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## Distributional Equity, Social Welfare

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**Distributional Equity, Social Welfare**Public Economics: University of Barcelona Frank Cowell http://darp.lse.ac.uk/ub June 2005**Onwards from welfare economics...**• We’ve seen the welfare-economics basis for redistribution as a public-policy objective • How to assess the impact and effectiveness of such policy? • We need appropriate criteria for comparing distributions of income and personal welfare • This requires a treatment of issues in distributional analysis.**Equity and social welfare**Overview... • Income distributions • Comparisons Welfare comparisons SWFs How to represent problems in distributional analysis Rankings Welfare and needs Compensation and responsibility**Representing a distribution**Recall our two standard approaches: • Irene and Janet • The F-form particularly appropriate in approaches to the subject based primarily upon individualistic welfare criteria especially useful in cases where it is appropriate to adopt a parametric model of income distribution**Pen's parade**• Plot income against proportion of population • Parade in ascending order of "income" / height x "income" (height) x0.8 Now for some formalisation: x0.2 q 1 0 0.2 0.8 proportion of the population**A distribution function**F(x) 1 F(x0) x 0 x0**The set of distributions**• We can imagine a typical distribution as belonging to some class F Î F • How should members of F be described or compared? • Sets of distributions are, in principle complicated entities • We need some fundamental principles**Equity and social welfare**Overview... • Income distributions • Comparisons Welfare comparisons SWFs Methods and criteria of distributional analysis Rankings Welfare and needs Compensation and responsibility**Comparing Income Distributions**• Consider the purpose of the comparison... • …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”. • The ethical basis rests on some aspects of the last lecture… • …and the practical implementation requires an comparison in terms of “inequality”. • Which is easy. Isn’t it?**P**R $ 10 0 1 2 3 4 5 6 7 8 9 R P $ 10 0 1 2 3 4 5 6 7 8 9 P R $ 10 0 1 2 3 4 5 6 7 8 9 R P $ 10 0 1 2 3 4 5 6 7 8 9 Some comparisons self-evident...**A fundamental issue...**• 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. • 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**Monday**High inequality Low inequality Q P R $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 Q P R Low inequality High inequality Tuesday $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 The 3-Person problem:two types of income difference • Which do you think is “better”? • Top Sensitivity • Bottom Sensitivity**Syldavia**Ruritania Arcadia Borduria Distributional Orderings and Rankings • In an ordering we unambiguously arrange distributions • But a ranking may include distributions that cannot be ordered more welfare • {Syldavia, Arcadia, Borduria} is an ordering. • {Syldavia, Ruritania, Borduria} is also an ordering. • But the ranking {Syldavia, Arcadia, Ruritania, Borduria} is not an ordering. less welfare**Comparing income distributions - 2**• 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. • Make precise “one distribution is better than another” • Axioms could be rooted in welfare economics • There are other logical bases. • Apply the approach to general ranking principles • Lorenz comparisons • Social-welfare rankings • Also to specific indices • Welfare functions • Inequality measures**The Basics: Summary**• Income distributions can be represented in two main ways • Irene-Janet • F-form • The F-form is characterised by Pen’s Parade • 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…**Equity and social welfare**Overview... Welfare comparisons SWFs • Axiomatic structure • Classes • Values How to incorporate fundamental principles Rankings Welfare and needs Compensation and responsibility**Social-welfare functions**• 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 • Simple example of a SWF: • Total income in the economy W = Sixi • Perhaps not very interesting • Consider principles on whichSWF could be based**Another fundamental question**• What makes a “good” set of principles? • There is no such thing as a “right” or “wrong” axiom. • 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 • 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)**Basic Axioms:**• Anonymity • Population principle • Monotonicity • Principle of Transfers • Scale / translation Invariance • Strong independence / Decomposability**Basic Axioms:**• Anonymity • Permute the individuals and social welfare does not change • Population principle • Monotonicity • Principle of Transfers • Scale / translation Invariance • Strong independence / Decomposability**x**$ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 x' $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 Anonymity W(x′) = W(x)**x**$ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 y $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 x' $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 y' Implication of anonymity End state principle: xy is equivalent to x′y .**Basic Axioms:**• Anonymity • Population principle • Scale up the population and social welfare comparisons remain unchanged • Monotonicity • Principle of Transfers • Scale / translation Invariance • Strong independence / Decomposability**$**10 0 1 2 3 4 5 6 7 8 9 $ 10 0 1 2 3 4 5 6 7 8 9 Population replication W(x) W(y) W(x,x,…,x) W(y,y,…,y)**A change of notation?