Water Example:
So, as I was saying in section, there are two groups of interest, the farmer and the ten sports fishermen.
Their benefits functions are given as follows:
2
2
05
.
0
18
)
(
2
240
)
(
s
s
s
si
f
f
f
f
Y
Y
Y
B
Y
Y
Y
B
−
=
−
=
The constraint here is that there are only 200 acre/feet of water, so the consumption of the two groups is
limited to this.
We want to know what the socially optimal level of water consumption is.
There are two
ways to solve this problem.
The first is the same method we used in the heterogeneous polluters case,
where we first summed the demands horizontally (after taking their inverses), figured out the MB where
the supply curve intersects the aggregate marginal benefits, and then figure out the consumption for both
groups.
The alternative method is to maximize the sum of the two benefits functions subject to the
constraint that the groups can only consume 200 acre/feet of water in total.
This problem can be
described by the following Lagrangian:
)
200
(
)
(
)
(
f
s
s
s
f
f
Y
Y
Y
B
Y
B
L
−
−
+
+
=
λ
where the demand for the sports fishermen is now their
aggregate demand, i.e. ten times the individual one.
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 Spring '07
 Wood
 Economics, Social welfare function, Foundations of Economic Analysis, marginal benefits

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