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
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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
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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
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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
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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
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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
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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
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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
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“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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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