1. Frank Cowell: TU Lisbon – Inequality & Poverty Income Distribution and Welfare July 2006 Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell http://darp.lse.ac.uk/lisbon2006

2. Frank Cowell: TU Lisbon – Inequality & Poverty Onwards from welfare economics... We’ve seen the welfare-economics basis for redistribution as a public-policy objective We’ve seen the welfare-economics basis for redistribution as a public-policy objective How to assess the impact and effectiveness of such policy? How to assess the impact and effectiveness of such policy? We need appropriate criteria for comparing distributions of income and personal welfare We need appropriate criteria for comparing distributions of income and personal welfare This requires a treatment of issues in distributional analysis. This requires a treatment of issues in distributional analysis.

3. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare How to represent problems in distributional analysis Income distributions Comparisons

4. Frank Cowell: TU Lisbon – Inequality & Poverty Representing a distribution Irene and Janet Irene and Janet The F-form The F-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

5. Frank Cowell: TU Lisbon – Inequality & Poverty x 0.20.8 1 0 x 0.8 q "income" (height) proportion of the population x 0.2 Pen’s parade (Pen, 1971) Now for some formalisation:   Plot income against proportion of population   Parade in ascending order of "income" / height

6. Frank Cowell: TU Lisbon – Inequality & Poverty 0 1 x F(x)F(x) x0x0 F(x0)F(x0) A distribution function

7. 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

8. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare Methods and criteria of distributional analysis Income distributions Comparisons

9. Frank Cowell: TU Lisbon – Inequality & Poverty 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 impact of government activity - taxes and benefits. …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”. This requires some concept of distributional “fairness” or “equity”. The ethical basis rests on some aspects of the last lecture… The ethical basis rests on some aspects of the last lecture… …and the practical implementation requires an comparison in terms of “inequality”. …and the practical implementation requires an comparison in terms of “inequality”. Which is easy. Isn’t it? Which is easy. Isn’t it?

10. Frank Cowell: TU Lisbon – Inequality & Poverty Some comparisons self-evident... 0123456789 10 $ PR 0123456789 $ P R 0123456789 $ PR 0123456789 $ R P

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

12. Frank Cowell: TU Lisbon – Inequality & Poverty The 3-Person problem: two types of income difference 0123456789 10111213 $ P Q R Tuesday 0123456789 10111213 $ P Q R Monday   Which do you think is “better”?   Top Sensitivity   Bottom Sensitivity Low inequality High inequality Low inequality High inequality

13. Frank Cowell: TU Lisbon – Inequality & Poverty 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.

14. Frank Cowell: TU Lisbon – Inequality & Poverty Comparing income distributions - 2 Distributional comparisons are more complex when more than two individuals are involved. 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. To make progress we need an axiomatic approach.  Make precise “one distribution is better than another” Axioms could be rooted in welfare economics Axioms could be rooted in welfare economics  There are other logical bases. Apply the approach to general ranking principles Apply the approach to general ranking principles  Lorenz comparisons  Social-welfare rankings Also to specific indices Also to specific indices  Welfare functions  Inequality measures

15. Frank Cowell: TU Lisbon – Inequality & Poverty The Basics: Summary Income distributions can be represented in two main ways Income distributions can be represented in two main ways  Irene-Janet  F-form The F-form is characterised by Pen’s Parade The F-form is characterised by Pen’s Parade Distributions are complicated entities: 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… A useful class of tools can be found from Welfare Functions with suitable properties…

16. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare How to incorporate fundamental principles Axiomatic structure Classes Values

17. Frank Cowell: TU Lisbon – Inequality & Poverty Social-welfare functions Basic tool is a social welfare function (SWF) 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 Properties will depend on economic principles Simple example of a SWF: Simple example of a SWF:  Total income in the economy W =   x i  Perhaps not very interesting Consider principles on which SWF could be based Consider principles on which SWF could be based

