1. Frank Cowell: EC426 Public Economics MSc Public Economics 2011/12 http://darp.lse.ac.uk/ec426 Equity, Social Welfare and Taxation Frank A. Cowell 10 October 2011

2. Frank Cowell: EC426 Public Economics Welfare and distributional analysis We’ve seen welfare-economics basis for redistribution as a policy objective We’ve seen welfare-economics basis for redistribution as a policy objective  how to assess the impact and effectiveness of such policy?  need appropriate criteria for comparing distributions of income  issues of distributional analysis (Cowell 2008, 2011) Cowell 20082011Cowell 20082011 Distributional analysis covers broad class of economic problems Distributional analysis covers broad class of economic problems  inequality  social welfare  poverty Similar techniques Similar techniques  rankings  measures Four basic components need to be clarified Four basic components need to be clarified  “income” concept  “income receiving unit” concept  a distribution  method of assessment or comparison

3. Frank Cowell: EC426 Public Economics 1: “Irene and Janet” approach 0 Irene's income Janet's income Karen's income i x k x x j ray of equality   A representation with 3 incomes   Income distributions with given total X x i + x j + x k = X   Feasible income distributions given X   Equal income distributions  particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria

4. Frank Cowell: EC426 Public Economics 2: The parade x 0.20.8 1 0 x 0.8 q "income" (height) proportion of the population x 0.2  especially useful in cases where it is appropriate to adopt a parametric model of income distribution  Pen (1971) 0 1 x F(x)F(x) x0x0 F(x0)F(x0) distribution function F(∙)   Plot income against proportion of population   Parade in ascending order of "income" / height   Related to familiar statistical concept

5. Frank Cowell: EC426 Public Economics Overview... Welfare axioms Rankings Equity and social welfare Taxation and sacrifice Equity, Social Welfare, Taxation Basic principles of distributional comparisons

6. Frank Cowell: EC426 Public Economics Social-welfare functions A standard approach to a method of assessment A standard approach to a method of assessment 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 1, x 2,…,x n ) 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

7. Frank Cowell: EC426 Public Economics The approach 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) Appendix A Follow the approach of Amiel-Cowell (1999) Appendix AAmiel-Cowell (1999)Amiel-Cowell (1999)

8. Frank Cowell: EC426 Public Economics SWF axioms Anonymity. Suppose x′ is a permutation of x. Then: W(x′) = W(x) Population principle. W(x)  W(y)  W(x,x,…,x)  W(y,y,…,y) Decomposability. Suppose x' is formed by joining x with z and y' is formed by joining y with z. Then : W(x)  W(y)  W(x')  W(y') Monotonicity. W(x 1,x 2..., x i + ,..., x n ) > W(x 1,x 2,..., x i,..., x n ) Transfer principle. (Dalton 1920) Suppose x i < x j then for small  :Dalton 1920 W(x 1,x 2..., x i + ,..., x j  ,..., x n ) > W(x 1,x 2,..., x i,..., x n ) Scale invariance. W(x)  W(y)  W( x)  W( y)

9. Frank Cowell: EC426 Public Economics Classes of SWFs (1) Use axioms to characterise important classes of SWF Use axioms to characterise important classes of SWF Anonymity and population principle imply we can write SWF in either Irene-Janet form or F form Anonymity and population principle imply we can write SWF in either Irene-Janet form or F form  most modern approaches use these assumptions  but may need to standardise for needs etc Introduce decomposability and you get class of Additive SWFs W : Introduce decomposability and you get class of Additive SWFs W :  W(x) =   u(x i )  or equivalently in F-form W(F) =  u(x) dF(x)  think of u() as social utility or social evaluation function The class W is of great importance The class W is of great importance  But W excludes some well-known welfare criteria

10. Frank Cowell: EC426 Public Economics Classes of SWFs (2) From W we get important subclasses If we impose monotonicity we get   W 1  W : u() increasing   subclass where marginal social utility always positive If we further impose the transfer principle we get   W 2  W 1 : u() increasing and concave   subclass where marginal social utility is positive and decreasing We often need to use these special subclasses Illustrate their properties with a simple example…

