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Welfare: The SocialWelfare Function
MICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Welfare: BasicsWelfare: Efficiency
Prerequisites
July 2015
1
Slide2Social Welfare Function
Limitations of the welfare analysis so far:Constitution approachArrow theorem – is the approach overambitious?General welfare criteriaefficiency – nice but indecisiveextensions – contradictory?SWF is our third attemptSomething like a simple utility function…?
Requirements
July 2015
2
Slide3Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
What is special about a socialwelfare function?
July 2015
3
Slide4The SWF approach
Restriction of “relevant” aspects of social state to each person (household)Knowledge of preferences of each person (household)Comparability of individual utilitiesutility levelsutility scalesAn aggregation function W for utilitiescontrast with constitution approachthere we were trying to aggregate orderings
A sketch of the approach
July 2015
4
Slide5Using a SWF
u
a
u
b
Take the utilitypossibility set
A socialwelfare optimum?
Social welfare contours
W
defined on utility
levels
Not on orderings
Imposes several restrictions…
..and
raises several questions
W
(
u
a, ub,... )
•
July 2015
5
Slide6Issues 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?
July 2015
6
Slide7Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
Where does the socialwelfare function come from?
July 2015
7
Slide8An individualistic SWF
The standard form expressed thus W(u1, u2, u3, ...)an ordinal functiondefined on space of individual utility levelsnot on profiles of orderingsBut where does W come from...?We'll check out two approaches:The equalignorance assumptionThe PLUM principle
July 2015
8
Slide91: 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 lotteriesdon't confuse w and q!Each individual has partial knowledge:knows the distribution of allocations in the populationknows the utility implications of the allocationsknows the alternatives in the Great Lottery of Lifedoes not know which lottery ticket he/she will receive
July 2015
9
Slide10“Equal ignorance”: formalisation
Individualistic welfare: W(u1, u2, u3, ...)
use theory of choice under uncertainty to find shape of W
vNM 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
Replace W by set of identities {1,2,...nh}: åh phuh
An additive form of the welfare function
July 2015
10
Slide11Questions about “equal ignorance”
ph
identity
nh
h

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:
p
h = 1/nhBut is this appropriate?
Or should we assume that people know their identities with certainty?
Or is the "truth" somewhere between...?
July 2015
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Slide122: The PLUM principle
Now 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 powerThere’s a whole branch of modern microeconomics that is a reinvention of classical “Political Economy”Concerned with the interaction of political decisionmaking and economic outcomes.But beyond the scope of this course
July 2015
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Slide13Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
Conditions for a welfare maximum
July 2015
13
Slide14The 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 sketch
July 2015
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Slide15From an allocation to social welfare
From the attainable set...
A
A
(
x
1
a
,
x
2
a
)(x1b, x2b)
...take an allocation
Evaluate utility for each agent
Plug into
W
to get social welfare
u
a=Ua(x1a, x2a)ub=Ub(x1b, x2b)
W(ua, ub)
But what happens to welfare if we vary the allocation in A?
July 2015
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Slide16Varying the allocation
Differentiate w.r.t. xih : duh = Uih(xh) dxih
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) dxih i=1
The effect on h if all commodities are changed
Differentiate W with respect to uh: nh dW = S Wh duh h=1
Changes in utility change social welfare .
Substitute for duh in the above: nh n dW = S Wh S Uih(xh) dxih 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
July 2015
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Slide17Use this to characterise a welfare optimum
Write down SWF, defined on individual utilitiesIntroduce feasibility constraints on overall consumptionsSet up the LagrangianSolve in the usual way
Now for the maths
July 2015
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Slide18The 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: nh F(x) £ 0, xi = S xih h=1
Feasibility constraint
The Socialwelfare Lagrangian: nh W(U1(x1), U2(x2),...)  lF (S xh ) h=1
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
All goods are private
July 2015
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Slide19Solution to SWF maximum problem
From FOCs: Uih(xh) Uiℓ(xℓ) ——— = ——— Ujh(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 Uih(xh) = Wℓ Uiℓ(xℓ)
social marginal utility of toothpaste equated across all h
Relate marginal utility to prices: Uih(xh) = Vyhpi
This is valid if all consumers optimise
Substituting into the above: Wh Vyh = 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
July 2015
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Slide20To focus on main result...
