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

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

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

A sketch of the approach 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

Using a SWF • U ub W(ua, ub,... ) ua W defined on utility levels 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

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?

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

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

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

“Equal ignorance”: formalisation payoffs I would get if I were assigned identity 1,2,3,... in the Great Lottery of Life The individualistic welfare model: W(u1, u2, u3, ...) use theory of choice under uncertainty to find the shape of SWF W vN-M form of the utility function: åwÎW pwu(xw) Equivalently: åwÎW pwuw 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}: åh phuh welfare is expected utility from a "lottery on identity“ A suitable assumption about “probabilities”? nh 1 W = — S uh nh h=1 An additive form of the welfare function

Questions about “equal ignorance” Construct a lottery on identity The “equal ignorance” assumption... ph Where people know their identity with certainty 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 | 2 | 3 | | nh identity h Or is the "truth" somewhere between...?

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

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

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

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 ua=Ua(x1a, x2a) ub=Ub(x1b, x2b) But what happens to welfare if we vary the allocation in A? W(ua, ub)

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 = S Uih(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 = S Wh duh 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 Wh SUih(xh) dxih h=1 i=1

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

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(S xh ) 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

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) Also from the FOCs: Wh Uih(xh) = Wℓ Uiℓ(xℓ) This is new! 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 value M Substituting into the above: Wh Vyh = Wℓ Vyℓ

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

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 Differentiate the SWF w.r.t. {yh}: nh dW = S Wh duh h=1 Changes in utility and change social welfare … nh = S WhVyh dyh h=1 Change in total incomes - i.e. change in “national income” ...related to income nh dW = M S dyh 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 S xihdpi h=1 ...related to prices . .

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?

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

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, … )

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

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

Welfare contours  Ey x y  yb ya x Ey An arbitrary income distribution Contours of W Swap identities Distributions with the same mean  equivalent in welfare terms Equally-distributed-equivalent income Anonymity implies symmetry of W. Ey is mean income Richer-to-poorer income transfers increase welfare. Ey higher welfare x  y 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 x Ey

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”

Special form of the SWF W = — S z(yh) nh h=1 z(y) = —— y1 – i 1 – i It can make sense to write W in the additive form nh 1 W = — S z(yh) nh h=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 = E z(yh) where the expectation is over all identities probability of identity h is the same, 1/nh , for all h Constant relative-inequality aversion: z(y) = —— y1 – i 1 – i where i is the index of inequality aversion works just like r,the index of relative risk aversion

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 aversion i ...and lower equally-distributed-equivalent income and more sharply curved contours z(y) z = φ(z) higher inequality aversion y income

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

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

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

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