18. Frank Cowell: TU Lisbon – Inequality & Poverty 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 inappropriate 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 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 := (x 1, x 2,…,x n ) Follow the approach of Amiel-Cowell (1999) Follow the approach of Amiel-Cowell (1999)

19. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle Monotonicity Monotonicity Principle of Transfers Principle of Transfers Scale / translation Invariance Scale / translation Invariance Strong independence / Decomposability Strong independence / Decomposability

20. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity  Permute the individuals and social welfare does not change Population principle Population principle Monotonicity Monotonicity Principle of Transfers Principle of Transfers Scale / translation Invariance Scale / translation Invariance Strong independence / Decomposability Strong independence / Decomposability

21. Frank Cowell: TU Lisbon – Inequality & Poverty 0123456789 10111213 $ x 0123456789 10111213 $ x' Anonymity W(x′) = W(x)

22. Frank Cowell: TU Lisbon – Inequality & Poverty 0123456789 10111213 0123456789 10111213 $ $ x y 0123456789 10111213 $ x' y' Implication of anonymity End state principle: x  y is equivalent to x′  y.

23. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle  Scale up the population and social welfare comparisons remain unchanged Monotonicity Monotonicity Principle of Transfers Principle of Transfers Scale / translation Invariance Scale / translation Invariance Strong independence / Decomposability Strong independence / Decomposability

24. Frank Cowell: TU Lisbon – Inequality & Poverty 0123456789 10 $ 0123456789 $ Population replication W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y)

25. Frank Cowell: TU Lisbon – Inequality & Poverty A change of notation? Using the first two axioms Using the first two axioms  Anonymity  Population principle We can write welfare using F –form We can write welfare using F –form Just use information about distribution Just use information about distribution Sometimes useful for descriptive purposes Sometimes useful for descriptive purposes Remaining axioms can be expressed in either form Remaining axioms can be expressed in either form

26. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle Monotonicity Monotonicity  Increase anyone’s income and social welfare increases Principle of Transfers Principle of Transfers Scale / translation Invariance Scale / translation Invariance Strong independence / Decomposability Strong independence / Decomposability

27. Frank Cowell: TU Lisbon – Inequality & Poverty Monotonicity x′x′ $ 02468 101214161820 x $ 02468 1012 14161820 W(x 1 + ,x 2,..., x n ) > W(x 1,x 2,..., x n )

28. Frank Cowell: TU Lisbon – Inequality & Poverty xx $ 02468 1012 14161820 Monotonicity W(x 1,x 2..., x i + ,..., x n ) > W(x 1,x 2,..., x i,..., x n ) x′x′ $ 02468 101214161820

29. Frank Cowell: TU Lisbon – Inequality & Poverty Monotonicity x′x′ $ 02468 1012 14161820 x′x′ $ 02468 101214161820 W(x 1,x 2,..., x n +  ) > W(x 1,x 2,..., x n )

30. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle Monotonicity Monotonicity Principle of Transfers Principle of Transfers  Poorer to richer transfer must lower social welfare Scale / translation Invariance Scale / translation Invariance Strong independence / Decomposability Strong independence / Decomposability

31. Frank Cowell: TU Lisbon – Inequality & Poverty Transfer principle: The Pigou (1912) approach: The Pigou (1912) approach:  Focused on a 2-person world  A transfer from poor P to rich R must lower social welfare The extension The Dalton (1920) extensionDalton (1920)  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… Although convenient, the extension is really quite strong…

32. Frank Cowell: TU Lisbon – Inequality & Poverty Which group seems to have the more unequal distribution? 0123456789 10111213 0123456789 10111213 $ $

33. Frank Cowell: TU Lisbon – Inequality & Poverty 0123456789 10111213 0123456789 10111213 $ $ The issue viewed as two groups

34. Frank Cowell: TU Lisbon – Inequality & Poverty Focus on just the affected persons 0123456789 10111213 0123456789 10111213 $ $

35. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle Monotonicity Monotonicity Principle of Transfers Principle of Transfers Scale Invariance Scale Invariance  Rescaling incomes does not affect welfare comparisons Strong independence / Decomposability Strong independence / Decomposability

36. Frank Cowell: TU Lisbon – Inequality & Poverty Scale invariance (homotheticity) x y $ 05 10 15 $ 05 10 15 W(x)  W(y)  W( x)  W( y) x $ 0500 1000 1500 $ 0500 1000 1500 y

37. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle Monotonicity Monotonicity Principle of Transfers Principle of Transfers Translation Invariance Translation Invariance  Adding a constant to all incomes does not affect welfare comparisons Strong independence / Decomposability Strong independence / Decomposability

38. Frank Cowell: TU Lisbon – Inequality & Poverty Translation invariance x y $ 05 10 15 $ 05 10 15 W(x)  W(y)  W(x  1)  W(y  1) x  1 $ 510 15 20 $ 510 15 20 y  1

39. Frank Cowell: TU Lisbon – Inequality & Poverty Basic Axioms: Anonymity Anonymity Population principle Population principle Monotonicity Monotonicity Principle of Transfers Principle of Transfers Scale / translation Invariance Scale / translation Invariance Strong independence / Decomposability Strong independence / Decomposability  merging with an “irrelevant” income distribution does not affect welfare comparisons

40. Frank Cowell: TU Lisbon – Inequality & Poverty 0123456789 10111213 012345678910111213 $ $ Before merger... x y After merger... 012345678910111213 0123456789 10111213 $ $ x'x' y'y' Decomposability / Independence W(x)  W(y)  W(x')  W(y')

41. Frank Cowell: TU Lisbon – Inequality & Poverty Using axioms Why the list of axioms? Why the list of axioms? We can use some, or all, of them to characterise particular classes of SWF 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 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 This technique can be applied to other types of tool  Inequality  Poverty  Deprivation.

42. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare Categorising important types Axiomatic structure Classes Values

43. 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   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)=   x i  W(x)=   u(x i )   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

44. Frank Cowell: TU Lisbon – Inequality & Poverty Classes of SWFs (2) From W we get important subclasses If we impose monotonicity we get   W 1  W : u() increasing If we further impose the transfer principle we get   W 2  W 1 : u() increasing and concave We often need to use these special subclasses Illustrate their behaviour with a simple example…

45. Frank Cowell: TU Lisbon – Inequality & Poverty The density function x f(x)f(x) x 0 x 1 x 0   Income growth at x 0   Welfare increases if W  W 1   A mean-preserving spread   Welfare decreases if W  W 2

46. Frank Cowell: TU Lisbon – Inequality & Poverty An important family Take the subclass and impose scale invariance. Take the W 2 subclass and impose scale invariance. Get the family of SWFs where u is iso-elastic: Get the family of SWFs where u is iso-elastic: x x  1 –  – 1 x u(x) = ————,   1 –  Same as that in lecture 2: Same as that in lecture 2:  individual utility represented by x.  also same form as CRRA utility function Parameter captures society’s inequality aversion. Parameter  captures society’s inequality aversion.  Similar interpretation to individual risk aversion  See  See Atkinson (1970)Atkinson (1970)

47. Frank Cowell: TU Lisbon – Inequality & Poverty Another important family Take the subclass and impose translation invariance. Take the W 2 subclass and impose translation invariance. Get the family of SWFs where u is iso-elastic: Get the family of SWFs where u is iso-elastic: x 1 – e –  x x u(x) = ———  Same form as CARA utility function Same form as CARA utility function Parameter captures society’s absolute inequality aversion. Parameter  captures society’s absolute inequality aversion.  Similar to individual absolute risk aversion

48. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare …Can we deduce how inequality- averse “society” is? Axiomatic structure Classes Values