**• Using the first two axioms • Anonymity • Population principle • We can write welfare using F –form • Just use information about distribution • Sometimes useful for descriptive purposes • Remaining axioms can be expressed in either form**Basic Axioms:**• Anonymity • Population principle • Monotonicity • Increase anyone’s income and social welfare increases • Principle of Transfers • Scale / translation Invariance • Strong independence / Decomposability**$**x 10 12 0 2 4 6 8 14 16 18 20 x′ $ 10 12 14 16 18 20 0 2 4 6 8 Monotonicity W(x1+,x2,..., xn) >W(x1,x2,..., xn)**x′**$ 10 12 14 16 18 20 0 2 4 6 8 x $ 10 12 0 2 4 6 8 14 16 18 20 Monotonicity W(x1,x2..., xi+,..., xn) >W(x1,x2,..., xi,..., xn)**x′**$ x′ 10 12 14 16 18 20 0 2 4 6 8 $ 10 12 14 16 18 20 0 2 4 6 8 Monotonicity W(x1,x2,..., xn+) >W(x1,x2,..., xn)**Basic Axioms:**• Anonymity • Population principle • Monotonicity • Principle of Transfers • Poorer to richer transfer must lower social welfare • Scale / translation Invariance • Strong independence / Decomposability**Transfer principle:**• The Pigou (1912) approach: • Focused on a 2-person world • A transfer from poor P to rich R must lower social welfare • The Dalton (1920) extension • 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…**$**10 11 12 13 0 1 2 3 4 5 6 7 8 9 $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 Which group seems to have the more unequal distribution?**$**10 11 12 13 0 1 2 3 4 5 6 7 8 9 $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 The issue viewed as two groups**$**10 11 12 13 0 1 2 3 4 5 6 7 8 9 $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 Focus on just the affected persons**Basic Axioms:**• Anonymity • Population principle • Monotonicity • Principle of Transfers • Scale Invariance • Rescaling incomes does not affect welfare comparisons • Strong independence / Decomposability**x**y $ $ 15 15 10 10 0 0 5 5 lx $ $ ly 1500 1500 1000 1000 0 0 500 500 Scale invariance (homotheticity) W(x) W(y) W(lx) W(ly)**Basic Axioms:**• Anonymity • Population principle • Monotonicity • Principle of Transfers • Translation Invariance • Adding a constant to all incomes does not affect welfare comparisons • Strong independence / Decomposability**x**y $ $ 15 15 10 10 0 0 5 5 $ $ 20 20 15 15 5 5 10 10 Translation invariance W(x) W(y) W(x+d1) W(y+d1) x+d1 y+d1**Basic Axioms:**• Anonymity • Population principle • Monotonicity • Principle of Transfers • Scale / translation Invariance • Strong independence / Decomposability • merging with an “irrelevant” income distribution does not affect welfare comparisons**x**$ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 Before merger... y $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 x' $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 After merger... y' $ 10 11 12 13 0 1 2 3 4 5 6 7 8 9 Decomposability / Independence W(x) W(y) W(x') W(y')**Using axioms**• Why the list of axioms? • 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 • 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 • Inequality • Poverty • Deprivation.**Equity and social welfare**Overview... Welfare comparisons SWFs • Axiomatic structure • Classes • Values Categorising important types Rankings Welfare and needs Compensation and responsibility**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)= Siu(xi) • or equivalently in F-form W(F) = òu(x) dF(x) • The class W is of great importance • Already seen this in lecture 1. • But W excludes some well-known welfare criteria**Classes of SWFs (2)**• From Wwe get important subclasses • If we impose monotonicity we get • W1 ÌW: u(•) increasing • If we further impose the transfer principle we get • W2 ÌW1: u(•) increasing and concave • We often need to use these special subclasses • Illustrate their behaviour with a simple example…**x**x x 1 0 0 The density function • Income growth at x0 f(x) • Welfare increases if WÎ W1 • A mean-preserving spread • Welfare decreases if WÎ W2 x**An important family**• Take the W2 subclass and impose scale invariance. • Get the family of SWFs where u is iso-elastic: x1 – e– 1 u(x) = ————, e ³ 0 1 – e • Same as that in lecture 1: • individual utility represented by x. • also same form as CRRA utility function • Parameter e captures society’s inequality aversion. • Similar interpretation to individual risk aversion • See Atkinson (1970)**Another important family**• Take the W2 subclass and impose translation invariance. • Get the family of SWFs where u is iso-elastic: 1 – exp–kx u(x) = ——— k • Same form as CARA utility function • Parameter k captures society’s absolute inequality aversion. • Similar to individual absolute risk aversion**Equity and social welfare**Overview... Welfare comparisons SWFs • Axiomatic structure • Classes • Values …Can we deduce how inequality-averse “society” is? Rankings Welfare and needs Compensation and responsibility**Values: the issues**• In previous lecture we saw the problem of adducing social values. • Here we will focus on two questions… • First: do people care about distribution? • Justify a motive for considering positive inequality aversion • Second: What is the shape of u? • What is the value of e? • Examine survey data and other sources**Happiness and welfare?**• 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 US Continental Europe Share of government in GDP 30% 45% Share of transfers in GDP 11% 18% • But does this reflect values? • Do people in Europe care more about inequality?