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Distributional Equity, Social Welfare
Public Economics: University of Barcelona Public Economics: University of Barcelona
Frank CowellFrank Cowell
http://darp.lse.ac.uk/ubhttp://darp.lse.ac.uk/ub
June 2005 June 2005
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Onwards from welfare economics...
We’ve seen the welfare-economics basis for We’ve seen the welfare-economics basis for redistribution as a public-policy objectiveredistribution as a public-policy objective
How to assess the impact and effectiveness of such How to assess the impact and effectiveness of such policy?policy?
We need appropriate criteria for comparing distributions We need appropriate criteria for comparing distributions of income and personal welfareof income and personal welfare
This requires a treatment of issues in distributional This requires a treatment of issues in distributional analysis.analysis.
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
How to represent problems in distributionalanalysis
•Income distributions•Comparisons
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Representing a distribution
Irene and JanetIrene and Janet
The The FF-form-form
particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria
Recall our two standard approaches:
especially useful in cases where it is appropriate to adopt a parametric model of income distribution
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x
0.2 0.8 10
x0.8
q
"in
com
e" (
heig
ht)
proportion of the population
x0.2
Pen's parade
Now for some formalisation:
Now for some formalisation:
Plot income against proportion of population
Parade in ascending order of "income" / height
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0
1
x
F(x)
x0
F(x0)
A distribution function
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The set of distributions
We can imagine a typical distribution as belonging to some class F F
How should members of F be described or compared? Sets of distributions are, in principle complicated
entities We need some fundamental principles
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
Methods and criteria of distributional analysis
•Income distributions•Comparisons
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Comparing Income Distributions Consider the purpose of the comparison...Consider the purpose of the comparison...
……in this case to get a handle on the redistributive in this case to get a handle on the redistributive impact of government activity - taxes and benefits.impact of government activity - taxes and benefits.
This requires some concept of distributional “fairness” This requires some concept of distributional “fairness” or “equity”.or “equity”.
The ethical basis rests on some aspects of the last The ethical basis rests on some aspects of the last lecture…lecture…
……and the practical implementation requires an and the practical implementation requires an comparison in terms of “inequality”.comparison in terms of “inequality”.
Which is easy. Isn’t it?Which is easy. Isn’t it?
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Some comparisons self-evident...
0 1 2 3 4 5 6 7 8 9 10$
P R
0 1 2 3 4 5 6 7 8 9 10$
P R
0 1 2 3 4 5 6 7 8 9 10$
P R
0 1 2 3 4 5 6 7 8 9 10$
RP
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A fundamental issue...
Can distributional orderings be modelled using the two-Can distributional orderings be modelled using the two-person paradigm?person paradigm?
If so then comparing distributions in terms of inequality If so then comparing distributions in terms of inequality or other concepts of equity will be almost trivial.or other concepts of equity will be almost trivial.
Then the comparison of tax systems in terms of Then the comparison of tax systems in terms of distributive effect presents no problemdistributive effect presents no problem
But, consider a simple example with three persons and But, consider a simple example with three persons and fixed incomesfixed incomes
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The 3-Person problem:two types of income difference
0 1 2 3 4 5 6 7 8 9 10 11 12 13$
P Q R
Tuesday
0 1 2 3 4 5 6 7 8 9 10 11 12 13
$
P Q RMonday
Which do you think is “better”? Top Sensitivity Bottom Sensitivity
Low inequality
High inequality
Low inequality
High inequality
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Distributional Orderings and Rankings
Arcadia
Borduria
Ruritania
more welfare
less welfare
Syldavia
In an ordering we unambiguously arrange distributions
But a ranking may include distributions that cannot be ordered
{Syldavia, Arcadia, Borduria} is an ordering.
{Syldavia, Ruritania, Borduria} is also an ordering.
But the ranking {Syldavia, Arcadia, Ruritania, Borduria} is not an ordering.
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Comparing income distributions - 2
Distributional comparisons are more complex when more than Distributional comparisons are more complex when more than two individuals are involved.two individuals are involved. P-Q and Q-R gaps importantP-Q and Q-R gaps important
To make progress we need an axiomatic approach.To make progress we need an axiomatic approach. Make precise “one distribution is better than another”Make precise “one distribution is better than another”
Axioms could be rooted in welfare economicsAxioms could be rooted in welfare economics There are other logical bases.There are other logical bases.
