Transcript Welfare: The Social-Welfare Function

Prerequisites
Almost essential
Welfare: Basics
Welfare: Efficiency
Frank Cowell: Microeconomics
December 2006
Welfare: The Social-Welfare
Function
MICROECONOMICS
Principles and Analysis
Frank Cowell
Social Welfare Function
Frank Cowell: Microeconomics
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Limitations of the welfare analysis so far:
Constitution approach
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General welfare criteria
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Arrow theorem – is the approach overambitious?
Efficiency – nice but indecisive
Extensions – contradictory?
SWF is our third attempt
Something like a simple utility function…?
Requirements
Overview...
Welfare: SWF
Frank Cowell: Microeconomics
The Approach
What is special
about a socialwelfare function?
SWF: basics
SWF: national
income
SWF: income
distribution
The SWF approach
Frank Cowell: Microeconomics
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Restriction of “relevant” aspects of social state to each
person (household)
Knowledge of preferences of each person (household)
Comparability of individual utilities
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utility levels
utility scales
An aggregation function W for utilities
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contrast with constitution approach
there we were trying to aggregate orderings
A sketch of the
approach
Using a SWF
Frank Cowell: Microeconomics
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
ua questions
Issues in SWF analysis
Frank Cowell: Microeconomics
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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?
Overview...
Welfare: SWF
Frank Cowell: Microeconomics
The Approach
Where does the
social-welfare
function come
from?
SWF: basics
SWF: national
income
SWF: income
distribution
An individualistic SWF
Frank Cowell: Microeconomics
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The standard form expressed thus
W(u1, u2, u3, ...)
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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:
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The equal-ignorance assumption
The PLUM principle
1: The equal ignorance approach
Frank Cowell: Microeconomics
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Suppose the SWF is based on individual preferences.
Preferences are expressed behind a “veil of ignorance”
It works like a choice amongst lotteries
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don't confuse w and q!
Each individual has partial knowledge:
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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
payoffs I would get if I were
“Equalassigned
ignorance”:
formalisation
identity 1,2,3,... in the
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Great
Lottery model:
of Life
The individualistic
welfare
use theory of choice under
uncertainty to find the
shape of SWF W
 vN-M form of the utility function:
pw: probability assigned to w
u: cardinal utility function,
independent of w
uw: utility payoff in state w
W(u1, u2, u3, ...)
w pwu(xw)
Equivalently:
w pwuw
 Replace  by the set of
identities {1,2,...nh}:
welfare is expected utility
from a "lottery on identity“
A suitable assumption about
“probabilities”?
An additive form of
the welfare function
h phuh
nh
W = — S uh
1
nh
h=1
Questions about “equal ignorance”
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 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?
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1 2 3
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identity
h
nh
Or should we assume
that people know their
identities with certainty?
Or is the "truth"
somewhere between...?
2: The PLUM principle
Frank Cowell: Microeconomics
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Now for the second  rather cynical approach
Acronym stands for People Like Us Matter
Whoever is in power may impute:
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...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”
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Concerned with the interaction of political decision-making
and economic outcomes.
But beyond the scope of this course
Overview...
Welfare: SWF
Frank Cowell: Microeconomics
The Approach
Conditions for a
welfare
maximum
SWF: basics
SWF: national
income
SWF: income
distribution
The SWF maximum problem
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Take the individualistic welfare model
Standard
assumption
Assume everyone is selfish:
my utility depends only
on my bundle
W(u1, u2, u3, ...)
uh = Uh(xh) , h=1,2,...nh
Substitute in the above:
W(U1(x1), U2(x2), U3(x3), ...)
Gives SWF in terms
of the allocation
a quick sketch
From an allocation to social welfare
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 From the attainable set...
(x1a, x2a)
(x1b, x2b)
A
 ...take an allocation
 Evaluate utility for each agent
 Plug into W to get social
welfare
A
ua=Ua(x1a, x2a)
ub=Ub(x1b, x2b)
W(ua, ub)
 But what happens to
welfare if we vary the
allocation in A?
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The
marginal
utility
Varyingderived
the
allocation
by household h
from good
h i
Differentiate w.r.t. xi :
duh = Uih(xh) dxih
The effect on h if
commodity i is changed
Sum over i:
n
h
du = S Uih(xh) dxih
The effect on h if all
commodities are changed
i=1
Differentiate W
nh
dW =
The marginal impact on
social welfare of
Changes in utility
h
with
respect
to u : change social welfare .
household
h’s utility
S Wh duh
h=1
Weights from the
Weights from the
h
So changes in allocation
Substitute
du
in
the
above:
SWF for
utility
function
nh
n
change welfare.
dW =
S Wh SUih(xh)
h=1
i=1
dxih
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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
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Now for the
maths
The SWF maximum problem
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Utility depends
on own
consumption
Individualistic
welfare function
First component of the problem:
W(U1(x1), U2(x2), U3(x3), ...)
The objective function
All goods are
private
Second component of the problem:
n
F(x)  0, xi = Sh=1 xih Usual
Feasibility constraint
The Social-welfare Lagrangean:
n
1
1
2
2
W(U (x ), U (x ), ...)  lF(Sh=1xh )
Note: 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
Lagrange
multiplier
h
Solution to SWF maximum problem
Any pair of goods, i,j
Any pair of households h, ℓ
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• From the first-order conditions :
