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**Distributional Equity, Social Welfare**

Public Economics: University of Barcelona Frank Cowell June 2005

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**Onwards from welfare economics...**

We’ve seen the welfare-economics basis for redistribution as a public-policy objective How to assess the impact and effectiveness of such policy? We need appropriate criteria for comparing distributions of income and personal welfare This requires a treatment of issues in distributional analysis.

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**Overview... How to represent problems in distributional analysis**

Equity and social welfare Overview... Welfare comparisons Income distributions Comparisons SWFs How to represent problems in distributional analysis Rankings Welfare and needs Compensation and responsibility

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**Representing a distribution**

Recall our two standard approaches: Irene and Janet The F-form particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria especially useful in cases where it is appropriate to adopt a parametric model of income distribution

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**Now for some formalisation:**

Pen's parade Plot income against proportion of population Parade in ascending order of "income" / height x "income" (height) x0.8 Now for some formalisation: x0.2 q 0.2 0.8 1 proportion of the population

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**A distribution function**

F(x) 1 F(x0) x x0

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**The set of distributions**

We can imagine a typical distribution as belonging to some class F Î 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... Methods and criteria of distributional analysis**

Equity and social welfare Overview... Welfare comparisons Income distributions Comparisons SWFs Methods and criteria of distributional analysis Rankings Welfare and needs Compensation and responsibility

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**Comparing Income Distributions**

Consider the purpose of the comparison... …in this case to get a handle on the redistributive impact of government activity - taxes and benefits. This requires some concept of distributional “fairness” or “equity”. The ethical basis rests on some aspects of the last lecture… …and the practical implementation requires an comparison in terms of “inequality”. Which is easy. Isn’t it?

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**Some comparisons self-evident...**

1 2 3 4 5 6 7 8 9 10 $ P R 1 2 3 4 5 6 7 8 9 10 $ P R 1 2 3 4 5 6 7 8 9 10 $ P R 1 2 3 4 5 6 7 8 9 10 $ R P

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A fundamental issue... Can distributional orderings be modelled using the two-person paradigm? If so then comparing distributions in terms of inequality or other concepts of equity will be almost trivial. Then the comparison of tax systems in terms of distributive effect presents no problem But, consider a simple example with three persons and fixed incomes

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**The 3-Person problem: two types of income difference**

Which do you think is “better”? Top Sensitivity Bottom Sensitivity 1 2 3 4 5 6 7 8 9 10 11 12 13 $ P Q R Monday Low inequality High inequality Low inequality High inequality 1 2 3 4 5 6 7 8 9 10 11 12 13 $ P Q R Tuesday

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**Distributional Orderings and Rankings**

In an ordering we unambiguously arrange distributions But a ranking may include distributions that cannot be ordered more welfare Syldavia {Syldavia, Arcadia, Borduria} is an ordering. {Syldavia, Ruritania, Borduria} is also an ordering. But the ranking {Syldavia, Arcadia, Ruritania, Borduria} is not an ordering. Arcadia Ruritania Borduria less welfare

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**Comparing income distributions - 2**

Distributional comparisons are more complex when more than two individuals are involved. P-Q and Q-R gaps important To make progress we need an axiomatic approach. Make precise “one distribution is better than another” Axioms could be rooted in welfare economics There are other logical bases. Apply the approach to general ranking principles Lorenz comparisons Social-welfare rankings Also to specific indices Welfare functions Inequality measures

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The Basics: Summary Income distributions can be represented in two main ways Irene-Janet F-form The F-form is characterised by Pen’s Parade Distributions are complicated entities: compare them using tools with appropriate properties. A useful class of tools can be found from Welfare Functions with suitable properties…

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**Overview... How to incorporate fundamental principles**

Equity and social welfare Overview... Welfare comparisons SWFs Axiomatic structure Classes Values How to incorporate fundamental principles Rankings Welfare and needs Compensation and responsibility

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**Social-welfare functions**

