Suppose all firms cut 100 trees. Then each is fined with prob
α
. Thus expected
profits are
Eπ
i
(100) = (1
−
α
)(
p
−
c
)100
−
αc
100. Individual deviation to 99
trees yields profits:
Eπ
i
(99) = 99(
p
−
c
). Hence individual deviation is strictly
profitable for all
α >
(
p
−
c
)
/
(100
p
), and the maximum value of
α
for which an
equil where all firms cut 100 trees exists is
α
= (
p
−
c
)
/
(100
p
)
7. (5 points) And what is the minimum value of
α
such that your answer to
(4) above do not change?
Since
α >
1
/n
, any firm that increases logging while all others are at
x
= 0 is
sure to be fined. Hence an equil with 0 trees cut exists for all values of
α
in the
specified interval (i.e. in (1
/n,
1)).
Our environmentalist mayor, sharply opposed by the tree-logging interests,
loses the next elections. The new mayor decides that some industrial growth is
necessary for the area, but should be balanced with its costs. He estimates the
total negative externality suffered by the community at
E
= (1
/
2)(
∑
n
i
=1
x
i
)
2
.
8. (5 points) Define total welfare as the sum of the firms’ total profits minus
the negative externality suffered by the community.
What is the total
number of trees (
∑
n
i
=1
x
i
) that maximizes total welfare?
Total profits Π =
∑
n
i
=1
π
i
=
∑
i
(
p
−
c
)
x
i
−
(1
/
2)(
∑
n
i
=1
x
i
)
2
= (
p
−
c
)
X
−
(1
/
2)
X
2
if we denote
X
≡
∑
n
i
=1
x
i
. Hence the answer is
X
= (
p
−
c
)
.
(Deriving
n
first
order conditions, in terms of
{
x
1
, x
2
, .., x
n
}
will yield the same result.
That
solution is fine too).
Now the problem is how to achieve just that total number. The mayor thinks
that this is a good case for a Pigouvian tax because there is no information
problem–
p,c
, and the function
E
are all known. However, even ignoring possible
difficulties of enforcement, there seems to be a problem.
9. (5+5 points) Try solving the individual firm’s problem at the correct level
of the Pigouvian tax
t
. (i) What is the correct level of
t
? (ii) Do you find
a well-identified solution?
The correct value of the Pigouvian tax is the marginal value of the externality
evaluated at the optimum. Hence
t
=
X
at
X
=
p
−
c
, or
t
=
p
−
c
. But then the
firm’s problem becomes:
Max
x
[(
p
−
c
)
x
−
tx
] or
Max
x
[(
p
−
c
)
x
−
(
p
−
c
)
x
] = 0
for all
x.
10.
(5 points) Why do you think there is a problem with answering (9) above?
Which assumption would you change in the set-up of the problem to re-
establish the good functioning of a Pigouvian tax?
The problem is that the cost function has a constant marginal cost.
(Notice
that for any
t
, there is no interior solution–either (
p
−
c
)
> t
, and the firm sets
x
= 100, or (
p
−
c
)
< t
and the firm sets
x
= 0). We need increasing marginal
costs, or the cost function to be convex.
4
