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Prerequisites Almost essential Welfare: Basics Welfare: Efficiency WELFARE: THE SOCIALWELFARE FUNCTION MICROECONOMICS Principles and Analysis Frank Cowell March 2012 Frank Cowell: Welfare - Social Welfare function 1 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 March 2012 Frank Cowell: Welfare - Social Welfare function 2 Overview... Welfare: SWF The Approach What is special about a social-welfare function? SWF: basics SWF: national income SWF: income distribution March 2012 Frank Cowell: Welfare - Social Welfare function 3 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 March 2012 Frank Cowell: Welfare - Social Welfare function 4 Using a SWF ub Take the utility-possibility set Social welfare contours A social-welfare optimum? W(ua, ub,... ) W defined on utility levels U Not on orderings • Imposes several restrictions… ..and raises several questions ua March 2012 Frank Cowell: Welfare - Social Welfare function 5 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? March 2012 Frank Cowell: Welfare - Social Welfare function 6 Overview... Welfare: SWF The Approach Where does the social-welfare function come from? SWF: basics SWF: national income SWF: income distribution March 2012 Frank Cowell: Welfare - Social Welfare function 7 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 March 2012 Frank Cowell: Welfare - Social Welfare function 8 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 March 2012 Frank Cowell: Welfare - Social Welfare function 9 “Equal ignorance”: formalisation Individualistic welfare: payoffs if assigned identity 1,2,3,... in the Lottery of Life W(u1, u2, u3, ...) vN-M form of utility function: w pwu(xw) Equivalently: w pwuw Replace by set of identities {1,2,...nh}: h phuh A suitable assumption about “probabilities”? nh W = — uh 1 nh March 2012 use theory of choice under uncertainty to find shape of W pw: probability assigned to w u : cardinal utility function, independent of w uw: utility payoff in state w welfare is expected utility from a "lottery on identity“ An additive form of the welfare function h=1 Frank Cowell: Welfare - Social Welfare function 10 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? | | | 1 2 3 | | identity h nh Or should we assume that people know their identities with certainty? Or is the "truth" somewhere between...? March 2012 Frank Cowell: Welfare - Social Welfare function 11 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 March 2012 Frank Cowell: Welfare - Social Welfare function 12 Overview... Welfare: SWF The Approach Conditions for a welfare maximum SWF: basics SWF: national income SWF: income distribution March 2012 Frank Cowell: Welfare - Social Welfare function 13 The SWF maximum problem Take the individualistic welfare model Standard assumption Assume everyone is selfish: my utility depends only on my bundle Substitute in the above: Gives SWF in terms of the allocation W(u1, u2, u3, ...) uh = Uh(xh) , h=1,2,...nh W(U1(x1), U2(x2), U3(x3), ...) a quick sketch March 2012 Frank Cowell: Welfare - Social Welfare function 14 From an allocation to social welfare From the attainable set... (x1a, x2a) (x1b, x2b) ...take an allocation Evaluate utility for each agent A A 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) March 2012 Frank Cowell: Welfare - Social Welfare function 15 Varying the allocation Differentiate w.r.t. xih : duh = Uih(xh) dxih The effect on h if commodity i is changed marginal utility derived by h from good i Sum over i: n h du = S Uih(xh) dxih i=1 Differentiate W with respect to uh: nh dW = SWh duh h in the above: Substitute for du n n dW = S Wh S Uih(xh) dxih h Weights from the SWF March 2012 Changes in utility change social welfare . marginal impact on social welfare of h’s utility h=1 h=1 The effect on h if all commodities are changed i=1 So changes in allocation change welfare. Weights from utility function Frank Cowell: Welfare - Social Welfare function 16 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 March 2012 Frank Cowell: Welfare - Social Welfare function 17 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: n F(x) 0, xi = Sh=1 xih Feasibility constraint The Social-welfare Lagrangean: n 1 1 2 2 W(U (x ), U (x ),...) - lF (Sh=1 xh ) 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 h All goods are private h March 2012 Frank Cowell: Welfare - Social Welfare