Transcript Extensive Form - The Subjective Approach to Inequality

Frank Cowell: TU Lisbon – Inequality & Poverty
July 2006
Welfare Analysis of Distribution
Inequality and Poverty Measurement
Technical University of Lisbon
Frank Cowell
http://darp.lse.ac.uk/lisbon2006
Frank Cowell: TU Lisbon – Inequality & Poverty
Introduction
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From introductory lecture…
…should be able to incorporate inequality and poverty
analysis into standard welfare economics
In this lecture, focus on the underlying principles
Examine the underlying motivation for concern with
redistribution
Why is this necessary?
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Welfare Analysis of
Distribution
Foundation
Roots in basic
microeconomics
Income,
welfare, utility
The basis for
redistribution
Risk and
welfare
Frank Cowell: TU Lisbon – Inequality & Poverty
Agenda

Briefly reconsider the rôle for government
Central problem is often simply depicted
We will examine the components of this problem
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Then consider how social values fit in.
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A policy trade-off…?
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 A classic trade-off
 Social values
 An optimum?
efficiency
Frank Cowell: TU Lisbon – Inequality & Poverty
A standard approach?
Need to define
terms...
What is
“efficiency”?
What is “equity”?

equity
Frank Cowell: TU Lisbon – Inequality & Poverty
Policy options
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“Equity-efficiency trade-off” idea raises some
serious questions.
Is a trade-off actually necessary?
If so, what does it mean?
And how to make the choice from the trade-off
options?
Frank Cowell: TU Lisbon – Inequality & Poverty
Efficiency-equity trade-off 1
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Is there necessarily a trade-off?
Not if we are inside the frontier:
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Not if we can redistribute resources without
transactions cost.
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Dismiss such cases as frivolous?
Not, perhaps in the world of practical politics
But this is only possible with lump-sum transfers
Encounter informational problems
So there is some meaning to the trade-off
Frank Cowell: TU Lisbon – Inequality & Poverty
Efficiency-equity trade-off 2
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Not clear what goes on the axes
What is efficiency?
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Standard approach to efficiency gains losses:
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Welfare economics provides a criterion for the goal of efficiency.
Pareto criterion gives no guidance away from efficient point.
Based on cost-benefit analysis
Used as a criterion for applications in Public Economics such as
tax design.
What is equity?
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Raises issues of definition.
Also of the case for egalitarianism (Putterman et al. - JEL98).
Frank Cowell: TU Lisbon – Inequality & Poverty
Components of the trade-off
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Specification of the technology
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A definition of equity
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Also related concepts such as inequality
See next lecture
An analysis of the nature of the trade-off
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Enables precise definition of efficiency
Informational problems
A statement of social preferences
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What is the basis for concern with distribution?
We deal with this in the current lecture
Review the alternative welfare approaches
Frank Cowell: TU Lisbon – Inequality & Poverty
Welfare approaches 1
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The constitutional approach
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Purely ordinal
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Uses peoples’ orderings of social states
Including attitude to redistribution
Motive for redistribution in terms of social states
must come from somewhere
This is extremely demanding
Run into the Arrow problem
Hence – hopelessly indecisive?
No clear imperative for action

