28
• Decreasing Temporal Dependence: FτT < 0 ⇒ TτH < 0
F (T,τ) r
Proof. i) Assume that F, > 0 ⇒ Wt, > 0 ⇔ τ-----------<-----—
τ Tτ T 1 - e-rT
∫ Fτ(s,τ)e -rs ds
0
FT < 0 ⇒
∫F (T,τ)e-rsds < ∫Fτ (s,τ)e-rsds
Fτ (T ,τ )(1 - e
⇔∙ ----------
,
rT ) T
< ∫Fτ (s,τ)e-rs ds
0
Fτ ( T T )
T
∫F (s,τ)e-rsds
0
,
<.
1 - e-,T
Hence, WT < 0 so that TτH < 0 .
ii) Assume that F < 0
⇒ Wττ < 0 ⇔ T F (Tτ)
∫Fτ(s,τ)e-rsds
0
,
> - ,T .
1 - e -,
FT < 0 ⇒
∫F (T,τ)e-rsds < ∫Fτ (s,τ)e -rs ds ⇔
Fτ (T ,τ )(1 - e -rT )
,
< T∫Fτ (s,τ)e-rsds
0
Fτ (T ,τ) > ,
rT . Hence, WTτ < 0 so that TτH < 0 . Q.E.D.
T 1 - e
∫Fτ (s,τ)e-,sds
0
Appendix 5. Optimal public harvesting
The optimal public rotation age τ* is implicitly defined by
A5.1 SWτ = Wτ* + ( n -1) Eτ + Vτg + nEg +{ ( n -1) Et + nETg } Tth = 0,
in which we have accounted for the fact that WT* = 0 due to the envelope theorem.
The individual terms in A5.1 are
H TH (τ,...)
Wτ= (1 - e - T τ,-))-1 ∫ Ft ( 5 ,τ ) e - rsds
0
Vτg = (1 -e-,τ)-1[pg'(τ)-,pg(τ)-,Vg]
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