Welfare Analysis of Distribution

1
Welfare Analysis of Distribution

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

July 2006
2
Introduction

• 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?

3
Overview...
Welfare Analysis of Distribution
Foundation
Roots in basic microeconomics
Income, welfare, utility
The basis for redistribution
Risk and welfare
4
Agenda

• Briefly reconsider the rôle for government
• Central problem is often simply depicted
• We will examine the components of this problem
• Then consider how social values fit in.
• A policy trade-off?

5
A standard approach?

• A classic trade-off

• Social values

• An optimum?

efficiency

• Need to define terms...
• What is efficiency?
• What is equity?

equity
6
Policy options

• 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?

7
Efficiency-equity trade-off 1

• Is there necessarily a trade-off?
• Not if we are inside the frontier
• Dismiss such cases as frivolous?
• Not, perhaps in the world of practical politics
• Not if we can redistribute resources without
  transactions cost.
• But this is only possible with lump-sum transfers
• Encounter informational problems
• So there is some meaning to the trade-off

8
Efficiency-equity trade-off 2

• Not clear what goes on the axes
• What is efficiency?
• Welfare economics provides a criterion for the
  goal of efficiency.
• Pareto criterion gives no guidance away from
  efficient point.
• Standard approach to efficiency gains losses
• Based on cost-benefit analysis
• Used as a criterion for applications in Public
  Economics such as tax design.
• What is equity?
• Raises issues of definition.
• Also of the case for egalitarianism (Putterman
  et al. - JEL98).

9
Components of the trade-off

• Specification of the technology
• Enables precise definition of efficiency
• A definition of equity
• Also related concepts such as inequality
• See next lecture
• An analysis of the nature of the trade-off
• Informational problems
• A statement of social preferences
• What is the basis for concern with distribution?
• We deal with this in the current lecture
• Review the alternative welfare approaches

10
Welfare approaches 1

• The constitutional approach
• Uses peoples orderings of social states
• Including attitude to redistribution
• Motive for redistribution in terms of social
  states must come from somewhere
• Purely ordinal
• This is extremely demanding
• Run into the Arrow problem
• Hence hopelessly indecisive?
• No clear imperative for action
• Difficulties of implementation

11
Welfare approaches 2

• Equity as a fundamental principle like
  efficiency
• Some attempts at formalising this in economics
• 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?
• Both traditional and modern approaches
• To be reviewed later

12
Welfare approaches 3

• 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

13
Overview...
Welfare Analysis of Distribution
Foundation
The basic units of analysis
Income, welfare, utility
The basis for redistribution
Risk and welfare
14
Ingredients of an approach

• A model of individual resources
• A measure of individual welfare
• A basis for interpersonal comparisons
• An intellectual base for state intervention
• We will deal with the first three of these now.

15
Individual resources and distribution

• We adopt two simple paradigms concerning
  resources
• The cake-sharing problem
• The general case with production
• Often distributional analysis can be conducted in
  terms of typical individuals i and j.
• In some cases one needs a more general
  distributional notation

Fixed total income
Incorporates incentive effects
Irene and Janet
The F-form approach
16
A simple model for the distributional problem

• Two persons

• The feasible set

• The interesting distributions

• The basic cake-sharing income-distribution
  problem

ray of equality
Janets income
45
0
Irenes income
17
Limitations of this basic model

• Just 2 persons
• n ³ 3 persons for the inequality problem
• Fixed-size cake
• Economic growth?
• Waste through distortion?
• Costlessly transferable incomes
• The leaky bucket problem
• (Okun 1975)
• Incomes or utilities?

Essential to first-best welfare economics
18
For welfare purposes we are concerned with
utility...
Example 1
Example 2

• Comparability without measurability
• Imagine a world where access to public services
  determines utility and the following ordering is
  recognised
• GasElectricity
• Electricity only
• Gas only
• Neither
• It makes no sense to say U(GE) 2U(E), but you
  could still compare individuals.

