1. Frank Cowell: HMRC-HMT Economics of Taxation HMRC-HMT Economics of Taxation 2011 http://darp.lse.ac.uk/HMRC-HMT 9.1 Distributional Analysis and Methods 14 December 2011

2. Frank Cowell: HMRC-HMT Economics of Taxation 2 Overview... The basics SWF and rankings Inequality measures Evidence Distributional Analysis and Methods How to represent problems in distributional analysis

3. Frank Cowell: HMRC-HMT Economics of Taxation 3 Distributional analysis Covers a broad class of economic problems Covers a 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  usually equivalised, disposable income  but could be other income types, consumption, wealth…  “income receiving unit” concept  usually the individual  but could also be family, household  a distribution  method of assessment or comparison See Cowell (2000, 2008, 2011), Sen and Foster (1997) See Cowell (2000, 2008, 2011), Sen and Foster (1997)Cowell (20002008Cowell (20002008

4. Frank Cowell: HMRC-HMT Economics of Taxation 4 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

5. Frank Cowell: HMRC-HMT Economics of Taxation 5 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 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  Pen (1971)

6. Frank Cowell: HMRC-HMT Economics of Taxation 6 Overview... The basics SWF and rankings Inequality measures Evidence Distributional Analysis and Methods How to incorporate fundamental principles

7. Frank Cowell: HMRC-HMT Economics of Taxation 7 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) 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

8. Frank Cowell: HMRC-HMT Economics of Taxation 8 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) Appendix A Follow the approach of Amiel-Cowell (1999) Appendix AAmiel-Cowell (1999)Amiel-Cowell (1999)

9. Frank Cowell: HMRC-HMT Economics of Taxation 9 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)

10. Frank Cowell: HMRC-HMT Economics of Taxation 10 Classes of SWFs Anonymity and population principle:   can write SWF in either Irene-Janet form or F form   may need to standardise for needs etc Introduce decomposability   get class of Additive SWFs W :  W(x) =   x i  W(x) =   u(x i )   or equivalently W(F) =  u(x) dF(x) 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

11. Frank Cowell: HMRC-HMT Economics of Taxation 11 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 –   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)

12. Frank Cowell: HMRC-HMT Economics of Taxation 12 Ranking and dominance 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 Need to generalise this a little Need to generalise this a little  represent distributions in F-form (anonymity, population principle)  q: population proportion (0 ≤ q ≤ 1)  F(x): proportion of population with incomes ≤ x   (F): mean of distribution F

13. Frank Cowell: HMRC-HMT Economics of Taxation 13 1 st -Order approach Basic tool is 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   interquartile range, decile-ratios, semi-decile ratios   graph of Q is Pen’s Parade Also to characterise the idea of 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:   G quantile-dominates F iff  W(G) > W(F) for all W  W 1 Illustrate using Parade:

14. Frank Cowell: HMRC-HMT Economics of Taxation 14 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: HMRC-HMT Economics of Taxation 15 2 nd -Order approach 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   the “shares” ranking, Gini coefficient   graph of C is the generalised Lorenz curve 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):Shorrocks 1983   G cumulant-dominates F iff  W(G) > W(F) for all W  W 2 Illustrate using GLC:

16. Frank Cowell: HMRC-HMT Economics of Taxation 16 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: HMRC-HMT Economics of Taxation 17 2 nd -Order approach (continued) A useful tool: the share of the proportion q of distribution F is A useful tool: the share of the proportion q of distribution F is L(F;q) := C(F;q) /  (F)   “income cumulation at q divided 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: HMRC-HMT Economics of Taxation 18 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: HMRC-HMT Economics of Taxation 19 Overview... The basics SWF and rankings Inequality measures Evidence Distributional Analysis and Methods Three ways of approaching an index

20. Frank Cowell: HMRC-HMT Economics of Taxation 20 1: Intuitive inequality measures Perhaps borrow from other disciplines… Perhaps borrow from other disciplines… A standard measure of spread… A standard measure of spread…  variance But maybe better to use a normalised version But maybe better to use a normalised version  coefficient of variation Comparison between these two is instructive Comparison between these two is instructive  Same iso-inequality contours for a given .  Different behaviour as  alters Alternative intuition based on Lorenz approach Alternative intuition based on Lorenz approach Lorenz comparisons (2 nd -order) may be indecisive Lorenz comparisons (2 nd -order) may be indecisive  problem is essentially one of aggregation of information  so use the diagram to “force a solution”

21. Frank Cowell: HMRC-HMT Economics of Taxation 21 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:

22. Frank Cowell: HMRC-HMT Economics of Taxation 22 2: 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

23. Frank Cowell: HMRC-HMT Economics of Taxation 23 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

24. Frank Cowell: HMRC-HMT Economics of Taxation 24 3: “Distance” and inequality SWF route provides a coherent approach to inequality SWF route provides a coherent approach to inequality But do we need to use an approach via social welfare? But do we need to use an approach via social welfare?  it’s indirect  maybe introduces unnecessary assumptions Alternative route: “distance” and inequality Alternative route: “distance” and inequality Can see inequality as a deviation from the norm Can see inequality as a deviation from the norm  norm in this case is perfect equality  …but what distance concept to use?

