Transcript No Slide Title

10 October 2011
Frank Cowell: EC426 Public Economics
MSc Public Economics
2011/12
http://darp.lse.ac.uk/ec426
Equity, Social Welfare and Taxation
Frank A. Cowell
Frank Cowell: EC426 Public Economics
Welfare and distributional analysis

We’ve seen welfare-economics basis for redistribution as a
policy objective
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Distributional analysis covers broad class of economic problems
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inequality
social welfare
poverty
Similar techniques
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how to assess the impact and effectiveness of such policy?
need appropriate criteria for comparing distributions of income
issues of distributional analysis (Cowell 2008, 2011)
rankings
measures
Four basic components need to be clarified

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“income” concept
“income receiving unit” concept
a distribution
method of assessment or comparison
xj
 A representation with 3 incomes
 Income distributions with given total X
 Feasible income distributions given X
 Equal income distributions
Janet's income
Frank Cowell: EC426 Public Economics
1: “Irene and Janet” approach
xi + xj + xk = X
particularly appropriate in
approaches to the subject
based primarily upon
individualistic welfare criteria
0
xi
Frank Cowell: EC426 Public Economics
2: The parade
 Plot income against proportion of
population
xF(x)
 Parade in ascending order of
"income" / height
1
 Related to familiar statistical
concept
distribution
function F(∙)
F(x0)
Pen (1971)
especially useful in cases
where it is appropriate to
adopt a parametric model
of income distribution
x0.8
x0.2
0
x
0
proportion of the population
0.2 x0
q
0.8
1
Frank Cowell: EC426 Public Economics
Overview...
Equity, Social
Welfare, Taxation
Welfare
axioms
Basic principles of
distributional
comparisons
Rankings
Equity and
social welfare
Taxation and
sacrifice
Frank Cowell: EC426 Public Economics
Social-welfare functions
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A standard approach to a method of assessment
Basic tool is a social welfare function (SWF)
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Properties will depend on economic principles
Simple example of a 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(x1, x2,…,xn )
Total income in the economy W = Si xi
Perhaps not very interesting
Consider principles on which SWF could be based
Frank Cowell: EC426 Public Economics
The approach
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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) Appendix A
Frank Cowell: EC426 Public Economics
SWF axioms

Anonymity. Suppose x′ is a permutation of x. Then:
W(x′) = W(x)
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Population principle.
W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y)

Decomposability. Suppose x' is formed by joining x with z and y'
is formed by joining y with z. Then :
W(x)  W(y)  W(x')  W(y')
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Monotonicity. W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)
Transfer principle. (Dalton 1920) Suppose xi< xj then for small :
W(x1,x2..., xi+ ,..., xj ,..., xn) > W(x1,x2,..., xi,..., xn)
Scale invariance. W(x)  W(y)  W(lx)  W(ly)
Frank Cowell: EC426 Public Economics
Classes of SWFs (1)
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Use axioms to characterise important classes of SWF
Anonymity and population principle imply we can
write SWF in either Irene-Janet 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 may need to standardise for needs etc
W(x) = Si u(xi)
or equivalently in F-form W(F) =  u(x) dF(x)
think of u(•) as social utility or social evaluation function
The class W is of great importance

But W excludes some well-known welfare criteria
Frank Cowell: EC426 Public Economics
Classes of SWFs (2)
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From W we get important subclasses
If we impose monotonicity we get
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If we further impose the transfer principle we get
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W1  W : u(•) increasing
subclass where marginal social utility always positive
W2  W1: u(•) increasing and concave
subclass where marginal social utility is positive and
decreasing
We often need to use these special subclasses
Illustrate their properties with a simple example…
Frank Cowell: EC426 Public Economics
Overview...
Equity, Social
Welfare, Taxation
Welfare
axioms
Classes of SWF
and distributional
comparisons
Rankings
Equity and
social welfare
Taxation and
sacrifice
Frank Cowell: EC426 Public Economics
Ranking and dominance


Consider problem of comparing distributions
Introduce two simple concepts
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First-order dominance:
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y(1) > x(1), y(2) > x(2), y(3) > x(3)
Each ordered income in y larger than that in x
Second-order dominance:
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first illustrate using the Irene-Janet representation
take income vectors x and y for a given n
y(1) > x(1), y(1)+y(2) > x(1)+x(2), y(1)+y(2) +…+ y(n) > x(1)+x(2) …+ x(n)
Each cumulated income sum in y larger than that in x
Weaker than first-order dominance
Need to generalise this a little
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given anonymity, population principle can represent distributions in F-form
q: population proportion (0 ≤ q ≤ 1)
F(x): proportion of population with incomes ≤ x
m(F): mean of distribution F
Frank Cowell: EC426 Public Economics
1st-Order approach

