Inequality: Advanced Topics

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Inequality Advanced Topics

• Inequality and Poverty Measurement
• Technical University of Lisbon
• Frank Cowell
• http//darp.lse.ac.uk/lisbon2006

July 2006
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Overview...
Inequality Advanced Topics
Introduction
Themes and methodology
Inequality responsibility
Deprivation
Complaints
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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

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Overview...
Inequality Advanced Topics
Introduction
An alternative approach
Inequality responsibility
Deprivation
Complaints
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Responsibility

• Standard approach to case for redistribution
• Use reference point of equality
• How effective is tax/benefit system in moving
  actual distribution toward reference point?
• Does not take account of individual
  responsibility
• Role of individual actions
• The responsibility cut
• Dworkin (1981a, 1981b)
• Distinguish between
• things that are your fault
• things for which you deserve compensation

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Responsibility and redistribution

• Should affect the evaluation of distributions
• Both case for redistribution...
• ... and effectiveness of taxation.
• Need to differentiate between
• characteristics for which people can be held
  responsible
• characteristics for which people should not
• Assume that these characteristics are known and
  agreed...

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Basic structure

• Each person i has a vector of attributes ai
• Attributes partitioned into two classes
• R-attributes for which the individual is
  responsible
• C-attributes for which the individual may be
  compensated
• Situation before intervention
• Determined by income function f
• f maps attributes into incomes f(ai)
• Only person is attributes involved
• Situation after intervention
• Determined by distribution rule F
• We need to compare fairness of outcomes from f
  and F.

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Distribution rule

• The rule F
• depends on whole profile of attributes
• maps the attributes into income of i.
• Assume feasibility

Profile of attributes

• Also assume that the rule F is anonymous

• But what other principles should the rule F
  satisfy?

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Responsibility Principle EIER

• Bossert and Fleurbaey (1996)
• Equal Income for Equal Responsibility
• Focus on distribution itself
• Full compensation

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Responsibility Principle ETEC

• Equal Transfers for Equal C-attributes
• Focus on changes in distribution
• Strict Compensation

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A difficulty

• For large populations...
• EIER and ETEC are incompatible except for...
• Additive separability

• Fleurbaey (1995a,b)
• In this special case...
• ...a natural redistribution mechanism

Consider two compromise approaches
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Compromise (1)

• Insist on Full compensation (EIER)
• Weaken ETEC
• Egalitarian-equivalent mechanisms

Reference profile

• Every agent has a post-tax income equal to
• the pre-tax income earned given reference
  compensation characteristics plus...
• a uniform transfer

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Compromise (2)

• Insist on strict compensation (ETEC)
• Weaken EIER
• Conditionally egalitarian mechanisms

Reference profile

• Every agent k is guaranteed the average income of
  a hypothetical economy
• In this economy all agents have characteristics
  equal to reference profile

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Application

• The responsibility approach gives a reference
  income distribution
• Exact version depends on balance of compensation
  rules
• And on income function f.
• Redefine inequality measurement
• not based on perfect equality as a norm
• use the norm income distribution from the
  responsibility approach
• Devooght (2005) bases this on Cowell (1985)
• Cowell approach based on Theils conditional
  entropy
• Instead of looking at information content in
  going from perfect equality to actual
  distribution...
• Start from the reference distribution

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Overview...
Inequality Advanced Topics
Introduction
An economic interpretation of a sociological
concept
Inequality responsibility
Deprivation
Complaints
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Themes

• Cross-disciplinary concepts
• Income differences
• Reference incomes
• Formal methodology

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Methodology

• Exploit common structure
• poverty
• deprivation
• complaints and inequality
• see Cowell (2005)
• Axiomatic method
• minimalist approach
• characterise structure
• introduce ethics

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Structural axioms

• Take some social evaluation function F...
• Continuity
• Linear homogeneity
• Translation invariance

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Common structure

• These assumptions underlie several problems
• Already seen this with poverty axiomatisation
• Ebert and Moyes (2002)
• Apply this to other issues in distributional
  analysis
• Individual deprivation
• Aggregate deprivation
• Inequality and complaints
• Need to endow each individual problem with
• Ethical assumptions
• Reference level of income

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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-DAmbrosio (2006)

• 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

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Aggregate deprivation (2)

• An ethically weighted relative index
• Chakravarty and Mukherjee (1999a)
• One based on the generalised-Gini
• Duclos and Grégoire (2002)

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Overview...
Inequality Advanced Topics
Introduction
Reference groups and distributional judgments
Inequality responsibility
Deprivation
Complaints

• Model
• Inequality results
• Rankings and welfare

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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?