Apply the approach to general ranking principlesApply the approach to general ranking principles Lorenz comparisonsLorenz comparisons Social-welfare rankingsSocial-welfare rankings
Also to specific indicesAlso to specific indices Welfare functionsWelfare functions Inequality measuresInequality measures
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The Basics: Summary
Income distributions can be represented in two main Income distributions can be represented in two main ways ways Irene-Janet Irene-Janet FF-form-form
The The FF-form is characterised by Pen’s Parade-form is characterised by Pen’s Parade Distributions are complicated entities: Distributions are complicated entities:
compare them using tools with appropriate properties.compare them using tools with appropriate properties.
A useful class of tools can be found from Welfare A useful class of tools can be found from Welfare Functions with suitable properties…Functions with suitable properties…
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
How to incorporate fundamental principles
•Axiomatic structure•Classes•Values
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Social-welfare functions
Basic tool is a Basic tool is a social welfare functionsocial welfare function (SWF) (SWF) Maps set of distributions into the real lineMaps set of distributions into the real line I.e. for each distribution we get one specific numberI.e. for each distribution we get one specific number In Irene-Janet notation In Irene-Janet notation W = WW = W((xx))
Properties will depend on economic principlesProperties will depend on economic principles Simple example of a SWF:Simple example of a SWF:
Total income in the economy Total income in the economy WW = = xxii
Perhaps not very interestingPerhaps not very interesting Consider principles on whichConsider principles on which SWF could be basedSWF could be based
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Another fundamental question What makes a “good” set of principles?What makes a “good” set of principles? There is no such thing as a “right” or “wrong” axiom.There is no such thing as a “right” or “wrong” axiom. However axioms could be appropriate or inappropriateHowever axioms could be appropriate or inappropriate
Need some standard of “reasonableness”Need some standard of “reasonableness” For example, how do people view income distribution For example, how do people view income distribution
comparisons?comparisons?
Use a simple framework to list some of the basic axiomsUse a simple framework to list some of the basic axioms Assume a fixed population of size n.Assume that individual Assume a fixed population of size n.Assume that individual
utility can be measured by utility can be measured by xx Income normalised by equivalence scalesIncome normalised by equivalence scales Rules out utility interdependenceRules out utility interdependence Welfare is just a function of the vector Welfare is just a function of the vector xx := ( := (xx11, , xx22,…,,…,xxnn ) )
Follow the approach of Amiel-Cowell (1999)Follow the approach of Amiel-Cowell (1999)
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability
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Basic Axioms:
AnonymityAnonymity Permute the individuals and social welfare does not changePermute the individuals and social welfare does not change
Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability
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0 1 2 3 4 5 6 7 8 9 10 11 12 13$
x
0 1 2 3 4 5 6 7 8 9 10 11 12 13$
x'
Anonymity
W(x′) = W(x)
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$
x
y
0 1 2 3 4 5 6 7 8 9 10 11 12 13$
x'
y'
Implication of anonymity
End state principle: xy is equivalent to x′y .
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle
Scale up the population and social welfare comparisons Scale up the population and social welfare comparisons remain unchangedremain unchanged
MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability
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0 1 2 3 4 5 6 7 8 9 10$
0 1 2 3 4 5 6 7 8 9 10$
Population replication
W(x) W(y) W(x,x,…,x) W(y,y,…,y)
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A change of notation?