11. Frank Cowell: EC426 Public Economics Overview... Welfare axioms Rankings Equity and social welfare Taxation and sacrifice Equity, Social Welfare, Taxation Classes of SWF and distributional comparisons

12. Frank Cowell: EC426 Public Economics Ranking and dominance Consider problem of comparing distributions Consider problem of comparing distributions Introduce two simple concepts Introduce two simple concepts  first illustrate using the Irene-Janet representation  take income vectors x and y for a given n First-order dominance: 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 Second-order dominance: Second-order dominance:  y (1) > x (1), y (1) +y (2) > x (1) +x (2), y (1) +y (2) +…+ y (n) > x (1) +x (2) …+ x (n)  Each cumulated income sum in y larger than that in x  Weaker than first-order dominance Need to generalise this a little Need to generalise this a little  given anonymity, population principle can represent distributions in F-form  q: population proportion (0 ≤ q ≤ 1)  F(x): proportion of population with incomes ≤ x   (F): mean of distribution F

13. Frank Cowell: EC426 Public Economics 1 st -Order approach Basic tool is the quantile, expressed as Basic tool is the quantile, expressed as Q(F; q) := inf {x | F(x)  q} = x q  “smallest income such that cumulative frequency is at least as great as q” Use this to derive a number of intuitive concepts Use this to derive a number of intuitive concepts  interquartile range, decile-ratios, semi-decile ratios  graph of Q is Pen’s Parade Also to characterise 1 st -order (quantile) dominance: Also to characterise 1 st -order (quantile) 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: A fundamental result:  G quantile-dominates F iff  W(G) > W(F) for all W  W 1 Illustrate using Parade:

14. Frank Cowell: EC426 Public Economics Parade and 1 st -order dominance F G Q(.; q) 1 0 q   Plot quantiles against proportion of population   Parade for distribution F again   Parade for distribution G  In this case G clearly quantile-dominates F  But (as often happens) what if it doesn’t?  Try second-order method

15. Frank Cowell: EC426 Public Economics 2 nd -Order approach Basic tool is the income cumulant, expressed as Basic tool is the income cumulant, expressed as C(F; q) := ∫ Q(F; q) x dF(x)  “The sum of incomes in the Parade, up to and including position q” Use this to derive a number of intuitive concepts Use this to derive a number of intuitive concepts  the “shares” ranking, Gini coefficient  graph of C is the generalised Lorenz curve Also to characterise the idea of 2 nd -order (cumulant) dominance: Also to characterise the idea of 2 nd -order (cumulant) 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 (Shorrocks 1983): A fundamental result (Shorrocks 1983):Shorrocks 1983Shorrocks 1983  G cumulant-dominates F iff  W(G) > W(F) for all W  W 2

16. Frank Cowell: EC426 Public Economics GLC and 2 nd -order dominance 1 0 0 C(G;. ) C(F;. ) C(.; q) (F)(F) (G)(G) q cumulative income   Plot cumulations against proportion of population   GLC for distribution F   GLC for distribution G  Intercept on vertical axis is at mean income

17. Frank Cowell: EC426 Public Economics 2 nd -Order approach (continued) A useful tool: the share of the proportion q of distribution F is L(F;q) := C(F;q) /  (F) A useful tool: the share of the proportion q of distribution F is L(F;q) := C(F;q) /  (F)  “income cumulation at divided q by total income” Yields Lorenz dominance, or the “shares” ranking: Yields Lorenz dominance, or the “shares” ranking:  “G Lorenz-dominates F”  means:  for every q, L(G;q)  L(F;q),  for some q, L(G;q) > L(F;q) Another fundamental result (Atkinson 1970): Another fundamental result (Atkinson 1970):Atkinson 1970Atkinson 1970  For given , G Lorenz-dominates F iff  W(G) > W(F) for all W  W 2 Illustrate using Lorenz curve:

18. Frank Cowell: EC426 Public Economics Lorenz curve and ranking 00.20.40.60.81 0 0.2 0.4 0.6 0.8 1 proportion of income proportion of population L(G;.) L(F;.) L(.; q) q   Plot shares against proportion of population   Lorenz curve for distribution F   Lorenz curve for distribution G  In this case G clearly Lorenz-dominates F  So F displays more inequality than G  But (as often happens) what if it doesn’t?  No clear statement about inequality (or welfare) is possible without further information   Perfect equality

19. Frank Cowell: EC426 Public Economics Overview... Welfare axioms Rankings Equity and social welfare Taxation and sacrifice Equity, Social Welfare, Taxation Quantifying social values

20. Frank Cowell: EC426 Public Economics Equity and social welfare So far we have just general principles So far we have just general principles  may lead to ambiguous results  need a tighter description of social-welfare? Consider “reduced form” of social welfare Consider “reduced form” of social welfare  W =  ( , I)   (F) is mean of distribution F  I  I(F) is inequality of distribution F   embodies trade-off between objectives I can be taken as an “equity” criterion I can be taken as an “equity” criterion  from same roots as SWF?  or independently determined? Consider a “natural” definition of inequality Consider a “natural” definition of inequality

21. Frank Cowell: EC426 Public Economics 01 1 L(.; q) q Gini coefficient Redraw Lorenz diagram Redraw Lorenz diagram A “natural” inequality measure…? A “natural” inequality measure…?  normalised area above Lorenz curve  can express this also in I-J terms Also (equivalently) represented as normalised difference between income pairs: Also (equivalently) represented as normalised difference between income pairs:  In F-form:  In Irene-Janet terms: Intuition is neat, but Intuition is neat, but  inequality index is clearly arbitrary  can we find one based on welfare criteria?

22. Frank Cowell: EC426 Public Economics SWF and inequality O xixi xjxj   The Irene &Janet diagram   A given distribution   Distributions with same mean   Contours of the SWF E (F)(F) (F)(F) F   Construct an equal distribution E such that W(E) = W(F)   Equally-Distributed Equivalent income   Social waste from inequality   Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality   contour: x values such that W(x) = const do this for special case

23. Frank Cowell: EC426 Public Economics An important family of SWF 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 –   has 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)

24. Frank Cowell: EC426 Public Economics Welfare-based inequality From the concept of social waste Atkinson (1970) suggested an inequality measure: From the concept of social waste Atkinson (1970) suggested an inequality measure:Atkinson (1970)Atkinson (1970)  (F)  (F) I(F) = 1 – ——  (F) Atkinson further assumed Atkinson further assumed  additive SWF, W(F) =  u(x) dF(x)  isoelastic u So inequality takes the form So inequality takes the form But what value of inequality aversion parameter  But what value of inequality aversion parameter 

25. Frank Cowell: EC426 Public Economics Values: the issues SWF is central to public policy making SWF is central to public policy making  Practical example in HM Treasury (2003) pp 93-94 HM Treasury (2003)HM Treasury (2003)  We need to focus on two questions… First: do people care about distribution? First: do people care about distribution?  Experiments suggest they do – Carlsson et al (2005) Carlsson et al (2005)Carlsson et al (2005)  Do social and economic factors make a difference? Second: What is the shape of u? (Cowell-Gardiner 2000) Second: What is the shape of u? (Cowell-Gardiner 2000)Cowell-Gardiner 2000Cowell-Gardiner 2000  Direct estimates of inequality aversion  Estimates of risk aversion as proxy for inequality aversion  Indirect estimates of risk aversion  Indirect estimates of inequality aversion from choices made by government

26. Frank Cowell: EC426 Public Economics Preferences, happiness and welfare? Ebert, U. and Welsch, H. (2009) Ebert, U. and Welsch, H. (2009) Ebert, U. and Welsch, H. (2009) Ebert, U. and Welsch, H. (2009)   Evaluate subjective well being as a function of personal and environmental data using  form   Examine which inequality index seems to fit preferences best Alesina et al (2004) 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 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% Do people in Europe care more about inequality? Do people in Europe care more about inequality?

