welfare: the social-welfare functiondarp.lse.ac.uk/presentations/mp2book/oup/welfareswf.pdf ·...
TRANSCRIPT
Frank Cowell: Welfare - Social Welfare function
WELFARE: THE SOCIAL-WELFARE FUNCTIONMICROECONOMICSPrinciples and AnalysisFrank Cowell
Almost essential Welfare: BasicsWelfare: Efficiency
Prerequisites
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Frank Cowell: Welfare - Social Welfare function
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 2
Frank Cowell: Welfare - Social Welfare function
Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
What is special about a social-welfare function?
April 2018 3
Frank Cowell: Welfare - Social Welfare function
The SWF approachRestriction 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 4
Frank Cowell: Welfare - Social Welfare function
Using a SWF
υa
υb
𝕌𝕌
Take the utility-possibility set
A social-welfare optimum? Social welfare contours
W defined on utility levels
Not on orderings
Imposes several restrictions…
..and raises several questions
W(υa, υb,... )
•
April 2018 5
Frank Cowell: Welfare - Social Welfare function
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 6
Frank Cowell: Welfare - Social Welfare function
Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
Where does the social-welfare function come from?
April 2018 7
Frank Cowell: Welfare - Social Welfare function
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 8
Frank Cowell: Welfare - Social Welfare function
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 9
Frank Cowell: Welfare - Social Welfare function
“Equal ignorance”: formalisation
Individualistic welfare:W(υ1, υ2, υ3, ...)
use theory of choice under uncertainty to find shape of W
vN-M form of utility function:∑ω∈Ω πω u(xω)
Equivalently:∑ω∈Ω πω υω
πω: probability assigned to ωu : cardinal utility function,
independent of ωυω: utility payoff in state ω
A suitable assumption about “probabilities”?
nh1W = — ∑ υhnh h=1
welfare is expected utility from a "lottery on identity“
payoffs if assigned identity 1,2,3,... in the Lottery of Life
Replace Ω by set of identities 1,2,...nh:
∑h πh υh
An additive form of the welfare function
April 2018 10
Frank Cowell: Welfare - Social Welfare function
Questions about “equal ignorance”
πh
identity|
nhh|
1
|
2
|
3
|
Construct a lottery on identity The “equal ignorance” assumption...Where people know identity with certainty Intermediate case
The “equal ignorance” assumption: πh = 1/nhBut is this appropriate?
Or should we assume that people know their identities with certainty?
Or is the "truth" somewhere between...?
April 2018 11
Frank Cowell: Welfare - Social Welfare function
2: The PLUM principleNow for the second − rather cynical − approachAcronym stands for People Like Us MatterWhoever 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 12
Frank Cowell: Welfare - Social Welfare function
Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
Conditions for a welfare maximum
April 2018 13
Frank Cowell: Welfare - Social Welfare function
The SWF maximum problem Take the individualistic welfare model
W(υ1, υ2, υ3, ...) Standard assumption
Assume everyone is selfish: υh = 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
April 2018 14
Frank Cowell: Welfare - Social Welfare function
From an allocation to social welfare
From the attainable set...
AA
(x1a, x2
a)(x1
b, x2b) ...take an allocation
Evaluate utility for each agent
Plug into W to get social welfare
υa=Ua(x1a, x2
a)υb=Ub(x1
b, x2b)
W(υa, υb)
But what happens to welfare if we vary the allocation in A?
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Frank Cowell: Welfare - Social Welfare function
Varying the allocation
Differentiate w.r.t. xih :
dυh = Uih(xh) dxi
h
marginal utility derived by h from good i
The effect on h if commodity i is changed
Sum over i: n
dυh = Σ Uih(xh) dxi
hi=1
The effect on h if all commodities are changed
Differentiate W with respect to υh:nh
dW = Σ Wh dυh
h=1
Changes in utility change social welfare .
Substitute for dυh in the above:nh n
dW = Σ Wh Σ Uih(xh) dxi
h
h=1 i=1
So changes in allocation change welfare.