Using the first two axiomsUsing the first two axioms AnonymityAnonymity Population principlePopulation principle
We can write welfare using We can write welfare using F F –form–form Just use information about distributionJust use information about distribution Sometimes useful for descriptive purposesSometimes useful for descriptive purposes Remaining axioms can be expressed in either formRemaining axioms can be expressed in either form
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity
Increase anyone’s income and social welfare increasesIncrease anyone’s income and social welfare increases Principle of TransfersPrinciple of Transfers Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability
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Monotonicity
x′$
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16 18 20
x
$0 2 4 6 8 10 12 1
416 18 20
W(x1+,x2,..., xn ) > W(x1,x2,..., xn )
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x$
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16 18 20
Monotonicity
W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)
x′$
0 2 4 6 8 10 12 14
16 18 20
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Monotonicity
x′$
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16 18 20
x′$
0 2 4 6 8 10 12 14
16 18 20
W(x1,x2,..., xn+) > W(x1,x2,..., xn )
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers
Poorer to richer transfer must lower social welfarePoorer to richer transfer must lower social welfare Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability
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Transfer principle:
The Pigou (1912) approach:The Pigou (1912) approach: Focused on a 2-person worldFocused on a 2-person world A transfer from poor P to rich R must lower social welfareA transfer from poor P to rich R must lower social welfare
The The Dalton (1920) extension extension Extended to an Extended to an nn-person world-person world A transfer from (any) poorer A transfer from (any) poorer ii to (any) richer to (any) richer jj must must
lower social welfarelower social welfare Although convenient, the extension is really Although convenient, the extension is really
quite strong…quite strong…
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Which group seems to have the more unequal distribution?
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0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 2 3 4 5 6 7 8 9 10 11 12 13
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$
The issue viewed as two groups
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Focus on just the affected persons
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Scale InvarianceScale Invariance
Rescaling incomes does not affect welfare comparisonsRescaling incomes does not affect welfare comparisons Strong independence / DecomposabilityStrong independence / Decomposability
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Scale invariance (homotheticity)x
y
$0 5 10 15
$0 5 10 15
W(x) W(y) W(x) W(y)
x
$0 500 1000 1500
$0 500 1000 1500
y
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Translation InvarianceTranslation Invariance
Adding a constant to all incomes does not affect welfare Adding a constant to all incomes does not affect welfare comparisonscomparisons
Strong independence / DecomposabilityStrong independence / Decomposability
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Translation invariance x
y
$0 5 10 15
$0 5 10 15
W(x) W(y) W(x1) W(y1)
x1
$5 10 15 20
$5 10 15 20
y1
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Basic Axioms:
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability
merging with an “irrelevant” income distribution does not merging with an “irrelevant” income distribution does not affect welfare comparisonsaffect welfare comparisons
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$
Before merger...
x
y
After merger...0 1 2 3 4 5 6 7 8 9 10 11 12 13
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$
x'
y'
Decomposability / Independence
W(x) W(y) W(x') W(y')
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Using axioms
Why the list of axioms?Why the list of axioms? We can use some, or all, of them to characterise We can use some, or all, of them to characterise
particular classes of SWFparticular classes of SWF More useful than picking individual functions More useful than picking individual functions WW ad hoc ad hoc
This then enables us to get fairly general results This then enables us to get fairly general results Depends on richness of the classDepends on richness of the class The more axioms we impose (perhaps) the less general the The more axioms we impose (perhaps) the less general the
resultresult
This technique will be applied to other types of toolThis technique will be applied to other types of tool InequalityInequality PovertyPoverty Deprivation.Deprivation.
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
Categorising important types
•Axiomatic structure•Classes•Values
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Classes of SWFs (1)
Anonymity and population principle imply we can write SWF in either I-J form or F form Most modern approaches use these assumptions But you may need to standardise for needs etc
Introduce decomposability and you get class of Additive SWFs W : WW((xx)= )= u(xxii) or equivalently in F-form W(F) = u(x) dF(x)
The class W is of great importance Already seen this in lecture 1. But W excludes some well-known welfare criteria
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Classes of SWFs (2)
From W we get important subclasses
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
We often need to use these special subclasses Illustrate their behaviour with a simple
example…
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The density function
x
f(x)
x0x1 x0
Income growth at x0
Welfare increases if WW1
A mean-preserving spread
Welfare decreases if WW2
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An important family
Take the Take the W2 subclass and impose subclass and impose scale invariance. scale invariance. Get the family of SWFs where Get the family of SWFs where uu is iso-elastic: is iso-elastic:
xx1 – – 1 u(xx) = ————, 1 –
Same as that in lecture 1:Same as that in lecture 1: individual utility represented by individual utility represented by xx. . also same form as CRRA utility functionalso same form as CRRA utility function
Parameter Parameter captures society’s inequality aversion. captures society’s inequality aversion. Similar interpretation to individual risk aversionSimilar interpretation to individual risk aversion See Atkinson (1970)See Atkinson (1970)
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Another important family
Take the Take the W2 subclass and impose subclass and impose translation invariancetranslation invariance. .
