Transcript No Slide Title

Frank Cowell: HMRC-HMT Economics of Taxation
14 December 2011
HMRC-HMT Economics of
Taxation 2011
http://darp.lse.ac.uk/HMRC-HMT
9.1 Distributional Analysis and Methods
Frank Cowell: HMRC-HMT Economics of Taxation
Distributional Analysis
and Methods
Overview...
The basics
How to represent
problems in
distributional
analysis
SWF and
rankings
Inequality
measures
Evidence
2
Frank Cowell: HMRC-HMT Economics of Taxation
Distributional analysis

Covers a broad class of economic problems
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Similar techniques
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inequality
social welfare
poverty
rankings
measures
Four basic components need to be clarified

“income” concept
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“income receiving unit” concept
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usually equivalised, disposable income
but could be other income types, consumption, wealth…
usually the individual
but could also be family, household
a distribution
method of assessment or comparison
See Cowell (2000, 2008, 2011), Sen and Foster (1997)
3
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: HMRC-HMT Economics of Taxation
1: “Irene and Janet” approach
xi + xj + xk = X
particularly appropriate in
approaches to the subject
based primarily upon
individualistic welfare criteria
0
xi
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Frank Cowell: HMRC-HMT Economics of Taxation
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
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Frank Cowell: HMRC-HMT Economics of Taxation
Distributional Analysis
and Methods
Overview...
The basics
How to
incorporate
fundamental
principles
SWF and
rankings
Inequality
measures
Evidence
6
Frank Cowell: HMRC-HMT Economics of Taxation
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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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:

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: HMRC-HMT Economics of Taxation
Another fundamental question
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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) Appendix A
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Frank Cowell: HMRC-HMT Economics of Taxation
SWF axioms
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Anonymity. Suppose x′ is a permutation of x. Then:
W(x′) = W(x)
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')
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)
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Frank Cowell: HMRC-HMT Economics of Taxation
Classes of SWFs
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Anonymity and population principle:
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Introduce decomposability
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can write SWF in either Irene-Janet form or F form
may need to standardise for needs etc
get class of Additive SWFs W :
W(x) = Si u(xi)
or equivalently W(F) =  u(x) dF(x)
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
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Frank Cowell: HMRC-HMT Economics of Taxation
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


has 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)
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Frank Cowell: HMRC-HMT Economics of Taxation
Ranking and dominance
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Introduce two simple concepts
 first illustrate using the Irene-Janet representation
 take income vectors x and y for a given n
First-order dominance:
 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:
 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
Need to generalise this a little
 represent distributions in F-form (anonymity, population principle)
 q: population proportion (0 ≤ q ≤ 1)
 F(x): proportion of population with incomes ≤ x
 m(F): mean of distribution F
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Frank Cowell: HMRC-HMT Economics of Taxation
1st-Order approach

Basic tool is the quantile, expressed as
Q(F; q) := inf {x | F(x) q} = xq
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“smallest income such that cumulative frequency is at least as great as q”
Use this to derive a number of intuitive concepts
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interquartile range, decile-ratios, semi-decile ratios
graph of Q is Pen’s Parade
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Also to characterise the idea of 1st-order (quantile) dominance:
 “G quantile-dominates F” means:
Illustrate using
 for every q, Q(G;q)  Q(F;q),
Parade:
 for some q, Q(G;q) > Q(F;q)

A fundamental result:
 G quantile-dominates F iff W(G) > W(F) for all WW1
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Frank Cowell: HMRC-HMT Economics of Taxation
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
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Frank Cowell: HMRC-HMT Economics of Taxation
2nd-Order approach

Basic tool is the income cumulant, expressed as
C(F; q) := ∫ Q(F; q) x dF(x)
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“The sum of incomes in the Parade, up to and including position q”
Use this to derive a number of intuitive concepts
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the “shares” ranking, Gini coefficient
graph of C is the generalised Lorenz curve
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Also to characterise the idea of 2nd-order (cumulant) dominance:
 “G cumulant-dominates F” means:
Illustrate using
 for every q, C(G;q)  C(F;q),
GLC:
 for some q, C(G;q) > C(F;q)
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A fundamental result (Shorrocks 1983):
 G cumulant-dominates F iff W(G) > W(F) for all WW2
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 Plot cumulations against
proportion of population
C(.; q)
 GLC for distribution F
 GLC for distribution G
m(G)
m(F)
C(G; . )
C(F; . )
0
q
Intercept on vertical axis
is at mean income
cumulative income
Frank Cowell: HMRC-HMT Economics of Taxation
GLC and 2nd-order dominance
0
1
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Frank Cowell: HMRC-HMT Economics of Taxation
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 q divided by total income”
Yields Lorenz dominance, or the “shares” ranking:
 “G Lorenz-dominates F” means:
 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
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Frank Cowell: HMRC-HMT Economics of Taxation
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
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
0
0
0.2
0.4
proportion of population
0.6
q
0.8
1
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Frank Cowell: HMRC-HMT Economics of Taxation
Distributional Analysis
and Methods
Overview...
The basics
Three ways of
approaching an
index
SWF and
rankings
Inequality
measures
Evidence
19
Frank Cowell: HMRC-HMT Economics of Taxation
1: Intuitive inequality measures
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Perhaps borrow from other disciplines…
A standard measure of spread…
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But maybe better to use a normalised version
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coefficient of variation
Comparison between these two is instructive
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variance
Same iso-inequality contours for a given m.
Different behaviour as m alters
Alternative intuition based on Lorenz approach
Lorenz comparisons (2nd-order) may be indecisive
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problem is essentially one of aggregation of information
so use the diagram to “force a solution”
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Frank Cowell: HMRC-HMT Economics of Taxation
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
1
Also (equivalently) represented as
normalised difference between income pairs:

In F-form:

In Irene-Janet terms:
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Frank Cowell: HMRC-HMT Economics of Taxation
2: 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)
•F
xi
 Curvature of contour
indicates society’s
willingness to tolerate
“efficiency loss” in
pursuit of greater
equality
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Frank Cowell: HMRC-HMT Economics of Taxation
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
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additive SWF, W(F) =  u(x) dF(x)
isoelastic u
So inequality takes the form
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Frank Cowell: HMRC-HMT Economics of Taxation
3: “Distance” and inequality
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SWF route provides a coherent approach to inequality
But do we need to use an approach via social welfare?
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it’s indirect
maybe introduces unnecessary assumptions
Alternative route: “distance” and inequality
Can see inequality as a deviation from the norm
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norm in this case is perfect equality
…but what distance concept to use?
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Frank Cowell: HMRC-HMT Economics of Taxation
Generalised Entropy measures

Defines a class of inequality measures, given parameter a :
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GE class is rich. Some important special cases
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for a < 1 it is ordinally equivalent to Atkinson (a= 1 – e)
a= 0: –  log (x / m(F)) dF(x) (mean logarithmic deviation)
a= 1:  [ x / m(F)] log (x / m(F)) dF(x) (the Theil index)

or a = 2 it is ordinally equivalent to (normalised) variance.
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Parameter a can be assigned any positive or negative value
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indicates sensitivity of each member of the class
a large and positive gives a “top-sensitive” measure
a negative gives a “bottom-sensitive” measure
each agives a specific distance concept
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Frank Cowell: HMRC-HMT Economics of Taxation
Inequality contours
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Each a defines a set contours in the I-J diagram
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each related to a different concept of distance
For example
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the Euclidian case
other types
a=.25
a=
a=−.25
a=−1
a=2
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Frank Cowell: HMRC-HMT Economics of Taxation
Distributional Analysis
and Methods
Overview...
The basics
Attitudes and
perceptions
SWF and
rankings
Inequality
measures
Evidence
27
Frank Cowell: HMRC-HMT Economics of Taxation
Views on distributions
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Do people make distributional comparisons in the same way as
economists?
Summarised from Amiel-Cowell (1999)
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examine proportion of responses in conformity with standard axioms
both directly in terms of inequality and in terms of social welfare
Anonymity
Population
Decomposability
Monotonicity
Transfers
Scale indep.
Inequality
Num
Verbal
SWF
Num
Verbal
83%
58%
57%
35%
51%
66%
66%
58%
54%
47%
-
54%
53%
37%
55%
33%
-
72%
66%
40%
31%
47%
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Frank Cowell: HMRC-HMT Economics of Taxation
Inequality aversion
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Are people averse to inequality?
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evidence of both inequality and risk aversion (Carlsson et al 2005)
risk-aversion may be used as proxy for inequality aversion? (Cowell and
Gardiner 2000)
What value for e?
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affected by way the question is put? (Pirttilä and Uusitalo 2010)
evidence on risk aversion is mixed
 high values from survey evidence (Barsky et al 1997)
 much lower from savings analysis (Blundell et al 1994)
from happiness studies 1.0 to 1.5 (Layard et al 2008)
related to the extent of inequality in the country? (Lambert et al 2003)
perhaps a value of around 0.7 – 2 is reasonable
see also HM Treasury (2003) page 94
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Frank Cowell: HMRC-HMT Economics of Taxation
Conclusion
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Axiomatisation of welfare can be accomplished using
just a few basic principles
Ranking criteria can provide broad judgments
But may be indecisive, so specific SWFs could be used
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What shape should they have?
How do we specify them empirically?
Several axioms survive scrutiny in experiment

but Transfer Principle often rejected
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Frank Cowell: HMRC-HMT Economics of Taxation
References (1)
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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
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 LifeCycle 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, 87166
* Cowell, F.A. (2008) “Inequality: measurement,” The New Palgrave, second edition
* Cowell, F.A. (2011) Measuring Inequality, Oxford University Press
Cowell, F.A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research
Paper 202, Office of Fair Training, Salisbury Square, London
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Frank Cowell: HMRC-HMT Economics of Taxation
References (2)
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Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic
Journal, 30, 348-361
HM Treasury (2003) The Green Book: Appraisal and Evaluation in Central
Government (and Technical Annex), TSO, London
Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) “Inequality aversion and
the natural rate of subjective inequality,” Journal of Public Economics, 87, 10611090.
Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of
income,” Journal of Public Economics, 92, 1846-1857.
Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London
Pirttilä, J. and Uusitalo, R. (2010) “A ‘Leaky Bucket’ in the Real World:
Estimating Inequality Aversion using Survey Data,” Economica, 77, 60–76
Sen, A. K. and Foster, J. E. (1997) On Economic Inequality (Second ed.).
Oxford: Clarendon Press.
Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17
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