Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

29

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

Substituting these explicit formulas into equation A5.1 yields

A5.2 SW_τ = (e^-rτ -1)^-1 [pg'(τ) - rpg(τ) -rV^g ]+ (e^rτ -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^-rτ)^-1nT_τ ^Hτ∫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

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

Fg⁽T,^τ)    > (≤)

τ                             ₍₁ e-rτ₎

∫F^g(T,x)e^-rxdx      ^{( -}e ⁾

0

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

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