• What is the relationship of utility to income?
• What properties does utility have?
• Is it measurable?
• Is it comparable?
• These properties are independent
• We usually need both

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 familys income and you
double each members utility but you cannot
compare utilities of persons from different
families.
We need a simple model of utility....
19
Ingredients

• a personal attributes
• Identity
• Needs
• Abilities
• Special merit or desert
• y income
• Could be exogenous
• Or you can model as a function of attributes
  yy(a)
• u individual utility
• Several ways of modelling this
• see below
• x equivalised income
• Dollar/Pound/Euro units
• Can be treated as a version of utility

20
Ingredients (2)

• F distribution function
• Standard tool borrowed from statistics
• U utility function
• A variety of specifications see below
• Gives indicator of how well-off a person of
  given attributes is
• c equivalisation function
• A simple way of accounting for differences in
  needs
• Perhaps too simple?
• We will try something different in the next
  lecture

21
Basic questions about income

• Is it unique?
• How comprehensive should it be?
• What is the relevant receiving unit?
• Is it comparable between persons?

22
Income Uniqueness?

• Should we use univariate or multivariate
  analysis?
• income and expenditure?
• income and wealth?
• income over time?
• A relationship between different types of
  income?
• covariance of earnings and asset income?
• conditional transfers?
• Several definitions may be relevant?
• gross income?
• disposable income?
• other concepts?

23
Income comprehensiveness?

• Is income full income?
• final income
• value of leisure ...?
• Is income a proxy for economic welfare?
• discount for risk?
• valuation over time?..
• Can income be zero?
• rental income?
• ... or less than zero?
• business losses?

24
Income Comparability?

• Price adjustment
• Normalise by price indices
• Adjustment for needs and household size
• Usual approach is to introduce equivalence scales
• The equivalence transformation is

x c ( y, a )
personal attributes
Equivalised income
nominal income

• Usually a simplifying assumption is made.
• Write transformation as an income-independent
  equivalence scale

Number of equivalent adults
x y / n (a)

• Where does the function c come from?

25
Equivalence Scales

• We will assume that there is an agreed method of
  determining equivalence scales.
• But there is a variety of possible sources of
  information for equivalence scales
• From official government sources
• From international bodies such as OECD
• From econometric models of household budgets.
• Consider an example of the last of these

26
A model of income and need

• Plot share of food in budget against household
  income

sfood

• A reference household type...

• Engel Equivalence Scale

childless couple
proxy for need
couple with children
xr º yr
From budget studies
x, y
0
yi
xi
income
27
Alternative models of utility

• u U (y)
• Inter-personally comparable utility
• u U (y a)
• Individualistic utility
• May not be comparable, depending on information
  about a.
• u U (y, F)
• Concern for distribution as a kind of externality
• Need not be benevolent concern
• Evidence that people are
• 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?

28
The relationship between utility and income
u
Increase concavity
u U(y)

u U(y)
y
29
A simple model

• As an example take the iso-elastic form
• y1 d 1
• U(y) , d ³ 0
• 1 d

• We can think of d as risk aversion
• But it may take on an additional welfare
  significance

30
What to do with this information?

• We need a method of appraising either the
  distribution of utilities
• or, the system by which they were produced
• This involves fundamentally different approaches
  to welfare judgments.

31
Overview...
Welfare Analysis of Distribution
Foundation
Philosophies, social welfare and the basis for
intervention
Income, welfare, utility
The basis for redistribution
Risk and welfare
32
Five intellectual bases for public action

• and five social philosophers
• Entitlement theories
• Nozick
• Unanimity
• Pareto
• Utilitarianism
• Bentham
• Concern with the least advantaged
• Rawls
• Egalitarianism
• Plato

33
A distributional outcome

• Standard cake-sharing model

• N stands for Nozick

ray of equality
Janets income

• N

implications for utility possibilities
45
0
Irenes income
34
Utility-possibility set

• Plot utility on the axes

• Simple cake-sharing

uj

• The effect of utility interdependence

ray of equality

• N

• N

Assuming that U is strictly concave...
and that U is the same function for both Irene
and Janet.
45
ui
0
35
Should we move from N?

