Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

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)^-1V^J and E=(1-e^-rT)^-1∫F(s,τ)e^-rsds.

0

The first-order condition W_T = V_T + E_T = 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 F_Tτ 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 F_τ 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.

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