Basic tool is a social welfare function (SWF) Maps set of distributions into the real line I.e. for each distribution we get one specific number In Irene-Janet notation W = W(x) Properties will depend on economic principles Simple example of a SWF: Total income in the economy W = Si xi Perhaps not very interesting Consider principles on which SWF could be based

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**Another fundamental question**

What makes a “good” set of principles? There is no such thing as a “right” or “wrong” axiom. However axioms could be appropriate or inappropriate Need some standard of “reasonableness” For example, how do people view income distribution comparisons? Use a simple framework to list some of the basic axioms Assume a fixed population of size n.Assume that individual utility can be measured by x Income normalised by equivalence scales Rules out utility interdependence Welfare is just a function of the vector x := (x1, x2,…,xn ) Follow the approach of Amiel-Cowell (1999)

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**Basic Axioms: Anonymity Population principle Monotonicity**

Principle of Transfers Scale / translation Invariance Strong independence / Decomposability

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**Basic Axioms: Anonymity Population principle Monotonicity**

Permute the individuals and social welfare does not change Population principle Monotonicity Principle of Transfers Scale / translation Invariance Strong independence / Decomposability

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**Anonymity x W(x′) = W(x) x' $ $ 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4**

1 2 3 4 5 6 7 8 9 10 11 12 13 $ x W(x′) = W(x) 1 2 3 4 5 6 7 8 9 10 11 12 13 $ x'

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**Implication of anonymity**

1 2 3 4 5 6 7 8 9 10 11 12 13 $ x y End state principle: xy is equivalent to x′y . 1 2 3 4 5 6 7 8 9 10 11 12 13 $ x' y'

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**Basic Axioms: Anonymity Population principle Monotonicity**

Scale up the population and social welfare comparisons remain unchanged Monotonicity Principle of Transfers Scale / translation Invariance Strong independence / Decomposability

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**Population replication**

1 2 3 4 5 6 7 8 9 10 $ W(x) W(y) W(x,x,…,x) W(y,y,…,y) 1 2 3 4 5 6 7 8 9 10 $

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**A change of notation? Using the first two axioms**

Anonymity Population principle We can write welfare using F –form Just use information about distribution Sometimes useful for descriptive purposes Remaining axioms can be expressed in either form

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**Basic Axioms: Anonymity Population principle Monotonicity**

Increase anyone’s income and social welfare increases Principle of Transfers Scale / translation Invariance Strong independence / Decomposability

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

Monotonicity x $ 2 4 6 8 10 12 14 16 18 20 x′ $ 2 4 6 8 10 12 14 16 18 20 W(x1+,x2,..., xn ) > W(x1,x2,..., xn )

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

Monotonicity x′ $ 2 4 6 8 10 12 14 16 18 20 x $ 2 4 6 8 10 12 14 16 18 20 W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn)

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

Monotonicity x′ $ 2 4 6 8 10 12 14 16 18 20 x′ $ 2 4 6 8 10 12 14 16 18 20 W(x1,x2,..., xn+) > W(x1,x2,..., xn )

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**Basic Axioms: Anonymity Population principle Monotonicity**

Principle of Transfers Poorer to richer transfer must lower social welfare Scale / translation Invariance Strong independence / Decomposability

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**Transfer principle: The Pigou (1912) approach:**

Focused on a 2-person world A transfer from poor P to rich R must lower social welfare The Dalton (1920) extension Extended to an n-person world A transfer from (any) poorer i to (any) richer j must lower social welfare Although convenient, the extension is really quite strong…

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**Which group seems to have the more unequal distribution?**

1 2 3 4 5 6 7 8 9 10 11 12 13 $

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**The issue viewed as two groups**

1 2 3 4 5 6 7 8 9 10 11 12 13 $

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**Focus on just the affected persons**

1 2 3 4 5 6 7 8 9 10 11 12 13 $

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**Basic Axioms: Anonymity Population principle Monotonicity**

Principle of Transfers Scale Invariance Rescaling incomes does not affect welfare comparisons Strong independence / Decomposability

