Income Distribution and Welfare

Income Distribution and Welfare
Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell July 2006

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.

Overview... How to represent problems in distributional analysis
Income Distribution and Welfare Overview... Welfare comparisons Income distributions Comparisons How to represent problems in distributional analysis SWFs Rankings Social welfare and needs

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

Now for some formalisation:
Pen’s parade (Pen, 1971) 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

A distribution function
F(x) 1 F(x0) x x0

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

Overview... Methods and criteria of distributional analysis
Income Distribution and Welfare Overview... Welfare comparisons Income distributions Comparisons Methods and criteria of distributional analysis SWFs Rankings Social welfare and needs

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?

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

A fundamental issue... Can distributional orderings be modelled using the two-person paradigm? If so then comparing distributions in terms of inequality will be almost trivial. Same applies to other equity criteria But, consider a simple example with three persons and fixed incomes

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

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

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

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…

Overview... How to incorporate fundamental principles
Income Distribution and Welfare Overview... Welfare comparisons How to incorporate fundamental principles SWFs Axiomatic structure Classes Values Rankings Social welfare and needs

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

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)

Basic Axioms: Anonymity Population principle Monotonicity
Principle of Transfers Scale / translation Invariance Strong independence / Decomposability

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

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'

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

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

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 $

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

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

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 )

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)

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 )

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

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…

Which group seems to have the more unequal distribution?
1 2 3 4 5 6 7 8 9 10 11 12 13 $

The issue viewed as two groups
1 2 3 4 5 6 7 8 9 10 11 12 13 $

Focus on just the affected persons
1 2 3 4 5 6 7 8 9 10 11 12 13 $

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

Scale invariance (homotheticity)
x y $ 5 10 15 W(x)  W(y)  W(lx)  W(ly) lx $ 500 1000 1500 ly

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

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

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

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'

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 can be applied to other types of tool Inequality Poverty Deprivation.

Overview... Categorising important types
Income Distribution and Welfare Overview... Welfare comparisons Categorising important types SWFs Axiomatic structure Classes Values Rankings Social welfare and needs

Classes of SWFs (1) Anonymity and population principle imply we can write SWF in either Irene-Janet 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 2. But W excludes some well-known welfare criteria

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…

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

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 2: 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)

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

Overview... …Can we deduce how inequality-averse “society” is?
Income Distribution and Welfare Overview... Welfare comparisons …Can we deduce how inequality-averse “society” is? SWFs Axiomatic structure Classes Values Rankings Social welfare and needs

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

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?

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

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

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 (for later…)

Direct evidence on risk aversion
Barsky et al (1997) estimated relative risk-aversion from survey evidence. Note dependence on how well-off people are.

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.

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.

Overview... ...general comparison criteria
Income Distribution and Welfare Overview... Welfare comparisons ...general comparison criteria SWFs Rankings Welfare comparisons Inequality comparisons Practical tools Social welfare and needs

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)

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.

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

Social-welfare criteria and dominance
Why are these concepts useful? Relate these dominance ideas to 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.

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…

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

First-order dominance
Q(.; q) G F q 1

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…

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 (Shorrocks 1983): G cumulant-dominates F Û W(G) > W(F) for all WÎW2 To illustrate, draw the GLC

Second order dominance
C(.; q) cumulative income m(G) C(G; . ) m(F) C(F; . ) q 1

UK “Final income” – GLC

“Original income” – GLC

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. Lorenz dominance equivalent to ranking by shares. Special case of GL-dominance normalised by means. Where Lorenz-curves intersect unambiguous inequality orderings are not possible. This makes inequality measures especially interesting.

Overview... Extensions of the ranking approach
Income Distribution and Welfare Overview... Welfare comparisons Extensions of the ranking approach SWFs Rankings Social welfare and needs

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

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) This makes distributional comparisons multidimensional Intrinsically very difficult (Atkinson and Bourguignon 1982) To make progress: We simplify the structure of the problem We again use results on ranking criteria

Alternative approach to needs
Based on Atkinson and Bourguignon (1982, 1987)  see also Cowell (2000) Sort individuals be into needs groups N1, N2 ,… Suppose a proportion pj are in group Nj . Then social welfare can be written: To make this operational… Utility people get from income depends on their needs:

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

Main result A UK example
Let F( j) denote distribution for all needs groups up to and including j. Distinguish this from the marginal distribution Theorem (Atkinson and Bourguignon 1987) So to examine if welfare is higher in F than in G… …we have a “sequential dominance” test. Check first the neediest group then the first two neediest groups then the first three… …etc A UK example

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

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

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

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: For example inequality and poverty… …in next lecture

References: Alesina, A., Di Tella, R. and MacCulloch, R (2004) “Inequality and happiness: are Europeans and Americans different?”, Journal of Public Economics, 88, Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality, Cambridge University Press Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, Atkinson, A. B. and Bourguignon, F. (1987) “Income distribution and differences in needs,” in Feiwel, G. R. (ed), Arrow and the Foundations of the Theory of Economic Policy, Macmillan, New York, chapter 12, pp Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multi-dimensional distributions of economic status,” Review of Economic Studies, 49, Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997) “Preference parameters and behavioral heterogeneity : An Experimental Approach in the Health and Retirement Survey,” Quarterly Journal of Economics,112,

References: Blundell, R., Browning, M. and Meghir, C. (1994) “Consumer Demand and the Life-Cycle Allocation of Household Expenditures,” Review of Economic Studies, 61, 57-80 Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) “Are people inequality averse or just risk averse?” Economica, 72, Cowell, F. A. (2000) “Measurement of Inequality,” in Atkinson, A. B. and Bourguignon, F. (eds) Handbook of Income Distribution, North Holland, Amsterdam, Chapter 2, Cowell, F. A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research Paper 202, Office of Fair Training, Salisbury Square, London Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London Pigou, A. C. (1912) Wealth and Welfare, Macmillan, London Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17

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