Get the family of SWFs where Get the family of SWFs where uu is iso-elastic: is iso-elastic: 1 – exp–xx
u(xx) = ——— Same form as CARA utility functionSame form as CARA utility function Parameter Parameter captures society’s absolute inequality captures society’s absolute inequality
aversion.aversion. Similar to individual absolute risk aversionSimilar to individual absolute risk aversion
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
…Can we deduce how inequality-averse “society” is?
•Axiomatic structure•Classes•Values
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Values: the issues
In previous lecture we saw the problem of adducing In previous lecture we saw the problem of adducing social values.social values.
Here we will focus on two questions…Here we will focus on two questions… First: do people care about distribution?First: do people care about distribution?
Justify a motive for considering positive inequality aversionJustify a motive for considering positive inequality aversion
Second: What is the shape of Second: What is the shape of uu?? What is the value of What is the value of ??
Examine survey data and other sourcesExamine survey data and other sources
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Happiness and welfare?
Alesina et al (2004)Alesina et al (2004) Use data on happiness from social surveyUse data on happiness from social survey Construct a model of the determinants of happinessConstruct a model of the determinants of happiness Use this to see if income inequality makes a differenceUse this to see if income inequality makes a difference Seems to be a difference in priorities between US and Seems to be a difference in priorities between US and
EuropeEuropeUSUS Continental EuropeContinental Europe
Share of government in GDP Share of government in GDP 30% 30% 45%45%Share of transfers in GDP Share of transfers in GDP 11% 11% 18%18%
But does this reflect values?But does this reflect values? Do people in Europe care more about inequality?Do people in Europe care more about inequality?
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The Alesina et al model
An ordered logitAn ordered logit ““Happy” is categorical; built from three (0,1) variables:Happy” is categorical; built from three (0,1) variables:
not too happynot too happy fairly happyfairly happy very happyvery happy
iindividual, ndividual, sstate, tate, ttime, ime, ggroup. roup. Macro variables include inflation, unemployment rateMacro variables include inflation, unemployment rate Micro variables include personal characteristicsMicro variables include personal characteristics are state, time dummiesare state, time dummies
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The Alesina et al. results
People tend to declare lower happiness levels when People tend to declare lower happiness levels when inequality is high. inequality is high.
Strong negative effects of inequality on happiness of the Strong negative effects of inequality on happiness of the European poor and leftists. European poor and leftists.
No effects of inequality on happiness of US poor and the No effects of inequality on happiness of US poor and the left-wingers are not affected by inequalityleft-wingers are not affected by inequality
Negative effect of inequality on happiness of US richNegative effect of inequality on happiness of US rich No differences across the American right and the No differences across the American right and the
European right. European right. No differences between the American rich and the No differences between the American rich and the
European richEuropean rich
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The shape of u: approaches
Direct estimates of inequality aversionDirect estimates of inequality aversion See Cowell-Gardiner (2000)See Cowell-Gardiner (2000) CarlssonCarlsson et al (2005) et al (2005)
Direct estimates of risk aversion Direct estimates of risk aversion Use as proxy for inequality aversionUse as proxy for inequality aversion Base this on Harsanyi arguments?Base this on Harsanyi arguments?
Indirect estimates of risk aversionIndirect estimates of risk aversion Indirect estimates of inequality aversionIndirect estimates of inequality aversion
From choices made by governmentFrom choices made by government
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Direct evidence on risk aversion
Barsky et al (1997) estimated relative risk-aversion from Barsky et al (1997) estimated relative risk-aversion from survey evidencesurvey evidence..
Note dependence on how well-off people are.Note dependence on how well-off people are.
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Indirect evidence on risk aversion Blundell et al (1994) inferred relative risk-aversion from Blundell et al (1994) inferred relative risk-aversion from
estimated parameter of savings using estimated parameter of savings using expenditure dataexpenditure data.. Use two models: second version includes variables to capture Use two models: second version includes variables to capture
anticipated income growth.anticipated income growth. Again note dependence on how well-off people are.Again note dependence on how well-off people are.