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**Scale invariance (homotheticity)**

x y $ 5 10 15 W(x) W(y) W(lx) W(ly) lx $ 500 1000 1500 ly

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**Basic Axioms: Anonymity Population principle Monotonicity**

Principle of Transfers Translation Invariance Adding a constant to all incomes does not affect welfare comparisons Strong independence / Decomposability

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**Translation invariance**

x y $ 5 10 15 W(x) W(y) W(x+d1) W(y+d1) x+d1 $ 5 10 15 20 y+d1 $ 5 10 15 20

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**Basic Axioms: Anonymity Population principle Monotonicity**

Principle of Transfers Scale / translation Invariance Strong independence / Decomposability merging with an “irrelevant” income distribution does not affect welfare comparisons

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**Decomposability / Independence**

1 2 3 4 5 6 7 8 9 10 11 12 13 $ Before merger... x y W(x) W(y) W(x') W(y') After merger... 1 2 3 4 5 6 7 8 9 10 11 12 13 $ x' y'

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**Using axioms Why the list of axioms?**

We can use some, or all, of them to characterise particular classes of SWF More useful than picking individual functions W ad hoc This then enables us to get fairly general results Depends on richness of the class The more axioms we impose (perhaps) the less general the result This technique will be applied to other types of tool Inequality Poverty Deprivation.

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**Overview... Categorising important types Equity and social welfare**

Welfare comparisons SWFs Axiomatic structure Classes Values Categorising important types Rankings Welfare and needs Compensation and responsibility

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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 : W(x)= Si u(xi) or equivalently in F-form W(F) = ò 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 Ì W : u(•) increasing If we further impose the transfer principle we get W2 Ì 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 x f(x) x 1 Income growth at x0**

Welfare increases if WÎ W1 A mean-preserving spread Welfare decreases if WÎ W2 x x x 1

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**An important family x 1 – e – 1**

Take the W2 subclass and impose scale invariance. Get the family of SWFs where u is iso-elastic: x 1 – e – 1 u(x) = ————, e ³ 0 1 – e Same as that in lecture 1: individual utility represented by x. also same form as CRRA utility function Parameter e captures society’s inequality aversion. Similar interpretation to individual risk aversion See Atkinson (1970)

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**Another important family**

Take the W2 subclass and impose translation invariance. Get the family of SWFs where u is iso-elastic: 1 – exp–kx u(x) = ——— k Same form as CARA utility function Parameter k captures society’s absolute inequality aversion. Similar to individual absolute risk aversion

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**Overview... …Can we deduce how inequality-averse “society” is?**

Equity and social welfare Overview... Welfare comparisons SWFs Axiomatic structure Classes Values …Can we deduce how inequality-averse “society” is? Rankings Welfare and needs Compensation and responsibility

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Values: the issues In previous lecture we saw the problem of adducing social values. Here we will focus on two questions… First: do people care about distribution? Justify a motive for considering positive inequality aversion Second: What is the shape of u? What is the value of e? Examine survey data and other sources

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**Happiness and welfare? Alesina et al (2004)**

Use data on happiness from social survey Construct a model of the determinants of happiness Use this to see if income inequality makes a difference Seems to be a difference in priorities between US and Europe US Continental Europe Share of government in GDP 30% 45% Share of transfers in GDP 11% 18% But does this reflect values? Do people in Europe care more about inequality?

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**The Alesina et al model An ordered logit**

“Happy” is categorical; built from three (0,1) variables: not too happy fairly happy very happy individual, state, time, group. Macro variables include inflation, unemployment rate Micro variables include personal characteristics h, m are state, time dummies

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**The Alesina et al. results**

People tend to declare lower happiness levels when inequality is high. Strong negative effects of inequality on happiness of the European poor and leftists. No effects of inequality on happiness of US poor and the left-wingers are not affected by inequality Negative effect of inequality on happiness of US rich No differences across the American right and the European right. No differences between the American rich and the European rich

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**The shape of u: approaches**

Direct estimates of inequality aversion See Cowell-Gardiner (2000) Carlsson et al (2005) Direct estimates of risk aversion Use as proxy for inequality aversion Base this on Harsanyi arguments? Indirect estimates of risk aversion Indirect estimates of inequality aversion From choices made by government

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**Direct evidence on risk aversion**

Barsky et al (1997) estimated relative risk-aversion from survey evidence. 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 estimated parameter of savings using expenditure data. Use two models: second version includes variables to capture anticipated income growth. Again note dependence on how well-off people are.