Difficulties of implementation
Frank Cowell: TU Lisbon – Inequality & Poverty
Welfare approaches 2
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Equity as a fundamental principle like “efficiency”
Some attempts at formalising this in economics
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E.g. Varian (1974) “no-envy” criterion
But these are usually very restrictive
Usually need to seek support on philosophical base outside
economics
But what?
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Both traditional and modern approaches
To be reviewed later
Frank Cowell: TU Lisbon – Inequality & Poverty
Welfare approaches 3
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Welfarism
Uses a cardinally measurable and interpersonally comparable
approach to welfare.
Usually based on individualism
Provides the basis for a coherent model
Need to examine the basic building blocks…
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Welfare Analysis of
Distribution
Foundation
The basic units
of analysis
Income,
welfare, utility
The basis for
redistribution
Risk and
welfare
Frank Cowell: TU Lisbon – Inequality & Poverty
Ingredients of an approach
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A model of individual resources
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A measure of individual welfare
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A basis for interpersonal comparisons
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An intellectual base for state intervention
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We will deal with the first three of these now.
Frank Cowell: TU Lisbon – Inequality & Poverty
Individual resources and
distribution
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We adopt two simple paradigms concerning
Fixed total
resources:
income
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The cake-sharing problem
The general case with production
Incorporates
incentive effects
Irene and
Janet
Often distributional analysis can be conducted
in terms of typical individuals i and j. The F-form
approach
In some cases one needs a more general
distributional notation
 Two persons
 The feasible set
 The interesting distributions
 The basic cake-sharing
income-distribution
problem
Janet’s income
Frank Cowell: TU Lisbon – Inequality & Poverty
A simple model for the
distributional problem
Income distributions
with given total
45°
0
Irene’s income
Frank Cowell: TU Lisbon – Inequality & Poverty
Limitations of this basic model
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Just 2 persons
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Fixed-size cake
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Economic growth?
Waste through distortion?
Costlessly transferable incomes
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n  3 persons for the inequality problem
The “leaky bucket” problem
(Okun 1975)
Incomes or utilities?
Essential to first-best
welfare economics
Frank Cowell: TU Lisbon – Inequality & Poverty
Example 1
Example 2
For welfare purposes we are
concerned with utility...
Comparability without measurability :
Imagine a world where access to public
services determines utility and the
following ordering is recognised:
•Gas+Electricity
•Electricity only
•Gas only
•Neither
It makes no sense to say “U(G+E)
=2U(E)”, but you could still compare
individuals.
Measurability without comparability:
Imagine a world where utility is
proportional to income, but the constant of
proportionality is known to depend on
family characteristics which may be
unobservable.
Double a family’s income and you double
each member’s utility; but you cannot
compare utilities of persons from different
families.
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What is the relationship of utility to
income?
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What properties does utility have?
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Is it measurable?
Is it comparable?
These properties are independent
We usually need both
We need a
simple model
of utility....
Frank Cowell: TU Lisbon – Inequality & Poverty
Ingredients
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a: personal attributes
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y: income
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Could be exogenous
Or you can model as a function of attributes: y=y(a)
u: individual utility
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Identity
Needs
Abilities
Special “merit” or “desert”
Several ways of modelling this…
…see below
x: “equivalised” income
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Dollar/Pound/Euro units…
Can be treated as a version of “utility”
Frank Cowell: TU Lisbon – Inequality & Poverty
Ingredients (2)
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F : distribution function
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U : utility function
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Standard tool borrowed from statistics
A variety of specifications – see below
Gives indicator of how “well-off” a person of given
attributes is
c : equivalisation function
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A simple way of accounting for differences in needs
Perhaps too simple?
We will try something different in the next lecture
Frank Cowell: TU Lisbon – Inequality & Poverty
Basic questions about income
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Is it unique?
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How comprehensive should it be?
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What is the relevant receiving unit?
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Is it comparable between persons?
Frank Cowell: TU Lisbon – Inequality & Poverty
Income: Uniqueness?
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Should we use univariate or multivariate analysis?
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A relationship between different types of “income”?
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income and expenditure?
income and wealth?
income over time?
covariance of earnings and asset income?
conditional transfers?
Several definitions may be relevant?
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gross income?
disposable income?
other concepts?
Frank Cowell: TU Lisbon – Inequality & Poverty
Income: comprehensiveness?
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Is income “full income”?
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Is income a proxy for economic welfare?
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discount for risk?
valuation over time?..
Can income be zero?
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final income +
value of leisure +...?
rental income?
... or less than zero?