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Indirect evidence on social values
Assume constant absolute sacrificeAssume constant absolute sacrifice
Assume isoelastic social utilityAssume isoelastic social utility Then estimate Then estimate from from
Results for UK:Results for UK:
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SWFs: Summary
A small number of key axiomsA small number of key axioms Generate an important class of SWFs with useful Generate an important class of SWFs with useful
subclasses.subclasses. Need to make a decision on the form of the SWFNeed to make a decision on the form of the SWF
Decomposable?Decomposable? Scale invariant?Scale invariant? Translation invariant?Translation invariant?
If we use the isoelastic model perhaps a value of If we use the isoelastic model perhaps a value of around 1.5 – 2 is reasonable.around 1.5 – 2 is reasonable.
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
...rankings, orderings and practical tools
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Ranking and dominance
We pick up on the problem of comparing distributionsWe pick up on the problem of comparing distributions Two simple concepts based on elementary axiomsTwo simple concepts based on elementary axioms
AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Transfer principleTransfer principle
Illustrate these tools with a simple exampleIllustrate these tools with a simple example Use the Irene-Janet representation of the distributionUse the Irene-Janet representation of the distribution Fixed population (so we don’t need pop principle)Fixed population (so we don’t need pop principle)
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First-order Dominance
y[1] > x[1], y[2] > x[2], y[3] > x[3]
Each ordered income in Each ordered income in yy larger than that in larger than that in xx..
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Second-order Dominance
y[1] > x[1], y[1]+y[2] > x[1]+x[2], y[1]+y[2] +y[3] > x[1]+x[2] +x[3]
Each cumulated income sum in Each cumulated income sum in yy larger than that in larger than that in xx..Weaker than first-order dominanceWeaker than first-order dominance
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Social-welfare criteria and dominance Why are these concepts useful? First these concepts and classes of SWF Recall the class of additive SWFs
W : W(F) = u(x) dF(x)
… and its important subclasses W1 W : u(•) increasing
W2 W1: u(•) increasing and concave
Now for the special relationship. We need to move on from the example by introducing
formal tools of distributional analysis.
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1st-Order approach
The basic tool is the The basic tool is the quantilequantile. This can be . This can be expressed in general as the functionalexpressed in general as the functional
Use this to derive a number of intuitive concepts Interquartile range Decile-ratios Semi-decile ratios
The graph of The graph of QQ is Pen’s Parade is Pen’s Parade
Extend it to characterise the idea of dominance…Extend it to characterise the idea of dominance…
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An important relationship
The idea of quantile (1The idea of quantile (1stst-order) dominance:-order) dominance:
G quantile-dominates FW(G) > W(F) for all WW1
A fundamental result:A fundamental result:
To illustrate, use Pen's paradeTo illustrate, use Pen's parade
G quantile-dominates Fmeans: for every q, Q(G;q) Q(F;q), for some q, Q(G;q) > Q(F;q)
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First-order dominance
F
G
Q(.; q)
10 q
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2nd-Order approachThe basic tool is the The basic tool is the income cumulantincome cumulant. This can . This can be expressed as the functionalbe expressed as the functional
Use this to derive three intuitive concepts The (relative) Lorenz curve The shares ranking Gini coefficient
The graph of The graph of CC is the is the generalised Lorenz curvegeneralised Lorenz curve
Again use it to characterise dominance…Again use it to characterise dominance…
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Another important relationship
The idea of cumulant (2The idea of cumulant (2ndnd-order) dominance:-order) dominance:
G cumulant-dominates FW(G) > W(F) for all WW2
A fundamental result:A fundamental result:
To illustrate, draw the GLCTo illustrate, draw the GLC
G cumulant-dominates Fmeans: for every q, C (G;q) C (F;q), for some q, C (G;q) > C (F;q)
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Second order dominance
10
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C(G; . )
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C(.; q)
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practical example, UK
practical example, UK
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Application of ranking
The tax and -benefit system maps one distribution into The tax and -benefit system maps one distribution into another...another...
Use ranking tools to assess the impact of this in welfare Use ranking tools to assess the impact of this in welfare terms.terms.
Typically this uses one or other concept of Lorenz Typically this uses one or other concept of Lorenz dominance.dominance.