49. Frank Cowell: TU Lisbon – Inequality & Poverty Values: the issues In previous lecture we saw the problem of adducing social values. In previous lecture we saw the problem of adducing 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 aversion Second: What is the shape of u? Second: What is the shape of u?  What is the value of ?  What is the value of  ? Examine survey data and other sources Examine survey data and other sources

50. Frank Cowell: TU Lisbon – Inequality & Poverty Happiness and welfare? Alesina et al (2004) Alesina et al (2004) Use data on happiness from social survey Use data on happiness from social survey Construct a model of the determinants of happiness Construct a model of the determinants of happiness Use this to see if income inequality makes a difference Use this to see if income inequality makes a difference Seems to be a difference in priorities between US and Europe Seems to be a difference in priorities between US and Europe USContinental Europe Share of government in GDP 30% 45% Share of transfers in GDP 11% 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?

51. Frank Cowell: TU Lisbon – Inequality & Poverty The Alesina et al model An ordered logit An ordered logit “Happy” is categorical; built from three (0,1) variables: “Happy” is categorical; built from three (0,1) variables:  not too happy  fairly happy  very happy individual, state, time, group. individual, state, time, group. Macro variables include inflation, unemployment rate Macro variables include inflation, unemployment rate Micro variables include personal characteristics Micro variables include personal characteristics  are state, time dummies  are state, time dummies

52. Frank Cowell: TU Lisbon – Inequality & Poverty The Alesina et al. results People tend to declare lower happiness levels when inequality is high. People tend to declare lower happiness levels when inequality is high. Strong negative effects of inequality on happiness of the European poor and leftists. 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 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 Negative effect of inequality on happiness of US rich No differences across the American right and the European right. No differences across the American right and the European right. No differences between the American rich and the European rich No differences between the American rich and the European rich

53. Frank Cowell: TU Lisbon – Inequality & Poverty The shape of u: approaches Direct estimates of inequality aversion Direct estimates of inequality aversion  See Cowell-Gardiner (2000) See Cowell-Gardiner (2000) See Cowell-Gardiner (2000)  Carlsson et al (2005) Carlsson et al (2005) Carlsson et al (2005) Direct estimates of risk aversion Direct estimates of risk aversion  Use as proxy for inequality aversion  Base this on Harsanyi arguments? Indirect estimates of risk aversion Indirect estimates of risk aversion Indirect estimates of inequality aversion Indirect estimates of inequality aversion  From choices made by government  (for later…)

54. Frank Cowell: TU Lisbon – Inequality & Poverty Direct evidence on risk aversion Barsky et al (1997) estimated relative risk-aversion from survey evidence. Barsky et al (1997) estimated relative risk-aversion from survey evidence. Note dependence on how well-off people are. Note dependence on how well-off people are.

55. 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. 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. Use two models: second version includes variables to capture anticipated income growth. Again note dependence on how well-off people are. Again note dependence on how well-off people are.

56. Frank Cowell: TU Lisbon – Inequality & Poverty SWFs: Summary A small number of key axioms A small number of key axioms Generate an important class of SWFs with useful subclasses. Generate an important class of SWFs with useful subclasses. Need to make a decision on the form of the SWF 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. If we use the isoelastic model perhaps a value of around 1.5 – 2 is reasonable.

57. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare...general comparison criteria Welfare comparisons Inequality comparisons Practical tools

58. Frank Cowell: TU Lisbon – Inequality & Poverty Ranking and dominance We pick up on the problem of comparing distributions We pick up on the problem of comparing distributions Two simple concepts based on elementary axioms Two simple concepts based on elementary axioms  Anonymity  Population principle  Monotonicity  Transfer principle Illustrate these tools with a simple example Illustrate these tools with a simple example  Use the Irene-Janet representation of the distribution  Fixed population (so we don’t need pop principle)

59. Frank Cowell: TU Lisbon – Inequality & Poverty 02468 1010 1212 1414 1616 1818 2020 02468 1010 1212 1414 1616 1818 x y 2020 $ $ First-order Dominance y [1] > x [1], y [2] > x [2], y [3] > x [3] Each ordered income in y larger than that in x. Each ordered income in y larger than that in x.

