#### Transcript Extensive Form

Frank Cowell: TU Lisbon – Inequality & Poverty July 2006 Income Distribution and Welfare Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell http://darp.lse.ac.uk/lisbon2006 Frank Cowell: TU Lisbon – Inequality & Poverty 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. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons How to represent problems in distributional analysis SWFs Rankings Social welfare and needs •Income distributions •Comparisons Frank Cowell: TU Lisbon – Inequality & Poverty Representing a distribution Recall our two standard approaches: Irene and Janet particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria The F-form especially useful in cases where it is appropriate to adopt a parametric model of income distribution Frank Cowell: TU Lisbon – Inequality & Poverty Pen’s parade (Pen, 1971) Plot income against proportion of population Parade in ascending order of "income" / height x x0.8 Now for some formalisation: x0.2 0 proportion ofqthe population 0.2 0.8 1 F(x) 1 F(x0) x x0 0 Frank Cowell: TU Lisbon – Inequality & Poverty A distribution function Frank Cowell: TU Lisbon – Inequality & Poverty 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 Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons Methods and criteria of distributional analysis SWFs Rankings Social welfare and needs •Income distributions •Comparisons Frank Cowell: TU Lisbon – Inequality & Poverty 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? Frank Cowell: TU Lisbon – Inequality & Poverty Some comparisons self-evident... P 0 1 2 3 4 R 5 6 1 2 3 4 5 6 7 P 0 1 2 8 9 10 8 9 10 9 10 $ R 3 4 5 6 7 8 $ R P 0 $ R P 0 7 1 2 3 4 5 6 7 8 9 10 $ Frank Cowell: TU Lisbon – Inequality & Poverty A fundamental issue... Can distributional orderings be modelled using the twoperson 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 Frank Cowell: TU Lisbon – Inequality & Poverty The 3-Person problem: two types of income difference Which do you think is “better”? Top Sensitivity Bottom Sensitivity Monday P Q High Low inequality R $ 0 1 2 3 4 5 6 P Q 5 6 7 8 9 10 11 12 R High Low inequality inequality Tuesday 13 $ 0 1 2 3 4 7 8 9 10 11 12 13 Frank Cowell: TU Lisbon – Inequality & Poverty Distributional Orderings and Rankings In an ordering we unambiguously arrange distributions But a ranking may include distributions that cannot be ordered more welfare Syldavia Ruritania Arcadia Borduria less welfare {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 Cowell: TU Lisbon – Inequality & Poverty Comparing income distributions - 2 Distributional comparisons are more complex when more than two individuals are involved. To make progress we need an axiomatic approach. There are other logical bases. Apply the approach to general ranking principles Make precise “one distribution is better than another” Axioms could be rooted in welfare economics P-Q and Q-R gaps important Lorenz comparisons Social-welfare rankings Also to specific indices Welfare functions Inequality measures Frank Cowell: TU Lisbon – Inequality & Poverty The Basics: Summary Income distributions can be represented in two main ways The F-form is characterised by Pen’s Parade Distributions are complicated entities: Irene-Janet F-form compare them using tools with appropriate properties. A useful class of tools can be found from Welfare Functions with suitable properties… Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons How to incorporate fundamental principles SWFs Rankings Social welfare and needs •Axiomatic structure •Classes •Values Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Use a simple framework to list some of the basic axioms Need some standard of “reasonableness” For example, how do people view income distribution comparisons? 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) Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Population principle Monotonicity Principle of Transfers Scale / translation Invariance Strong independence / Decomposability Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Permute the individuals and social welfare does not change Population principle Monotonicity Principle of Transfers Scale / translation Invariance Strong independence / Decomposability Frank Cowell: TU Lisbon – Inequality & Poverty Anonymity x $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 W(x′) = W(x) x' $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Frank Cowell: TU Lisbon – Inequality & Poverty Implication of anonymity x $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 8 9 10 11 12 13 y $ 0 1 2 3 4 5 6 7 End state principle: xy is equivalent to x′y . x' $ 0 1 2 3 4 5 6 7 y' 8 9 10 11 12 13 Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Population principle Scale up the population and social welfare comparisons remain unchanged Monotonicity Principle of Transfers Scale / translation Invariance Strong independence / Decomposability Frank Cowell: TU Lisbon – Inequality & Poverty Population replication $ 0 1 2 3 4 5 6 7 8 9 10 W(x) W(y) W(x,x,…,x) W(y,y,…,y) $ 0 1 2 3 4 5 6 7 8 9 10 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Population principle Monotonicity