Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

16

Proposition 3.If private and public stands are independents in the valuation of
amenity services (F_τ = 0), allowing for free access to a public forest
lengthens (shortens or leaves unchanged), the public rotation age if the
valuation of the amenity services increases (decreases, or remains unchanged)
with the age of the public forest.

What happens if private and public stands are interdependent in the marginal
valuation of amenity services, but this interdependence does not change with the age
of the private stand? Under these circumstances, public harvesting has no effect on
private harvesting age (T_τ^H = 0 ), and the first-order condition for the socially optimal
public rotation age can be written as

_H              T^H (τ,...)

[15]    SW Fτ ≠ o,Fττ =0,n >1 = (1 - e - ^rT"(^τ^')^-1 n ∫ FT ( s ^,τ ) e ^-rsds

0

+ (e^rτ -1)^-1 { [pg^f(τ) -rpg(τ) -rV^g]+ n[F^g(T,τ) -rE^g ] }= 0.

Utilizing equation [14] yields the following connection

[16]    SW_τ∣p∙^ π∙ ∩ . ₁ = 0 ⇔

^{1 j}           τ F_τ ≠o,F_ττ =0,n>1

₍erτ _{- 1)}      T ^H (τ )          _rs

SWτ∣Fτ =0,Fττ =0,n>1 ⁺~    _-rTH(_τ ) n ∫Fτ ^(s,τ)^e d^s = 0.

(1 - e )     0

Equation [16] gives a generalized Hartman rule under temporal independence
between public and private forest stands.

Proposition 4.Compared with independent stands, temporal independence between
stands (F_τT = 0) implies a longerpublic rotation age whenpublic andprivate
forests are complements (F_τ > 0), and a shorter public rotation age when
they are substitutes (F_τ < 0).

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