business losses?
Frank Cowell: TU Lisbon – Inequality & Poverty
Income: Comparability?
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Price adjustment
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Adjustment for needs and household size
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Normalise by price indices
Usual approach is to introduce equivalence scales
The equivalence transformation is
x = c ( y, a )
Equivalised
income
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personal attributes
nominal income
Usually a simplifying assumption is made.
Write transformation as an income-independent
Number of
equivalence scale:
equivalent adults
x = y / n (a)
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Where does the function c come from?
Frank Cowell: TU Lisbon – Inequality & Poverty
Equivalence Scales
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We will assume that there is an agreed
method of determining equivalence scales.
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But there is a variety of possible sources of
information for equivalence scales:
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From official government sources
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From international bodies such as OECD
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From econometric models of household budgets.
Consider an example of the last of these:
 Plot share of food in
budget against household
income
A reference household type...
sfood
 Engel Equivalence Scale
proxy for “need”
Frank Cowell: TU Lisbon – Inequality & Poverty
A model of income and need
childless
couple
couple with
children
xr  yr
From budget
studies
x, y
0
xi
yi
income
Frank Cowell: TU Lisbon – Inequality & Poverty
Alternative models of utility
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u = U (y)
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u = U (y; a)
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Inter-personally comparable utility
Individualistic utility
May not be comparable, depending on information about a.
u = U (y, F)
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Concern for distribution as a kind of externality
Need not be benevolent concern
Evidence that people are

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Concerned about relative incomes
“upward looking” in their comparisons.
Ferrar-i-Carbonell (2005)
x = c(y ; a) = y / n(a)

A comparable money-metric utility?
Frank Cowell: TU Lisbon – Inequality & Poverty
The relationship between utility
and income:
u
Increase
concavity
u = U(y)
^
u = U(y)
y
Frank Cowell: TU Lisbon – Inequality & Poverty
A simple model

As an example take the iso-elastic form:
y1 – d – 1
U(y) = ———— , d  
1 –d
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We can think of d as risk aversion
But it may take on an additional welfare significance
Frank Cowell: TU Lisbon – Inequality & Poverty
What to do with this information?

We need a method of appraising either the
distribution of utilities…
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…or, the system by which they were
produced
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This involves fundamentally different
approaches to welfare judgments.
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Welfare Analysis of
Distribution
Foundation
Philosophies,
social welfare
and the basis for
intervention
Income,
welfare, utility
The basis for
redistribution
Risk and
welfare
Frank Cowell: TU Lisbon – Inequality & Poverty
Five intellectual bases for public
action
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…and five social philosophers
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Entitlement theories
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Unanimity
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Bentham
Concern with the least advantaged
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Pareto
Utilitarianism
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Nozick
Rawls
Egalitarianism

Plato
 Standard cake-sharing model
 N stands for “Nozick”
Janet’s income
Frank Cowell: TU Lisbon – Inequality & Poverty
A distributional outcome

N
implications for
utility
possibilities
45°
0
Irene’s income
Frank Cowell: TU Lisbon – Inequality & Poverty
Utility-possibility set
 Plot utility on the axes
 Simple cake-sharing
uj
 The effect of utility
interdependence
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N
N
Assuming that U is strictly
concave...
…and that U is the same
function for both Irene
and Janet.
45°
0
ui
Frank Cowell: TU Lisbon – Inequality & Poverty
Should we move from N?

What is the case for shifting from the status-quo
point?
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Answer differs dramatically according to social
philosophy:

Entitlement approach is concerned with process

Other approaches concerned with end-states
Frank Cowell: TU Lisbon – Inequality & Poverty
Entitlement approach
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Focus on Nozick (1974)
Answer depends crucially on how N came about
Distinguish three key issues:
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fairness in original acquisition
fair transfers
rectification of past injustice
Presumption is that there will be little or no role
for the State