• What is the case for shifting from the status-quo
  point?
• Answer differs dramatically according to social
  philosophy
• Entitlement approach is concerned with process
• Other approaches concerned with end-states

36
Entitlement approach

• Focus on Nozick (1974)
• Answer depends crucially on how N came about
• Distinguish three key issues
• 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

37
Pareto Criterion

• Pareto unanimity criterion is an end-state
  principle
• Approve the move from N to another point
• if at least one person gains
• and no-one loses
• Individualistic
• Based on utilities
• But utility may have a complicated relationship
  with income
• May depend on the income of others
• See how Pareto applies in the simple example

38
Pareto improvement simple case

• The utility-possibility set again

• The initial point

uj

• Pareto superior points

ray of equality

• N

• No case for intervention?

45
ui
0
39
End-state approaches beyond Pareto

• Pareto criterion can be indecisive
• Alternative end state approaches use a social
  welfare function
• Typically get unique solution
• What principles should this embody?
• Individualism?
• The Pareto principle?
• Additivity?
• Take a simple example that combines them all...

40
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 ò u dF(u)

41
Distributional implications of utilitarianism

• Much of public economics uses utilitarianism.
• Efficiency criteria
• Sacrifice theories in taxation
• But does utilitarianism provide a basis for
  egalitarian transfers?
• Sen has argued that this is a common fallacy
• Sen and Foster (1997)
• Again look at this within the simple model

42
Benthamite redistribution?

• Take a symmetric utility-possibility set

uj

• The initial distribution

• Benthamite welfare contour

• Maximise welfare

ray of equality

• Optimum in this case

• Implied tax/transfer

• N

• B

The general case?
45
ui
0
43
The general case...

• N

uj

• C

• Incorporates differential incentive effects etc.

• B

• N. The status quo

• Pareto improvements

• Points that Pareto-dominate N

• C The voluntary solution?

• Anywhere above C might be a candidate

• B. Benthamite solution

• Paretianism leads to multiple solutions
• Benthamite utilitarianism leads to a unique,
  possibly different, solution.

ui
0
44
General case discussion

• A motive for changing distribution?
• Nozickians might still insist that no move from N
  is justified
• unless it came through private voluntary action
• Applies even to C
• Implementation
• 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

45
Rawls (1971)

• Rawls distributional philosophy is based on two
  fundamental principles
• 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
• Economic focus has usually been on 2
• 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

46
The Rawls approach?

• What is meant by the difference principle?
• This is typically interpreted as maximising the
  welfare of the worst-off person.
• 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)gt0

47
Egalitarianism?

• 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

48
General case (2)
uj

• N

• A 'Rawlsian' solution

• Superegalitarianism

Contours of max min function

• R

ray of equality

• Maxi-min does not imply equality

• E

• Superegalitaranism implies equality

ui
0
49
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)

50
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 ò u(u) dF(u)
• This is clear for Bentham where u(u) u. But

51
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 ³ 0
• 1 e
• Bentham corresponds to the case e 0.
• Max-min (Rawls) corresponds to the case e?.
• Intermediate cases (0ltelt?) are interesting too

52
General case (closeup)

• B. Benthamite (e 0)

• W. Intermediate (e 1)

• R. 'Rawlsian' ( e ?)

• B

• E. Superegalitarianism' (no e value)

• W

• R

• E

53
A brief summary

• Entitlement theories
• Thatcherism?
• Unanimity
• Blairism?
• Utilitarianism
• A basis for egalitarianism?
• Concern with the least advantaged
• How to be interpreted?
• (Super)-egalitarianism
• Out of fashion in UK.

54
Overview...
Welfare Analysis of Distribution
Foundation
A reinvention of utilitarianism?
Income, welfare, utility
The basis for redistribution
Risk and welfare
55
But where do the values in the SWF come from...?

