Welfare Analysis of Distribution

1. Welfare Analysis of Distribution Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell http://darp.lse.ac.uk/lisbon2006 July 2006

2. Introduction • From introductory lecture… • …should be able to incorporate inequality and poverty analysis into standard welfare economics • In this lecture, focus on the underlying principles • Examine the underlying motivation for concern with redistribution • Why is this necessary?

3. Overview... Welfare Analysis of Distribution Foundation Roots in basic microeconomics Income, welfare, utility The basis for redistribution Risk and welfare

4. Agenda • Briefly reconsider the rôle for government • Central problem is often simply depicted • We will examine the components of this problem • Then consider how social values fit in. • A policy trade-off…?

5. A standard approach? • A classic trade-off • Social values • An optimum? efficiency • Need to define terms... • What is “efficiency”? • What is “equity”? equity

6. Policy options • “Equity-efficiency trade-off” idea raises some serious questions. • Is a trade-off actually necessary? • If so, what does it mean? • And how to make the choice from the trade-off options?

7. Efficiency-equity trade-off 1 • Is there necessarily a trade-off? • Not if we are inside the frontier: • Dismiss such cases as frivolous? • Not, perhaps in the world of practical politics • Not if we can redistribute resources without transactions cost. • But this is only possible with lump-sum transfers • Encounter informational problems • So there is some meaning to the trade-off

8. Efficiency-equity trade-off 2 • Not clear what goes on the axes • What is efficiency? • PE provides a criterion for the goal of efficiency itself. • Pareto criterion gives no guidance away from efficient point. • Standard approach to efficiency gains losses: • Based on cost-benefit analysis • Used as a criterion for applications in Public Economics such as tax design. • What is equity? • Raises issues of definition. • Also of the case for egalitarianism (Putterman et al. - JEL98).

9. Components of the trade-off • Specification of the technology • Enables precise definition of efficiency • A definition of equity • Also related concepts such as inequality • See next lecture • An analysis of the nature of the trade-off • Informational problems • A statement of social preferences • What is the basis for concern with distribution? • We deal with this in the current lecture • Review the alternative welfare approaches

10. Welfare approaches 1 • The constitutional approach • Uses peoples’ orderings of social states • Including attitude to redistribution • Motive for redistribution in terms of social states must come from somewhere • Purely ordinal • This is extremely demanding • Run into the Arrow problem • Hence – hopelessly indecisive? • No clear imperative for action • Difficulties of implementation

11. Welfare approaches 2 • Equity as a fundamental principle like “efficiency” • Some attempts at formalising this in economics • E.g. Varian (1974) “no-envy” criterion • But these are usually very restrictive • Usually need to seek support on philosophical base outside economics • But what? • Both traditional and modern approaches • To be reviewed later

12. Welfare approaches 3 • Welfarism • Uses a cardinally measurable and interpersonally comparable approach to welfare. • Usually based on individualism • Provides the basis for a coherent model • Need to examine the basic building blocks…

13. Overview... Welfare Analysis of Distribution Foundation The basic units of analysis Income, welfare, utility The basis for redistribution Risk and welfare

14. Ingredients of an approach • A model of individual resources • A measure of individual welfare • A basis for interpersonal comparisons • An intellectual base for state intervention • We will deal with the first three of these now.

15. Individual resources and distribution • We adopt two simple paradigms concerning resources: • The cake-sharing problem • The general case with production • Often distributional analysis can be conducted in terms of typical individuals i and j. • In some cases one needs a more general distributional notation Fixed total income Incorporates incentive effects Irene and Janet The F-form approach

16. Income distributions with given total A simple model for the distributional problem • Two persons • The feasible set • The interesting distributions • The basic cake-sharing income-distribution problem ray of equality Janet’s income 45° 0 Irene’s income

17. Limitations of this basic model • Just 2 persons • n³ 3 persons for the inequality problem • Fixed-size cake • Economic growth? • Waste through distortion? • Costlessly transferable incomes • The “leaky bucket” problem (Okun 1975) • Analysed further in discussion of incentives • Incomes or utilities? Essential to first-best welfare economics

18. Example 1 For welfare purposes we are concerned with utility... Example 2 • What is the relationship of utility to income? • What properties does utility have? • Is it measurable? • Is it comparable? • These properties are independent • We usually need both Comparability without measurability : Imagine a world where access to public services determines utility and the following ordering is recognised: • Gas+Electricity • Electricity only • Gas only • Neither It makes no sense to say “U(G+E) =2U(E)”, but you could still compare individuals. Measurability without comparability: Imagine a world where utility is proportional to income, but the constant of proportionality is known to depend on family characteristics which may be unobservable. Double a family’s income and you double each member’s utility; but you cannot compare utilities of persons from different families. We need a simple model of utility....