Weights from the SWF
Weights from utility function
marginal impact on social welfare of h’s utility
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Frank Cowell: Welfare - Social Welfare function
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 17
Frank Cowell: Welfare - Social Welfare function
The SWF maximum problem First component of the problem:
W(U1(x1), U2(x2), U3(x3), ...)Individualistic welfare Utility depends on
own consumption
The objective function
Second component of the problem: nhΦ(x) ≤ 0, xi = Σ xi
hh=1
Feasibility constraint
The Social-welfare Lagrangian:nhW(U1(x1), U2(x2),...) - λΦ (Σ xh )
h=1
Constraint subsumes technological feasibility and materials balance
FOCs for an interior maximum:Wh (...) Ui
h(xh) − λΦi(x) = 0From differentiating Lagrangean with respect to xi
h
And if xih = 0 at the optimum:
Wh (...) Uih(xh) − λΦi(x) ≤ 0
Usual modification for a corner solution
All goods are private
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Frank Cowell: Welfare - Social Welfare function
Solution to SWF maximum problem From FOCs:
Uih(xh) Ui
ℓ(xℓ)——— = ———Uj
h(xh) Ujℓ(xℓ)
Any pair of goods, i,jAny pair of households h, ℓ
MRS equated across all h
We’ve met this condition before - Pareto efficiency
Also from the FOCs: Wh Ui
h(xh) = Wℓ Uiℓ(xℓ)
social marginal utility of toothpaste equated across all h
Relate marginal utility to prices:Ui
h(xh) = Vyhpi
This is valid if all consumers optimise
Substituting into the above:Wh Vy
h = Wℓ Vyℓ
At optimum the welfare value of $1 is equated across all h. Call this common value M
Marginal utility of money
Social marginal utility of income
April 2018 19
Frank Cowell: Welfare - Social Welfare function
To focus on main result... Look what happens in neighbourhood of optimumAssume 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 20
Frank Cowell: Welfare - Social Welfare function
Differentiate the SWF w.r.t. yh:nh
dW = Σ Wh dυhh=1
Changes in income, social welfare
nh
dW = M Σ dyhh=1
nh
= Σ WhVyh dyh
h=1
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
Changes in utility and change social welfare …
...related to incomechange in “national income”
Differentiate the SWF w.r.t. pi :nh
dW = Σ WhVihdpi
h=1
.
Changes in utility and change social welfare …nh
= – ΣWhVyh xi
hdpih=1 from Roy’s
identitynh
dW = – M Σ xihdpi
h=1...related to prices
Change in total expenditure
.
April 2018 21
Frank Cowell: Welfare - Social Welfare function
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 22
Frank Cowell: Welfare - Social Welfare function
Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
A lesson from risk and uncertainty
April 2018 23
Frank Cowell: Welfare - Social Welfare function
Derive a SWF in terms of incomesWhat 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 incomesDo 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 asW(ya, yb, yc, … )
April 2018 24
Frank Cowell: Welfare - Social Welfare function
Income-distribution space: nh=2
Bill'sincom
e
Alf'sincome
O
The income space: 2 persons
An income distribution
• y
45°
Note the similarity with a diagram used in the analysis of uncertainty
April 2018 25
Alf'sincome
Frank Cowell: Welfare - Social Welfare function
Extension to nh = 3
Here we have 3 persons
Charlie's
income
O
•y
An income distribution.
April 2018 26
Frank Cowell: Welfare - Social Welfare function
Welfare contours
ξ Ey
ya
yb
ξEy
• y
An arbitrary income distribution Contours of W Swap identities Distributions with the same mean
Anonymity implies symmetry of W
Equally-distributed-equivalent income
E y is mean income Richer-to-poorer income transfers increase welfare
•
equivalent in welfare terms
ξ is income that, if received uniformly by all, would yield same level of social welfare as y
higher welfare
E y −ξ is income that society would give up to eliminate inequality
April 2018 27
Frank Cowell: Welfare - Social Welfare function
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 28
Frank Cowell: Welfare - Social Welfare function
Special form of the SWF It can make sense to write W in the additive form
nh1W = — 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:1ζ(y) = —— y1 – ι
1 – ι
• where ι is the index of inequality aversion• works just like ρ,the index of relative risk aversion
April 2018 29
Frank Cowell: Welfare - Social Welfare function
Concavity and inequality aversion
W
ζ(y)
incomey
ζ°(y)
The social evaluation function Let values change: φ is a concave transformation.
More concave ζ(•) implies higher inequality aversion ι
...and lower equally-distributed-equivalent income
and more sharply curved contours
lower inequality aversion
higher inequality aversion
ζ° = φ(ζ)
April 2018 30
Frank Cowell: Welfare - Social Welfare function
Social views: inequality aversion
ι = ½
yb
yaO
ι = 0
yb
yaO
ι = 2
yb
yaO
ι = ∞
Indifference to inequality
Mild inequality aversionyb
yaO
Strong inequality aversion Priority to poorest
“Benthamite” case (ι = 0): nh
W= Σ yh
h=1
General case (0< ι< ∞): nh
W = Σ [yh]1-ι/ [1-i]h=1
“Rawlsian” case (ι = ∞): W = min yh
h
April 2018 31
Frank Cowell: Welfare - Social Welfare function
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 curiosityOr 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 32
Frank Cowell: Welfare - Social Welfare function
Social values and welfare optimum
ya
yb The income-possibility set Y
Welfare contours ( ι = ½)Welfare contours ( ι = 0)
Welfare contours ( ι = ∞)
Y derived from set A
Nonconvexity, asymmetry come from heterogeneity of households
y* maximises total income irrespective of distribution
• y*** gives priority to equality; then maximises income subject to that
•Y
y*
y***
• y** y** trades off some income for greater equality
April 2018 33
Frank Cowell: Welfare - Social Welfare function
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 values2. 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 34