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**Indirect evidence on social values**

Assume constant absolute sacrifice Assume isoelastic social utility Then estimate e from Results for UK:

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**SWFs: Summary A small number of key axioms**

Generate an important class of SWFs with useful subclasses. Need to make a decision on the form of the SWF Decomposable? Scale invariant? Translation invariant? If we use the isoelastic model perhaps a value of around 1.5 – 2 is reasonable.

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**Overview... ...rankings, orderings and practical tools**

Equity and social welfare Overview... Welfare comparisons SWFs ...rankings, orderings and practical tools Rankings Welfare and needs Compensation and responsibility

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Ranking and dominance We pick up on the problem of comparing distributions Two simple concepts based on elementary axioms Anonymity Population principle Monotonicity Transfer principle Illustrate these tools with a simple example Use the Irene-Janet representation of the distribution Fixed population (so we don’t need pop principle)

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**First-order Dominance**

x $ 2 4 6 8 10 12 14 16 18 20 y[1] > x[1], y[2] > x[2], y[3] > x[3] y $ 2 4 6 8 10 12 14 16 18 20 Each ordered income in y larger than that in x.

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**Second-order Dominance**

x $ 2 4 6 8 10 12 14 16 18 20 y[1] > x[1], y[1]+y[2] > x[1]+x[2], y[1]+y[2] +y[3] > x[1]+x[2] +x[3] y $ 2 4 6 8 10 12 14 16 18 20 Each cumulated income sum in y larger than that in x. Weaker 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) = ò u(x) dF(x) … and its important subclasses W1 Ì W : u(•) increasing W2 Ì 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 quantile. This can be expressed in general as the functional Use this to derive a number of intuitive concepts Interquartile range Decile-ratios Semi-decile ratios The graph of Q is Pen’s Parade Extend it to characterise the idea of dominance…

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**An important relationship**

The idea of quantile (1st-order) dominance: G quantile-dominates F means: for every q, Q(G;q) ³ Q(F;q), for some q, Q(G;q) > Q(F;q) A fundamental result: G quantile-dominates F Û W(G) > W(F) for all WÎW1 To illustrate, use Pen's parade

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**First-order dominance**

Q(.; q) G F q 1

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2nd-Order approach The basic tool is the income cumulant. This can be expressed as the functional Use this to derive three intuitive concepts The (relative) Lorenz curve The shares ranking Gini coefficient The graph of C is the generalised Lorenz curve Again use it to characterise dominance…

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**Another important relationship**

The idea of cumulant (2nd-order) dominance: G cumulant-dominates F means: for every q, C (G;q) ³ C (F;q), for some q, C (G;q) > C (F;q) A fundamental result: G cumulant-dominates F Û W(G) > W(F) for all WÎW2 To illustrate, draw the GLC

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**Second order dominance**

C(.; q) cumulative income m(G) C(G; . ) m(F) practical example, UK C(F; . ) q 1

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**Application of ranking**

The tax and -benefit system maps one distribution into another... Use ranking tools to assess the impact of this in welfare terms. Typically this uses one or other concept of Lorenz dominance.

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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 by quantiles. A strong result. Where Parades cross, second-order methods may be appropriate. Second-order (GL)-dominance is equivalent to ranking by cumulations. Another strong result.

