Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

Eτ^g =(e^r^τ-1)^-¹[F^g(T^H(τ,...),τ)-rE^g]

E_T =(e^rTH⁽^τ^,...)-1)^-¹[F(T^H(τ,...),τ)-rE], and

E_T^g =(1-e^-^r^τ)^-¹^τ∫F_T^g(T^H(τ,...),τ)e^-^rxdx
0

Substituting these explicit formulas into equation A5.1 yields

A5.2 SW_τ = (e^-^r^τ -1)^-¹ [pg'(τ) - rpg(τ) -rV^g ]+ (e^r^τ -1)^-¹ n[f(T^h (τ,...),τ) - rE^g ]

T^H (τ,...)

+(1-e^-^rTH⁽^τ^,...))^-¹n ∫Fτ(s,τ)e^-^rsds+(n-1)Tτ^H(e^rT-1)^-¹[F(T^H(τ,...),τ)-rE]
0

(1-e^-^r^τ)^-¹nT_τ ^H^τ∫F_T(T,x)e^-^rxdx =0,
0

This first-order condition A5.2 is a combination of direct and indirect effects on social
welfare from private and public forest stands over infinite cycles of rotations.

Appendix 6. Proof of Lemma 2

Note first that we can write

A6.1

F^g(T,τ)-rE^g =

^τ

∫ F^g (T, x ) e ^-^rxdx
о

F^g (T T
^τ∫F^g(T,x)e^-^rxdx

(1 - e^-^r^τ)

If F_τ^g >(≤) 0, then ^τ∫F^g(T,τ)e^-^rxdx>(≤)^τ∫F^g(T,x)e^-^rxdx ⇔
00

Fg ⁽T^,^T ) (1 - e -^rτ ) > (≤)∫ F^g (T, τ)e ~^rxdx ⇔
r₀

Fg⁽T,^τ) > (≤)

τ ₍₁ e-rτ₎

∫F^g(T,x)e^-^rxdx ⁽^-e ⁾

Hence, F^g(T,τ)-rE^g >(≤) 0 as F_τ^g(T,τ)>(≤) 0. Q.E.D.

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Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

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