WELFARE: THE SOCIAL- WELFARE FUNCTION

Transcription

1 Prerequisites Almost essential Welfare: Basics Welfare: Efficiency WELFARE: THE SOCIAL- WELFARE FUNCTION MICROECONOMICS Principles and Analysis Frank Cowell July

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 July

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

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 July

5 Using a SWF υ b W(υ a, υ b,... ) Take the utility-possibility set Social welfare contours A social-welfare optimum? W defined on utility levels UU Not on orderings Imposes several restrictions..and raises several questions υ a July

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? July

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

8 An individualistic SWF The standard form expressed thus W(υ 1, υ 2, υ 3,...) 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 July

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 ω and θ! 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 July

10 Equal ignorance : formalisation Individualistic welfare: W(υ 1, υ 2, υ 3,...) vn-m form of utility function: ω Ω π ω u(x ω ) Equivalently: ω Ω π ω υ ω Replace Ω by set of identities {1,2,...n h }: h π h υ h A suitable assumption about probabilities? n h 1 W = υ h n h h=1 payoffs if assigned identity 1,2,3,... in the Lottery of Life use theory of choice under uncertainty to find shape of W π ω : probability assigned to ω u : cardinal utility function, independent of ω υ ω : utility payoff in state ω welfare is expected utility from a "lottery on identity An additive form of the welfare function July

11 Questions about equal ignorance π h Construct a lottery on identity The equal ignorance assumption... Where people know identity with certainty Intermediate case The equal ignorance assumption: π h = 1/n h But is this appropriate? identity h n h Or should we assume that people know their identities with certainty? Or is the "truth" somewhere between...? July

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 July

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

14 The SWF maximum problem Take the individualistic welfare model W(υ 1, υ 2, υ 3,...) Standard assumption Assume everyone is selfish: υ h = U h (x h ), h = 1,2,..., n h my utility depends only on my bundle Substitute in the above: W(U 1 (x 1 ), U 2 (x 2 ), U 3 (x 3 ),...) Gives SWF in terms of the allocation a quick sketch July

15 From an allocation to social welfare A A (x 1a, x 2a ) (x 1b, x 2b )...take an allocation From the attainable set... Evaluate utility for each agent Plug into W to get social welfare υ a =U a (x 1a, x 2a ) υ b =U b (x 1b, x 2b ) But what happens to welfare if we vary the allocation in A? W(υ a, υ b ) July

16 Varying the allocation Differentiate w.r.t. x ih : dυ h = U ih (x h ) dx i h marginal utility derived by h from good i Sum over i: n dυ h = Σ U ih (x h ) dx i h i=1 Differentiate W with respect to υ h : n h dw = Σ W h dυ h h=1 Substitute for dυ h in the above: n h n dw = Σ W h Σ U ih (x h ) dx h i Weights from the SWF h=1 i=1 marginal impact on social welfare of h s utility Weights from utility function The effect on h if commodity i is changed The effect on h if all commodities are changed Changes in utility change social welfare. So changes in allocation change welfare. July

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

18 The SWF maximum problem First component of the problem: W(U 1 (x 1 ), U 2 (x 2 ), U 3 (x 3 ),...) Individualistic welfare Utility depends on own consumption Second component of the problem: n Φ(x) 0, x h i = Σ x h i h=1 The Social-welfare Lagrangian: n h W(U 1 (x 1 ), U 2 (x 2 ),...) - λφ (Σ x h ) h=1 FOCs for an interior maximum: W h (...) U ih (x h ) λφ i (x) = 0 And if x ih = 0 at the optimum: W h (...) U ih (x h ) λφ i (x) 0 All goods are private The objective function Feasibility constraint Constraint subsumes technological feasibility and materials balance From differentiating Lagrangean with respect to x i h Usual modification for a corner solution July

19 Solution to SWF maximum problem Any pair of goods, i,j Any pair of households h, l From FOCs: U ih (x h ) U il (x l ) = U jh (x h ) U jl (x l ) Also from the FOCs: W h U ih (x h ) = W l U il (x l ) MRS equated across all h We ve met this condition before - Pareto efficiency social marginal utility of toothpaste equated across all h Relate marginal utility to prices: U ih (x h ) = V yh p i Substituting into the above: W h V yh = W l V y l Marginal utility of money Social marginal utility of income This is valid if all consumers optimise At optimum the welfare value of $1 is equated across all h. Call this common value M July

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 July

21 Changes in income, social welfare Social welfare can be expressed as: W(U 1 (x 1 ), U 2 (x 2 ),...) = W(V 1 (p,y 1 ), V 2 (p,y 2 ),...) Differentiate the SWF w.r.t. {y h }: n h dw = Σ W h dυ h h=1 n h dw = M Σ dy h h=1 n h = Σ W h V yh dy h h=1 change in national income Differentiate the SWF w.r.t. p i : n h dw = Σ W h V ih dp i = ΣW h V yh x ih dp i h=1 n h dw = M Σ x ih dp i h=1 n h h=1 Change in total expenditure from Roy s identity SWF in terms of direct utility. Using indirect utility function Changes in utility and change social welfare...related to income Changes in utility and change social welfare...related to prices.. July

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? July

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

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(y a, y b, y c, ) July

25 Income-distribution space: n h =2 The income space: 2 persons Bill's income An income distribution Note the similarity with a diagram used in the analysis of uncertainty y O 45 Alf's income Alf's income July

26 Extension to n h =3 Charlie's income Here we have 3 persons An income distribution. y O July

27 Welfare contours y b 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 E y is mean income ξ E y higher welfare y Richer-to-poorer income transfers increase welfare ξ is income that, if received uniformly by all, would yield same level of social welfare as y y a E y ξ is income that society would give up to eliminate inequality ξ E y July

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 July

29 Special form of the SWF It can make sense to write W in the additive form n h 1 W = Σ ζ(y h ) n h h=1 where the function ζ is the social evaluation function (the 1/n h term is unnecessary arbitrary normalisation) Counterpart of u-function in choice under uncertainty Can be expressed equivalently as an expectation: W = E ζ(y h ) where the expectation is over all identities probability of identity h is the same, 1/n h, for all h Constant relative-inequality aversion: 1 ζ(y) = y 1 ι 1 ι where ι is the index of inequality aversion works just like ρ,the index of relative risk aversion July

30 Concavity and inequality aversion W The social evaluation function Let values change: φ is a concave transformation. lower inequality aversion higher inequality aversion ζ(y) ζ (y) ζ = φ(ζ) y income More concave ζ( ) implies higher inequality aversion ι...and lower equally-distributedequivalent income and more sharply curved contours July

31 Social views: inequality aversion y b y b Indifference to inequality ι = 0 ι = ½ Mild inequality aversion Strong inequality aversion Priority to poorest Benthamite case (ι = 0): O y b y a O y b y a n h W= Σ y h h=1 ι = 2 ι = General case (0< ι< ): n h W = Σ [y h ] 1-ι / [1-i] h=1 O y a O y a Rawlsian case (ι = ): W = min y h h July

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 ζ should be taken as identical Optimal social state depends crucially on shape of W In other words the shape of ζ Or the value of ι Three examples July

33 Social values and welfare optimum y b The income-possibility set Y Welfare contours ( ι = 0) Welfare contours ( ι = ½) Welfare contours ( ι = ) Y derived from set A Nonconvexity, asymmetry come from heterogeneity of households Y y *** y** y* maximises total income irrespective of distribution y** trades off some income for greater equality y * y a y*** gives priority to equality; then maximises income subject to that July

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: 1. Information on social values 2. 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 July