EEP101/ECON125 Spring 00
Prof.: D. Zilberman
GSIs: Malick/McGregor/StPierre
PROBLEM SET 5 Solutions
1.
Suppose r=0.1, price (P) of timber is $10 per boardfoot, and the volume of trees in a stand obeys the
function Q(T) = 12 T
2
 1/3 T
3
.
a)
Consider a single rotation.
Set up the profit maximization problem and derive the equilibrium
condition.
Solve for the optimal rotation length (T*).
Be sure to check to see if your answer makes
sense.
The problem assumed no harvesting costs.
The grower’s objective is to choose the rotation length to
maximize profits (or revenues in this case).
Of course, since this profit is realized T years from now, we
need to discount this future profit to the present. This objective is expressed as
rT
T
e
T
PQ
V
P
MAX

=
Π
)
(
.
.
Maximizing this objective with respect to T gives one first order condition which implicitly defines T*:
*
*
*
*
*)
(
*)
(
'
0
*)
(
*)
(
'
0
:
rT
rT
rT
rT
e
T
rPQ
e
T
PQ
e
T
rPQ
e
T
PQ
dT
d
FOC




=
?
=

?
=
Π
Since e
rt
and P cancel, this leaves us with
or,
( 29
r
T
Q
T
Q
=
)
(
'
(1)
This says that the profit maximizing rotation length is such that the growth of the tree volume is equal to
the interest rate.
Notice that Q’(T) refers to change in growth from one period to the next and represents
the change in volume of trees.
The growth rate of trees multiplied by 100 is the percentage growth of
trees from one period to the next.
Using the information given in the description of the problem, and
Q’(T) = 24TT
2
, equation 1 is rewritten as
1
.
0
T
1/3

T
12
24
3
2
2
=

T
T
(2)
To find T*, we need to solve equation (2) for T.
After factoring out a T and rearranging terms, equation
(2) can be rewritten as
*)
(
*)
(
'
T
rQ
T
Q
=
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