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10 October 2011 Frank Cowell: EC426 Public Economics MSc Public Economics 2011/12 http://darp.lse.ac.uk/ec426 Equity, Social Welfare and Taxation Frank A. Cowell Frank Cowell: EC426 Public Economics Welfare and distributional analysis We’ve seen welfare-economics basis for redistribution as a policy objective Distributional analysis covers broad class of economic problems inequality social welfare poverty Similar techniques 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) rankings measures Four basic components need to be clarified “income” concept “income receiving unit” concept a distribution method of assessment or comparison xj A representation with 3 incomes Income distributions with given total X Feasible income distributions given X Equal income distributions Janet's income Frank Cowell: EC426 Public Economics 1: “Irene and Janet” approach xi + xj + xk = X particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria 0 xi Frank Cowell: EC426 Public Economics 2: The parade Plot income against proportion of population xF(x) Parade in ascending order of "income" / height 1 Related to familiar statistical concept distribution function F(∙) F(x0) Pen (1971) especially useful in cases where it is appropriate to adopt a parametric model of income distribution x0.8 x0.2 0 x 0 proportion of the population 0.2 x0 q 0.8 1 Frank Cowell: EC426 Public Economics Overview... Equity, Social Welfare, Taxation Welfare axioms Basic principles of distributional comparisons Rankings Equity and social welfare Taxation and sacrifice Frank Cowell: EC426 Public Economics Social-welfare functions A standard approach to a method of assessment Basic tool is a social welfare function (SWF) Properties will depend on economic principles Simple example of a 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(x1, x2,…,xn ) Total income in the economy W = Si xi Perhaps not very interesting Consider principles on which SWF could be based Frank Cowell: EC426 Public Economics The approach There is no such thing as a “right” or “wrong” axiom However axioms could be appropriate or inappropriate Use a simple framework to list some of the basic axioms Need some standard of “reasonableness” For example, how do people view income distribution comparisons? Assume a fixed population of size n Assume that individual utility can be measured by x Income normalised by equivalence scales Rules out utility interdependence Welfare is just a function of the vector x := (x1, x2,…,xn ) Follow the approach of Amiel-Cowell (1999) Appendix A 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(x1,x2..., xi+,..., xn) > W(x1,x2,..., xi,..., xn) Transfer principle. (Dalton 1920) Suppose xi< xj then for small : W(x1,x2..., xi+ ,..., xj ,..., xn) > W(x1,x2,..., xi,..., xn) Scale invariance. W(x) W(y) W(lx) W(ly) Frank Cowell: EC426 Public Economics Classes of SWFs (1) 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 Introduce decomposability and you get class of Additive SWFs W : most modern approaches use these assumptions but may need to standardise for needs etc W(x) = Si u(xi) 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 But W excludes some well-known welfare criteria Frank Cowell: EC426 Public Economics Classes of SWFs (2) From W we get important subclasses If we impose monotonicity we get If we further impose the transfer principle we get W1 W : u(•) increasing subclass where marginal social utility always positive W2 W1: 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… Frank Cowell: EC426 Public Economics Overview... Equity, Social Welfare, Taxation Welfare axioms Classes of SWF and distributional comparisons Rankings Equity and social welfare Taxation and sacrifice Frank Cowell: EC426 Public Economics Ranking and dominance Consider problem of comparing distributions Introduce two simple concepts 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: first illustrate using the Irene-Janet representation take income vectors x and y for a given n 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 given anonymity, population principle can represent distributions in F-form q: population proportion (0 ≤ q ≤ 1) F(x): proportion of population with incomes ≤ x m(F): mean of distribution F Frank Cowell: EC426 Public Economics 1st-Order approach Basic tool is the quantile, expressed as Q(F; q) := inf {x | F(x) q} = xq Use this to derive a number of intuitive concepts “smallest income such that cumulative frequency is at least