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The Social-Welfare Function

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The Social-Welfare Function Prerequisites Almost essential Welfare: Basics Welfare: Efficiency WELFARE: THE SOCIAL- WELFARE FUNCTION MICROECONOMICS Principles and Analysis Frank Cowell April 2018 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 April 2018 Frank Cowell: Welfare - Social Welfare function 2 Overview Welfare: SWF The Approach What is special about a social-welfare SWF: basics function? SWF: national income SWF: income distribution April 2018 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 April 2018 Frank Cowell: Welfare - Social Welfare function 4 Using a SWF υb . Take the utility-possibility set . Social welfare contours . A social-welfare optimum? W(υa, υb,... ) .W defined on utility levels .Not on orderings • .Imposes several restrictions… ...and raises several questions υa April 2018 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? April 2018 Frank Cowell: Welfare - Social Welfare function 6 Overview Welfare: SWF The Approach Where does the social-welfare SWF: basics function come from? SWF: national income SWF: income distribution April 2018 Frank Cowell: Welfare - Social Welfare function 7 An individualistic SWF . The standard form expressed thus W(υ1, υ2, υ3, ...) • 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 April 2018 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 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 April 2018 Frank Cowell: Welfare - Social Welfare function 9 “Equal ignorance”: formalisation payoffs if assigned identity 1,2,3,... in the Lottery of Life . Individualistic welfare: use theory of choice under W(υ1, υ2, υ3, ...) uncertainty to find shape of W . vN-M form of utility function: π : probability assigned to ω ∑ π u(x ) ω ω∈Ω ω ω u : cardinal utility function, Equivalently: independent of ω ∑ω∈Ω πω υω υω: utility payoff in state ω . Replace Ω by set of identities welfare is expected utility {1,2,...nh}: h from a "lottery on identity“ ∑h πh υ . A suitable assumption about “probabilities”? An additive form of the nh welfare function 1 W = — ∑ υh nh h=1 April 2018 Frank Cowell: Welfare - Social Welfare function 10 Questions about “equal ignorance” . Construct a lottery on identity . The “equal ignorance” assumption... Where people know identity with certainty . Intermediate case πh .The “equal ignorance” assumption: πh = 1/nh But is this appropriate? .Or should we assume that | | | | | people know their identities 1 2 3 identity h nh with certainty? .Or is the "truth" somewhere between...? April 2018 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 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 our present course April 2018 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 April 2018 Frank Cowell: Welfare - Social Welfare function 13 The SWF maximum problem . Take the individualistic welfare model Standard W(υ1, υ2, υ3, ...) assumption . Assume everyone is selfish: my utility depends h h h υ = U (x ) , h = 1,2, ..., nh only on my bundle . Substitute in the above: Gives SWF in terms 1 1 2 2 3 3 W(U (x ), U (x ), U (x ), ...) of the allocation a quick sketch April 2018 Frank Cowell: Welfare - Social Welfare function 14 From an allocation to social welfare a a . (x1 , x2 ) From the attainable set... b b (x1 , x2 ) . ...take an allocation . Evaluate utility for each agent A A . Plug into W to get social welfare a a a a υ =U (x1 , x2 ) b b b b υ =U (x1 , x2 ) . But what happens to welfare if we vary the allocation in A? W(υa, υb) April 2018 Frank Cowell: Welfare - Social Welfare function 15 Varying the allocation . Differentiate w.r.t. x h : i h h h h h The effect on if dυ = Ui (x ) dxi commodity i is changed marginal utility derived . Sum over i: by h from good i n The effect on h if all h h h h dυ = Σ Ui (x ) dxi commodities are changed i=1 . Differentiate W with respect to υh: nh Changes in utility h change social welfare . dW = Σ Wh dυ h=1 marginal impact on social welfare of h’s utility . Substitute for dυh in the above: nh n So changes in allocation h h h change welfare. dW = Σ Wh Σ Ui (x ) dxi h=1 i=1 Weights from Weights from utility function the SWF April 2018 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 Lagrangian . Solve in the usual way Now for the maths April 2018 Frank Cowell: Welfare - Social Welfare function 17 The SWF maximum problem . First component of the problem: The objective function W(U1(x1), U2(x2), U3(x3), ...) Individualistic welfare Utility depends on own consumption . Second component of the problem: n Feasibility constraint Φ(x) ≤ 0, x = Σ h x h i h=1 i All goods are private . The Social-welfare Lagrangian: Constraint subsumes W(U1(x1), U2(x2),...) - λΦ (Σnh xh ) technological feasibility and h=1 materials balance . FOCs for an interior maximum: From differentiating h h x h Wh (...) Ui (x ) − λΦi(x) = 0 Lagrangean with respect to i h . And if xi = 0 at the optimum: Usual modification for a h h corner solution Wh (...) Ui (x ) − λΦi(x) ≤ 0 April 2018 Frank Cowell: Welfare - Social Welfare function 18 Solution to SWF maximum problem Any pair of goods, i,j Any pair of households h, ℓ . From FOCs: h h ℓ ℓ MRS equated across all h Ui (x ) Ui (x ) ——— = ——— h h ℓ ℓ We’ve met this condition Uj (x ) Uj (x ) before - Pareto efficiency . Also from the FOCs: social marginal utility of h h ℓ ℓ toothpaste equated across all h Wh Ui (x ) = Wℓ Ui (x ) . Relate marginal utility to prices: h h h This is valid if all consumers Ui (x ) = Vy pi optimise Marginal utility of money . Substituting into the above: At optimum the welfare value of h ℓ $1 is equated across all h. Call Wh Vy = Wℓ Vy Social marginal this common value M utility of income April 2018 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 April 2018 Frank Cowell: Welfare - Social Welfare function 20 Changes in income, social welfare . Social welfare can be expressed as: W(U1(x1), U2(x2),...) SWF in terms of direct utility. = W(V1(p,y1), V2(p,y2),...) Using indirect utility function . Differentiate the SWF w.r.t. {yh}: Changes in utility and change nh nh social welfare … h h h dW = Σ Wh dυ = Σ WhVy dy h=1 h=1 nh Σ h ...related to income dW = M dy change in “national income” h=1 . Differentiate the SWF w.r.t. pi : Changes in utility and change n nh h social welfare … h h h dW = Σ WhVi dpi= – ΣWhVy xi dpi from Roy’s h=1 h=1 nh identity h dW = – M Σ xi dpi Change in total expenditure ...related to prices. h=1 . April 2018 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? April 2018 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 April 2018 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 .
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