Welfare: The Social-Welfare Function

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

2. Social Welfare Function Requirements • 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…?

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 A sketch of the 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

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 if assigned identity 1,2,3,... in the Lottery of Life use theory of choice under uncertainty to find shape of W • Individualistic welfare: W(u1, u2,u3, ...) • vN-M form of 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 Wby set of identities {1,2,...nh}: • åhphuh welfare is expected utility from a "lottery on identity“ • A suitable assumption about “probabilities”? • nh • 1 • W = — åuh • 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 (x1a, x2a) (x1b, x2b) • From the attainable set... • ...take an allocation • Evaluate utility for each agent • Plug into W to get social welfare ua=Ua(x1a, x2a) ub=Ub(x1b, x2b) • But what happens to welfare if we vary the allocation in A? W(ua, ub)

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

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

18. The SWF maximum problem • First component of the problem: W(U1(x1), U2(x2), U3(x3), ...) The objective function Utility depends on own consumption Individualistic welfare • Second component of the problem: nh • F(x) £0, xi =Sxih • h=1 Feasibility constraint All goods are private • The Social-welfare Lagrangean: nh • W(U1(x1), U2(x2),...) - lF(Sxh) h=1 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 FOCs: Uih(xh) Uiℓ(xℓ) ——— = ——— Ujh(xh) Ujℓ(xℓ) • Also from the FOCs: WhUih(xh) = Wℓ Uiℓ(xℓ) • social marginal utility of toothpaste equated across all h. • Relate marginal utility to prices: Uih(xh) = Vyhpi • This is valid if all consumers optimise Marginal utility of money • Substituting into the above: WhVyh = Wℓ Vyℓ • At optimum the welfare value of $1 is equated across all h. Call this common value M Social marginal utility of income

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 = S Whduh h=1 nh • = S WhVyhdyh h=1 • ...related to income nh • dW = M S dyh h=1 change in “national income” • Differentiate the SWF w.r.t. pi : nh • dW = S WhVihdpi h=1 • Changes in utility and change social welfare … nh • = – SWhVyhxihdpi h=1 from Roy’s identity nh • dW = – M S xihdpi h=1 • ...related to prices Change in total expenditure . .

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 function V • Express utility in terms of prices p and income y • Assume prices p are given • “Equivalise” (i.e. rescale) each income 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 Charlie's income • An income distribution. line of perfect equality • y O

27. equivalent in welfare terms  y  Welfare contours • An arbitrary income distribution yb • 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 income that, if received uniformly by all, would yield same level of social welfare as y. • Eyx is income that society would give up to eliminate inequality ya 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 W • The social evaluation function • 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 • Mild inequality aversion i = 0 i = ½ • 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 Three examples • 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

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 • Nonconvexity, asymmetry come from heterogeneity of households • 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 y*  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 • Item 1 might be considered as beyond the scope of simple microeconomics • Item 2 requires modelling of what is possible in the underlying structure of the economy... • ...which is what microeconomics is all about