Transcript Extensive Form

Frank Cowell: TU Lisbon – Inequality & Poverty
July 2006
Income Distribution and Welfare
Inequality and Poverty Measurement
Technical University of Lisbon
Frank Cowell
http://darp.lse.ac.uk/lisbon2006
Frank Cowell: TU Lisbon – Inequality & Poverty
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.
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
How to represent
problems in
distributional
analysis
SWFs
Rankings
Social welfare
and needs
•Income distributions
•Comparisons
Frank Cowell: TU Lisbon – Inequality & Poverty
Representing a distribution
Recall our two standard
approaches:

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: TU Lisbon – Inequality & Poverty
Pen’s parade (Pen, 1971)
 Plot income against proportion
of population
 Parade in ascending order of
"income" / height
x
x0.8
Now for some
formalisation:
x0.2
0
proportion ofqthe population
0.2
0.8
1
F(x)
1
F(x0)
x
x0
0
Frank Cowell: TU Lisbon – Inequality & Poverty
A distribution function
Frank Cowell: TU Lisbon – Inequality & Poverty
The set of distributions




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: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
Methods and
criteria of
distributional
analysis
SWFs
Rankings
Social welfare
and needs
•Income distributions
•Comparisons
Frank Cowell: TU Lisbon – Inequality & Poverty
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?
Frank Cowell: TU Lisbon – Inequality & Poverty
Some comparisons self-evident...
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Frank Cowell: TU Lisbon – Inequality & Poverty
A fundamental issue...

Can distributional orderings be modelled using the twoperson paradigm?

If so then comparing distributions in terms of inequality
will be almost trivial.

Same applies to other equity criteria

But, consider a simple example with three persons and
fixed incomes
Frank Cowell: TU Lisbon – Inequality & Poverty
The 3-Person problem:
two types of income difference
 Which do you think is “better”?
 Top Sensitivity
 Bottom Sensitivity
Monday
P
Q
High
Low
inequality
R
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High
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inequality
inequality
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Frank Cowell: TU Lisbon – Inequality & Poverty
Distributional Orderings and
Rankings
 In an ordering we unambiguously arrange
distributions
 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: TU Lisbon – Inequality & Poverty
Comparing income distributions - 2

Distributional comparisons are more complex when more
than two individuals are involved.


To make progress we need an axiomatic approach.


There are other logical bases.
Apply the approach to general ranking principles



Make precise “one distribution is better than another”
Axioms could be rooted in welfare economics


P-Q and Q-R gaps important
Lorenz comparisons
Social-welfare rankings
Also to specific indices


Welfare functions
Inequality measures
Frank Cowell: TU Lisbon – Inequality & Poverty
The Basics: Summary

Income distributions can be represented in two main ways




The F-form is characterised by Pen’s Parade
Distributions are complicated entities:


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: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
How to
incorporate
fundamental
principles
SWFs
Rankings
Social welfare
and needs
•Axiomatic structure
•Classes
•Values
Frank Cowell: TU Lisbon – Inequality & Poverty
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


Frank Cowell: TU Lisbon – Inequality & Poverty
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



Use a simple framework to list some of the basic axioms






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: TU Lisbon – Inequality & Poverty
Basic Axioms:






Anonymity
Population principle
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell: TU Lisbon – Inequality & Poverty
Basic Axioms:

Anonymity






Permute the individuals and social welfare does not change
Population principle
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell: TU Lisbon – Inequality & Poverty
Anonymity
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W(x′) = W(x)
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Frank Cowell: TU Lisbon – Inequality & Poverty
Implication of anonymity
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End state principle: xy is
equivalent to x′y .
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Frank Cowell: TU Lisbon – Inequality & Poverty
Basic Axioms:


Anonymity
Population principle





Scale up the population and social welfare comparisons remain
unchanged
Monotonicity
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell: TU Lisbon – Inequality & Poverty
Population replication
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W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y)
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Frank Cowell: TU Lisbon – Inequality & Poverty
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
Frank Cowell: TU Lisbon – Inequality & Poverty
Basic Axioms:



Anonymity
Population principle
Monotonicity




Increase anyone’s income and social welfare increases
Principle of Transfers
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell: TU Lisbon – Inequality & Poverty
Monotonicity
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W(x1+,x2,..., xn ) > W(x1,x2,..., xn )
Frank Cowell: TU Lisbon – Inequality & Poverty
Monotonicity
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W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)
Frank Cowell: TU Lisbon – Inequality & Poverty
Monotonicity
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W(x1,x2,..., xn+) > W(x1,x2,..., xn )
Frank Cowell: TU Lisbon – Inequality & Poverty
Basic Axioms:




Anonymity
Population principle
Monotonicity
Principle of Transfers



Poorer to richer transfer must lower social welfare
Scale / translation Invariance
Strong independence / Decomposability
Frank Cowell: TU Lisbon – Inequality & Poverty
Transfer principle:

The Pigou (1912) approach:



The Dalton (1920) extension



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: TU Lisbon – Inequality & Poverty
Which group seems to have the
more unequal distribution?
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Frank Cowell: TU Lisbon – Inequality & Poverty
The issue viewed as two groups
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Frank Cowell: TU Lisbon – Inequality & Poverty
Focus on just the affected persons
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Frank Cowell: TU Lisbon – Inequality & Poverty
Basic Axioms:





Anonymity
Population principle
Monotonicity
Principle of Transfers
Scale Invariance


Rescaling incomes does not affect welfare comparisons
Strong independence / Decomposability
Frank Cowell: TU Lisbon – Inequality & Poverty
Scale invariance (homotheticity)
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Frank Cowell: TU Lisbon – Inequality & Poverty
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
Frank Cowell: TU Lisbon – Inequality & Poverty
Translation invariance
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W(x)  W(y)  W(x+1)  W(y+1)
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Frank Cowell: TU Lisbon – Inequality & Poverty
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
Frank Cowell: TU Lisbon – Inequality & Poverty
Decomposability / Independence
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Frank Cowell: TU Lisbon – Inequality & Poverty
Using axioms


Why the list of axioms?
We can use some, or all, of them to characterise particular
classes of SWF


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 can be applied to other types of tool



Inequality
Poverty
Deprivation.
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
Categorising
important types
SWFs
Rankings
Social welfare
and needs
•Axiomatic structure
•Classes
•Values
Frank Cowell: TU Lisbon – Inequality & Poverty
Classes of SWFs (1)

Anonymity and population principle imply we can write
SWF in either I-J form or F form



Introduce decomposability and you get class of Additive
SWFs W :



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 2.
But W excludes some well-known welfare criteria
Frank Cowell: TU Lisbon – Inequality & Poverty
Classes of SWFs (2)





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: TU Lisbon – Inequality & Poverty
The density function
 Income growth at x0
f(x)
 Welfare increases if WW1
 A mean-preserving spread
 Welfare decreases if WW2
x
x1
x0
Frank Cowell: TU Lisbon – Inequality & Poverty
An important family

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 2:




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)
Frank Cowell: TU Lisbon – Inequality & Poverty
Another important family




Take the W2 subclass and impose translation invariance.
Get the family of SWFs where u is iso-elastic:
1 – e–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: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
…Can we deduce
how inequalityaverse “society”
is?
SWFs
Rankings
Social welfare
and needs
•Axiomatic structure
•Classes
•Values
Frank Cowell: TU Lisbon – Inequality & Poverty
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?


Second: What is the shape of u?


Justify a motive for considering positive inequality aversion
What is the value of e?
Examine survey data and other sources
Frank Cowell: TU Lisbon – Inequality & Poverty
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
Share of government in GDP
Share of transfers in GDP


US
30%
11%
Continental Europe
45%
18%
But does this reflect values?
Do people in Europe care more about inequality?
Frank Cowell: TU Lisbon – Inequality & Poverty
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
Frank Cowell: TU Lisbon – Inequality & Poverty
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
Frank Cowell: TU Lisbon – Inequality & Poverty
The shape of u: approaches

Direct estimates of inequality aversion



Direct estimates of risk aversion




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
(for later…)
Frank Cowell: TU Lisbon – Inequality & Poverty
Direct evidence on risk aversion


Barsky et al (1997) estimated relative risk-aversion from survey
evidence.
Note dependence on how well-off people are.
Frank Cowell: TU Lisbon – Inequality & Poverty
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.
Frank Cowell: TU Lisbon – Inequality & Poverty
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.
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
...general
comparison
criteria
SWFs
Rankings
Social welfare
and needs
•Welfare comparisons
•Inequality comparisons
•Practical tools
Frank Cowell: TU Lisbon – Inequality & Poverty
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)
Frank Cowell: TU Lisbon – Inequality & Poverty
First-order Dominance
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Each ordered income in y larger than that in x.
Frank Cowell: TU Lisbon – Inequality & Poverty
Second-order Dominance
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Each cumulated income sum in y larger than that in x.
Weaker than first-order dominance

Frank Cowell: TU Lisbon – Inequality & Poverty
Social-welfare criteria and
dominance



Why are these concepts useful?
Relate these dominance ideas to classes of SWF
Recall the class of additive SWFs


