Social welfare functions
Autumn term 2011
Introduction
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Last week:
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Interest group behavior
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Economic man versus political man
Political contributions fundamental in electoral outcome
Today:
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Positive versus normative theory
Normative: need to develop theory based on generally
accepted postulates
Social welfare function
the Bergson-Samuelson social welfare function
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Values of the community:
W = W (z1 , z2 , ..., zn )
(1)
W is real valued function and zs are ethical values of society
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General agreement on the Pareto postulate: necessary
conditon for maximization of welfare
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Production conditions
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Consumption conditions
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Can define a Pareto-possiblity frontier
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Which point on this frontier should be chosen:
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W = W (U1 , U2 , ..., Us )
(2)
6 apple share
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Consider:
W = U1 + U2
(3)
U1 (3) + U2 (3) > U1 (4) + U2 (2)
(4)
U2 (3) − U2 (2) > U1 (4) − U1 (3)
(5)
U1 (3)U2 (3) > U1 (4)U2 (2)
(6)
Wish to select:
implying:
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Next:
which gives:
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U1 (4)
U2 (3)
>
(7)
U2 (2)
U1 (3)
Ordinal utility function incapable of yielding a maximum at
(3,3)
Only way to get a determinant outcome is to define ordinal
utility indicators lexicographically
Axiomatic social welfare functions
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what sort of axioms must we impose to obtain a reasonable
SWF
Fleming’s social welfare function:
W = f1 (U1 ) + f2 (U2 ) + ... + fs (Us )
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(8)
satisfying Pareto principle and indifferent individuals axiom
Elimination of indifferent individuals axiom: Given at least 3
individuals suppose that i and j are indifferent between x and
x 0 , and between y and y 0 but i prefers x to y and j prefers y
to x . Suppose that all individuals are indifferent between x
and y , and x 0 and y 0 . Then social preferences must always go
in the same way between x and y as they do with x 0 and y 0
EII:
1. Eliminates individuals who are indifferent between x and y
2. it requires that whatever convention is used to decide
preferences regarding x and y override j 0 s, it must also decide
the pair (x 0 , y 0 ) given i and j’s indifference between x and x 0
Harsanyi’s social welfare function
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Three assumptions:
1. Individual personal preferences satisfy the von
Neumann-Morgenstern-Marschak axioms of choice involving
risk
2. Individual ethical preferences satisfy the same axioms
3. If two propsects P and Q are indifferent from the standpoint of
every individual, they are indifferent from a social standpoint
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Social welfare:
W = a1 U1 + a2 U2 + ... + as Us
(9)
where ai stands for the value that W takes when Uj = 0 for
all j 6= i
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first two assumptions effectively introduce interpersonally
comparable, cardinal utilities into the SWF
Two criticisms of Harsanyi’s social welfare function
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Should individual attiudes toward risk count?
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whether social choices should depend on individual attitudes is
a difficult question
Can individuals agree on a value of W ?
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It does not allow for differences in risk aversion among
impartial observers who determine the SWF values. If they
differ in their preferences towards risk, so too will their
evaluations of social welfare under the possible states of the
world, and unanimous agreement on the SWF will not be
possible
Ng’s social welfare function
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finite sensibility units: Individuals are capable of perceiving
changes in x only for discreet intervals of x
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Weak majority preference criterion: if a majority prefers x to
y and all members of the minority are indifferent, then society
prefers x to y
Multiplicative SWFs
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Nash bargaining game:
W = (U1 − U1∗ )(U2 − U2∗ )...(Us − Us∗ )
where Uj∗ is the status quo point
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Property α
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Symmetry
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Reasonable way to view writing of constitution or social
contract
(10)