Increase anyone’s income and social welfare increases Principle of Transfers Scale / translation Invariance Strong independence / Decomposability Frank Cowell: TU Lisbon – Inequality & Poverty Monotonicity x $ 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 x′ $ 0 2 4 6 8 10 W(x1+,x2,..., xn ) > W(x1,x2,..., xn ) Frank Cowell: TU Lisbon – Inequality & Poverty Monotonicity x′ $ 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 x $ 0 2 4 6 8 10 W(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn) Frank Cowell: TU Lisbon – Inequality & Poverty Monotonicity x′ $ 0 2 4 6 8 10 12 14 16 18 20 14 16 18 20 x′ $ 0 2 4 6 8 10 12 W(x1,x2,..., xn+) > W(x1,x2,..., xn ) Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Population principle Monotonicity Principle of Transfers Poorer to richer transfer must lower social welfare Scale / translation Invariance Strong independence / Decomposability Frank Cowell: TU Lisbon – Inequality & Poverty Transfer principle: The Pigou (1912) approach: The Dalton (1920) extension Focused on a 2-person world A transfer from poor P to rich R must lower social welfare 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… Frank Cowell: TU Lisbon – Inequality & Poverty 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 Cowell: TU Lisbon – Inequality & Poverty The issue viewed as two groups $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 9 10 11 12 13 $ 0 1 2 3 4 5 6 7 8 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Population principle Monotonicity Principle of Transfers Scale Invariance Rescaling incomes does not affect welfare comparisons Strong independence / Decomposability Frank Cowell: TU Lisbon – Inequality & Poverty Scale invariance (homotheticity) x 0 5 10 15 10 15 $ y 0 5 $ W(x) W(y) W(lx) W(ly) lx $ 0 1000 500 1500 ly $ 0 500 1000 1500 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty Translation invariance x 0 5 10 15 10 15 $ y 0 5 $ W(x) W(y) W(x+1) W(y+1) x+1 5 15 10 20 $ y+1 5 10 15 20 $ Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty Decomposability / Independence x $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 9 10 11 12 13 Before merger... y $ 0 1 2 3 4 5 6 7 8 W(x) W(y) W(x') W(y') x' 0 1 2 3 4 y' 0 1 2 3 4 $ 5 6 7 8 9 10 11 12 13 5 6 7 8 9 10 11 12 13 After merger... $ Frank Cowell: TU Lisbon – Inequality & Poverty Using axioms Why the list of axioms? We can use some, or all, of them to characterise particular classes of SWF This then enables us to get fairly general results More useful than picking individual functions W ad hoc 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. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons Categorising important types SWFs Rankings Social welfare and needs •Axiomatic structure •Classes •Values Frank Cowell: TU Lisbon – Inequality & Poverty Classes of SWFs (1) Anonymity and population principle imply we can write SWF in either I-J form or F form Introduce decomposability and you get class of Additive SWFs W : Most modern approaches use these assumptions But you may need to standardise for needs etc W(x)= Si u(xi) or equivalently in F-form W(F) = 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 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Cowell: TU Lisbon – Inequality & Poverty The density function Income growth at x0 f(x) Welfare increases if WW1 A mean-preserving spread Welfare decreases if WW2 x x1 x0 Frank Cowell: TU Lisbon – Inequality & Poverty An important family 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 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) Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons …Can we deduce how inequalityaverse “society” is? SWFs Rankings Social welfare and needs •Axiomatic structure •Classes •Values Frank Cowell: TU Lisbon – Inequality & Poverty 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? Second: What is the shape of u? Justify a motive for considering positive inequality aversion What is the value of e? Examine survey data and other sources Frank Cowell: TU Lisbon – Inequality & Poverty 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 Share of government in GDP Share of transfers in GDP US 30% 11% Continental Europe 45% 18% But does this reflect values? Do people in Europe care more about inequality? Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty The shape of u: approaches Direct estimates of inequality aversion Direct estimates of risk aversion See Cowell-Gardiner (2000) Carlsson et al (2005) 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…) Frank Cowell: TU Lisbon – Inequality & Poverty Direct evidence on risk aversion Barsky et al (1997) estimated relative risk-aversion from survey evidence. Note dependence on how well-off people are. Frank Cowell: TU Lisbon – Inequality & Poverty 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. Frank Cowell: TU Lisbon – Inequality & Poverty 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. