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Frank Cowell: UB Public Economics Distributional Equity, Social Welfare Public Economics: University of Barcelona Frank Cowell

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1 Frank Cowell: UB Public Economics Distributional Equity, Social Welfare Public Economics: University of Barcelona Frank Cowell http://darp.lse.ac.uk/ub June 2005

2 Frank Cowell: UB Public Economics 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: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare How to represent problems in distributional analysis Income distributions Comparisons

4 Frank Cowell: UB Public Economics 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: UB Public Economics x 0.20.8 1 0 x 0.8 q "income" (height) proportion of the population x 0.2 Pen's parade Now for some formalisation:   Plot income against proportion of population   Parade in ascending order of "income" / height

6 Frank Cowell: UB Public Economics 0 1 x F(x)F(x) x0x0 F(x0)F(x0) A distribution function

7 Frank Cowell: UB Public Economics 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: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare Methods and criteria of distributional analysis Income distributions Comparisons

9 Frank Cowell: UB Public Economics 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: UB Public Economics Some comparisons self-evident... 0123456789 10 $ PR 0123456789 $ P R 0123456789 $ PR 0123456789 $ R P

11 Frank Cowell: UB Public Economics 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 or other concepts of equity will be almost trivial. 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 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 But, consider a simple example with three persons and fixed incomes

12 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare How to incorporate fundamental principles Axiomatic structure Classes Values

17 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 0123456789 10111213 $ x 0123456789 10111213 $ x' Anonymity W(x′) = W(x)

22 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 0123456789 10 $ 0123456789 $ Population replication W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y)

25 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics Which group seems to have the more unequal distribution? 0123456789 10111213 0123456789 10111213 $ $

33 Frank Cowell: UB Public Economics 0123456789 10111213 0123456789 10111213 $ $ The issue viewed as two groups

34 Frank Cowell: UB Public Economics Focus on just the affected persons 0123456789 10111213 0123456789 10111213 $ $

35 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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 will be applied to other types of tool This technique will be applied to other types of tool  Inequality  Poverty  Deprivation.

42 Frank Cowell: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare Categorising important types Axiomatic structure Classes Values

43 Frank Cowell: UB Public Economics 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 1.   But W excludes some well-known welfare criteria

44 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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 1: Same as that in lecture 1:  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 Atkinson (1970)

47 Frank Cowell: UB Public Economics 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 – exp –  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: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare …Can we deduce how inequality- averse “society” is? Axiomatic structure Classes Values

49 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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: UB Public Economics 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

54 Frank Cowell: UB Public Economics 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: UB Public Economics 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: UB Public Economics Indirect evidence on social values Assume constant absolute sacrifice Assume constant absolute sacrifice Assume isoelastic social utility Assume isoelastic social utility Then estimate  from Then estimate  from Results for UK: Results for UK:

57 Frank Cowell: UB Public Economics 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.

58 Frank Cowell: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare...rankings, orderings and practical tools

59 Frank Cowell: UB Public Economics 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)

60 Frank Cowell: UB Public Economics 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.

61 Frank Cowell: UB Public Economics 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

62 Frank Cowell: UB Public Economics Social-welfare criteria and dominance Why are these concepts useful? First these concepts and classes of SWF Recall the class of additive SWFs   W : W(F) =  u(x) dF(x) … and its important subclasses   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.

63 Frank Cowell: UB Public Economics 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…

64 Frank Cowell: UB Public Economics 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)

65 Frank Cowell: UB Public Economics First-order dominance F G Q(.; q) 1 0 q

66 Frank Cowell: UB Public Economics 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   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…

67 Frank Cowell: UB Public Economics 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)

68 Frank Cowell: UB Public Economics Second order dominance 1 0 0 C(G;. ) C(F;. ) C(.; q) (F)(F) (G)(G) q cumulative income practical example, UK

69 Frank Cowell: UB Public Economics Application of ranking The tax and -benefit system maps one distribution into another... The tax and -benefit system maps one distribution into another... Use ranking tools to assess the impact of this in welfare terms. Use ranking tools to assess the impact of this in welfare terms. Typically this uses one or other concept of Lorenz dominance. Typically this uses one or other concept of Lorenz dominance.