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**Overview... Extensions of the ranking approach**

Equity and social welfare Overview... Welfare comparisons SWFs Extensions of the ranking approach Rankings Welfare and needs Compensation and responsibility

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**Difficulties with needs**

Why equivalence scales? Need a way of making welfare comparisons Should be coherent Take account of differing family size Take account of needs But there are irreconcilable difficulties: Logic Source information Estimation problems Perhaps a more general approach “Needs” seems an obvious place for explicit welfare analysis

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**Income and needs reconsidered**

Standard approach uses "equivalised income" The approach assumes: Given, known welfare-relevant attributes a A known relationship n = n(a) Equivalised income given by x = y / n n is the "exchange-rate" between income types x, y Set aside the assumption that we have a single n(•). Get a general result on joint distribution of (y, a) To 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) = ò u(x) dF(x) With principal subclasses W1 Ì W : u(•) increasing W2 Ì W1: u(•) increasing and concave Recall the second-order result G cumulant-dominates F Û W(G) > W(F) for all WÎW2 Make progress by further restricting subclasses

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**Alternative approach to needs**

Sort individuals into needs groups N1, N2 ,… Suppose a proportion pj 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**

Suppose we want j=1,2,… to reflect decreasing order of need. Consider need and the marginal utility of income: “Need” reflected in high MU of income? If need falls with j then the above should be positive. Let W3 Ì W2 be the subclass of welfare functions for which the above is positive and decreasing in y

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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 distribution Theorem: A UK example

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**Household types in Economic Trends**

2+ads,3+chn/3+ads,chn 2 adults with 2 children 1 adult with children 2 adults with 1 child 2+ adults 0 children 1 adult, 0 children

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**Impact of Taxes and Benefits. UK 1991. Sequential GLCs (1)**

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

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**Needs: summary Doing without equivalence scales seems attractive**

Removes a level of arbitrariness Simplifies computation? But the sequential dominance principle is problematic Demands one-dimensional needs categorisation It is often indecisive May get even more complicated for comparisons over time. Can the approach be rescued? Perhaps one is trying to do too much May make sense to put upper and lower bounds on equivalence scales

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**Overview... What should be equalised? Equity and social welfare**

Welfare comparisons SWFs What should be equalised? Rankings Welfare and needs Compensation and responsibility

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**Responsibility (1) Standard approach to case for redistribution**

Use reference point of equality How effective is tax/benefit system in moving actual distribution toward reference point? Does not take account of individual responsibility The Responsibility “cut” of Dworkin (1981a, 1981b) Distinguish between things that are your fault and things for which you deserve compensation May need to revise our concept of “equality” or “equal treatment”

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Responsibility (2) Responsibility should affect the evaluation of redistribution Both case for redistribution... ... and effectiveness of taxation. Differentiate between characteristics for which people can be held responsible characteristics for which people should not Assume that these characteristics are known and agreed Follow the approach of Fleurbaey (1995a), (1995b), (1995c)

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**Basic structure Anonymity Each person i has a vector of attributes ai:**

Attributes partitioned into two classes R-attributes: for which the individual is responsible C-attributes: for which the individual may be compensated The income function f maps attributes into incomes f(ai) A distribution rule F: Profile of attributes Anonymity

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**Responsibility: Rules**

Bossert and Fleurbaey (1996) Equal Income for Equal Responsibility Focus on distribution itself Full compensation Equal Transfers for Equal C-attributes Focus on changes in distribution Strict Compensation

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**Consider two compromise approaches**

A difficulty For large populations... EIER and ETEC are incompatible except for... Additive separability: Fleurbaey (1995a,b) In this special case... ...a natural redistribution mechanism Consider two compromise approaches

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**Compromise (1) Insist on Full compensation (EIER) Weaken ETEC**

Egalitarian-equivalent mechanisms Reference profile Every agent has a post-tax income equal to the pre-tax income earned given reference compensation characteristics plus... a uniform transfer

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**Compromise (2) Insist on strict compensation (ETEC) Weaken EIER**

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

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Conclusion Axiomatisation of welfare can be accomplished using just a few basic principles Ranking criteria can be used to provide broad judgments These may be indecisive, so specific SWFs could be used What shape should they have? How do we specify them empirically? The same basic framework of distributional analysis can be extended to a number of related problems: Move on to consider inequality and poverty… …in the next lecture component.

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