“Night watchman”
Frank Cowell: TU Lisbon – Inequality & Poverty
Pareto Criterion
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Pareto unanimity criterion is an end-state principle
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Individualistic
Based on utilities
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Approve the move from N to another point…
…if at least one person gains
…and no-one loses
But utility may have a complicated relationship with
income
May depend on the income of others
See how Pareto applies in the simple example
Frank Cowell: TU Lisbon – Inequality & Poverty
Pareto improvement: simple case
 The utility-possibility
set again
 The initial point
uj
 Pareto superior points

N
No case for
intervention?
0
45°
ui
Frank Cowell: TU Lisbon – Inequality & Poverty
End-state approaches: beyond Pareto
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Pareto criterion can be indecisive
Alternative end state approaches use a social
welfare function

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What principles should this embody?

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Typically get unique solution
Individualism?
The Pareto principle?
Additivity?
Take a simple example that combines them all...
Frank Cowell: TU Lisbon – Inequality & Poverty
Benthamite approach

General principle is “Seek the greatest good of the
greatest number”

This is typically interpreted as maximising the
sum of individual welfare.

In Irene-Janet terms: u1 + u2 + ...+ un

More generally the SWF is:
WB =  u dF(u)
Frank Cowell: TU Lisbon – Inequality & Poverty
Distributional implications of
utilitarianism

Much of public economics uses utilitarianism.

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But does utilitarianism provide a basis for
egalitarian transfers?

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Efficiency criteria
Sacrifice theories in taxation
Sen has argued that this is a common fallacy
Sen and Foster (1997)
Again look at this within the simple model
Frank Cowell: TU Lisbon – Inequality & Poverty
Benthamite redistribution?
 Take a symmetric utilitypossibility set
uj
 The initial distribution
 Benthamite welfare
contour
 Maximise welfare
 Optimum in this case
 Implied tax/transfer

N

B
ui+uj = constant
0
45°
ui
The general
case?
Frank Cowell: TU Lisbon – Inequality & Poverty
The general case...
uj

N

C

 Incorporates differential
incentive effects etc.
 N. The status quo
B
 Pareto improvements
 Points that Paretodominate N
 C The voluntary solution?
 Anywhere above C
might be a candidate
 B. Benthamite solution
ui
0
Paretianism leads to
multiple solutions
Benthamite
utilitarianism leads to
a unique, possibly
different, solution.
Frank Cowell: TU Lisbon – Inequality & Poverty
General case: discussion

A motive for changing distribution?
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Nozickians might still insist that no move from N is justified
unless it came through private voluntary action
Applies even to C
Implementation:

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Private voluntary action might not be able to implement C
Could arise if there were many individuals

Case for egalitarianism?

Clearly Bentham approach does not usually imply egalitarian
outcome.
Consider two further alternative approaches:



Concern for the least advantaged (Rawls)
Egalitarianism
Frank Cowell: TU Lisbon – Inequality & Poverty
Rawls (1971)

Rawls’ distributional philosophy is based on two
fundamental principles:
1.
2.

Economic focus has usually been on 2
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each person has equal right to the most extensive scheme of
equal basic liberties compatible with a similar scheme of
liberties for all
society should so order its decisions as to secure the best
outcome for the least advantaged
Argument based on reasoning behind a “veil of ignorance”
I do not know which position in society I have when making
social judgment
Needs careful interpretation

Avoid confusion with probabilistic approach later
Frank Cowell: TU Lisbon – Inequality & Poverty
The Rawls approach…?


What is meant by the difference principle?
This is typically interpreted as maximising the welfare of
the worst-off person.

