P pe
C=
2γ
-rT T
b (∞) - b ( t ) 1 e - r∙dt Г
b (T ) ) J
(25)
The first order condition for the optimal rotation period depends linearly on γ and PC, and
PC depends inversely on γ. Plugging the condition for optimal percentage commercial use into
equation (24) shows that the optimal rotation period does not depend on γ:
1 ∂ G ( T, PC ) _ r
G (T) ∂ T ~ 1 - e -rT
-rT T
-1 T
∫(b(∞)-b(t))e-rtdt 11 ∫(b(t) -b(T))e-rtdt (26)
J0
For any given PC, γ increases the optimal rotation period. However, γ also decreases the
optimal PC. These effects cancel out exactly and the optimal T is left unaffected by γ. The
tax on the commercial use fraction will not affect the firm’s decision on rotation length.
From equation (25), it is apparent that an internal solution, 0 < PC < 1, will exist if:
pe
-rT
2γ
≤ ∫ r b(∞) - b(t) ']e -rtdt
∫ к b ( T ) )
(27)
This is true for a sufficiently small p or sufficiently large γ. There exists an internal
solution as long as the ratio of marginal timber value to marginal amenity value is sufficiently
small.
The clear-cut tax is equal to the square of PC: θ(PC)= PC2. This penalizes the firm relatively
more for a commercial use percentage closer to 100%. This reflects the idea that the planner is
22
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