Equation (6) gives the conditions for optimal rotation time given a certain commercial
percentage per acre. The social planner will allow harvest when the growth rate of the forest in
net timber value is equal to a discount factor plus the externalities balance divided by the timber
value at T and harvesting fraction PC.
Equation (7) gives the first order condition for the optimal percentage of commercial use per
acre given a certain rotation period. The social planner will choose PC such that the marginal
net timber value with respect to PC is equal to the marginal value of the lost forest amenity with
respect to PC. Intuitively, the social benefits of a marginal increase in PC, derived from an
increased amount of timber harvested, must equal the social costs of a marginal increase in PC,
derived from a decrease in the amenity value of the standing forest. By definition, the timber
value of the forest is increasing in PC. Therefore, the social planner must choose a percentage
commercial use such that the amenity value of the forest is marginally decreasing. An internal
solution to the problem requires that F(t, PC) be decreasing in PC, but monotonicity is not
necessary. Nonmonotonicity in PC may occur if the amenity is fire prevention.
3. PRIVATE OPTIMIZATION, TAXATION, AND SOCIALLY OPTIMAL OUTCOMES
This section examines the private logging firm’s problem, where the firm faces taxes aimed
at controlling both the percentage of commercial use on each acre and rotation time. If feasible
to implement, a Pigovian tax or subsidy on the externality would directly force the firm to
internalize the externality and could induce optimal behavior in both choices. However, a
Pigovian tax or subsidy is not feasible in this case. If the externality in question were erosion
control for example, a tax on the units of dirt that travel from one acre to the next, or into a
By definition, 0 ≥ PC ≥ 1. PC = 1 would be a clear-cut.
10
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