27. Frank Cowell: EC426 Public Economics The Alesina et al model Ordered logit model Ordered logit model  “Happy” is categorical: “ not too happy”, “fairly happy”, “very happy”  individual, state, time, group.  Macro variables include inflation, unemployment rate  Micro variables include personal characteristics   are state, time dummies Results Results  People declare lower happiness levels when inequality is high.  Strong negative effect of inequality on happiness of the European poor and leftists  No effects of inequality on happiness of US poor and left-wingers  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

28. Frank Cowell: EC426 Public Economics Inequality aversion and Elasticity of MU Consistent inequality preferences? Consistent inequality preferences?  Preference reversals (Amiel et al 2008) Amiel et al 2008Amiel et al 2008 What value for  ? What value for  ?  from happiness studies 1.0 to 1.5 (Layard et al 2008) Layard et al 2008Layard et al 2008  related to extent of inequality in the country? (Lambert et al 2003) Lambert et al 2003Lambert et al 2003  affected by way the question is put? (Pirttilä and Uusitalo 2010) Pirttilä and Uusitalo 2010Pirttilä and Uusitalo 2010 Evidence on risk aversion is mixed Evidence on risk aversion is mixed  direct survey evidence suggests estimated relative risk-aversion 3.8 to 4.3 (Barsky et al 1997) Barsky et al 1997Barsky et al 1997  indirect evidence (from estimated life-cycle consumption model) suggests 0.4 to 1.4 (Blundell et al 1994) Blundell et al 1994Blundell et al 1994  in each case depends on how well-off people are

29. Frank Cowell: EC426 Public Economics Overview... Welfare axioms Rankings Equity and social welfare Taxation and sacrifice Equity, Social Welfare, Taxation Does structure of taxation make sense in welfare terms?

30. Frank Cowell: EC426 Public Economics Another application of ranking Tax and benefit system maps one distribution into another Tax and benefit system maps one distribution into another  c = y  T(y)  y: pre-tax income c: post-tax income 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 Linked to effective tax progression Linked to effective tax progression  T is progressive if c Lorenz-dominates y  see Jakobsson (1976) Jakobsson (1976)Jakobsson (1976) What Lorenz ranking would we expect to apply to these 5 concepts? What Lorenz ranking would we expect to apply to these 5 concepts? original income + cash benefits gross income - direct taxes disposable income - indirect taxes post-tax income + non-cash benefits final income

31. Frank Cowell: EC426 Public Economics Impact of Taxes and Benefits. UK 2006/7. Lorenz Curve   + cash benefits    direct taxes    indirect taxes   + noncash benefits  Big effect from benefits side  Modest impact of taxes  Direct and indirect taxes work in opposite directions

32. Frank Cowell: EC426 Public Economics Implied tax rates in Economic and Labour Market Review  Formerly Economic Trends. Taxes as proportion of gross income – see Jones, (2008) Jones, (2008)Jones, (2008)

33. Frank Cowell: EC426 Public Economics Tax-benefit example: implications We might have guessed the outcome of example We might have guessed the outcome of example In most countries: In most countries:  income tax progressive  so are public expenditures  but indirect tax is regressive  so Lorenz-dominance is not surprising. In many countries there is a move toward greater reliance on indirect taxation In many countries there is a move toward greater reliance on indirect taxation  political convenience?  administrative simplicity? Structure of taxation should not be haphazard Structure of taxation should not be haphazard How should the y → c mapping be determined? How should the y → c mapping be determined?

34. Frank Cowell: EC426 Public Economics Tax Criteria What broad principles should tax reflect? What broad principles should tax reflect?  “benefit received:” pay for benefits you get through public sector  “sacrifice:” some principle of equality applied across individuals  see for a reconciliation of these two  see Neill (2000) for a reconciliation of these twoNeill (2000) The sacrifice approach The sacrifice approach  Reflect views on equity in society.  Analyse in terms of income distribution  Compare distribution of y with distribution of y  T? Three types of question: Three types of question: 1. what’s a “fair” or “neutral” way to reduce incomes…? 2. when will a tax system induce L-dominance? 3. what is implication of imposing uniformity of sacrifice?

