present the parametric specifications of the amenity valuation function F(T,τ) ,
which produce all the possibilities given in Definition 2.8
3. Optimal Private Rotation Age under Ongoing Rotations
In the Hartman model, extended to an exogenous adjacent stand, the private forest
owner begins with bare land, plants trees and clear-cuts so as to maximize the present
value from future harvest revenue and the utility of amenity services over an infinite
cycle of rotations. This is given by
[5] max W = V + E ,
{T}
T
where V =(1-e-rT)-1VJ and E=(1-e-rT)-1∫F(s,τ)e-rsds.
0
The first-order condition WT = VT + ET = 0 for the maximization of [5] can be
expressed as follows:
[6] Wt = Pf '(T ) — rpf (T ) — rV + F (T T) - rE = 0.
The second-order condition is
[7] WTT = Pf"( T ) — rpf( T ) + FT ( T T ) < 0,
which we assume to hold. According to [6] the private forest owner chooses the
rotation age so as to equate the marginal benefit to delaying the harvest to age T,
defined bypf'(T) +F(T,τ), to the marginal opportunity cost of delaying the harvest,
defined by rpf(T) + r(V + E) .
8 Swallow and Wear (1993) say that the stands are substitutes or complements when FTτ is
negative or positive, respectively (see 1993, p. 108). This corresponds to our definition 2 of
decreasing or increasing temporal dependence, when the valuation function of amenities is
continuously differentiable. Later on, Swallow and Wear (1993) say that the substitutability
exists when Fτ is negative (p. 109), which corresponds to our definition 1 of ALEP
substitutes. We know from the previous Result that Definition 1 matters only for the case of a
single rotation, but - as we will show -- not for the case of ongoing rotations.