as great as q” interquartile range, decile-ratios, semi-decile ratios graph of Q is Pen’s Parade Also to characterise 1st-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) Illustrate using Parade: A fundamental result: G quantile-dominates F iff W(G) > W(F) for all WW1 Frank Cowell: EC426 Public Economics Parade and 1st-order dominance Plot quantiles against proportion of population Q(.; q) Parade for distribution F again Parade for distribution G G In this case G clearly quantile-dominates F But (as often happens) what if it doesn’t? Try second-order method F 0 q 1 Frank Cowell: EC426 Public Economics 2nd-Order approach Basic tool is the income cumulant, expressed as C(F; q) := ∫ Q(F; q) x dF(x) Use this to derive a number of intuitive concepts “The sum of incomes in the Parade, up to and including position q” the “shares” ranking, Gini coefficient graph of C is the generalised Lorenz curve Also to characterise the idea of 2nd-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): G cumulant-dominates F iff W(G) > W(F) for all WW2 Frank Cowell: EC426 Public Economics GLC and 2nd-order dominance Plot cumulations against proportion of population C(.; q) GLC for distribution F GLC for distribution G m(G) m(F) cumulative income C(G; . ) C(F; . ) 0 q 0 1 Intercept on vertical axis is at mean income Frank Cowell: EC426 Public Economics 2nd-Order approach (continued) A useful tool: the share of the proportion q of distribution F is L(F;q) := C(F;q) / m(F) “income cumulation at divided q by total income” 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) Illustrate using Lorenz curve: Another fundamental result (Atkinson 1970): For given m, G Lorenz-dominates F iff W(G) > W(F) for all W W2 Frank Cowell: EC426 Public Economics Lorenz curve and ranking Plot shares against proportion of population 1 Perfect equality Lorenz curve for distribution F 0.8 Lorenz curve for distribution G L(.; q) 0.6 L(G;.) 0.4 L(F;.) 0.2 0 0 0.2 0.4 proportion of population 0.6 q 0.8 1 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 Frank Cowell: EC426 Public Economics Overview... Equity, Social Welfare, Taxation Welfare axioms Quantifying social values Rankings Equity and social welfare Taxation and sacrifice Frank Cowell: EC426 Public Economics Equity and social welfare So far we have just general principles Consider “reduced form” of social welfare W = W(m, I) m=m(F) is mean of distribution F I =I(F) is inequality of distribution F W embodies trade-off between objectives I can be taken as an “equity” criterion may lead to ambiguous results need a tighter description of social-welfare? from same roots as SWF? or independently determined? Consider a “natural” definition of inequality Frank Cowell: EC426 Public Economics 1 Gini coefficient Redraw Lorenz diagram A “natural” inequality measure…? L(.; q) normalised area above Lorenz curve can express this also in I-J terms q 0 Also (equivalently) represented as normalised difference between income pairs: In F-form: In Irene-Janet terms: Intuition is neat, but inequality index is clearly arbitrary can we find one based on welfare criteria? 1 Frank Cowell: EC426 Public Economics SWF and inequality The Irene &Janet diagram A given distribution Distributions with same mean xj Contours of the SWF Construct an equal distribution E such that W(E) = W(F) Equally-Distributed Equivalent income Social waste from inequality contour: x values such that W(x) = const •E O x(F) m(F) Curvature of contour indicates society’s willingness to tolerate “efficiency loss” in pursuit of greater equality •F xi do this for special case Frank Cowell: EC426 Public Economics An important family of SWF Take the W2 subclass and impose scale invariance. Get the family of SWFs where u is iso-elastic: x 1–e – 1 u(x) = ————— , e 1–e has same form as CRRA utility function Parameter e captures society’s inequality aversion. Similar interpretation to individual risk aversion See Atkinson (1970) Frank Cowell: EC426 Public Economics Welfare-based inequality From the concept of social waste Atkinson (1970) suggested an inequality measure: x(F) I(F) = 1 – —— m(F) Atkinson further assumed additive SWF, W(F) = u(x) dF(x) isoelastic u So inequality takes the form But what value of inequality aversion parameter e? Frank Cowell: EC426 Public Economics Values: the issues SWF is central to public policy making First: do people care about distribution? Practical example in HM Treasury (2003) pp 93-94 We need to focus on two questions… Experiments suggest they do – Carlsson