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 𝕌𝕌
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 . 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, … )
April 2018 Frank Cowell: Welfare - Social Welfare function 24 Income-distribution space: nh=2
. The income space: 2 persons income Bill's
.An income distribution
. Note the similarity with a diagram used in the analysis of uncertainty
• y
45° O Alf's Alf's income income
April 2018 Frank Cowell: Welfare - Social Welfare function 25 Extension to nh = 3 income Charlie's . Here we have 3 persons .An income distribution.
•y
O
April 2018 Frank Cowell: Welfare - Social Welfare function 26 Welfare contours yb . An arbitrary income distribution . Contours of W . Swap identities . Distributions with the same mean . Equally-distributed-equivalent income equivalent in welfare terms . • Anonymity implies symmetry of W . E y is mean income . Richer-to-poorer income Ey higher transfers increase welfare ξ welfare . ξ is income that, if received • y uniformly by all, would yield same level of social welfare as y . E −ξ ya y is income that society would give up to eliminate inequality ξ Ey
April 2018 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”
April 2018 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 Σ ζ(yh) nh h=1 • where the function ζ 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 ζ(yh) • where the expectation is over all identities • probability of identity h is the same, 1/nh , for all h . Constant relative-inequality aversion: ζ(y) = ——1 y1 – ι 1 – ι • where ι is the index of inequality aversion • works just like ρ,the index of relative risk aversion
April 2018 Frank Cowell: Welfare - Social Welfare function 29 Concavity and inequality aversion
W .The social evaluation function . Let values change: φ is a concave transformation.
lower inequality ζ(y) aversion . More concave ζ(•) implies higher ζ°(y) inequality aversion ι
....and lower equally-distributed- ζ° = φ(ζ) higher inequality equivalent income aversion .and more sharply curved contours y income
April 2018 Frank Cowell: Welfare - Social Welfare function 30 Social views: inequality aversion
yb yb . Indifference to inequality ι = 0 ι = ½ . Mild inequality aversion . Strong inequality aversion . Priority to poorest
. “Benthamite” case (ι = 0):
nh O ya O ya h b b W= Σ y y y h=1 ι = 2 ι = ∞ . General case (0< ι< ∞):
nh W = Σ [yh]1-ι/ [1-i] h=1
. “Rawlsian” case (ι = ∞): a a O y O y W = min yh h
April 2018 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 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
April 2018 Frank Cowell: Welfare - Social Welfare function 32 Social values and welfare optimum
b y . 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* maximises total income Y y*** irrespective of distribution • . y** trades off some income for ** • y greater equality y* . y*** gives priority to equality; then • ya maximises income subject to that
April 2018 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. 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
April 2018 Frank Cowell: Welfare - Social Welfare function 34