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Proportion of population
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UK “Final income” – GLC
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“Original income” – GLC
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Ranking Distributions: Summary
First-order (Parade) dominance is equivalent to ranking First-order (Parade) dominance is equivalent to ranking by quantiles. by quantiles. A strong result.A strong result.
Where Parades cross, second-order methods may be Where Parades cross, second-order methods may be appropriate.appropriate.
Second-order (GL)-dominance is equivalent to ranking Second-order (GL)-dominance is equivalent to ranking by cumulations. by cumulations. Another strong result.Another strong result.
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
Extensions of the ranking approach
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Difficulties with needs
Why equivalence scales?Why equivalence scales? Need a way of making welfare comparisonsNeed a way of making welfare comparisons
Should be coherentShould be coherent Take account of differing family sizeTake account of differing family size Take account of needsTake account of needs
But there are irreconcilable difficulties:But there are irreconcilable difficulties: LogicLogic Source informationSource information Estimation problemsEstimation problems
Perhaps a more general approach Perhaps a more general approach ““Needs” seems an obvious place for explicit welfare Needs” seems an obvious place for explicit welfare
analysisanalysis
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Income and needs reconsidered
Standard approach uses "equivalised income"Standard approach uses "equivalised income" The approach assumes:The approach assumes:
Given, known welfare-relevant attributes Given, known welfare-relevant attributes aa A known relationshipA known relationship((aa)) Equivalised income given by Equivalised income given by x x = = y y / / is the "exchange-rate" between income types is the "exchange-rate" between income types xx, , yy
Set aside the assumption that we have a single Set aside the assumption that we have a single ((••).). Get a general result on joint distribution of (Get a general result on joint distribution of (yy, , aa)) To do this need to recall results on ranking criteriaTo do this need to recall results on ranking criteria
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Social-welfare criteria
Recall the standard classes of SWF Additive SWFs
W : W(F) = u(x) dF(x) With principal subclasses
W1 W : u(•) increasing W2 W1: u(•) increasing and concave
Recall the second-order resultG cumulant-dominates FW(G) > W(F) for all WW2
Make progress by further restricting subclasses
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Alternative approach to needs
Sort individuals into needs groups N1, N2 ,…
Suppose a proportion j are in group Nj .
Then social welfare can be written:
To make this operational… The utility people get from income depends on their
needs:
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A needs-related class of SWFs
“Need” reflected in high MU of income? If need falls with j then the above should be positive. Let W3 W2 be the subclass of welfare functions for
which the above is positive and decreasing in y
Suppose we want Suppose we want jj=1,2,… to reflect decreasing order of =1,2,… to reflect decreasing order of need.need.
Consider need and the marginal utility of income:Consider need and the marginal utility of income:
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Atkinson-Bourguignon result
Let F( j) denote distribution for all needs groups up to and including j..
Distinguish this from the marginal distributionDistinguish this from the marginal distribution Theorem:Theorem:
A UK example
A UK example
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Household types in Economic Trends
2+ads,3+chn/3+ads,chn2+ads,3+chn/3+ads,chn 2 adults with 2 children2 adults with 2 children 1 adult with children1 adult with children 2 adults with 1 child2 adults with 1 child 2+ adults 0 children2+ adults 0 children 1 adult, 0 children1 adult, 0 children
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Proportion of population
Type 1 originalType 1 finalTypes 1,2 originalTypes 1,2 finalTypes 1-3 originalTypes 1-3 final
Impact of Taxes and Benefits. UK 1991. Sequential GLCs (1)
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Proportion of population
Types 1-4 orig Types 1-4 final
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All types orig All types final
Impact of Taxes and Benefits. UK 1991. Sequential GLCs (2)
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Needs: summary Doing without equivalence scales seems attractiveDoing without equivalence scales seems attractive
Removes a level of arbitrarinessRemoves a level of arbitrariness Simplifies computation?Simplifies computation?
But the sequential dominance principle is problematicBut the sequential dominance principle is problematic Demands one-dimensional needs categorisationDemands one-dimensional needs categorisation It is often indecisiveIt is often indecisive May get even more complicated for comparisons over time.May get even more complicated for comparisons over time.
