Welfare: The Social-Welfare Function

1. Prerequisites Almost essential Welfare: Basics Welfare: Efficiency Welfare: The Social-Welfare Function MICROECONOMICS Principles and Analysis Frank Cowell December 2006

2. Social Welfare Function • Limitations of the welfare analysis so far: • Constitution approach • Arrow theorem – is the approach overambitious? • General welfare criteria • Efficiency – nice but indecisive • Extensions – contradictory? • SWF is our third attempt • Something like a simple utility function…? Requirements

3. Overview... Welfare: SWF The Approach What is special about a social-welfare function? SWF: basics SWF: national income SWF: income distribution

4. The SWF approach • Restriction of “relevant” aspects of social state to each person (household) • Knowledge of preferences of each person (household) • Comparability of individual utilities • utility levels • utility scales • An aggregation function W for utilities • contrast with constitution approach • there we were trying to aggregate orderings A sketch of the approach

5. Using a SWF ub • Take the utility-possibility set • Social welfare contours • A social-welfare optimum? W(ua,ub,... ) • W defined on utility levels • Not on orderings • Imposes several restrictions… • …and raises several questions • U ua

6. Issues in SWF analysis • What is the ethical basis of the SWF? • What should be its characteristics? • What is its relation to utility? • What is its relation to income?

7. Overview... Welfare: SWF The Approach Where does the social-welfare function come from? SWF: basics SWF: national income SWF: income distribution

8. An individualistic SWF • The standard form expressed thus W(u1, u2,u3, ...) • an ordinal function • defined on space of individual utility levels • not on profiles of orderings • But where does W come from...? • We'll check out two approaches: • The equal-ignorance assumption • The PLUM principle

9. 1: The equal ignorance approach • Suppose the SWF is based on individual preferences. • Preferences are expressed behind a “veil of ignorance” • It works like a choice amongst lotteries • don't confuse w and q! • Each individual has partial knowledge: • knows the distribution of allocations in the population • knows the utility implications of the allocations • knows the alternatives in the Great Lottery of Life • does not know which lottery ticket he/she will receive

10. “Equal ignorance”: formalisation payoffs I would get if I were assigned identity 1,2,3,... in the Great Lottery of Life use theory of choice under uncertainty to find the shape of SWF W • The individualistic welfare model: W(u1, u2,u3, ...) • vN-M form of the utility function: åwÎWpwu(xw) Equivalently: åwÎWpwuw pw: probability assigned to w u: cardinal utility function, independent of w uw: utility payoff in state w • Replace W by the set of identities {1,2,...nh}: åhphuh welfare is expected utility from a "lottery on identity“ • A suitable assumption about “probabilities”? nh 1 W = — S uh nhh=1 An additive form of the welfare function

11. Questions about “equal ignorance” • Construct a lottery on identity • The “equal ignorance” assumption... • Where people know their identity with certainty ph • Intermediate case • The “equal ignorance” assumption: ph = 1/nh • But is this appropriate? • Or should we assume that people know their identities with certainty? | 1 | | nh | 3 | 2 h identity • Or is the "truth" somewhere between...?

12. 2: The PLUM principle • Now for the second  rather cynical approach • Acronym stands for People Like Us Matter • Whoever is in power may impute: • ...either their own views, • ... or what they think “society’s” views are, • ... or what they think “society’s” views ought to be, • ...probably based on the views of those in power • There’s a whole branch of modern microeconomics that is a reinvention of classical “Political Economy” • Concerned with the interaction of political decision-making and economic outcomes. • But beyond the scope of this course

13. Overview... Welfare: SWF The Approach Conditions for a welfare maximum SWF: basics SWF: national income SWF: income distribution

14. The SWF maximum problem • Take the individualistic welfare model W(u1, u2,u3, ...) Standard assumption • Assume everyone is selfish: uh = Uh(xh) , h=1,2,...nh my utility depends only on my bundle • Substitute in the above: W(U1(x1), U2(x2), U3(x3), ...) Gives SWF in terms of the allocation a quick sketch

