Frank Cowell: Welfare - Social Welfare function
WELFARE: THE SOCIAL-WELFARE FUNCTIONMICROECONOMICSPrinciples and Analysis
Frank Cowell
Almost essential
Welfare: Basics
Welfare: Efficiency
Almost essential
Welfare: Basics
Welfare: Efficiency
PrerequisitesPrerequisites
March 2012 1
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 attemptSomething like a simple utility function…?
RequirementsRequirements
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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?
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Frank Cowell: Welfare - Social Welfare function
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
A sketch of the approach
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Frank Cowell: Welfare - Social Welfare function
Using a SWF
ua
ub
U
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(ua, ub,... )W(ua, ub,... )
•
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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?
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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?
March 2012 7
Frank Cowell: Welfare - Social Welfare function
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 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 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 9
Frank Cowell: Welfare - Social Welfare function
“Equal ignorance”: formalisation
Individualistic welfare: W(u1, u2, u3, ...)
use theory of choice under uncertainty to find shape of W
vN-M form of utility function: åwÎW pwu(xw) Equivalently: åwÎW pwuw
pw: probability assigned to wu : cardinal utility function,
independent of wuw: utility payoff in state w
A suitable assumption about “probabilities”? nh
1 W = — å uh
nh h=1
welfare is expected utility from a "lottery on identity“
payoffs if assigned identity 1,2,3,... in the Lottery of Life
payoffs if assigned identity 1,2,3,... in the Lottery of Life
Replace W by set of identities {1,2,...nh}:
åh phuh
An additive form of the welfare function
March 2012 10
Frank Cowell: Welfare - Social Welfare function
Questions about “equal ignorance”
p h
identity
|
nhh |
1
|
2
|
3
|
Construct a lottery on identity The “equal ignorance” assumption...
Where people know their identity with certainty
Intermediate case
The “equal ignorance” assumption: ph = 1/nh
But is this appropriate?
Or should we assume that people know their identities with certainty?
Or is the "truth" somewhere between...?
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Frank Cowell: Welfare - Social Welfare function
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 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
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Frank Cowell: Welfare - Social Welfare function
The SWF maximum problem
Take the individualistic welfare model
W(u1, u2, u3, ...) Standard assumption
Assume everyone is selfish:
uh = 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 sketcha quick sketch
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Frank Cowell: Welfare - Social Welfare function
From an allocation to social welfare
From the attainable set...
AA
(x1a, x2
a)(x1
b, x2b)
(x1a, x2
a)(x1
b, x2b) ...take an allocation
Evaluate utility for each agent
Plug into W to get social welfare
ua=Ua(x1a, x2
a)ub=Ub(x1
b, x2b)
ua=Ua(x1a, x2
a)ub=Ub(x1
b, x2b)
W(ua, ub)W(ua, ub)
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 :
duh = Uih(xh) dxi
h
marginal utility derived by h from good i
marginal utility derived by h from good i
The effect on h if commodity i is changed
Sum over i: n
duh = S Uih(xh) dxi
h
i=1
The effect on h if all commodities are changed
Differentiate W with respect to uh: nh dW = SWh
duh
h=1
Changes in utility change social welfare .
Substitute for duh in the above: nh n dW = S Wh
S Uih(xh) dxi
h
h=1 i=1
So changes in allocation change welfare.
Weights from the SWF
Weights from the SWF
Weights from utility function
Weights from utility function
marginal impact on social welfare of h’s utility
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 Lagrangean. Solve in the usual way
Now for the maths
Now for the maths
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Frank Cowell: Welfare - Social Welfare function
The SWF maximum problem
First component of the problem: W(U1(x1), U2(x2), U3(x3), ...)