Basic tool is the quantile, expressed as
Q(F; q) := inf {x | F(x) q} = xq
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Use this to derive a number of intuitive concepts
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“smallest income such that cumulative frequency is at least as great as q”
interquartile range, decile-ratios, semi-decile ratios
graph of Q is Pen’s Parade
Also to characterise 1st-order (quantile) dominance:

“G quantile-dominates F” means:


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for every q, Q(G;q)  Q(F;q),
for some q, Q(G;q) > Q(F;q)
Illustrate using
Parade:
A fundamental result:

G quantile-dominates F iff W(G) > W(F) for all WW1
Frank Cowell: EC426 Public Economics
Parade and 1st-order dominance
 Plot quantiles against proportion
of population
Q(.; q)
 Parade for distribution F again
 Parade for distribution G
G
In this case G clearly
quantile-dominates F
But (as often happens)
what if it doesn’t?
Try second-order method
F
0
q
1
Frank Cowell: EC426 Public Economics
2nd-Order approach

Basic tool is the income cumulant, expressed as
C(F; q) := ∫ Q(F; q) x dF(x)

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Use this to derive a number of intuitive concepts
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“The sum of incomes in the Parade, up to and including position q”
the “shares” ranking, Gini coefficient
graph of C is the generalised Lorenz curve
Also to characterise the idea of 2nd-order (cumulant)
dominance:

“G cumulant-dominates F” means:


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for every q, C(G;q)  C(F;q),
for some q, C(G;q) > C(F;q)
A fundamental result (Shorrocks 1983):

G cumulant-dominates F iff W(G) > W(F) for all WW2
Frank Cowell: EC426 Public Economics
GLC and 2nd-order dominance
 Plot cumulations against
proportion of population
C(.; q)
 GLC for distribution F
 GLC for distribution G
m(G)
m(F)
cumulative income
C(G; . )
C(F; . )
0
q
0
1
Intercept on vertical axis
is at mean income
Frank Cowell: EC426 Public Economics
2nd-Order approach (continued)

A useful tool: the share of the proportion q of
distribution F is L(F;q) := C(F;q) / m(F)

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“income cumulation at divided q by total income”
Yields Lorenz dominance, or the “shares” ranking:

“G Lorenz-dominates F” means:


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for every q, L(G;q)  L(F;q),
for some q, L(G;q) > L(F;q)
Illustrate using
Lorenz curve:
Another fundamental result (Atkinson 1970):

For given m, G Lorenz-dominates F iff W(G) > W(F) for
all W W2
Frank Cowell: EC426 Public Economics
Lorenz curve and ranking
 Plot shares against proportion of
population
1
Perfect equality
Lorenz curve for distribution F
0.8
Lorenz curve for distribution G
L(.; q)
0.6
L(G;.)
0.4
L(F;.)
0.2
0
0
0.2
0.4
proportion of population
0.6
q
0.8
1
In this case G clearly
Lorenz-dominates F
So F displays more
inequality than G
But (as often happens)
what if it doesn’t?
No clear statement about
inequality (or welfare) is
possible without further
information
Frank Cowell: EC426 Public Economics
Overview...
Equity, Social
Welfare, Taxation
Welfare
axioms
Quantifying social
values
Rankings
Equity and
social welfare
Taxation and
sacrifice
Frank Cowell: EC426 Public Economics
Equity and social welfare

So far we have just general principles
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Consider “reduced form” of social welfare
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W = W(m, I)
m=m(F) is mean of distribution F
I =I(F) is inequality of distribution F
W embodies trade-off between objectives
I can be taken as an “equity” criterion

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may lead to ambiguous results
need a tighter description of social-welfare?
from same roots as SWF?
or independently determined?
Consider a “natural” definition of inequality
Frank Cowell: EC426 Public Economics
1
Gini coefficient
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Redraw Lorenz diagram
A “natural” inequality measure…?
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L(.; q)
normalised area above Lorenz curve
can express this also in I-J terms
q
0
Also (equivalently) represented as
normalised difference between income pairs:

In F-form:
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In Irene-Janet terms:
Intuition is neat, but
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inequality index is clearly arbitrary
can we find one based on welfare criteria?
1
Frank Cowell: EC426 Public Economics
SWF and inequality
 The Irene &Janet diagram
 A given distribution
 Distributions with same mean
xj
 Contours of the SWF
 Construct an equal distribution E
such that W(E) = W(F)
 Equally-Distributed Equivalent
income
Social waste from inequality
 contour: x values such that
W(x) = const
•E
O
x(F)
m(F)
 Curvature of contour
indicates society’s willingness
to tolerate “efficiency loss” in
pursuit of greater equality
•F
xi
do this for
special case
Frank Cowell: EC426 Public Economics
An important family of SWF


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


has same form as CRRA utility function
Parameter e captures society’s inequality aversion.