Uih(xh)
Uiℓ(xℓ)
——— = ———
Ujh(xh)
Ujℓ(xℓ)
• Also from the FOCs:
MRS equated across all h.
We’ve met this condition
before - Pareto efficiency
This is new!
Wh Uih(xh) = Wℓ Uiℓ(xℓ)
social marginal utility of
toothpaste equated across all h.
Marginal
utility of
money
• Relate marginal utility to prices:
Uih(xh) = Vyhpi
Social marginal
utility of income
• Substituting into the above:
Wh Vyh = Wℓ Vyℓ
This is valid if all consumers
optimise
At the optimum the welfare
value of a $ of income is
equated across all h. Call this
common value M
To focus on main result...
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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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Changes in income, social welfare
• Social welfare can be expressed as:
W(U1(x1), U2(x2),...)
= W(V1(p,y1), V2(p,y2),...)
• Differentiate the SWF w.r.t. {y }:
h
nh
nh
dW = S Wh duh =
h=1
nh
dW = M S dyh
S WhVyh dyh
SWF in terms of direct utility.
Using indirect utility function
Changes in utility and change
social welfare …
h=1 in total incomes - i.e.
Change
change in “national income”
...related to income
• Differentiate the SWF w.r.t. p :
h=1
Follows from
i
Roy’s
identity
n
nh
h x hdp
– SinWtotal
V
dW = SWhVihdpi =Change
h y i
i
h
Changes in utility and change
social welfare …
expenditure
h=1
h=1
nh
dW = – M S xihdpi
h=1
...related to prices.
.
An attractive result?
Frank Cowell: Microeconomics
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Summarising the results of the previous slide we
have:
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THEOREM: in the neighbourhood of a welfare
optimum welfare changes are measured by
changes in national income / national
expenditure
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But what if we are not in an ideal world?
Overview...
Welfare: SWF
Frank Cowell: Microeconomics
The Approach
A lesson from
risk and
uncertainty
SWF: basics
SWF: national
income
SWF: income
distribution
Derive a SWF in terms of incomes
Frank Cowell: Microeconomics
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What happens if the distribution of income is not ideal?
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Useful to express social welfare in terms of incomes
Do this by using indirect utility functions V
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Express utility in terms of prices p and incomes y
Assume prices p are given
“Equivalise” (i.e. rescale) incomes y
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M is no longer equal for all h
allow for differences in people’s needs
allow for differences in household size
Then you can write welfare as
W(ya, yb, yc, … )
Income-distribution space: nh=2
An income distribution
Bill's
income
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 The income space: 2
persons
 Note the similarity with
a diagram used in the
analysis of uncertainty
y
45°
O
Alf's
income
Extension to nh=3
Charlie's
income
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 Here we have 3 persons
An income distribution.
•y
O
Welfare contours
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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.
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 Ey is mean income
x
Ey
higher
welfare
y
x Ey
 Richer-to-poorer income
transfers increase welfare.
 x is the income that, if
received uniformly by all,
would yield same level of
ya social welfare as y.
 Ey x is the income that,
society would give up to
eliminate inequality
A result on inequality aversion
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Principle of Transfers : “a mean-preserving
redistribution from richer to poorer should
increase social welfare”
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THEOREM: Quasi-concavity of W implies
that social welfare respects the “Transfer
Principle”
Special form of the SWF
Frank Cowell: Microeconomics
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It can make sense to write W in the additive form
nh
W = — S z(yh)
1
nh
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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)
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h=1
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
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where i is the index of inequality aversion
works just like r,the index of relative risk aversion
Concavity and inequality aversion
Frank Cowell: Microeconomics
The social evaluation function
W
 Let values change: φ is a
concave transformation.
z(y)
lower inequality
aversion
z(y)
higher inequality
aversion
z = φ(z)
y
income
 More concave z(•)
implies higher inequality
aversion i
...and lower equallydistributed-equivalent
income
and more sharply
curved contours
Social views: inequality aversion
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yb
yb
 Indifference to inequality
i=½
i=0
 Mild inequality aversion
 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
Frank Cowell: Microeconomics
Inequality, welfare, risk and
uncertainty
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There is a similarity of form between…
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Likewise a logical link between risk and inequality.
This could be seen as just a curiosity
Or as an essential component of welfare economics
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personal judgments under uncertainty
social judgments about income distributions.
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
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In other words the shape of z
Or the value of i
Three examples
Social values and welfare optimum
Frank Cowell: Microeconomics
yb
 The income-possibility set Y
 Welfare contours ( i = 0)
 Welfare contours ( i = ½)
 Welfare contours ( i = )
Y derived from set A or U
Y
Nonconvexity, asymmetry
come from heterogeneity of
households
 y* maximises total income
irrespective of distribution
y***

**
y

y*

ya
 y** trades off some
income for greater equality
 y*** gives priority to
equality; then maximises
income subject to that
Summary
Frank Cowell: Microeconomics
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The standard SWF is an ordering on utility levels
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2.
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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:
Information on social values
Determining the income frontier
This requires a modelling of what is possible in the
underlying structure of the economy...
...which is what Micro-Economic principles is all
about