some q, C (G;q) > C (F;q)
 A fundamental
result:
G cumulant-dominates F

W(G) > W(F) for all WW2

To illustrate, draw the GLC
C(.; q)
UB Public Economics
m(G)
m(F)
C(G; . )
C(F; . )
0
q
cumulative income
Frank Cowell:
Second order dominance
practical
example, UK
0
1
Frank Cowell:
Application of ranking
UB Public Economics



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.
Frank Cowell:
UK “Final income” – GLC
£25,000
UB Public Economics
£20,000
£15,000
1993
2000-1
£10,000
£5,000
£0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Proportion of population
0.7
0.8
0.9
1.0
Frank Cowell:
“Original income” – GLC
£25,000
UB Public Economics
£20,000
£15,000
1993
2000-1
£10,000
£5,000
£0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Proportion of population
0.7
0.8
0.9
1.0
Frank Cowell:
Ranking Distributions: Summary

UB Public Economics


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.
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
Extensions of the
ranking approach
Rankings
Welfare and
needs
Compensation and
responsibility
Frank Cowell:
Difficulties with needs

UB Public Economics

Why equivalence scales?
Need a way of making welfare comparisons




But there are irreconcilable difficulties:





Should be coherent
Take account of differing family size
Take account of needs
Logic
Source information
Estimation problems
Perhaps a more general approach
“Needs” seems an obvious place for explicit welfare
analysis
Frank Cowell:
Income and needs reconsidered
UB Public Economics


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
Frank Cowell:
Social-welfare criteria
UB Public Economics



Recall the standard classes of SWF
Additive SWFs
 W : W(F) =  u(x) dF(x)
With principal subclasses




W1  W : u(•) increasing
W2  W1: u(•) increasing and concave
Recall the second-order result
G cumulant-dominates F

W(G) > W(F) for all WW2
Make progress by further restricting subclasses
Frank Cowell:
Alternative approach to needs

UB Public Economics




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:
Frank Cowell:
A needs-related class of SWFs
UB Public Economics





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  W2 be the subclass of welfare functions for
which the above is positive and decreasing in y
Frank Cowell:
Atkinson-Bourguignon result
UB Public Economics



Let F( j) denote distribution for all needs groups up to
and including j.
Distinguish this from the marginal distribution
Theorem:
A UK
example
Frank Cowell:
Household types in Economic
Trends
UB Public Economics
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

Frank Cowell:
Impact of Taxes and Benefits.
UK 1991. Sequential GLCs (1)
£4,500
UB Public Economics
Type 1 original
Type 1 final
Types 1,2 original
Types 1,2 final
Types 1-3 original
Types 1-3 final
£4,000
£3,500
£3,000
£2,500
£2,000
£1,500
£1,000
£500
0.00
0.05
0.10
0.15
Proportion of population
0.20
£0
0.25
Frank Cowell:
Impact of Taxes and Benefits.
UK 1991. Sequential GLCs (2)
£14,000
Types 1-4 orig
Types 1-4 final
UB Public Economics
£12,000
Types 1-5 orig
Types 1-5 final
All types orig
All types final
£10,000
£8,000
£6,000
£4,000
£2,000
0.00
0.10
0.20
0.30
0.40
0.50
0.60
Proportion of population
0.70
0.80
0.90
£0
1.00
Frank Cowell:
Needs: summary

Doing without equivalence scales seems attractive

UB Public Economics


But the sequential dominance principle is problematic




Removes a level of arbitrariness
Simplifies computation?
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
Frank Cowell:
Overview...
Equity and
social welfare
Welfare
comparisons
UB Public Economics
SWFs
What should be
equalised?
Rankings
Welfare and
needs
Compensation and
responsibility
Frank Cowell:
Responsibility (1)
UB Public Economics

Standard approach to case for redistribution



Does not take account of individual responsibility



Use reference point of equality
How effective is tax/benefit system in moving actual
distribution toward reference point?
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”
Frank Cowell:
Responsibility (2)
UB Public Economics

Responsibility should affect the evaluation of
redistribution



Differentiate between



Both case for redistribution...
... and effectiveness of taxation.
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)
Frank Cowell:
Basic structure

Each person i has a vector of attributes ai:
UB Public Economics





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
Frank Cowell:
Responsibility: Rules

UB Public Economics

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
Frank Cowell:
A difficulty

UB Public Economics





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
Frank Cowell:
Compromise (1)
UB Public Economics

Insist on Full compensation (EIER)
Weaken ETEC
Egalitarian-equivalent mechanisms

Every agent has a post-tax income equal to




Reference profile
the pre-tax income earned given reference compensation
characteristics plus...
a uniform transfer
Frank Cowell:
Compromise (2)
UB Public Economics



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
Frank Cowell:
Conclusion

UB Public Economics


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.