Transcript Document
7 March 2006
Frank Cowell: EC513 Public Economics
EC513 PhD Public Economics
2005/6
http://darp.lse.ac.uk/EC513.htm
Deprivation, Complaints and Inequality
Frank Cowell: EC513 Public Economics
Overview...
Deprivation,
complaints, inequality
Introduction
Themes and
methodology
Poverty
Deprivation
Complaints
Frank Cowell: EC513 Public Economics
Purpose of lecture
We will look at recent theoretical developments
in distributional analysis
Consider some linked themes
alternative approaches to inequality
related welfare concepts
Use ideas from sociology and philosophy
Focus on the way modern methodology is
applied
Frank Cowell: EC513 Public Economics
Themes
Cross-disciplinary concepts
Income differences
Reference incomes
Formal methodology
Frank Cowell: EC513 Public Economics
Methodology
Exploit common structure
poverty
deprivation
complaints and inequality
see Cowell (2005)
Axiomatic method
minimalist approach
characterise structure
introduce ethics
Frank Cowell: EC513 Public Economics
“Structural” axioms
Take some social evaluation function F...
Continuity
Linear homogeneity
Translation invariance
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Structural axioms: illustration
x2
x*
D for n=3
An income distribution
Perfect equality
Contours of “Absolute” Gini
Continuity
Continuous approach to I = 0
Linear homogeneity
Proportionate increase in I
Translation invariance
I constant
•
1
0
•
x1
x3
Frank Cowell: EC513 Public Economics
Overview...
Deprivation,
complaints, inequality
Introduction
An alternative
approach
Poverty
Deprivation
Complaints
Frank Cowell: EC513 Public Economics
Poverty concepts
Given poverty line z
Headcount
p(x,z)/n
Poverty gap
a reference point
fundamental income difference
Foster et al (1984) poverty index
Cumulative poverty gap
Frank Cowell: EC513 Public Economics
TIP / Poverty profile
G(x,z)
Cumulative gaps versus
population proportions
Proportion of poor
TIP curve
TIP curves have
same
interpretation as
GLC (Shorrocks
1983)
i/n
0
p(x,z)/n
TIP dominance
implies
unambiguously
greater poverty
Frank Cowell: EC513 Public Economics
Poverty: Axiomatic approach
Characterise an ordinal poverty index P(x ,z)
See Ebert and Moyes (2002)
Use some of the standard axioms we introduced for
analysing social welfare
Apply them to n+1 incomes – those of the n individuals
and the poverty line
Show that
given just these axioms…
…you are bound to get a certain type of poverty measure.
Frank Cowell: EC513 Public Economics
Poverty: The key axioms
Adapt standard axioms from social welfare
Strengthen two other axioms
anonymity
independence
monotonicity
income increments reduce poverty
scale invariance
translation invariance
Also need continuity
Plus a focus axiom
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A closer look at the axioms
Let D denote the set of ordered income vectors
The focus axiom is
Scale invariance now becomes
Define the number of the poor as
Independence means:
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Ebert-Moyes (2002)
Gives two types of FGT measures
“relative” version
“absolute” version
Additivity follows from the independence axiom
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Brief conclusion
Poverty indexes can be constructed from
scratch
Use standard axioms
Exploit the poverty line as a reference point
Impose structure
Frank Cowell: EC513 Public Economics
Overview...
Deprivation,
complaints, inequality
Introduction
An economic
interpretation of
a sociological
concept
Poverty
Deprivation
Complaints
Frank Cowell: EC513 Public Economics
Individual deprivation
The Yitzhaki (1979) definition
Equivalent form
In present notation
Use the conditional mean
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Deprivation: Axiomatic approach 1
The Better-than set for i
Focus
works like the poverty concept
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Deprivation: Axiomatic approach 2
Normalisation
Additivity
works like the independence axiom
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Bossert-D’Ambrosio (2004)
This is just the Yitzhaki individual deprivation
index
There is an alternative axiomatisation
Ebert-Moyes (Economics Letters 2000)
Different structure of reference group
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Aggregate deprivation
Simple approach: just sum individual deprivation
Could consider an ethically weighted variant
Chakravarty and Chakraborty (1984)
Chakravarty and Mukherjee (1999b)
As with poverty consider relative as well as
absolute indices…
Frank Cowell: EC513 Public Economics
Aggregate deprivation (2)
An ethically weighted relative index
Chakravarty and Mukherjee (1999a)
One based on the generalised-Gini
Duclos and Grégoire (2002)
Frank Cowell: EC513 Public Economics
Overview...