… 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: TU Lisbon – Inequality & Poverty
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…
Frank Cowell: TU Lisbon – Inequality & Poverty
An important relationship
The
idea of quantile (1st-order) dominance:
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
Q(.; q)
G
F
1
q
0
Frank Cowell: TU Lisbon – Inequality & Poverty
First-order dominance
Frank Cowell: TU Lisbon – Inequality & Poverty
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…
Frank Cowell: TU Lisbon – Inequality & Poverty
Another important relationship
The
idea of cumulant (2nd-order) dominance:
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)
m(G)
m(F)
C(G; . )
C(F; . )
0
q
0
1
cumulative income
Frank Cowell: TU Lisbon – Inequality & Poverty
Second order dominance
Frank Cowell: TU Lisbon – Inequality & Poverty
UK “Final income” – GLC
£25,000
£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: TU Lisbon – Inequality & Poverty
“Original income” – GLC
£25,000
£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: TU Lisbon – Inequality & Poverty
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.
Lorenz dominance equivalent to ranking by shares.
 Special case of GL-dominance normalised by means.
Where Lorenz-curves intersect unambiguous inequality
orderings are not possible.
This makes inequality measures especially interesting.
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Income Distribution
and Welfare
Welfare
comparisons
Extensions of the
ranking approach
SWFs
Rankings
Social welfare
and needs
Frank Cowell: TU Lisbon – Inequality & Poverty
Difficulties with needs


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: TU Lisbon – Inequality & Poverty
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)
 This makes distributional comparisons multidimensional
 Intrinsically very difficult (Atkinson and Bourguignon 1982)
To make progress:
 We simplify the structure of the problem
 We again use results on ranking criteria
Frank Cowell: TU Lisbon – Inequality & Poverty
Alternative approach to needs






Based on Atkinson and Bourguignon (1982, 1987)  see also Cowell
(2000)
Sort individuals be into needs groups N1, N2 ,…
Suppose a proportion pj are in group Nj .
Then social welfare can be written:
To make this operational…
Utility people get from income depends on their needs:
Frank Cowell: TU Lisbon – Inequality & Poverty
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  W2 be the subclass of welfare functions for
which the above is positive and decreasing in y
Frank Cowell: TU Lisbon – Inequality & Poverty
Main result

Let F( j) denote distribution for all needs groups up to and
including j.

Distinguish this from the marginal distribution

Theorem (Atkinson and Bourguignon 1987)

So to examine if welfare is higher in F than in G…
…we have a “sequential dominance” test.
A UK
 Check first the neediest group
 then the first two neediest groups
example



then the first three…
…etc
Frank Cowell: TU Lisbon – Inequality & Poverty
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

Frank Cowell: TU Lisbon – Inequality & Poverty
Impact of Taxes and Benefits.
UK 1991. Sequential GLCs (1)
£4,500
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: TU Lisbon – Inequality & Poverty
Impact of Taxes and Benefits.
UK 1991. Sequential GLCs (2)
£14,000
Types 1-4 orig
Types 1-4 final
£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: TU Lisbon – Inequality & Poverty
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:
For example inequality and poverty…

…in next lecture
Frank Cowell: TU Lisbon – Inequality & Poverty
References:






Alesina, A., Di Tella, R. and MacCulloch, R (2004) “Inequality and happiness:
are Europeans and Americans different?”, Journal of Public Economics, 88,
2009-2042
Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality, Cambridge
University Press
Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic
Theory, 2, 244-263
Atkinson, A. B. and Bourguignon, F. (1987) “Income distribution and differences
in needs,” in Feiwel, G. R. (ed), Arrow and the Foundations of the Theory of
Economic Policy, Macmillan, New York, chapter 12, pp 350-370
Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multidimensional distributions of economic status,” Review of Economic Studies, 49,
183-201
Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997) “Preference
parameters and behavioral heterogeneity : An Experimental Approach in the
Health and Retirement Survey,” Quarterly Journal of Economics,112, 537-579
Frank Cowell: TU Lisbon – Inequality & Poverty
References:







Blundell, R., Browning, M. and Meghir, C. (1994) “Consumer Demand
and the Life-Cycle Allocation of Household Expenditures,” Review of
Economic Studies, 61, 57-80
Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) “Are
people inequality averse or just risk averse?” Economica, 72,
Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and
Bourguignon, F. (eds) Handbook of Income Distribution, North Holland,
Amsterdam, Chapter 2, 87-166
Cowell, F. A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT
Economic Research Paper 202, Office of Fair Training, Salisbury Square,
London
Dalton, H. (1920) “Measurement of the inequality of incomes,” The
Economic Journal, 30, 348-361
Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press,
London
Pigou, A. C. (1912) Wealth and Welfare, Macmillan, London