35. Frank Cowell: EC426 Public Economics Amiel-Cowell (1999) approach Irene's income Janet's income xixi xjxj 0 ray of equality   The Irene &Janet diagram   Initial distribution   Possible directions for a “fair” tax ll ll   An iso-inequality path?   1 Scale-independent iso-inequality   1 Proportionate reductions are “fair”   2 Translation-independent iso-inequality   2 Absolute reductions are “fair”   3 “Intermediate” iso-inequality   3 Affine reductions are “fair”   4 No iso-inequality direction   4 Amiel-Cowell showed that individuals perceive inequality comparisons this way.   Based on Dalton (1920) conjectureDalton (1920)

36. Frank Cowell: EC426 Public Economics Tax and the Lorenz curve Assume a tax function T(∙) based on income Assume a tax function T(∙) based on income  a person with income y, pays an amount T(y)  disposable income given by c := y  T(y) Compare distribution of y with that of c Compare distribution of y with that of c  will c be more equally distributed than y?  will T(∙) produce more equally distributed c than T*(∙)?  if so under what conditions? Define residual progression of T Define residual progression of T  [dc/dy][y/c] = [1  T(y)] /[1  y /T(y)] Disposable income under T* L-dominates that under T* iff Disposable income under T* L-dominates that under T* iff  residual progression of T is higher than that of T*…  …for all y  see  see Jakobsson (1976)Jakobsson (1976)

37. Frank Cowell: EC426 Public Economics A sacrifice approach Suppose utility function U is continuous and increasing Suppose utility function U is continuous and increasing Definition of absolute sacrifice: Definition of absolute sacrifice:  (y, T) displays at least as much (absolute) sacrifice as (y *, T * )…  …if U(y)  U(y  T) ≥ U(y * )  U(y *  T * ) Equal absolute sacrifice is scale invariant iff Equal absolute sacrifice is scale invariant iff y 1 –  – 1 U(y) = ————  1 –   …or a linear transformation of this  (  (Young 1987)Young 1987 Apply sacrifice criterion to actual tax schedules T(∙) Apply sacrifice criterion to actual tax schedules T(∙)  does this for US tax system  Young (1990) does this for US tax system Young (1990)  Consider a UK application

38. Frank Cowell: EC426 Public Economics Applying sacrifice approach Assume simple form of social welfare function Assume simple form of social welfare function  social welfare defined in terms of income  drop distinction between U and u Use the equal absolute sacrifice principle Use the equal absolute sacrifice principle  u(y)  u(y  T(y)) = const If insist on scale invariance u is isoelastic If insist on scale invariance u is isoelastic  y 1 –   [y  T(y)] 1 –  = const   is again inequality aversion Differentiate and rearrange: Differentiate and rearrange:  y –   [1  T(y)] [y  T(y)] –  = 0  log[1  T(y)] =  log ([ y  T(y) / y]) Estimate  from actual structure T(∙) Estimate  from actual structure T(∙)  get implied inequality aversion  For income tax (1999/2000):  =   For IT and NIC (1999/2000):  =   see Cowell-Gardiner (2000) Cowell-Gardiner (2000)Cowell-Gardiner (2000)

39. Frank Cowell: EC426 Public Economics Conclusion Axiomatisation of welfare needs just a few basic principles Axiomatisation of welfare needs just a few basic principles  anonymity  population principle  decomposability  monotonicity  principle of transfers Basic framework of distributional analysis can be extended to related problems: Basic framework of distributional analysis can be extended to related problems:  for example inequality and poverty…  …in second term 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  scale invariant?  perhaps a value for  of around 0.7 – 2 Welfare criteria can be used to provide taxation principles Welfare criteria can be used to provide taxation principles

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