Deprivation,
complaints, inequality
Introduction
Reference
groups and
distributional
judgments
Poverty
Deprivation
Complaints
•Model
•Inequality results
•Rankings and welfare
Frank Cowell: EC513 Public Economics
The Temkin approach
Larry Temkin (1986, 1993) approach to inequality
Unconventional
Not based on utilitarian welfare economics
But not a complete “outlier”
Common ground with other distributional analysis
Poverty
deprivation
Contains the following elements:
Concept of a complaint
The idea of a reference group
A method of aggregation
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What is a “complaint?”
Individual’s relationship with the income
distribution
The complaint exists independently
does not depend on how people feel
does not invoke “utility” or (dis)satisfaction
Requires a reference group
effectively a reference income
a variety of specifications
see also Devooght (2003)
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Types of reference point
BOP
AVE
The Best-Off Person
Possible ambiguity if there is more than one
By extension could consider the best-off group
The AVErage income
Obvious tie-in with conventional inequality measures
A conceptual difficulty for those above the mean?
ATBO
All Those Better Off
A “conditional” reference point
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Aggregation
The complaint is an individual phenomenon.
How to make the transition from this to society as
a whole?
Temkin makes two suggestions:
Simple sum
Just add up the complaints
Weighted sum
Introduce distributional weights
Then sum the weighted complaints
Frank Cowell: EC513 Public Economics
The BOP Complaint
Let r(x) be the first richest person you find in N.
Person r (and higher) has income xn.
For “lower” persons, natural definition of complaint:
Similar to fundamental difference for poverty:
Now we replace “p” with “r”
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BOP-Complaint: Axiomatisation
Use same structural axioms as before. Plus…
Monotonicity: income increments reduce complaint
Independence
Normalisation
Frank Cowell: EC513 Public Economics
Overview...
Deprivation,
complaints, inequality
Introduction
A new approach
to inequality
Poverty
Deprivation
Complaints
•Model
•Inequality results
•Rankings and welfare
Frank Cowell: EC513 Public Economics
Implications for inequality
Broadly two types of axioms with different roles.
Axioms on structure:
use these to determine the “shape” of the measures.
Transfer principles and properties of measures:
use these to characterise ethical nature of measures
Frank Cowell: EC513 Public Economics
A BOP-complaint class
The Cowell-Ebert (SCW 2004) result
Similarity of form to FGT
Characterises a family of distributions …
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The transfer principle
Do BOP-complaint measures satisfy the transfer
principle?
If transfer is from richest, yes
But if transfers are amongst hoi polloi, maybe not
Cowell-Ebert (SCW 2004):
Look at some examples that satisfy this
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Inequality contours
To examine the properties of the derived indices…
…take the case n = 3
Draw contours of T–inequality
Note that both the sensitivity parameter and the weights
w are of interest…
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Inequality contours (=2)
•Now change the weights…
w1=0.5
w2=0.5
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Inequality contours (=2)
w1=0.75
w2=0.25
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Inequality contours ( = 1)
w1=0.75
w2=0.25
Frank Cowell: EC513 Public Economics
By contrast: Gini contours
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Inequality contours ( = 0)
Again change the weights…
w1=0.5
w2=0.5
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Inequality contours ( = –1)
w1=0.75
w2=0.25
Frank Cowell: EC513 Public Economics
Inequality contours ( = –1)
w1=0.5
w2=0.5
Frank Cowell: EC513 Public Economics
Special cases
“triangles”
If then inequality just becomes the range, xn–
x1 .
“Y-shapes”
If – then inequality just becomes the “uppermiddle class” complaint: xn–xn-1 .
If = 1 then inequality becomes a generalised
absolute Gini.
Hexagons
0
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0
Which is more unequal?
A
2
4
6
8
10 12 14 16 18 20 22
24 26 28 30
B
2
4
6
8
10 12 14 16 18 20 22
24 26 28 30
0
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0
Focus on one type of BOP
complaint
A
2
4
6
8
10 12 14 16 18 20 22
24 26 28 30
B
2
4
6
8
10 12 14 16 18 20 22
24 26 28 30
0
Frank Cowell: EC513 Public Economics
0
Orthodox approach
A
2
4
6
8
10 12 14 16 18 20 22
24 26 28 30
B
2
4
6
8
10 12 14 16 18 20 22
24 26 28 30
inequality
Frank Cowell: EC513 Public Economics
T – inequality
22
21
20
19
A: (2,5,9,20,30)
B: (2,6,9,19,30)
18
17
16
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Frank Cowell: EC513 Public Economics
The “sequence”
Temkin’s seminal contributions offer an intuitive approach
to considering changes in inequality.
Take a simple model of a ladder with just two rungs.
The rungs are fixed, but the numbers on them are not.
Initially everyone is on the upper rung.
Then, one by one, people are transferred to the lower rung.
Start with m = 0 on lower rung
Carry on until m = n on lower rung
What happens to inequality?