function 18 Solution to SWF maximum problem From FOCs: Any pair of goods, i,j Any pair of households h, ℓ MRS equated across all h. Uih(xh) Uiℓ(xℓ) ——— = ——— Ujh(xh) Ujℓ(xℓ) We’ve met this condition before - Pareto efficiency Also from the FOCs: Wh Uih(xh) = Wℓ Uiℓ(xℓ) Relate marginal utility to prices: Uih(xh) = Vy hp i social marginal utility of toothpaste equated across all h. This is valid if all consumers optimise Marginal utility of money Substituting into the above: Wh Vyh = Wℓ Vyℓ March 2012 Social marginal utility of income At optimum the welfare value of $1 is equated across all h. Call this common value M Frank Cowell: Welfare - Social Welfare function 19 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 March 2012 Frank Cowell: Welfare - Social Welfare function 20 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}: Changes in utility and change social welfare … nh nh h=1 h=1 dW = S Wh duh = S WhVyh dyh nh dW = M S dyh h=1 ...related to income change in “national income” Differentiate the SWF w.r.t. pi : nh nh h=1 h=1 dW = S WhVihdpi= – SWhVyh xihdpi nh dW = – M S xihdpi h=1 March 2012 Changes in utility and change social welfare … from Roy’s identity Change in total expenditure ...related to prices . . Frank Cowell: Welfare - Social Welfare function 21 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? March 2012 Frank Cowell: Welfare - Social Welfare function 22 Overview... Welfare: SWF The Approach A lesson from risk and uncertainty SWF: basics SWF: national income SWF: income distribution March 2012 Frank Cowell: Welfare - Social Welfare function 23 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, … ) March 2012 Frank Cowell: Welfare - Social Welfare function 24 Income-distribution space: nh=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 45° O March 2012 Alf's income Frank Cowell: Welfare - Social Welfare function 25 Extension to nh=3 Charlie's income Here we have 3 persons An income distribution. •y O March 2012 Frank Cowell: Welfare - Social Welfare function 26 Welfare contours An arbitrary income distribution Contours of W Swap identities yb Distributions with the same mean Equally-distributed-equivalent income equivalent in welfare terms Anonymity implies symmetry of W x Ey Ey is mean income Richer-to-poorer income transfers increase welfare. higher welfare x is income that, if received uniformly by all, would yield same level of social welfare as y. y ya Ey x is income that society would give up to eliminate inequality x Ey March 2012 Frank Cowell: Welfare - Social Welfare function 27 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” March 2012 Frank Cowell: Welfare - Social Welfare function 28 Special form of the SWF It can make sense to write W in the additive form nh W= 1 — S nh h=1 z(yh) • 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: 1 1–i z(y) = —— y 1–i • where i is the index of inequality aversion • works just like r,the index of relative risk aversion March 2012 Frank Cowell: Welfare - Social Welfare function 29 Concavity and inequality aversion W The social evaluation function Let values change: φ is a concave transformation. z(y) lower inequality aversion z(y) higher inequality aversion z = φ(z) More concave z(•) implies higher inequality aversion i ...and lower equally-distributedequivalent income and more sharply curved contours y income March 2012 Frank Cowell: Welfare - Social Welfare function 30 Social views: inequality aversion yb Indifference to inequality yb Mild inequality aversion i=½ i=0 Strong inequality aversion Priority to poorest “Benthamite” case (i = 0): nh ya O yb ya O yb i=2 W= S yh h=1 i= General case (0< i< ): nh W = S [yh]1-i/ [1-i] h=1 O ya O ya “Rawlsian” case (i = ): W = min yh h March 2012 Frank Cowell: Welfare - Social Welfare function 31 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 March 2012 Frank Cowell: Welfare - Social Welfare function Three examples 32 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 y* maximises total income irrespective of distribution y*** y** trades off some income for greater equality y** y* ya March 2012 y*** gives priority to equality; then maximises income subject to that Frank Cowell: Welfare - Social Welfare function 33 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. 2. March 2012 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 Frank Cowell: Welfare - Social Welfare function 34