Look what happens in neighbourhood of optimumAssume that everyone is acting as a maximiserfirmshouseholdsCheck what happens to the optimum if we alter incomes or prices a littleSimilar to looking at comparative statics for a single agent
July 2015
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Slide21Differentiate the SWF w.r.t. {yh}: nh dW = S Wh duh h=1
Changes in income, social welfare
nh dW = M S dyh 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 income
change in “national income”
Differentiate the SWF w.r.t. pi : nh dW = S WhVihdpi h=1
.
Changes in utility and change social welfare …
nh = – SWhVyh xihdpi h=1
from Roy’s identity
nh dW = – M S xihdpi h=1
...related to prices
Change in total expenditure
.
July 2015
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Slide22An 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 expenditureBut what if we are not in an ideal world?
July 2015
22
Slide23Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
A lesson from risk and uncertainty
July 2015
23
Slide24Derive a SWF in terms of incomes
What happens if the distribution of income is not ideal? M is no longer equal for all hUseful to express social welfare in terms of incomesDo this by using indirect utility function V Express utility in terms of prices p and income yAssume prices p are given“Equivalise” (i.e. rescale) each income yallow for differences in people’s needsallow for differences in household sizeThen you can write welfare as W(ya, yb, yc, … )
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Slide25Incomedistribution space:
n
h
=2
Bill's
income
Alf's
income
O
The income space: 2 persons
An income distribution
y
45
°
line of perfect equality
Note the similarity with a diagram used in the analysis of uncertainty
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Alf's
income
Slide26Extension to
nh=3
Here we have 3 persons
Charlie's income
Alf's income
Bill's income
O
line of perfect equality
•
y
An income distribution.
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Slide27Welfare contours
x
E
y
y
a
y
b
x
E
y
y
An arbitrary income distribution
Contours of
W
Swap identities
Distributions with the same mean
Anonymity implies symmetry of
W
Equallydistributedequivalent income
E
y
is mean income Richertopoorer income transfers increase welfare
equivalent in
welfare terms
x
is
income that, if received uniformly by all, would yield same level of social welfare as y
higher welfare
E y x is income that society would give up to eliminate inequality
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Slide28A result on inequality aversion
Principle of Transfers : “a meanpreserving redistribution from richer to poorer should increase social welfare”THEOREM: Quasiconcavity of W implies that social welfare respects the “Transfer Principle”
July 2015
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Slide29Special form of the SWF
It can make sense to write W in the additive form nh 1 W = — S z(yh) nh h=1where the function z is the social evaluation function(the 1/nh term is unnecessary – arbitrary normalisation)Counterpart of ufunction in choice under uncertaintyCan be expressed equivalently as an expectation: W = E z(yh)where the expectation is over all identitiesprobability of identity h is the same, 1/nh , for all hConstant relativeinequality aversion: 1 z(y) = —— y1 – i 1 – iwhere i is the index of inequality aversionworks just like r,the index of relative risk aversion
July 2015
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Slide30Concavity 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 equallydistributedequivalent incomeand more sharply curved contours
lower inequality aversion
higher inequality aversion
z = φ(z)
July 2015
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Slide31Social views: inequality aversion
i =
½
y
b
y
a
O
i = 0
y
b
y
a
O
i = 2
y
b
y
a
O
i =
Indifference to inequality
Mild inequality aversion
y
b
y
a
O
Strong inequality aversion
Priority to poorest
“
Benthamite
” case
(
i
= 0
)
:
n
h
W= S yh h=1
General case (0< i< ): nh W = S [yh]1i/ [1i] h=1
“Rawlsian” case (i = ): W = min yh h
July 2015
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Slide32Inequality, 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 economicsUses 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 WIn other words the shape of zOr the value of i
Three examples
July 2015
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Slide33Social values and welfare optimum
y
a
y
b
The incomepossibility set
Y
Welfare contours (
i
= ½)
Welfare contours ( i = 0)
Welfare contours ( i = )
Y derived from set ANonconvexity, 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
July 2015
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Slide34Summary
The standard SWF is an ordering on utility levels Analogous to an individual's ordering over lotteriesInequality and riskaversion are similar conceptsIn ideal conditions SWF is proxied by national incomeBut for realistic cases two things are crucial:Information on social valuesDetermining the income frontierItem 1 might be considered as beyond the scope of simple microeconomicsItem 2 requires modelling of what is possible in the underlying structure of the economy......which is what microeconomics is all about
July 2015
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