60. Frank Cowell: TU Lisbon – Inequality & Poverty 02468 1010 1212 1414 1616 1818 2020 02468 1010 1212 1414 1616 1818 x y 2020 $ $ 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 y larger than that in x. Each cumulated income sum in y larger than that in x. Weaker than first-order dominance Weaker than first-order dominance

61. 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   W : W(F) =  u(x) dF(x) … and its important subclasses   W 1  W : u() increasing   W 2  W 1 : u() increasing and concave Now for the special relationship. We need to move on from the example by introducing formal tools of distributional analysis.

62. Frank Cowell: TU Lisbon – Inequality & Poverty 1 st -Order approach The basic tool is the quantile. This can be expressed in general as the functional 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 The graph of Q is Pen’s Parade Extend it to characterise the idea of dominance… Extend it to characterise the idea of dominance…

63. Frank Cowell: TU Lisbon – Inequality & Poverty An important relationship The idea of quantile (1 st -order) dominance: The idea of quantile (1 st -order) dominance: G quantile-dominates F  W(G) > W(F) for all W  W 1 A fundamental result: A fundamental result: To illustrate, use Pen's parade To illustrate, use Pen's parade G quantile-dominates F  means:   for every q, Q(G;q)  Q(F;q),   for some q, Q(G;q) > Q(F;q)

64. Frank Cowell: TU Lisbon – Inequality & Poverty First-order dominance F G Q(.; q) 1 0 q

65. Frank Cowell: TU Lisbon – Inequality & Poverty 2 nd -Order approach The basic tool is the income cumulant. This can be expressed as the functional The basic tool is the income cumulant. This can be expressed as the functional Use this to derive three intuitive concepts 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 The graph of C is the generalised Lorenz curve Again use it to characterise dominance… Again use it to characterise dominance…

66. Frank Cowell: TU Lisbon – Inequality & Poverty Another important relationship The idea of cumulant (2 nd -order) dominance: The idea of cumulant (2 nd -order) dominance: G cumulant-dominates F  W(G) > W(F) for all W  W 2 A fundamental result: A fundamental result: To illustrate, draw the GLC To illustrate, draw the GLC G cumulant-dominates F  means:   for every q, C (G;q)  C (F;q),   for some q, C (G;q) > C (F;q)

67. Frank Cowell: TU Lisbon – Inequality & Poverty Second order dominance 1 0 0 C(G;. ) C(F;. ) C(.; q) (F)(F) (G)(G) q cumulative income

68. Frank Cowell: TU Lisbon – Inequality & Poverty UK “Final income” – GLC

69. Frank Cowell: TU Lisbon – Inequality & Poverty “Original income” – GLC

70. Frank Cowell: TU Lisbon – Inequality & Poverty Ranking Distributions: Summary First-order (Parade) dominance is equivalent to ranking by quantiles. First-order (Parade) dominance is equivalent to ranking by quantiles.  A strong result. Where Parades cross, second-order methods may be appropriate. Where Parades cross, second-order methods may be appropriate. Second-order (GL)-dominance is equivalent to ranking by cumulations. Second-order (GL)-dominance is equivalent to ranking by cumulations.  Another strong result. Lorenz dominance equivalent to ranking by shares. 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. Where Lorenz-curves intersect unambiguous inequality orderings are not possible. This makes inequality measures especially interesting. This makes inequality measures especially interesting.