Individualistic welfare Individualistic welfare Utility depends on own consumption Utility depends on own consumption
The objective function
Second component of the problem: nh F(x) £ 0, xi = S xi
h h=1
Feasibility constraint
The Social-welfare Lagrangean: nh W(U1(x1), U2(x2),...) - lF (S xh ) h=1
Constraint subsumes technological feasibility and materials balance
FOCs for an interior maximum: Wh (...) Ui
h(xh) − lFi(x) = 0From differentiating Lagrangean with respect to xi
h
And if xih = 0 at the optimum:
Wh (...) Uih(xh) − lFi(x) £ 0
Usual modification for a corner solution
All goods are privateAll goods are private
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Frank Cowell: Welfare - Social Welfare function
Solution to SWF maximum problem
From FOCs: Ui
h(xh) Uiℓ(xℓ)
——— = ———Uj
h(xh) Ujℓ(xℓ)
Any pair of goods, i,jAny pair of households h, ℓ
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 moneyMarginal utility of money
Social marginal utility of income
Social marginal utility of income
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Frank Cowell: Welfare - Social Welfare function
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
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Frank Cowell: Welfare - Social Welfare function
Differentiate the SWF w.r.t. {yh}: nh dW = S Wh
duh
h=1
Changes in income, social welfare
nh dW = M Sdyh
h=1
nh = S 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”change in “national income”
Differentiate the SWF w.r.t. pi : nh dW = S WhVi
hdpi h=1
.
Changes in utility and change social welfare … nh = – SWhVy
h xihdpi h=1
from Roy’s identityfrom Roy’s identity
nh dW = – M S xi
hdpi
h=1
...related to pricesChange in total expenditureChange in total expenditure
.
March 2012 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?
March 2012 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
March 2012 23
Frank Cowell: Welfare - Social Welfare function
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 24
Frank Cowell: Welfare - Social Welfare function
Income-distribution space: nh=2
Bill's
income
Alf'sincome
O
The income space: 2 persons
An income distribution
· y
45°
line o
f perf
ect e
quali
ty
Note the similarity with a diagram used in the analysis of uncertainty
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Frank Cowell: Welfare - Social Welfare function
Extension to nh=3
Here we have 3 persons
Charlie's
income
Alf's income
Bill's income
O
line o
f perf
ect
equality
• y
An income distribution.
March 2012 26
Frank Cowell: Welfare - Social Welfare function
Welfare contours
x Ey
ya
yb
xEy
y
An arbitrary income distribution Contours of W Swap identities Distributions with the same mean
Anonymity implies symmetry of W
Equally-distributed-equivalent income
Ey is mean income Richer-to-poorer income transfers increase welfare.
equivalent in welfare termsequivalent in welfare terms
x is income that, if received uniformly by all, would yield same level of social welfare as y.
higher welfarehigher welfare
Ey x is income that society would give up to eliminate inequality
March 2012 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”
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Frank Cowell: Welfare - Social Welfare function
Special form of the SWF It can make sense to write W in the additive form nh 1 W = — S (z yh) nh h=1
• 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 (z y) = —— y1 – i 1 – i
• where i is the index of inequality aversion• works just like r,the index of relative risk aversion
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Frank Cowell: Welfare - Social Welfare function
Concavity and inequality aversion
W
z(y)
income
y
z(y)
The social evaluation function
Let values change: φ is a concave transformation.
More concave z(•) implies higher inequality aversion i
...and lower equally-distributed-equivalent income
and more sharply curved contours
lower inequality aversionlower inequality aversion
higher inequality aversionhigher inequality aversion
z = φ(z)
March 2012 30
Frank Cowell: Welfare - Social Welfare function
Social views: inequality aversion
= i½
yb
yaO
= 0i
yb
yaO
= 2i
yb
yaO
= i
Indifference to inequality
Mild inequality aversion
yb
yaO
Strong inequality aversion
Priority to poorest
“Benthamite” case ( i = 0): nh
W= S yh
h=1 General case (0< < i ): nh
W = S [yh]1-i/ [1-i] h=1 “Rawlsian” case ( = i ): W = min yh
h March 2012 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 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
Three examplesThree examples
March 2012 32
Frank Cowell: Welfare - Social Welfare function
Social values and welfare optimum
ya
yb The income-possibility set Y
Welfare contours ( i = ½)
Welfare contours ( i = 0)
Welfare contours ( i = )
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
March 2012 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
March 2012 34
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