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Based on simplistic interpretation of veil of ignorance argument
Rawls interpreted it differently
But rather vaguely
In Irene-Janet terms: min {u1 , u2 , ..., un}
So the suggested SWF is:
WR = {min u: F(u)>0}
Frank Cowell: TU Lisbon – Inequality & Poverty
Egalitarianism?
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Origin goes back to Plato…
…but reinterpreted by Meade (1974).
 “Superegalitarianism”
Welfare is perceived in terms of pairwise differences:
[ui - uj]...
Welfare might not be expressible as a neat additive
expression involving individual utilities.
 Finds an echo in more recent welfare developments
 Covered in a later lecture
Frank Cowell: TU Lisbon – Inequality & Poverty
General case (2)
uj

N
 A 'Rawlsian' solution
 Superegalitarianism
Contours of max
min function


R
Maxi-min does not
imply equality
E
Superegalitaranism
implies equality
ui
0
Frank Cowell: TU Lisbon – Inequality & Poverty
Bergson-Samuelson approach

But why an additive form of the SWF?

We could just use a weaker individualistic form.

This is the basis of the Bergson-Samuelson
formulation

A generalisation

Subsumes several welfare concepts

In Irene-Janet terms: W(u1 , u2 , ..., un)

More generally the SWF is: WBS = W(F)
Frank Cowell: TU Lisbon – Inequality & Poverty
General individualistic welfare

The specific welfare functions are special cases of
Bergson-Samuelson.

Most satisfy the principle of additivity

Except for the last one (utility differences)

In Irene-Janet terms this means we can write:
u(u1) + u(u2) + ... + u(un)

More generally the SWF is:
WBSa =  u(u) dF(u)

This is clear for Bentham where u(u)= u. But…
Frank Cowell: TU Lisbon – Inequality & Poverty
General individualistic welfare (2)

…we can say more

Again take the iso-elastic form, this time of the
(social) u-function:

u 1–e – 1
u(u) = ————, e  
1–e
Bentham corresponds to the case e = .

Max-min (“Rawls”) corresponds to the case e=.

Intermediate cases (0<e<) are interesting too
Frank Cowell: TU Lisbon – Inequality & Poverty
General case (closeup)
 B. Benthamite (e = 0)
 W. Intermediate (e = 1)
 R. 'Rawlsian' ( e =)
B
 ‘E. Superegalitarianism'
(no e value)


W


E
R
Frank Cowell: TU Lisbon – Inequality & Poverty
A brief summary

Entitlement theories


Unanimity


A basis for egalitarianism?
Concern with the least advantaged


Blairism?
Utilitarianism


Thatcherism?
How to be interpreted?
(Super)-egalitarianism

Out of fashion in UK.
Frank Cowell: TU Lisbon – Inequality & Poverty
Overview...
Welfare Analysis of
Distribution
Foundation
A reinvention of
utilitarianism?
Income,
welfare, utility
The basis for
redistribution
Risk and
welfare
Frank Cowell: TU Lisbon – Inequality & Poverty
But where do the values in the SWF
come from...?

Consensus


High-minded idealism


Social and private values...?
The PLUM principle


Again the problem of the “Arrow Theorem...”
“People Like Us Matter” – a cynical approach
The Harsanyi approach

Based on individual rationality under uncertainty
take another
look...
Frank Cowell: TU Lisbon – Inequality & Poverty
High-minded idealism?


Do people care about inequality or other distributional
issues?
Multiple values argument



Externality argument



Suppose that people are “schizophrenic”
They have two sets of values, private and public.
People treat the income distribution as a “public good”
Hochman and Rodgers (AER 1969)
Motivates the formulation u = U (y, F)

Individuals care about the income distribution F
Frank Cowell: TU Lisbon – Inequality & Poverty
The PLUM principle

Interest groups may determine what the SWF is




Champernowne and Cowell (1998)
No reason to suppose that it has a direct
connection with individual utilities
However we may still be able to say something
about how values are/should be determined
For example they should at least be consistent
Frank Cowell: TU Lisbon – Inequality & Poverty
An approach based on risk analysis

Social welfare is based individual utility




Each citizen ranks social states on the basis of expected
utility
These utilities concern life prospects




Utility is of a representative person
Harsanyi (1953, 1955)
made behind a “veil of ignorance” similar to Rawls
Ignorance concerns income, wealth, social position etc
But what of personal values?
We need to reconsider and reinterpret the sum-of-utilities
approach.
Frank Cowell: TU Lisbon – Inequality & Poverty
Reinterpret sum-of-utilities