71. Frank Cowell: TU Lisbon – Inequality & Poverty Overview... Welfare comparisons SWFs Rankings Social welfare and needs Income Distribution and Welfare Extensions of the ranking approach

72. Frank Cowell: TU Lisbon – Inequality & Poverty Difficulties with needs Why equivalence scales? Why equivalence scales? Need a way of making welfare comparisons 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: But there are irreconcilable difficulties:  Logic  Source information  Estimation problems Perhaps a more general approach Perhaps a more general approach “Needs” seems an obvious place for explicit welfare analysis “Needs” seems an obvious place for explicit welfare analysis

73. Frank Cowell: TU Lisbon – Inequality & Poverty 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 a  A known relationship  (a)  Equivalised income given by x = y /  Equivalised income given by x = y /  is the "exchange-rate" between income types x, y 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 (y, a) Get a general result on joint distribution of (y, a)  This makes distributional comparisons multidimensional  Intrinsically very difficult (Atkinson and Bourguignon 1982) Atkinson and Bourguignon 1982Atkinson and Bourguignon 1982 To make progress: To make progress:  We simplify the structure of the problem  We again use results on ranking criteria

74. Frank Cowell: TU Lisbon – Inequality & Poverty Alternative approach to needs Atkinson and Bourguignon (1982, 1987)Cowell (2000) Based on Atkinson and Bourguignon (1982, 1987)  see also Cowell (2000)1982Cowell (2000)1982Cowell (2000) Sort individuals be into needs groups N 1, N 2,… Suppose a proportion  j are in group N j. Then social welfare can be written: To make this operational… Utility people get from income depends on their needs:

75. Frank Cowell: TU Lisbon – Inequality & Poverty 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 W 3  W 2 be the subclass of welfare functions for which the above is positive and decreasing in y Suppose we want j=1,2,… to reflect decreasing order of need. Suppose we want j=1,2,… to reflect decreasing order of need. Consider need and the marginal utility of income: Consider need and the marginal utility of income:

76. 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) Theorem (Atkinson and Bourguignon 1987) A UK example 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

77. Frank Cowell: TU Lisbon – Inequality & Poverty Household types in Economic Trends 2+ads,3+chn/3+ads,chn 2+ads,3+chn/3+ads,chn 2 adults with 2 children 2 adults with 2 children 1 adult with children 1 adult with children 2 adults with 1 child 2 adults with 1 child 2+ adults 0 children 2+ adults 0 children 1 adult, 0 children 1 adult, 0 children

78. Frank Cowell: TU Lisbon – Inequality & Poverty Impact of Taxes and Benefits. UK 1991. Sequential GLCs (1)

79. Frank Cowell: TU Lisbon – Inequality & Poverty Impact of Taxes and Benefits. UK 1991. Sequential GLCs (2)

80. Frank Cowell: TU Lisbon – Inequality & Poverty Conclusion Axiomatisation of welfare can be accomplished using just a few basic principles Axiomatisation of welfare can be accomplished using just a few basic principles Ranking criteria can be used to provide broad judgments Ranking criteria can be used to provide broad judgments These may be indecisive, so specific SWFs could be used 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: The same basic framework of distributional analysis can be extended to a number of related problems: For example inequality and poverty… For example inequality and poverty…  …in next lecture

81. 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 Alesina, A., Di Tella, R. and MacCulloch, R (2004) 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. (1970) 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. (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 multi- dimensional distributions of economic status,” Review of Economic Studies, 49, 183-201 Atkinson, A. B. and Bourguignon, F. (1982) “The comparison of multi- dimensional distributions of economic status,” Review of Economic Studies, 49, 183-201 Atkinson, A. B. and Bourguignon, F. (1982) Atkinson, A. B. and Bourguignon, F. (1982) 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 Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997)

82. 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 Blundell, R., Browning, M. and Meghir, C. (1994) Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) “Are people inequality averse or just risk averse?” Economica, 72, Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) 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. (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. (2000) Cowell, F. A. (2000) Cowell, F. A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research Paper 202, Office of Fair Training, Salisbury Square, London Cowell, F. A. and Gardiner, K.A. (2000) Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, 348-361 Dalton, H. (1920) Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London Pigou, A. C. (1912) Wealth and Welfare, Macmillan, London