Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

17

Proposition 4 has a natural interpretation. For complements, compared to
independence, it is optimal for the Forest Service to provide a longer public rotation
age, because increased public harvesting induces higher private harvesting, which
would reinforce the decrease in the amenity benefits of harvesting for citizens.
Analogously, for substitutes the public rotation age will be shortened, because private
forest owners will lengthen their rotation age, which yields amenity benefits to
citizens.

Finally, we consider the general case [13], which allows either substitutability or
complementarity to evolve over time, so that F_τ ≠ 0 and F_τT ≠ 0 . Rearranging
equation [13] (see Appendix 5) yields

[17]    SWTFr ≠0,Fττ ≠0,n>1 = 0 ⇔   SWTFr ≠o,Fττ =O,n>1

Comparing equations [16] and [17] allows us to infer how temporal interdependence
in the valuation of amenity services affects optimal public harvesting. Recall first
from the theorem presented in Section 3 that T_r^H > 0 for increasing and T_r^H < 0 for
decreasing temporal dependence and that according to Lemma 1 it holds that
F(T.r) - rE ≥ (<) 0 as F_T ≥ (<) 0 . Finally. the sign of the last term depends on the
sign of F_T^g . i.e.. on whether the private stand is an independent. a substitute or a
complement to the public stand in the marginal valuation of amenities from public
stand. On the basis of these considerations we can see that equation [17] gives rise to
several cases. In the following we characterize two alternatives by providing
sufficient conditions for them.

Proposition 5.Compared with temporally independent stands, increasing temporal
interdependence (F_rT > 0) implies that the public rotation age is a) longer if
F_T > 0 and F_T^g ≥ 0, and b) shorter if F_T < 0 and F_T^g ≤ 0 .

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