25. Frank Cowell: HMRC-HMT Economics of Taxation 25 Generalised Entropy measures Defines a class of inequality measures, given parameter  : Defines a class of inequality measures, given parameter  : GE class is rich. Some important special cases GE class is rich. Some important special cases  for  < 1 it is ordinally equivalent to Atkinson (  = 1 –  )   = 0: –  log (x /  (F)) dF(x) (mean logarithmic deviation)   = 1:  [ x /  (F)] log (x /  (F)) dF(x) (the Theil index)  or  = 2 it is ordinally equivalent to (normalised) variance. Parameter  can be assigned any positive or negative value Parameter  can be assigned any positive or negative value  indicates sensitivity of each member of the class   large and positive gives a “top-sensitive” measure   negative gives a “bottom-sensitive” measure  each  gives a specific distance concept

26. Frank Cowell: HMRC-HMT Economics of Taxation 26 Inequality contours Each  defines a set contours in the I-J diagram Each  defines a set contours in the I-J diagram  each related to a different concept of distance For example For example  the Euclidian case  other types  25   −   −   2

27. Frank Cowell: HMRC-HMT Economics of Taxation 27 Overview... The basics SWF and rankings Inequality measures Evidence Distributional Analysis and Methods Attitudes and perceptions

28. Frank Cowell: HMRC-HMT Economics of Taxation 28 Views on distributions Do people make distributional comparisons in the same way as economists? Do people make distributional comparisons in the same way as economists? Summarised from Amiel-Cowell (1999) Summarised from Amiel-Cowell (1999)Amiel-Cowell (1999)Amiel-Cowell (1999)  examine proportion of responses in conformity with standard axioms  both directly in terms of inequality and in terms of social welfare InequalitySWF NumVerbalNumVerbal NumVerbalNumVerbal Anonymity83%72%66%54% Population58%66%66%53% Decomposability57%40%58%37% Monotonicity--54%55% Transfers35%31%47%33% Scale indep.51%47%--

29. Frank Cowell: HMRC-HMT Economics of Taxation 29 Inequality aversion Are people averse to inequality? Are people averse to inequality?  evidence of both inequality and risk aversion (  evidence of both inequality and risk aversion (Carlsson et al 2005)Carlsson et al 2005  risk-aversion may be used as proxy for inequality aversion? ( )  risk-aversion may be used as proxy for inequality aversion? (Cowell and Gardiner 2000)Cowell and Gardiner 2000 What value for  ? What value for  ?   affected by way the question is put? (Pirttilä and Uusitalo 2010)Pirttilä and Uusitalo 2010  evidence on risk aversion is mixed  high values from survey evidence (Barsky et al 1997) Barsky et al 1997Barsky et al 1997  much lower from savings analysis (Blundell et al 1994) Blundell et al 1994Blundell et al 1994  from happiness studies 1.0 to 1.5 (Layard et al 2008) Layard et al 2008Layard et al 2008  related to the extent of inequality in the country? (Lambert et al 2003) Lambert et al 2003Lambert et al 2003  perhaps a value of around 0.7 – 2 is reasonable  see also HM Treasury (2003) page 94 HM Treasury (2003)HM Treasury (2003)

30. Frank Cowell: HMRC-HMT Economics of Taxation 30 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 provide broad judgments Ranking criteria can provide broad judgments But may be indecisive, so specific SWFs could be used But may be indecisive, so specific SWFs could be used  What shape should they have?  How do we specify them empirically? Several axioms survive scrutiny in experiment Several axioms survive scrutiny in experiment  but Transfer Principle often rejected

31. Frank Cowell: HMRC-HMT Economics of Taxation 31 References (1) Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality, Cambridge University Press Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality, Cambridge University Press Amiel, Y. and Cowell, F.A. (1999) Amiel, Y. and Cowell, F.A. (1999) Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, 244-263 Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, 244-263 Atkinson, A. B. (1970) Atkinson, A. B. (1970) 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) “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) Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997) 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) “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) 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) “Are people inequality averse or just risk averse?” Economica, 72, Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) 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. (2008) “Inequality: measurement,” The New Palgrave, second edition * Cowell, F.A. (2008) “Inequality: measurement,” The New Palgrave, second edition * Cowell, F.A. (2008) * Cowell, F.A. (2008) * Cowell, F.A. (2011) Measuring Inequality, Oxford University Press * Cowell, F.A. (2011) Measuring Inequality, Oxford University Press * Cowell, F.A. (2011) * Cowell, F.A. (2011) 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) “Welfare Weights”, OFT Economic Research Paper 202, Office of Fair Training, Salisbury Square, London Cowell, F.A. and Gardiner, K.A. (2000) Cowell, F.A. and Gardiner, K.A. (2000)

32. Frank Cowell: HMRC-HMT Economics of Taxation 32 References (2) Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, 348-361 Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, 348-361 Dalton, H. (1920) Dalton, H. (1920) HM Treasury (2003) The Green Book: Appraisal and Evaluation in Central Government (and Technical Annex), TSO, London HM Treasury (2003) The Green Book: Appraisal and Evaluation in Central Government (and Technical Annex), TSO, London HM Treasury (2003)Technical Annex HM Treasury (2003)Technical Annex Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) “Inequality aversion and the natural rate of subjective inequality,” Journal of Public Economics, 87, 1061- 1090. Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) “Inequality aversion and the natural rate of subjective inequality,” Journal of Public Economics, 87, 1061- 1090. Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) Lambert, P. J., Millimet, D. L. and Slottje, D. J. (2003) Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of income,” Journal of Public Economics, 92, 1846-1857. Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of income,” Journal of Public Economics, 92, 1846-1857. Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London Pen, J. (1971) Income Distribution, Allen Lane, The Penguin Press, London Pirttilä, J. and Uusitalo, R. (2010) “A ‘Leaky Bucket’ in the Real World: Estimating Inequality Aversion using Survey Data,” Economica, 77, 60–76 Pirttilä, J. and Uusitalo, R. (2010) Sen, A. K. and Foster, J. E. (1997) On Economic Inequality (Second ed.). Oxford: Clarendon Press. Sen, A. K. and Foster, J. E. (1997) On Economic Inequality (Second ed.). Oxford: Clarendon Press. Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Shorrocks, A. F. (1983) Shorrocks, A. F. (1983)