Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

15

If there is no interdependence between private and public forest stands in the
provision of amenity services (i.e. since the amenity valuation of the private
landowner holds that F_τ = 0), then public harvesting does not affect private rotation
(Case 1). As one can see from the Appendix 5, the optimal public harvesting is given
by

[14]   SWτ∣F_τ=0,_n>1 = 0 ⇔    [pg'(τ)-rpg(τ) — rV^g]+ nF^g(T,τ)-rE^g ] = 0.

The terms in the first RHS brackets give the familiar Faustmann rule for the rotation
of the public stand in the absence of amenity valuation. Recognizing, however, the
social benefits of amenity services from the public stand introduces the second RHS
bracket term into [14] to characterize the socially optimal public harvesting.
According to the first-order condition [14], the Forest Service equates the marginal
benefit of delaying public harvest until age τ (pg'(τ) +nF^g(T,τ)) with the marginal

opportunity cost of delaying public harvest (rpg(τ) +r(V^g + nE^g ) ).

How does equation [14] relate to the Faustmann rule? The answer depends on
whether the marginal valuation of the public stand at harvest time dominates the
opportunity cost of the public stand or not. The following Lemma provides the
answer.

Proof. See Appendix 6.

Applying Lemma 2 to [14] shows that if the valuation of the public forest is merely
site-specific so that F_τ^g = 0 , the optimal public rotation age is equal to the Faustman
rotation age. If the marginal valuation of public stand increases (decreases) with its
age (F_τ^g > 0(<) ), then the optimal public rotation age is longer (shorter) than the
Faustmann rotation age. Therefore, one gets from equation [14]

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