et al (2005) Do social and economic factors make a difference? Second: What is the shape of u? (Cowell-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 Frank Cowell: EC426 Public Economics Preferences, happiness and welfare? Ebert, U. and Welsch, H. (2009) Alesina et al (2004) Evaluate subjective well being as a function of personal and environmental data using W form Examine which inequality index seems to fit preferences best Use data on happiness from social survey Construct a model of the determinants of happiness Use this to see if income inequality makes a difference Seems to be a difference in priorities between US and Europe Share of government in GDP Share of transfers in GDP US 30% 11% Continental Europe 45% 18% Do people in Europe care more about inequality? Frank Cowell: EC426 Public Economics The Alesina et al 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 h,m are state, time dummies 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 Frank Cowell: EC426 Public Economics Inequality aversion and Elasticity of MU Consistent inequality preferences? What value for e? Preference reversals (Amiel et al 2008) from happiness studies 1.0 to 1.5 (Layard et al 2008) related to extent of inequality in the country? (Lambert et al 2003) affected by way the question is put? (Pirttilä and Uusitalo 2010) Evidence on risk aversion is mixed direct survey evidence suggests estimated relative risk-aversion 3.8 to 4.3 (Barsky et al 1997) indirect evidence (from estimated life-cycle consumption model) suggests 0.4 to 1.4 (Blundell et al 1994) in each case depends on how well-off people are Frank Cowell: EC426 Public Economics Overview... Equity, Social Welfare, Taxation Welfare axioms Does structure of taxation make sense in welfare terms? Rankings Equity and social welfare Taxation and sacrifice Frank Cowell: EC426 Public Economics Another application of ranking Tax and benefit system maps one distribution into another Use ranking tools to assess the impact of this in welfare terms Typically this uses one or other concept of Lorenz dominance Linked to effective tax progression c = y T(y) y: pre-tax income c: post-tax income T is progressive if c Lorenz-dominates y see Jakobsson (1976) 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 1.0 + cash benefits (Equality Line) Original Income Gross Income Disposable Income After Tax Income Final Income 0.9 indirect taxes 0.7 + noncash benefits 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 direct taxes 0.8 Proportion of Income Frank Cowell: EC426 Public Economics Impact of Taxes and Benefits. UK 2006/7. Lorenz Curve 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Proportion of population 0.8 0.9 1.0 Big effect from benefits side Modest impact of taxes Direct and indirect taxes work in opposite directions 0.4 direct 1992 1998/9 2006/7 indirect 1992 1998/9 2006/7 0.3 0.2 0.1 10 th h p 9t h 8t h 7t h 6t h 5t h 4t 3r d 2n d To to m 10 t h 0 Bo t 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) Frank Cowell: EC426 Public Economics Tax-benefit example: implications We might have guessed the outcome of example In most countries: In many countries there is a move toward greater reliance on indirect taxation income tax progressive so are public expenditures but indirect tax is regressive so Lorenz-dominance is not surprising. political convenience? administrative simplicity? Structure of taxation should not be haphazard How should the y → c mapping be determined? Frank Cowell: EC426 Public Economics Tax Criteria What broad principles should tax reflect? The sacrifice approach “benefit received:” pay for benefits you get through public sector “sacrifice:” some principle of equality applied across individuals see Neill (2000) for a reconciliation of these two 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: 1. 2. 3. what’s a “fair” or “neutral” way to reduce incomes…? when will a tax system induce L-dominance? what is implication of imposing uniformity of sacrifice? Frank Cowell: EC426 Public Economics Janet's income Amiel-Cowell (1999) approach The Irene &Janet diagram Initial distribution Possible directions for a “fair” tax xj 1 Scale-independent iso-inequality 2 Translation-independent iso-inequality 3 “Intermediate” iso-inequality 4 No iso-inequality direction An iso-inequality path? 1 Proportionate reductions are “fair” 2 Absolute reductions are “fair” 3 Affine reductions are “fair” xi 0 Irene's income 4 Amiel-Cowell showed that individuals perceive inequality comparisons this