• Individuals 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
• The Best-Off Person
• Possible ambiguity if there is more than one
• By extension could consider the best-off group
• AVE
• 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

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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

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Overview...
Inequality Advanced Topics
Introduction
A new approach to inequality
Inequality responsibility
Deprivation
Complaints

• Model
• Inequality results
• Rankings and welfare

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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

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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 (e2)

• Now change the weights

w10.5 w20.5
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Inequality contours (e2)
w10.75 w20.25
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Inequality contours (e 1)
w10.75 w20.25
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By contrast Gini contours
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Inequality contours (e 0)

• Again change the weights

w10.5 w20.5
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Inequality contours (e 1)
w10.75 w20.25
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Inequality contours (e 1)
w10.5 w20.5
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Special cases
triangles

• If ? ? ? then inequality just becomes the range,
  xnx1 .
• If ? ? ? then inequality just becomes the
  upper-middle class complaint xnxn-1 .
• If ? 1 then inequality becomes a generalised
  absolute Gini.

Y-shapes
Hexagons
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Which is more unequal?
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Focus on one type of BOP complaint
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Orthodox approach
B
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2
4
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10
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Te inequality
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The sequence

• Temkins 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?

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The sequence (2)

• For the case of T?inequality we have

• This is increasing in m if ? gt 0
• For other cases there is a degenerate sequence in
  the same direction

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Overview...
Inequality Advanced Topics
Introduction
A replacement for the Lorenz order?
Inequality responsibility
Deprivation
Complaints

• Model
• Inequality results
• Rankings and welfare

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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.

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Cumulative complaints

• Define cumulative complaints
• Gives the CCC
• cumulative-complaint contour
• Just like TIP / Poverty profile
• Use this to get a ranking principle

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Complaint-ranking

• The class of BOP-complaint indices
• Define complaint ranking
• Like the generalised-Lorenz result

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Social welfare again

• Temkins 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

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Welfare contours (f1)
Janets income
Irenes income
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Welfare contours (flt1)
Janets income
Irenes income
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Welfare contours (fgt1)
Janets income
Meades superegalitarianism
Irenes 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

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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

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References (1)

• Bossert, W. and C. DAmbrosio (2006) Reference
  groups and individual deprivation, Economics
  Letters, 90, 421-426
• Bossert, W. and M. Fleurbaey (1996)
  Redistribution and compensation, Social Choice
  and Welfare, 13, 343-355.
• 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. (1985) The measurement of
  distributional change an axiomatic approach.
  Review of Economic Studies, 52, 135.151.
• 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,

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References (2)

• Devooght, K. (2005) To each the same and to each
  his own. A proposal to measure responsibility-sens
  itive income inequality, Working paper,
  University of Kortrijk.
• Duclos, J.-Y. and P. Grégoire (2002) Absolute
  and relative deprivation and the measurement of
  poverty, Review of Income and Wealth 48,
  471-492.
• Dworkin, R. (1981a) What is equality? Part I
  Equality of welfare. Philosophy and Public
  Affairs, 10, 185- 246.
• Dworkin, R. (1981b) What is equality? Part I
  Equality of resources. Philosophy and Public
  Affairs, 10, 283-345.
• Dutta, B. and D. Ray (1989) A concept of
  egalitarianism under participation constraints
  Econometrica, 57, 615.635.
• Ebert, U. and P. Moyes (2000). An axiomatic
  characterization of Yitzhakis index of
  individual deprivation. Economics Letters 68,
  263-270.
• Ebert, U. and P. Moyes (2002) A simple
  axiomatization of the Foster-Greer-Thorbecke
  poverty orderings, Journal of Public Economic
  Theory 4, 455-473.
• Fleurbaey, M. (1995a) Equal opportunity or equal
  social outcome? Economics and Philosophy 11,
  25-55.

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References (3)

• Fleurbaey, M. (1995b) Equality and
  responsibility, European Economic Review, 39,
  683-689.
• Fleurbaey, M. (1995c) Three solutions to the
  compensation problem, Journal of Economic
  Theory, 65, 505-521.
• 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
  Is 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.