15. A A From an allocation to social welfare • From the attainable set... (x1a, x2a) (x1b, x2b) • ...take an allocation • Evaluate utility for each agent • Plug into W to get social welfare • But what happens to welfare if we vary the allocation in A? ua=Ua(x1a, x2a) ub=Ub(x1b, x2b) W(ua, ub)

16. Varying the allocation The marginal utility derived by household h from good i • Differentiate w.r.t. xih : duh = Uih(xh) dxih The effect on h if commodity i is changed • Sum over i: n duh = SUih(xh) dxih i=1 The effect on h if all commodities are changed The marginal impact on social welfare of household h’s utility Changes in utility change social welfare . • Differentiate W with respect to uh: nh dW = SWhduh h=1 Weights from the SWF Weights from the utility function So changes in allocation change welfare. • Substitute for duh in the above: nh n dW = S WhSUih(xh) dxih h=1 i=1

17. Use this to characterise a welfare optimum • Write down SWF, defined on individual utilities. • Introduce feasibility constraints on overall consumptions. • Set up the Lagrangean. • Solve in the usual way Now for the maths

18. The SWF maximum problem Utility depends on own consumption Individualistic welfare function • First component of the problem: W(U1(x1), U2(x2), U3(x3), ...) The objective function All goods are private Feasibility constraint • Second component of the problem: nh F(x) £ 0, xi =S xih h=1 Usual Lagrange multiplier • The Social-welfare Lagrangean: nh W(U1(x1), U2(x2), ...) - lF(Sxh ) h=1 Note: constraint subsumes technological feasibility and materials balance • FOCs for an interior maximum: Wh (...)Uih(xh) - lFi(x) = 0 From differentiating Lagrangean with respect to xih • And if xih =0 at the optimum: Wh (...)Uih(xh) - lFi(x) £ 0 Usual modification for a corner solution

19. Solution to SWF maximum problem Any pair of goods, i,j Any pair of households h, ℓ MRS equated across all h. We’ve met this condition before - Pareto efficiency • From the first-order conditions : Uih(xh) Uiℓ(xℓ) ——— = ——— Ujh(xh) Ujℓ(xℓ) This is new! • Also from the FOCs: Wh Uih(xh) = Wℓ Uiℓ(xℓ) social marginal utility of toothpaste equated across all h. Marginal utility of money This is valid if all consumers optimise • Relate marginal utility to prices: Uih(xh) = Vyhpi Social marginal utility of income At the optimum the welfare value of a $ of income is equated across all h. Call this common valueM • Substituting into the above: Wh Vyh = Wℓ Vyℓ

20. To focus on main result... • Look what happens in neighbourhood of optimum • Assume that everyone is acting as a maximiser • firms • households… • Check what happens to the optimum if we alter incomes or prices a little • Similar to looking at comparative statics for a single agent

21. Changes in income, social welfare • Social welfare can be expressed as: W(U1(x1), U2(x2),...) = W(V1(p,y1), V2(p,y2),...) SWF in terms of direct utility. Using indirect utility function Changes in utility and change social welfare … • Differentiate the SWF w.r.t. {yh}: nh dW = SWhduh h=1 nh = SWhVyh dyh h=1 Change in total incomes - i.e. change in “national income” ...related to income nh dW = MSdyh h=1 • Differentiate the SWF w.r.t. pi : nh dW = SWhVihdpi h=1 Follows from Roy’s identity Changes in utility and change social welfare … nh = – SWhVyh xihdpi h=1 Change in total expenditure nh dW = –M Sxihdpi h=1 ...related to prices . .

22. An attractive result? • Summarising the results of the previous slide we have: • THEOREM: in the neighbourhood of a welfare optimum welfare changes are measured by changes in national income / national expenditure • But what if we are not in an ideal world?