way. Based on Dalton (1920) conjecture Frank Cowell: EC426 Public Economics Tax and the Lorenz curve Assume a tax function T(∙) based on income 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 a person with income y, pays an amount T(y) disposable income given by c := y T(y) [dc/dy][y/c] = [1 T(y)] /[1 y /T(y)] Disposable income under T* L-dominates that under T* iff residual progression of T is higher than that of T*… …for all y see Jakobsson (1976) Frank Cowell: EC426 Public Economics A sacrifice approach Suppose utility function U is continuous and increasing Definition of absolute sacrifice: (y, T) displays at least as much (absolute) sacrifice as (y*, T*)… …if U(y) U(yT) ≥ U(y*) U(y*T*) Equal absolute sacrifice is scale invariant iff y1 – e – 1 U(y) = ———— 1 –e …or a linear transformation of this (Young 1987) Apply sacrifice criterion to actual tax schedules T(∙) Young (1990) does this for US tax system Consider a UK application Frank Cowell: EC426 Public Economics Applying sacrifice approach Assume simple form of social welfare function Use the equal absolute sacrifice principle y1 – e [y T(y)]1 – e = const e is again inequality aversion Differentiate and rearrange: u(y) u(y T(y)) = const If insist on scale invariance u is isoelastic social welfare defined in terms of income drop distinction between U and u y – e [1 T(y)] [y T(y)]– e = 0 log[1 T(y)] = e log ([ y T(y) / y]) Estimate e from actual structure T(∙) get implied inequality aversion For income tax (1999/2000): e = 1.414 For IT and NIC (1999/2000): e = 1.214 see Cowell-Gardiner (2000) Frank Cowell: EC426 Public Economics Conclusion Axiomatisation of welfare needs just a few basic principles 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 These may be indecisive, so specific SWFs could be used anonymity population principle decomposability monotonicity principle of transfers scale invariant? perhaps a value for eof around 0.7 – 2 Welfare criteria can be used to provide taxation principles Frank Cowell: EC426 Public Economics References (1) Alesina, A., Di Tella, R. and MacCulloch, R (2004) “Inequality and happiness: are Europeans and Americans different?”, Journal of Public Economics, 88, 2009-2042 Amiel, Y. and Cowell, F.A. (1999) Thinking about Inequality, Cambridge University Press Amiel, Y., Cowell, F.A., Davidovitz, L. and Polovin, A. (2008) “Preference reversals and the analysis of income distributions,” Social Choice and Welfare, 30, 305-330 Atkinson, A. B. (1970) “On the Measurement of Inequality,” Journal of Economic Theory, 2, 244-263 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 Blundell, R., Browning, M. and Meghir, C. (1994) “Consumer Demand and the Life-Cycle Allocation of Household Expenditures,” Review of Economic Studies, 61, 57-80 Carlsson, F., Daruvala, D. and Johansson-Stenman, O. (2005) “Are people inequality averse or just risk averse?” Economica, 72, 375-396 Cowell, F.A. (2008) “Inequality: measurement,” The New Palgrave, second edition * Cowell, F.A. (2011) Measuring Inequality, Oxford University Press Cowell, F. A. and Gardiner, K.A. (2000) “Welfare Weights”, OFT Economic Research Paper 202, Office of Fair Trading, Salisbury Square, London Dalton, H. (1920) “Measurement of the inequality of incomes,” The Economic Journal, 30, 348-361 Frank Cowell: EC426 Public Economics References (2) Ebert, U. and Welsch, H. (2009) “How Do Europeans Evaluate Income Distributions? An Assessment Based on Happiness Surveys,” Review of Income Wealth, 55, 803-819 HM Treasury (2003) The Green Book: Appraisal and Evaluation in Central Government (and Technical Annex), TSO, London Jakobsson, U. (1976) “On the measurement of the degree of progression,” Journal of Public Economics, 5, 161-168 Jones, F. (2008) “The effects of taxes and benefits on household income, 2006/07,” Economic and Labour Market Review, 2, 37-47. 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. Layard, P. R. G., Mayraz, G. and Nickell S. J. (2008) “The marginal utility of income,” Journal of Public Economics, 92, 1846-1857. * Neill, J. R. (2000) “The benefit and sacrifice principles of taxation: A synthesis,” Social Choice and Welfare, 17, 117-124 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 Shorrocks, A. F. (1983) “Ranking Income Distributions,” Economica, 50, 3-17 Young, H. P. (1987) “Progressive taxation and the equal sacrifice principle,” Journal of Public Economics, 32, 203-212. Young, H. P. (1990) “Progressive taxation and equal sacrifice,” American Economic Review, 80, 253-266