Similar interpretation to individual risk aversion
See Atkinson (1970)
Frank Cowell: EC426 Public Economics
Welfare-based inequality

From the concept of social waste Atkinson (1970)
suggested an inequality measure:
x(F)
I(F) = 1 – ——
m(F)

Atkinson further assumed


additive SWF, W(F) =  u(x) dF(x)
isoelastic u

So inequality takes the form

But what value of inequality aversion parameter e?
Frank Cowell: EC426 Public Economics
Values: the issues
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SWF is central to public policy making
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First: do people care about distribution?
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Practical example in HM Treasury (2003) pp 93-94
We need to focus on two questions…
Experiments suggest they do – Carlsson et al (2005)
Do social and economic factors make a difference?
Second: What is the shape of u? (Cowell-Gardiner 2000)
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Direct estimates of inequality aversion
Estimates of risk aversion as proxy for inequality aversion
Indirect estimates of risk aversion
Indirect estimates of inequality aversion from choices made by
government
Frank Cowell: EC426 Public Economics
Preferences, happiness and welfare?
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Ebert, U. and Welsch, H. (2009)
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Alesina et al (2004)
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Evaluate subjective well being as a function of personal and
environmental data using W form
Examine which inequality index seems to fit preferences best
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%
Do people in Europe care more about inequality?
Frank Cowell: EC426 Public Economics
The Alesina et al model

Ordered logit model
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“Happy” is categorical: “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
Results
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People declare lower happiness levels when inequality is high.
Strong negative effect of inequality on happiness of the European poor
and leftists
No effects of inequality on happiness of US poor and left-wingers
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: EC426 Public Economics
Inequality aversion and Elasticity
of MU
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Consistent inequality preferences?
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What value for e?
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Preference reversals (Amiel et al 2008)
from happiness studies 1.0 to 1.5 (Layard et al 2008)
related to extent of inequality in the country? (Lambert et al 2003)
affected by way the question is put? (Pirttilä and Uusitalo 2010)
Evidence on risk aversion is mixed
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
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direct survey evidence suggests estimated relative risk-aversion
3.8 to 4.3 (Barsky et al 1997)
indirect evidence (from estimated life-cycle consumption model)
suggests 0.4 to 1.4 (Blundell et al 1994)
in each case depends on how well-off people are
Frank Cowell: EC426 Public Economics
Overview...
Equity, Social
Welfare, Taxation
Welfare
axioms
Does structure of
taxation make
sense in welfare
terms?
Rankings
Equity and
social welfare
Taxation and
sacrifice
Frank Cowell: EC426 Public Economics
Another application of ranking

Tax and benefit system maps one
distribution into another
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Use ranking tools to assess the impact of
this in welfare terms
Typically this uses one or other concept of
Lorenz dominance
Linked to effective tax progression



c = y  T(y)
y: pre-tax income c: post-tax income
T is progressive if c Lorenz-dominates y
see Jakobsson (1976)
What Lorenz ranking would we expect to
apply to these 5 concepts?
original income
+ cash benefits
gross income
- direct taxes
disposable income
- indirect taxes
post-tax income
+ non-cash benefits
final income
1.0
 + cash benefits
(Equality Line)
Original Income
Gross Income
Disposable Income
After Tax Income
Final Income
0.9
  indirect taxes
0.7
 + noncash benefits
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
  direct taxes
0.8
Proportion of Income
Frank Cowell: EC426 Public Economics
Impact of Taxes and Benefits.
UK 2006/7. Lorenz Curve
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Proportion of population
0.8
0.9
1.0
Big effect from benefits side
Modest impact of taxes
Direct and indirect taxes work
in opposite directions
0.4
direct 1992
1998/9
2006/7
indirect 1992
1998/9
2006/7
0.3
0.2
0.1
10
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h
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9t
h
8t
h
7t
h
6t
h
5t
h
4t
3r
d
2n
d
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to
m
10
t
h
0
Bo
t
Frank Cowell: EC426 Public Economics
Implied tax rates in Economic and
Labour Market Review
Formerly
Economic Trends. Taxes as proportion of gross income – see Jones, (2008)
Frank Cowell: EC426 Public Economics
Tax-benefit example: implications


