Chapter 33
NAME
Welfare
Introduction.
Here you will look at various ways of determining social
preferences.
You will check to see which of the Arrow axioms for ag
gregating individual preferences are satisfied by these welfare relations.
You will also try to find optimal allocations for some given social welfare
functions.
The method for solving these last problems is analogous to
solving for a consumer’s optimal bundle given preferences and a budget
constraint.
Two hints. Remember that for a Pareto optimal allocation
inside the Edgeworth box, the consumers’ marginal rates of substitution
will be equal. Also, in a “fair allocation,” neither consumer prefers the
other consumer’s bundle to his own.
Example:
A social planner has decided that she wants to allocate income
between 2 people so as to maximize
√
Y
1
+
√
Y
2
where
Y
i
is the amount of
income that person
i
gets. Suppose that the planner has a fixed amount
of money to allocate and that she can enforce any income distribution
such that
Y
1
+
Y
2
=
W
, where
W
is some fixed amount.
This planner
would have ordinary convex indifference curves between
Y
1
and
Y
2
and
a “budget constraint” where the “price” of income for each person is 1.
Therefore the planner would set her marginal rate of substitution between
income for the two people equal to the relative price which is 1. When you
solve this, you will find that she sets
Y
1
=
Y
2
=
W/
2. Suppose instead
that it is “more expensive” for the planner to give money to person 1 than
to person 2. (Perhaps person 1 is forgetful and loses money, or perhaps
person 1 is frequently robbed.) For example, suppose that the planner’s
budget is 2
Y
1
+
Y
2
=
W
. Then the planner maximizes
√
Y
1
+
√
Y
2
subject
to 2
Y
1
+
Y
2
=
W
. Setting her MRS equal to the price ratio, we find that
√
Y
2
√
Y
1
= 2.
So
Y
2
= 4
Y
1
.
Therefore the planner makes
Y
1
=
W/
5 and
Y
2
= 4
W/
5.
33.1 (2)
One possible method of determining a social preference relation
is the
Borda count
, also known as rankorder voting. Each voter is asked
to rank all of the alternatives. If there are 10 alternatives, you give your
first choice a 1, your second choice a 2, and so on. The voters’ scores for
each alternative are then added over all voters.
The total score for an
alternative is called its Borda count. For any two alternatives,
x
and
y
,
if the Borda count of
x
is smaller than or the same as the Borda count
for
y
, then
x
is “socially at least as good as”
y
. Suppose that there are
a finite number of alternatives to choose from and that every individual
has complete, reﬂexive, and transitive preferences.
For the time being,
let us also suppose that individuals are never indifferent between any two
different alternatives but always prefer one to the other.
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410
WELFARE
(Ch.
33)
(a)
Is the social preference ordering defined in this way complete?
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 Summer '09
 JOHNG.SESSIONS
 Microeconomics, Utility, Social Choice and Individual Values

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