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons ...general comparison criteria SWFs Rankings Social welfare and needs •Welfare comparisons •Inequality comparisons •Practical tools Frank Cowell: TU Lisbon – Inequality & Poverty 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) Frank Cowell: TU Lisbon – Inequality & Poverty First-order Dominance x $ 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 1 2 1 4 1 6 1 8 2 0 y[1] > x[1], y[2] > x[2], y[3] > x[3] y $ 0 2 4 6 8 1 0 Each ordered income in y larger than that in x. Frank Cowell: TU Lisbon – Inequality & Poverty Second-order Dominance x $ 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 1 2 1 4 1 6 1 8 2 0 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 $ 0 2 4 6 8 1 0 2 0 Each cumulated income sum in y larger than that in x. Weaker than first-order dominance Frank Cowell: TU Lisbon – Inequality & Poverty Social-welfare criteria and dominance Why are these concepts useful? Relate these dominance ideas to classes of SWF Recall the class of additive SWFs … and its important subclasses W : W(F) = u(x) dF(x) 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 Cowell: TU Lisbon – Inequality & Poverty 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… Frank Cowell: TU Lisbon – Inequality & Poverty An important relationship The idea of quantile (1st-order) dominance: G quantile-dominates F means: for every q, Q(G;q) Q(F;q), for some q, Q(G;q) > Q(F;q) A fundamental result: G quantile-dominates F W(G) > W(F) for all WW1 To illustrate, use Pen's parade Q(.; q) G F 1 q 0 Frank Cowell: TU Lisbon – Inequality & Poverty First-order dominance Frank Cowell: TU Lisbon – Inequality & Poverty 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… Frank Cowell: TU Lisbon – Inequality & Poverty Another important relationship The idea of cumulant (2nd-order) dominance: G cumulant-dominates F means: for every q, C (G;q) C (F;q), for some q, C (G;q) > C (F;q) A fundamental result: G cumulant-dominates F W(G) > W(F) for all WW2 To illustrate, draw the GLC C(.; q) m(G) m(F) C(G; . ) C(F; . ) 0 q 0 1 cumulative income Frank Cowell: TU Lisbon – Inequality & Poverty Second order dominance Frank Cowell: TU Lisbon – Inequality & Poverty UK “Final income” – GLC £25,000 £20,000 £15,000 1993 2000-1 £10,000 £5,000 £0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Proportion of population 0.7 0.8 0.9 1.0 Frank Cowell: TU Lisbon – Inequality & Poverty “Original income” – GLC £25,000 £20,000 £15,000 1993 2000-1 £10,000 £5,000 £0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Proportion of population 0.7 0.8 0.9 1.0 Frank Cowell: TU Lisbon – Inequality & Poverty 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. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Income Distribution and Welfare Welfare comparisons Extensions of the ranking approach SWFs Rankings Social welfare and needs Frank Cowell: TU Lisbon – Inequality & Poverty Difficulties with needs Why equivalence scales? Need a way of making welfare comparisons But there are irreconcilable difficulties: Should be coherent Take account of differing family size Take account of needs Logic Source information Estimation problems Perhaps a more general approach “Needs” seems an obvious place for explicit welfare analysis Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty 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: Frank Cowell: TU Lisbon – Inequality & Poverty 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 W2 be the subclass of welfare functions for which the above is positive and decreasing in y Frank Cowell: TU Lisbon – Inequality & Poverty Main result 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. A UK Check first the neediest group then the first two neediest groups example then the first three… …etc Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty Impact of Taxes and Benefits. UK 1991. Sequential GLCs (1) £4,500 Type 1 original Type 1 final Types 1,2 original Types 1,2 final Types 1-3 original Types 1-3 final £4,000 £3,500 £3,000 £2,500 £2,000 £1,500 £1,000 £500 0.00 0.05 0.10 0.15 Proportion of population 0.20 £0 0.25 Frank Cowell: TU Lisbon – Inequality & Poverty Impact of Taxes and Benefits. UK 1991. Sequential GLCs (2) £14,000 Types 1-4 orig Types 1-4 final £12,000 Types 1-5 orig Types 1-5 final All types orig All types final £10,000 £8,000 £6,000 £4,000 £2,000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Proportion of population 0.70 0.80 0.90 £0 1.00 Frank Cowell: TU Lisbon – Inequality & Poverty 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 Frank Cowell: TU Lisbon – Inequality & Poverty References: Alesina, A., Di Tella, R. and MacCulloch, R (2004) “Inequality and happiness: are Europeans and Americans different?”, Journal of Public Economics, 88, 2009-2042 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, 244-263 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 350-370 Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multidimensional distributions of economic status,” Review of Economic Studies, 49, 183-201 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, 537-579 Frank Cowell: TU Lisbon – Inequality & Poverty 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, 87-166 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, 348-361 Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London Pigou, A. C. (1912) Wealth and Welfare, Macmillan, London