23. Overview... Welfare: SWF The Approach A lesson from risk and uncertainty SWF: basics SWF: national income SWF: income distribution

24. Derive a SWF in terms of incomes • What happens if the distribution of income is not ideal? • M is no longer equal for all h • Useful to express social welfare in terms of incomes • Do this by using indirect utility functions V • Express utility in terms of prices p and incomes y • Assume prices p are given • “Equivalise” (i.e. rescale) incomes y • allow for differences in people’s needs • allow for differences in household size • Then you can write welfare as W(ya, yb, yc, … )

25. Income-distribution space: nh=2 • The income space: 2 persons • An income distribution Bill's income line of perfect equality • Note the similarity with a diagram used in the analysis of uncertainty • y 45° Alf's income O

26. Bill's income Alf's income Extension to nh=3 • Here we have 3 persons • An income distribution. Charlie's income line of perfect equality • y O

27. equivalent in welfare terms  y  Welfare contours yb • An arbitrary income distribution • Contours of W • Swap identities • Distributions with the same mean • Equally-distributed-equivalent income • Anonymity implies symmetry of W. • Eyis mean income • Richer-to-poorer income transfers increase welfare. Ey higher welfare x • x is the income that, if received uniformly by all, would yield same level of social welfare as y. ya • Eyx is the income that, society would give up to eliminate inequality Ey x

28. A result on inequality aversion • Principle of Transfers : “a mean-preserving redistribution from richer to poorer should increase social welfare” • THEOREM: Quasi-concavity of W implies that social welfare respects the “Transfer Principle”

29. Special form of the SWF • It can make sense to write W in the additive form nh 1 W = — Sz(yh) nhh=1 • where the function z is the social evaluation function • (the 1/nh term is unnecessary – arbitrary normalisation) • Counterpart of u-function in choice under uncertainty • Can be expressed equivalently as an expectation: W = Ez(yh) • where the expectation is over all identities • probability of identity h is the same, 1/nh , for all h • Constant relative-inequality aversion: 1 z(y) = —— y1 – i 1 – i • where i is the index of inequality aversion • works just like r,the index of relative risk aversion

30. Concavity and inequality aversion • The social evaluation function W • Let values change: φ is a concave transformation. lower inequality aversion z(y) • More concave z(•)implies higher inequality aversioni • ...and lower equally-distributed-equivalent income • and more sharply curved contours z(y) z =φ(z) higher inequality aversion y income

31. yb yb ya ya O O yb yb ya ya O O Social views: inequality aversion • Indifference to inequality i = 0 i = ½ • Mild inequality aversion • Strong inequality aversion • Priority to poorest • “Benthamite” case (i=0): • nh W= Syh h=1 i = 2 i =  • General case (0<i<): • nh W = S [yh]1-i/ [1-i] h=1 • “Rawlsian” case (i=): W= min yh h

32. Inequality, welfare, risk and uncertainty • There is a similarity of form between… • personal judgments under uncertainty • social judgments about income distributions. • Likewise a logical link between risk and inequality. • This could be seen as just a curiosity • Or as an essential component of welfare economics • Uses the “equal ignorance argument” • In the latter case the functions u and z should be taken as identical • “Optimal” social state depends crucially on shape of W • In other words the shape of z • Or the value of i Three examples

33. Social values and welfare optimum yb • The income-possibility set Y • Welfare contours ( i = 0) • Welfare contours ( i = ½) • Welfare contours ( i = ) • Y derived from set A or U • Nonconvexity, asymmetry come from heterogeneity of households Y • y* maximises total income irrespective of distribution y***  y**  • y** trades off some income for greater equality y*  • y*** gives priority to equality; then maximises income subject to that ya

34. Summary • The standard SWF is an ordering on utility levels • Analogous to an individual's ordering over lotteries • Inequality- and risk-aversion are similar concepts • In ideal conditions SWF is proxied by national income • But for realistic cases two things are crucial: • Information on social values • Determining the income frontier • This requires a modelling of what is possible in the underlying structure of the economy... • ...which is what Micro-Economic principles is all about