Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

How to represent problems in distributionalanalysis

•Income distributions•Comparisons

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

x

0.2 0.8 10

x0.8

q

"inc

ome"

(hei

ght)

proportion of the population

x0.2

Pen's parade

Now for some formalisation:

Plot income against proportion of population

Parade in ascending order of "income" / height

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

0

1

x

F(x)

x0

F(x0)

A distribution function

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

Methods and criteria of distributional analysis

•Income distributions•Comparisons

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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?

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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 RTuesday

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Comparing income distributions - 2

Distributional comparisons are more complex when more than two Distributional comparisons are more complex when more than two individuals are involved.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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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…

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

How to incorporate fundamental principles

•Axiomatic structure•Classes•Values

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Basic Axioms: AnonymityAnonymity Population principlePopulation principle MonotonicityMonotonicity Principle of TransfersPrinciple of Transfers Scale / translation InvarianceScale / translation Invariance Strong independence / DecomposabilityStrong independence / Decomposability

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

$

$

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 .

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Monotonicity

x′$

0 2 4 6 8 10 12 14

16 18 20

x$

0 2 4 6 8 10 12 14

16 18 20

W(x1+,x2,..., xn ) > W(x1,x2,..., xn )

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

x$

0 2 4 6 8 10 12 14

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Monotonicity

x′$

0 2 4 6 8 10 12 14

16 18 20

x′$

0 2 4 6 8 10 12 14

16 18 20

W(x1,x2,..., xn+) > W(x1,x2,..., xn )

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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…

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Which group seems to have the more unequal distribution?

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

$

$

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

$

$

The issue viewed as two groups

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Focus on just the affected persons

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

$

$

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

$

$

Before merger...

x

y

After merger...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

$

$

x'

y'

Decomposability / Independence

W(x) W(y) W(x') W(y')

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

Categorising important types

•Axiomatic structure•Classes•Values

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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…

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

The density function

x

f(x)

x0x1 x0

Income growth at x0

Welfare increases if WW1

A mean-preserving spread Welfare decreases if WW2

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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?

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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:

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

...rankings, orderings and practical tools

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

0 2 4 6 8 10

12

14

16

18

20

0 2 4 6 8 10

12

14

16

18

x

y

20

$

$

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..

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

0 2 4 6 8 10

12

14

16

18

20

0 2 4 6 8 10

12

14

16

18

x

y

20

$

$

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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 ParadeExtend it to characterise the idea of dominance…Extend it to characterise the idea of dominance…

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

An important relationshipThe 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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

First-order dominance

F

G

Q(.; q)

10 q

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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…

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Another important relationshipThe 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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Second order dominance

10

0

C(G; . )

C(F; . )

C(.; q)

(F)

(G)

q

cum

ulat

ive

inco

me

practical example, UK

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

£0

£5,000

£10,000

£15,000

£20,000

£25,000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Proportion of population

19932000-1

UK “Final income” – GLC

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

£0

£5,000

£10,000

£15,000

£20,000

£25,000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Proportion of population

19932000-1

“Original income” – GLC

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

Extensions of the ranking approach

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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:

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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:

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

£0

£500

£1,000

£1,500

£2,000

£2,500

£3,000

£3,500

£4,000

£4,500

0.00 0.05 0.10 0.15 0.20 0.25

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

£0

£2,000

£4,000

£6,000

£8,000

£10,000

£12,000

£14,000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Proportion of population

Types 1-4 orig Types 1-4 final

Types 1-5 orig Types 1-5 final

All types orig All types final

Impact of Taxes and Benefits. UK 1991. Sequential GLCs (2)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

Overview...Welfare comparisons

SWFs

Rankings

Welfare and needs

Compensation and responsibility

Equity and social welfare

What should be equalised?

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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”

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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)

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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

Frank Cow

ell: Frank C

owell: U

B Public Economics

UB Public Econom

ics

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.