The Irene-Janet version: u1 + u2 + ...+ un

This is equivalent to:
(1/n)u1 + (1/n)u2
+ ...+ (1/n)un
 Reinterpreted as:
p1u1 + p2u2 + ...+ pnun , where pi := 1/n

Which is simply E ui
Frank Cowell: TU Lisbon – Inequality & Poverty
Reinterpret sum-of-utilities (2)

The formal utility function:  u dF(u)

This is equivalent to:  U(y) f(y)dy

Reinterpreted as: U(y(a)) p(a) da

Which is simply E U(y(a))

How do we reach this conclusion…?
Frank Cowell: TU Lisbon – Inequality & Poverty
Welfare and Risk?

Expect links between welfare and risk analysis


Argument by analogy
See Atkinson (1970) on inequality

The Harsanyi paradigm (1953, 1955)

Harsanyi’s contribution is fundamental
Consists of two strands.
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
See Amiel et al (2005)
Frank Cowell: TU Lisbon – Inequality & Poverty
Harsanyi 1
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Aggregation theorem
Consider preferences over set of lotteries L
Individuals’ preferences Vi satisfy EU axioms
i=1,…,n
Social preference V satisfies EU axioms
Assume Pareto indifference is satisfied
Then there are numbers ai and b such that, for all
pL
Frank Cowell: TU Lisbon – Inequality & Poverty
Harsanyi 1 (contd)
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Powerful result
Does not assume interpersonal utility comparisons.
If such comparisons ruled out, the ai are based on the
evaluator’s value judgments (Harsanyi 1978, p. 227)
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personal?
arbitrary?
the evaluator?
“Judges and other public officials” (1978, p. 226)
Need not be a member of the society
Must satisfy some consistency requirements
Frank Cowell: TU Lisbon – Inequality & Poverty
Harsanyi 2
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Impartial observer theorem.
Basic idea already in Vickrey (1945).
Assumes interpersonal comparisons of utility.
An impartial observer sympathetic to the interests of
each member of society makes value judgments.
The observer is to imagine himself being person i.
 i’s objective circumstances
 i’s preferences
Frank Cowell: TU Lisbon – Inequality & Poverty
Harsanyi 2 (contd)
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How to get a representative person?
Thought experiment
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Evaluator imagines he has an equal chance of being any
person in society
Equal consideration to each person’s interests.
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Impartial observer calculates average expected
utility of each lottery in L:
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I.e. person j’s expected utility
Frank Cowell: TU Lisbon – Inequality & Poverty
Implications of Harsanyi approach
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The aggregation theorem gives an argument for
additivity
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The “representative person” induces a
probabilistic approach
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Then social welfare is found to be inherited from
individual expected utility
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But on what basis do we get the probabilities
here?
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And is “expectations” an appropriate basis for
social choice?
Frank Cowell: TU Lisbon – Inequality & Poverty
Harsanyi: Some difficulties
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Are preferences known behind the “Veil of ignorance”?
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Not in the Rawls approach
But Harsanyi assumes that representative person knows others
utilities
Is it useful to suppose equal ignorance?
Subjective probabilities may be inconsistent
Should we be concerned only with expected utility?
It is not clear that individuals view risk-choices and
distributional choices in the same way
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Cowell and Schokkaert (2001).
Carlsson et al (2005)
Kroll and Davidovitz (2003)
the veil of ignorance
the cynical approach
a general view
probability
Frank Cowell: TU Lisbon – Inequality & Poverty
Identity
|
| |
1 2 3
|
|
i
n
identity
Frank Cowell: TU Lisbon – Inequality & Poverty
A difficulty with expected utility?
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Suppose the outcomes depend on uncertain events