70 Frank Cowell: UB Public Economics UK “Final income” – GLC

71 Frank Cowell: UB Public Economics “Original income” – GLC

72 Frank Cowell: UB Public Economics 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.

73 Frank Cowell: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare Extensions of the ranking approach

74 Frank Cowell: UB Public Economics 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

75 Frank Cowell: UB Public Economics 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) To do this need to recall results on ranking criteria To do this need to recall results on ranking criteria

76 Frank Cowell: UB Public Economics Social-welfare criteria Recall the standard classes of SWF Additive SWFs   W : W(F) =  u(x) dF(x) With principal subclasses   W 1  W : u() increasing   W 2  W 1 : u() increasing and concave Recall the second-order result G cumulant-dominates F  W(G) > W(F) for all W  W 2 Make progress by further restricting subclasses

77 Frank Cowell: UB Public Economics Alternative approach to needs Sort individuals 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… The utility people get from income depends on their needs:

78 Frank Cowell: UB Public Economics 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:

79 Frank Cowell: UB Public Economics Atkinson-Bourguignon result. Let F (  j) denote distribution for all needs groups up to and including j. Distinguish this from the marginal distribution Distinguish this from the marginal distribution Theorem: Theorem: A UK example

80 Frank Cowell: UB Public Economics 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

81 Frank Cowell: UB Public Economics Impact of Taxes and Benefits. UK 1991. Sequential GLCs (1)

82 Frank Cowell: UB Public Economics Impact of Taxes and Benefits. UK 1991. Sequential GLCs (2)

83 Frank Cowell: UB Public Economics Needs: summary Doing without equivalence scales seems attractive Doing without equivalence scales seems attractive  Removes a level of arbitrariness  Simplifies computation? But the sequential dominance principle is problematic 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? 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

84 Frank Cowell: UB Public Economics Overview... Welfare comparisons SWFs Rankings Welfare and needs Compensation and responsibility Equity and social welfare What should be equalised?

85 Frank Cowell: UB Public Economics Responsibility (1) Standard approach to case for redistribution 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 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” May need to revise our concept of “equality” or “equal treatment”

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

87 Frank Cowell: UB Public Economics Basic structure Anonymity Anonymity Each person i has a vector of attributes a i : Each person i has a vector of attributes a i :  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(a i ) The income function f maps attributes into incomes f(a i ) A distribution rule F: A distribution rule F: Profile of attributes

88 Frank Cowell: UB Public Economics Responsibility: Rules Bossert and Fleurbaey (1996) Bossert and Fleurbaey (1996) Equal Income for Equal Responsibility Equal Income for Equal Responsibility  Focus on distribution itself  Full compensation Equal Transfers for Equal C-attributes Equal Transfers for Equal C-attributes  Focus on changes in distribution  Strict Compensation

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

90 Frank Cowell: UB Public Economics Compromise (1) Insist on Full compensation (EIER) Weaken ETEC Egalitarian-equivalent mechanisms Every agent has a post-tax income equal to   the pre-tax income earned given reference compensation characteristics plus...   a uniform transfer Reference profile

91 Frank Cowell: UB Public Economics Compromise (2) Insist on strict compensation (ETEC) Weaken EIER Conditionally egalitarian mechanisms Conditionally egalitarian mechanisms Every agent k is guaranteed the average income of a hypothetical economy Every agent k is guaranteed the average income of a hypothetical economy  In this economy all agents have characteristics equal to reference profile Reference profile

92 Frank Cowell: UB Public Economics 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: Move on to consider inequality and poverty… Move on to consider inequality and poverty… …in the next lecture component. …in the next lecture component.


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