• Consensus
• Again the problem of the Arrow Theorem...
• High-minded idealism
• Social and private values...?
• The PLUM principle
• People Like Us Matter a cynical approach
• The Harsanyi approach
• Based on individual rationality under uncertainty

take another look...
56
High-minded idealism?

• Do people care about inequality or other
  distributional issues?
• Multiple values argument
• Suppose that people are schizophrenic
• They have two sets of values, private and public.
• Externality argument
• 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

57
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

58
An approach based on risk analysis

• Social welfare is based individual utility
• Utility is of a representative person
• Harsanyi (1953, 1955)
• Each citizen ranks social states on the basis of
  expected utility
• These utilities concern life prospects
• 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.

59
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

60
Reinterpret sum-of-utilities (2)

• The formal utility function ò u dF(u)
• This is equivalent to ò U(y) f(y)dy
• Reinterpreted as òU(y(a)) p(a) da
• Which is simply E U(y(a))
• How do we reach this conclusion?

61
Welfare and Risk?

• Expect links between welfare and risk analysis
• Argument by analogy
• See Atkinson (1970) on inequality
• The Harsanyi paradigm (1953, 1955)

• Harsanyis contribution is fundamental
• Consists of two strands.
• See Amiel et al (2005)

62
Harsanyi 1

• Aggregation theorem
• Consider preferences over set of lotteries L
• Individuals preferences Vi satisfy EU axioms
  i1,,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

63
Harsanyi 1 (contd)

• Powerful result
• Does not assume interpersonal utility
  comparisons.
• If such comparisons ruled out, the ai are based
  on the evaluators value judgments (Harsanyi
  1978, p. 227)
• 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

64
Harsanyi 2

• 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.
• is objective circumstances
• is preferences

65
Harsanyi 2 (contd)

• How to get a representative person?
• Thought experiment
• Evaluator imagines he has an equal chance of
  being any person in society
• Equal consideration to each persons interests.
• Impartial observer calculates average expected
  utility of each lottery in L

• I.e. person js expected utility

66
Implications of Harsanyi approach

• The aggregation theorem gives an argument for
  additivity
• The representative person induces a
  probabilistic approach
• Then social welfare is found to be inherited from
  individual expected utility
• But on what basis do we get the probabilities
  here?
• And is expectations an appropriate basis for
  social choice?

67
Harsanyi Some difficulties

• Are preferences known behind the Veil of
  ignorance?
• 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
• Cowell and Schokkaert (2001).
• Carlsson et al (2005)
• Kroll and Davidovitz (2003)

68
Identity
probability
the veil of ignorance
the cynical approach
a general view
1
n
2
3

identity
i
69
A difficulty with expected utility?

• Suppose the outcomes depend on uncertain events
• probabilities of Events 1,2 are (p, 1? p)
• Payoffs for persons (i,j) under two policies are

Policy Event 1 Event 2 a (1,0) (1,0) b (1,0)
(0,1)

• Consider choice between policies a and b
• Expected payoffs are
• 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
• See Diamond (Journal of Political Economy 1967).

70
Views on redistribution

• Source Ravallion and Lokshin (J. Pub. E. 2000)
• Clearly views on distribution depend on (i) your
  current position and (ii) your expectations

71
Concluding remarks

• 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?
• What if people like seeing the poor..?
• Should it rest on individual attitudes to risk?
• What if people are not risk-averse?
• We will come back to consider the implications of
  these questions

72
References 1

• Amiel, Y., Cowell, F.A. and Gaertner, W. (2005)
  To Be or not To Be Involved A
  Questionnaire-Experimental View on Harsanyis
  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.

73
References 2

• Harsanyi, J. (1978) Bayesian decision theory and
  utilitarian ethics, American Economic Review, 68,
  223-228
• 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
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• Sen, A.K. and Foster, J. (1997) On Economic
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• 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.