We might have guessed the outcome of example
In most countries:
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In many countries there is a move toward greater reliance on
indirect taxation

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income tax progressive
so are public expenditures
but indirect tax is regressive
so Lorenz-dominance is not surprising.
political convenience?
administrative simplicity?
Structure of taxation should not be haphazard
How should the y → c mapping be determined?
Frank Cowell: EC426 Public Economics
Tax Criteria
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What broad principles should tax reflect?
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The sacrifice approach
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“benefit received:” pay for benefits you get through public sector
“sacrifice:” some principle of equality applied across individuals
see Neill (2000) for a reconciliation of these two
Reflect views on equity in society.
Analyse in terms of income distribution
Compare distribution of y with distribution of y  T?
Three types of question:
1.
2.
3.
what’s a “fair” or “neutral” way to reduce incomes…?
when will a tax system induce L-dominance?
what is implication of imposing uniformity of sacrifice?
Frank Cowell: EC426 Public Economics
Janet's income
Amiel-Cowell (1999) approach
 The Irene &Janet diagram
 Initial distribution
 Possible directions for a “fair” tax
xj
 1 Scale-independent iso-inequality
 2 Translation-independent iso-inequality
 3 “Intermediate” iso-inequality
 4 No iso-inequality direction
 An iso-inequality path?
1 Proportionate reductions are “fair”
2 Absolute reductions are “fair”
3 Affine reductions are “fair”

xi
0
Irene's income
4 Amiel-Cowell showed that
individuals perceive inequality
comparisons this way.
Based on Dalton (1920) conjecture
Frank Cowell: EC426 Public Economics
Tax and the Lorenz curve

Assume a tax function T(∙) based on income


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Compare distribution of y with that of c



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will c be more equally distributed than y?
will T(∙) produce more equally distributed c than T*(∙)?
if so under what conditions?
Define residual progression of T


a person with income y, pays an amount T(y)
disposable income given by c := y  T(y)
[dc/dy][y/c] = [1  T(y)] /[1  y /T(y)]
Disposable income under T* L-dominates that under T* iff



residual progression of T is higher than that of T*…
…for all y
see Jakobsson (1976)
Frank Cowell: EC426 Public Economics
A sacrifice approach


Suppose utility function U is continuous and increasing
Definition of absolute sacrifice:



(y, T) displays at least as much (absolute) sacrifice as (y*, T*)…
…if U(y)  U(yT) ≥ U(y*)  U(y*T*)
Equal absolute sacrifice is scale invariant iff
y1 – e – 1
U(y) = ————
1 –e



…or a linear transformation of this
(Young 1987)
Apply sacrifice criterion to actual tax schedules T(∙)


Young (1990) does this for US tax system
Consider a UK application
Frank Cowell: EC426 Public Economics
Applying sacrifice approach

Assume simple form of social welfare function



Use the equal absolute sacrifice principle



y1 – e  [y  T(y)]1 – e = const
e is again inequality aversion
Differentiate and rearrange:



u(y)  u(y  T(y)) = const
If insist on scale invariance u is isoelastic


social welfare defined in terms of income
drop distinction between U and u
y – e [1  T(y)] [y  T(y)]– e = 0
log[1  T(y)] = e log ([ y  T(y) / y])
Estimate e from actual structure T(∙)




get implied inequality aversion
For income tax (1999/2000): e = 1.414
For IT and NIC (1999/2000): e = 1.214
see Cowell-Gardiner (2000)
Frank Cowell: EC426 Public Economics
Conclusion

Axiomatisation of welfare needs just a few basic principles






Basic framework of distributional analysis can be extended to
related problems:




for example inequality and poverty…
…in second term
Ranking criteria can be used to provide broad judgments
These may be indecisive, so specific SWFs could be used



anonymity
population principle
decomposability
monotonicity
principle of transfers
scale invariant?
perhaps a value for eof around 0.7 – 2
Welfare criteria can be used to provide taxation principles
Frank Cowell: EC426 Public Economics
References (1)
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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
Amiel, Y., Cowell, F.A., Davidovitz, L. and Polovin, A. (2008) “Preference reversals and
the analysis of income distributions,” Social Choice and Welfare, 30, 305-330
Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory,
2, 244-263
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
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, 375-396
Cowell, F.A. (2008) “Inequality: measurement,” The New Palgrave, second edition
* Cowell, F.A. (2011) Measuring Inequality, Oxford University Press
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Paper 202, Office of Fair Trading, Salisbury Square, London
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