Obviously zero at the two endpoints of the sequence
But in between?
Frank Cowell: EC513 Public Economics
The “sequence” (2)
For the case of T–inequality we have
This is increasing in m if > 0
For other cases there is a degenerate sequence in the
same direction
Frank Cowell: EC513 Public Economics
Overview...
Deprivation,
complaints, inequality
Introduction
A replacement
for the Lorenz
order?
Poverty
Deprivation
Complaints
•Model
•Inequality results
•Rankings and welfare
Frank Cowell: EC513 Public Economics
Rankings
Move beyond simple inequality measures
The notion of complaint can also be used to generate a
ranking principle that can be applied quite generally.
This is rather like the use of Lorenz curves to specify a
Lorenz ordering that characterises inequality comparisons.
Also similar to poverty rankings with arbitrary poverty
lines.
Frank Cowell: EC513 Public Economics
Cumulative complaints
Define cumulative complaints
Gives the CCC
cumulative-complaint contour
Just like TIP / Poverty profile
K(x)
Use this to get a ranking
principle
i/n
r(x) / n
Frank Cowell: EC513 Public Economics
Complaint-ranking
The class of BOP-complaint indices
Define complaint ranking
Like the generalised-Lorenz result
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Social welfare again
Temkin’s complaints approach to income
distribution was to be viewed in terms of “better”
or “worse”
Not just “less” or “more” inequality.
Can incorporate the complaint-inequality index in a
welfare-economic framework:
Total income
Inequality
Linear approximation:
B’s income
Frank Cowell: EC513 Public Economics
Welfare contours (φ=1)
A’s income
B’s income
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Welfare contours (φ<1)
A’s income
B’s income
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Welfare contours (φ>1)
Meade’s
“superegalitarianism”
A’s income
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The ATBO Complaint
Again, a natural definition of complaint:
Similar to fundamental difference for deprivation:
Use this complaint in the Temkin class
Get a form similar to Chakravarty deprivation
Frank Cowell: EC513 Public Economics
Summary: complaints
“Complaints” provide a useful basis for inequality
analysis.
Intuitive links with poverty and deprivation as
well as conventional inequality.
BOP extension provides an implementable
inequality measure.
CCCs provide an implementable ranking principle
Frank Cowell: EC513 Public Economics
References (1)
Bossert, W. and C. D’Ambrosio (2004) “Reference groups and individual
deprivation”. Working Paper 2004-10, Département de sciences économiques,
Université de Montréal, C.P. 6128, succursale Centre-Ville, Montréal (Québec)
H3C 3J7, Canada.
Chakravarty, S. R. and A. B. Chakraborty (1984) “On indices of relative
deprivation,” Economics Letters, 14, 283-287
Chakravarty, S. R. and D. Mukherjee (1999a) “Measures of deprivation and
their meaning in terms of social satisfaction.” Theory and Decision 47, 89-100
Chakravarty, S. R. and D. Mukherjee (1999b) “Ranking income distributions
by deprivation orderings,” Social Indicators Research 46, 125-135..
Cowell, F. A. (2005) “Gini, Deprivation and Complaints,” Distributional
Analysis Discussion Paper, 84, STICERD, LSE, Houghton St., London,
WC2A 2AE.
Cowell, F. A. and U. Ebert (2004) “Complaints and inequality,” Social Choice
and Welfare 23, 71-89.
Devooght, K. (2003) “Measuring inequality by counting ‘complaints:’ theory
and empirics,” Economics and Philosophy, 19, 241 - 263,
Duclos, J.-Y. and P. Grégoire (2002) “Absolute and relative deprivation and
the measurement of poverty,” Review of Income and Wealth 48, 471-492.
Frank Cowell: EC513 Public Economics
References (2)
Ebert, U. and P. Moyes (2000). An axiomatic characterization of Yitzhaki’s
index of individual deprivation. Economics Letters 68, 263-270.
Ebert, U. and P. Moyes (2002) “A simple axiomatization of the Foster-GreerThorbecke poverty orderings,” Journal of Public Economic Theory 4, 455473.
Foster, J. E., Greer, J. and Thorbecke, E. (1984) “A class of decomposable
poverty measures,” Econometrica, 52, 761-776
Jenkins, S. P. and Lambert, P. J. (1997) “Three ‘I’s of poverty curves, with an
analysis of UK poverty trends,” Oxford Economic Papers, 49, 317-327.
Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17
Temkin, L. S. (1986) “Inequality.” Philosophy and Public Affairs 15, 99-121.
Temkin, L. S. (1993) Inequality. Oxford: Oxford University Press.
Yitzhaki, S. (1979) “Relative deprivation and the Gini coefficient,” Quarterly
Journal of Economics 93, 321.324.