ECON101 Homework 5 Suggested Solutions
3/10/2011
1
Question:
Nancy lives in a gated community of 100 homes (99 + Nancy) that is separated from Stone
Canyon Reservoir by 1,000 feet of fencing.
The fencing is old, and coyotes are wandering in to the
communityl this is not good. The community association wants to repair the fence to prevent coyotes
from coming in. The cost of repairing the fence is $10/foot. All the residents are just like Nancy: if
F
feet of fence are repaired and a given resident pays
c
towards the repair, then that resident derives
utility 1
,
000

(1
,
000

F
)
1
/
2

c
.
(Notice that the benefit comes from the total
amount of fence
repaired, that benefit is increasing in amount of fence repaired, and that repairing the last foot of
fencing is more important than repairing the first foot.)
a Suppose the community association can decide to repair some or all of the fence and divide the
cost equally among all 100 homes. What is the optimal amount of fence repair? (Remember that
the maximum is 1,000 feet.)
b Suppose instead that the community association asks for voluntary
contributions. If each resident
acts perfectly selfishly, how much will each resident contribute?
How much total fence will be
repaired?
Solution:
a The community association wishes to maximize everybody’s utility.
We will assume that the
association views each resident the same, and therefore wishes to solve:
max
100
X
j
=1
(1000

(1000

F
)
1
/
2

c
j
)
Since the association is going to split the total cost of
F
,
C
= 10
F
evenly among the 100 residents,
the problem then becomes:
max
100
X
j
=1
(1000

(1000

F
)
1
/
2

10
F
100
) = 100
,
000

100(1000

F
)
1
/
2

10
F
Taking the first order condition with respect to F yields:
F.O.C.: 100
1
2
(1000

F
)

1
/
2

10 = 0
→
5 = (1000

F
)
1
/
2
→
25 = 1000

F
→
F
=
975
This implies that the optimal amount of fencing repair is 975 feet and that each consumer will
pay
10
*
975
100
= 97
.
5 and derive a utility of 1
,
000

(1
,
000

975)
1
/
2

97
.
5 = 897
.
5.
However, for the result of the first order condition to be a solution to the utility maximization
problem, the second order condition must hold, or:
∂
2
U
∂F
2
F
=975
<
0
must hold. Computing the second derivative yields:
∂
2
U
∂F
2
= 100
1
4
(1000

F
)

3
/
2
>
0
,
∀
F <
1000
1
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so
F
= 975 as a result of the first order condition does not solve the maximization problem. We
have instead found a minimizer of the utility function. Since the second order condition does not
hold for the only solution that satisfies the first order condition, we look towards the boundaries.
Checking the boundaries, we set
F
= 0 and therefore
c
i
= 0
,
∀
i
to find that the utility for any one
resident is:
U
i
(0) = 1000

√
1000
≈
968
.
38
>
897
.
5
If we set
F
= 1000, then
c
i
= 100 and the utility derived is:
U
i
(1000) = 1000

sqrt
(1000

1000)

100 = 900
<
968
.
38
Therefore, the community utility is higher when
F
= 0, so the
total amount of fencing that
should be repaired is 0
.
b Now we suppose that the association asks for voluntary contributions. Consumer
i
is interested
in solving:
max 1
,
000

(1
,
000

F
)
1
/
2

c
i
Here,
F
=
∑
100
j
=1
c
j
/
10 is the amount of fencing that can be financed given all the contributions
of the residents. Consumer
i
takes the contributions of the others as given and solves:
max
U
i
(
c
1
, ..., c
1
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 Winter '08
 Buddin
 Utility, , 000

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