Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

• Decreasing Temporal Dependence: F_τ_T < 0 ⇒ T_τ^H < 0

F (T,τ) r

Proof. i) Assume that F, > 0 ⇒ W_t, > 0 ⇔ ^τ-----------<-----—

^τ^T^τ^T 1 - e^-^rT

∫ F_τ(s,τ)e ^-^rs ds
0

FT < 0 ⇒

∫F (T,τ)e^-^rsds < ∫F_τ (s,τ)e^-^rsds

Fτ (T ,τ )(1 - e

⇔∙ ----------

rT ₎ T

< ∫F_τ (s,τ)e^-^rs ds
0

F_τ ( T T )

∫F (s,τ)e^-^rsds
0

,
<.
1 - e^-^,T

Hence, W_T < 0 so that T_τ^H < 0 .

ii) Assume that F < 0

⇒ Wττ < 0 ⇔ T ^F (T^τ)

∫F_τ(s,τ)e^-^rsds
0

^> - ,T ^.

1 - e ^-^,

FT < 0 ⇒

∫F (T,τ)e^-^rsds < ∫F_τ (s,τ)e ^-^rs ds ⇔

F_τ (T ,τ )(1 - e -^rT )

< ^T∫Fτ (s,τ)e^-^rsds
0

F_τ (T ,τ) > ,

_rT . Hence, W_T_τ < 0 so that T_τ^H < 0 . Q.E.D.

T 1 - e

∫F_τ (s,τ)e^-^,sds

Appendix 5. Optimal public harvesting

The optimal public rotation age τ* is implicitly defined by

A5.1 SW_τ = W_τ* + ( n -1) E_τ + V_τ^g + nE^g +{ ( n -1) E_t + nET^g } T_t^h = 0,
in which we have accounted for the fact that W_T* = 0 due to the envelope theorem.

The individual terms in A5.1 are

_H T^H (τ,...)

W_τ= (1 - e ^- T ^τ^,-))^-¹ ∫ Ft ( 5 ,τ ) e ^-^rsds

V_τ^g = (1 -e^-^,^τ)^-¹[pg'(τ)-,pg(τ)-,V^g]

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