19. Ingredients • a: personal attributes • Identity • Needs • Abilities • Special “merit” or “desert” • y: income • Could be exogenous • Or you can model as a function of attributes: y=y(a) • u: individual utility • Several ways of modelling this… • …see below • x: “equivalised” income • Dollar/Pound/Euro units… • Can be treated as a version of “utility”

20. Ingredients (2) • F : distribution function • Standard tool borrowed from statistics • U : utility function • A variety of specifications – see below • Gives indicator of how “well-off” a person of given attributes is • c : equivalisation function • A simple way of accounting for differences in needs • Perhaps too simple? • We will try something different in the next lecture

21. Basic questions about income • Is it unique? • How comprehensive should it be? • What is the relevant receiving unit? • Is it comparable between persons?

22. Income: Uniqueness? • Should we use univariate or multivariate analysis? • income and expenditure? • income and wealth? • income over time? • A relationship between different types of “income”? • covariance of earnings and asset income? • conditional transfers? • Several definitions may be relevant? • gross income? • disposable income? • other concepts?

23. Income: comprehensiveness? • Is income “full income”? • final income + • value of leisure +...? • Is income a proxy for economic welfare? • discount for risk? • valuation over time?.. • Can income be zero? • rental income? • ... or less than zero? • business losses?

24. Income: Comparability? • Price adjustment • Normalise by price indices • Adjustment for needs and household size • Usual approach is to introduce equivalence scales • The equivalence transformation is x = c( y, a ) personal attributes Equivalised income nominal income • Usually a simplifying assumption is made. • Write transformation as an income-independent equivalence scale: Number of equivalent adults x = y / n (a) • Where does the function c come from?

25. Equivalence Scales • We will assume that there is an agreed method of determining equivalence scales. • But there is a variety of possible sources of information for equivalence scales: • From official government sources • From international bodies such as OECD • From econometric models of household budgets. • Consider an example of the last of these:

26. A model of income and need • Plot share of food in budget against household income sfood • A reference household type... • Engel Equivalence Scale childless couple proxy for “need” couple with children xr ºyr From budget studies x, y 0 xi yi income

27. Alternative models of utility • u = U (y) • Inter-personally comparable utility • u = U (y;a) • Individualistic utility • May not be comparable, depending on information about a. • u = U (y, F) • Concern for distribution as a kind of externality • Need not be benevolent concern • Evidence that people are • Concerned about relative incomes • “upward looking” in their comparisons. • Ferrar-i-Carbonell (2005) • x= c(y ;a) = y /n(a) • A comparable money-metric utility?

28. The relationship between utility and income: u Increase concavity u = U(y) ^ u = U(y) y

29. A simple model • As an example take the iso-elastic form: y1 –d– 1 U(y) = ———— , d ³ 0 1 –d • We can think of d as risk aversion • But it may take on an additional welfare significance

30. What to do with this information? • We need a method of appraising either the distribution of utilities… • …or, the system by which they were produced • This involves fundamentally different approaches to welfare judgments.

31. Overview... Welfare Analysis of Distribution Foundation Philosophies, social welfare and the basis for intervention Income, welfare, utility The basis for redistribution Risk and welfare

32. Five intellectual bases for public action • …and five social philosophers • Entitlement theories • Nozick • Unanimity • Pareto • Utilitarianism • Bentham • Concern with the least advantaged • Rawls • Egalitarianism • Plato

33. A distributional outcome • Standard cake-sharing model • N stands for “Nozick” ray of equality Janet’s income • N implications for utility possibilities 45° 0 Irene’s income

34. Utility-possibility set • Plot utility on the axes • Simple cake-sharing uj • The effect of utility interdependence ray of equality • N • N Assuming that U is strictly concave... …and that U is the same function for both Irene and Janet. 45° ui 0

35. Should we move from N? • What is the case for shifting from the status-quo point? • Answer differs dramatically according to social philosophy: • Entitlement approach is concerned with process • Other approaches concerned with end-states