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probabilities of Events 1,2 are (p, 1- p)
Payoffs for persons (i,j) under two policies are
Policy
a
b
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Event 2
(1,)
(,1)
Consider choice between policies a and b
Expected payoffs are:
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Event 1
(1,)
(1,)
Under a: (1,0)
Under b: (p, 1- p)
Should society be indifferent between a and b?
Mobility may be important as well as expected outcome
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See Diamond (Journal of Political Economy 1967).
Frank Cowell: TU Lisbon – Inequality & Poverty
Views on redistribution
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Source: Ravallion and Lokshin (J. Pub. E. 2000)
Clearly views on distribution depend on (i) your current
position and (ii) your expectations
Frank Cowell: TU Lisbon – Inequality & Poverty
Concluding remarks
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We can construct a model with an individualistic base for
welfare comparisons.
The alternative social philosophies may give support to
redistributive arguments,
But it raises some awkward questions...
Should the social basis for redistribution rest on private
tastes for equality or aversion to misery?
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Should it rest on individual attitudes to risk?
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What if people like seeing the poor..?
What if people are not risk-averse?
We will come back to consider the implications of these
questions
Frank Cowell: TU Lisbon – Inequality & Poverty
References 1
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Amiel, Y., Cowell, F.A. and Gaertner, W. (2005) “To Be or not To Be
Involved: A Questionnaire-Experimental View on Harsanyi’s Utilitarian
Ethics, Distributional Analysis research Programme Discussion Paper,
STICERD, LSE.
Arrow, K. J. (1951) Social Choice and Individual Values ,Wiley, New York
Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of
Economic Theory, 2, 244-263
Carlsson et al (2005) “Are people inequality averse or just risk averse?”
Economica, 72
Cowell, F. A. and Schokkaert, E. (2001), “Risk Perceptions and Distributional
Judgments”, European Economic Review, 42, 941-952.
Diamond, P.A. (1967) “Cardinal welfare, individualistic ethics and
interpersonal comparison of utility: comment,” Journal of Political Economy,
75, 765-766.
Ferrer-i-Carbonell, A. (2005) “Income and well-being: an empirical analysis of
the comparison income effect”, Journal of Public Economics, 89, 997-1019
Harsanyi, J. (1953) “Cardinal utility in welfare economics and in the theory of
risk-taking”, Journal of Political Economy, 61, 434-435
Harsanyi, J. (1955) “Cardinal welfare, individualistic ethics and interpersonal
comparison of utility,” Journal of Political Economy, 63, 309-321.
Frank Cowell: TU Lisbon – Inequality & Poverty
References 2
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Harsanyi, J. (1978) Bayesian decision theory and utilitarian ethics, American
Economic Review, 68, 223-228
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Hochman, H. and Rodgers, J.D. (1969) Pareto-optimal redistribution,
American Economic Review, 59, 542-557
Kroll, Y. and Davidovitz, L. (2003) “Inequality aversion versus risk aversion.”
Economica, 70, 19-29
Meade, J.E. (1976) The Just Economy, Allen and Unwin, London
Nozick (1974) Anarchy, State and Utopia, Basic Books, New York
Okun, A. M. (1975) Equality and Efficiency: the Big Trade-off, Brookings
Institution, Washington.
Putterman, L. and Roemer, J. and Silvestre, J. (1998) “Does egalitarianism
have a future?”, Journal of Economic Literature, 36, 861-902 .
Ravallion, M. and Lokshin, M (2000) “Who wants to redistribute? The tunnel
effect in 1990s Russia,” Journal of Public Economics, 76, 87-104.
Rawls, J. (1971) A Theory of Justice, Harvard University Press
Sen, A.K. and Foster, J. (1997) On Economic Inequality, Clarendon Press,
Oxford
Varian, H. R. (1974) “Equity, Envy and Efficiency”, Journal of Economic
Theory, 9, 3-91
Vickrey, W.S. (1945) “Measuring marginal utility by reaction to risk,”
Econometrica, 13, 319-333.
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