Can the approach be rescued?Can the approach be rescued? Perhaps one is trying to do too muchPerhaps one is trying to do too much May make sense to put upper and lower bounds on May make sense to put upper and lower bounds on
equivalence scalesequivalence scales
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Overview...Welfare comparisons
SWFs
Rankings
Welfare and needs
Compensation and responsibility
Equity and social welfare
What should be equalised?
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Responsibility (1)
Standard approach to case for redistributionStandard approach to case for redistribution Use reference point of equalityUse reference point of equality How effective is tax/benefit system in moving actual How effective is tax/benefit system in moving actual
distribution toward reference point?distribution toward reference point?
Does not take account of individual responsibility Does not take account of individual responsibility The Responsibility “cut” of Dworkin (1981a, 1981b)The Responsibility “cut” of Dworkin (1981a, 1981b) Distinguish between things that are your fault and things for Distinguish between things that are your fault and things for
which you deserve compensationwhich you deserve compensation May need to revise our concept of “equality” or “equal May need to revise our concept of “equality” or “equal
treatment”treatment”
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Responsibility (2)
Responsibility should affect the evaluation of Responsibility should affect the evaluation of redistributionredistribution Both case for redistribution...Both case for redistribution... ... and effectiveness of taxation.... and effectiveness of taxation.
Differentiate between Differentiate between characteristics for which people can be held responsiblecharacteristics for which people can be held responsible characteristics for which people should notcharacteristics for which people should not
Assume that these characteristics are known and agreedAssume that these characteristics are known and agreed Follow the approach of Fleurbaey (1995a), Follow the approach of Fleurbaey (1995a), (1995b)(1995b), , (1995c)(1995c)
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Basic structure
AnonymityAnonymity
Each person Each person ii has a vector of attributes has a vector of attributes a aii:: Attributes partitioned into two classesAttributes partitioned into two classes RR-attributes: for which the individual is responsible-attributes: for which the individual is responsible CC-attributes: for which the individual may be compensated-attributes: for which the individual may be compensated
The income function The income function ff maps attributes into incomes maps attributes into incomes ff((aaii))
A distribution rule A distribution rule FF::Profile of attributes
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Responsibility: Rules Bossert and Fleurbaey (1996)Bossert and Fleurbaey (1996) EEqual qual IIncome for ncome for EEqual qual RResponsibilityesponsibility
Focus on distribution itselfFocus on distribution itself Full compensationFull compensation
EEqual qual TTransfers for ransfers for EEqual qual CC-attributes-attributes Focus on Focus on changeschanges in distribution in distribution Strict CompensationStrict Compensation
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A difficulty
Fleurbaey (1995a,b) In this special case... ...a natural redistribution mechanism
For large populations...For large populations... EIER and ETEC are incompatible except for...EIER and ETEC are incompatible except for... Additive separability:Additive separability:
Consider two compromise approaches
Consider two compromise approaches
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Compromise (1)
Insist on Full compensation (EIER) Weaken ETEC Egalitarian-equivalent mechanisms
Every agent has a post-tax income equal to the pre-tax income earned given reference compensation
characteristics plus... a uniform transfer
Reference profile
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Compromise (2)
Insist on strict compensation (ETEC) Weaken EIER Conditionally egalitarian mechanismsConditionally egalitarian mechanisms
Every agent Every agent kk is guaranteed the average income of a is guaranteed the average income of a hypothetical economyhypothetical economy In this economy all agents have characteristics equal In this economy all agents have characteristics equal
to reference profileto reference profile
Reference profile
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Conclusion Axiomatisation of welfare can be accomplished using Axiomatisation of welfare can be accomplished using
just a few basic principlesjust a few basic principles Ranking criteria can be used to provide broad Ranking criteria can be used to provide broad
judgmentsjudgments These may be indecisive, so specific SWFs could be These may be indecisive, so specific SWFs could be
usedused What shape should they have?What shape should they have? How do we specify them empirically?How do we specify them empirically?
The same basic framework of distributional analysis The same basic framework of distributional analysis can be extended to a number of related problems:can be extended to a number of related problems:
Move on to consider inequality and poverty…Move on to consider inequality and poverty… ……in the next lecture component.in the next lecture component.