36. Entitlement approach • Focus on Nozick (Anarchy, State and Utopia, 1974). • Answer depends crucially on how N came about • Distinguish three key issues: • fairness in original acquisition • fair transfers • rectification of past injustice • Presumption is that there will be little or no role for the State • “Night watchman”

37. Pareto Criterion • Pareto unanimity criterion is an end-state principle • Approve the move from N to another point… • …if at least one person gains • …and no-one loses • Individualistic • Based on utilities • But utility may have a complicated relationship with income • May depend on the income of others • See how Pareto applies in the simple example

38. Pareto improvement: simple case • The utility-possibility set again • The initial point uj • Pareto superior points ray of equality • N • No case for intervention? 45° ui 0

39. End-state approaches: beyond Pareto • Pareto criterion can be indecisive • Alternative end state approaches use a social welfare function • Typically get unique solution • What principles should this embody? • Individualism? • The Pareto principle? • Additivity? • Take a simple example that combines them all...

40. Benthamite approach • General principle is “Seek the greatest good of the greatest number” • This is typically interpreted as maximising the sum of individual welfare. • In Irene-Janet terms: u1 + u2 + ...+ un • More generally the SWF is: WB=ò udF(u)

41. Distributional implications of utilitarianism • Much of public economics uses utilitarianism. • Efficiency criteria • Sacrifice theories in taxation • But does utilitarianism provide a basis for egalitarian transfers? • Sen has argued that this is a common fallacy • Sen and Foster (1997) • Again look at this within the simple model

42. ui+uj = constant Benthamite redistribution? • Take a symmetric utility-possibility set uj • The initial distribution • Benthamite welfare contour • Maximise welfare ray of equality • Optimum in this case • Implied tax/transfer • N • B The general case? 45° ui 0

43. The general case... uj • N • C • Incorporates differential incentive effects etc. • B • N. The status quo • Pareto improvements • Points that Pareto-dominate N • C The voluntary solution? • Anywhere above C might be a candidate • B. Benthamite solution • Paretianism leads to multiple solutions • Benthamite utilitarianism leads to a unique, possibly different, solution. ui 0

44. General case: discussion • A motive for changing distribution? • Nozickians might still insist that no move from N is justified • unless it came through private voluntary action • Applies even to C • Implementation: • Private voluntary action might not be able to implement C • Could rise if there were many individuals • Case for egalitarianism? • Clearly Bentham approach does not usually imply egalitarian outcome. • Consider two further alternative approaches: • Concern for the least advantaged (Rawls) • Egalitarianism

45. Rawls (1971) • Rawls’ distributional philosophy is based on two fundamental principles: • each person has equal right to the most extensive scheme of equal basic liberties compatible with a similar scheme of liberties for all • society should so order its decisions as to secure the best outcome for the least advantaged • Economic focus has usually been on 2 • Argument based on reasoning behind a “veil of ignorance” • I do not know which position in society I have when making social judgment • Needs careful interpretation • Avoid confusion with probabilistic approach later

46. The Rawls approach…? • What is meant by the difference principle? • This is typically interpreted as maximising the welfare of the worst-off person. • Based on simplistic interpretation of veil of ignorance argument • Rawls interpreted it differently • But rather vaguely • In Irene-Janet terms: min {u1 , u2 , ..., un} • So the suggested SWF is: WR= {minu: F(u)>0}

47. Egalitarianism? • Origin goes back to Plato… • …but reinterpreted by Meade (1974). • “Superegalitarianism” • Welfare is perceived in terms of pairwise differences: [ui - uj]... • Welfare might not be expressible as a neat additive expression involving individual utilities. • Finds an echo in more recent welfare developments • Covered in a later lecture

48. General case (2) uj • N • A 'Rawlsian' solution • Superegalitarianism Contours of max min function • R ray of equality • Maxi-min does not imply equality • E • Superegalitaranism implies equality ui 0

49. Bergson-Samuelson approach • But why an additive form of the SWF? • We could just use a weaker individualistic form. • This is the basis of the Bergson-Samuelson formulation • A generalisation • Subsumes several welfare concepts • In Irene-Janet terms: W(u1 , u2 , ..., un) • More generally the SWF is: WBS= W(F)

50. General individualistic welfare • The specific welfare functions are special cases of Bergson-Samuelson. • Most satisfy the principle of additivity • Except for the last one (utility differences) • In Irene-Janet terms this means we can write: u(u1) + u(u2) + ... + u(un) • More generally the SWF is: WBSa=òu(u)dF(u) • This is clear for Bentham where u(u)=u. But…