Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

A4.1 W_T_τ = ^T∫F_τ(s,τ)e^-^rsds

F- ( T ,τ) r

T 1 - e -^rT

∫F_τ (s,τ)e^-^rs ds
0

• Temporal Independence: F_τ_T = 0 ⇒ T_τ^H = 0

Proof. If F_τ_T = 0 , equation [10] reduces to

W_T_τ = F_τ(T,τ) - (1 -e^-^rT )^-¹ [F_τ(0,τ) - F_τ(T,τ)e^-^rT ]. There are two possibilities.

F_τ = 0, then trivially W_T_τ = 0 . Under F_τ ≠ 0, F_τ_T = 0 implies that

[Fτ(0,τ)-Fτ(T,τ)e^-^rT]=Fτ(1-e^-^rT)⇒

WTτ =Fτ(T,τ)-(1-e^-^rT)^-¹Fτ(T,τ)(1-e^-^rT)=0. Hence, Tτ^H =0.

• Increasing Temporal Dependence: F_τ_T > 0 ⇒ T_τ^H > 0
F (T τ) r

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

τ Tτ T _-_rT

∫1-e

F_τ (s,τ)e^-^rsds

FτT > 0 ⇒

^T∫Fτ(T,τ)e^-^rsds

> ∫F_τ (s,τ)e ^-^rs ds ⇔

F- (T ,τ )(1 - e —^rT )
r

> ^T∫Fτ(s,τ)e^-^rsds

F (T τ) r

⇔ ----^τ-(-^-)--->------ . Hence, WT_τ > 0 so that TH > 0.

T ₁_- e-rT Tτ τ

∫F_τ (s,τ)e^-^rsds

F (T τ) r

ii) Assume thatF, < 0 ⇒ W_t < 0 ⇔ ^----^τ-------<-----—

^τ^T^τ^T 1 - e^-^rT

∫F_τ (s,τ)e^-^rs ds
0

FτT > 0 ⇒

^T∫Fτ(T,τ)e^-^rsds

> ∫F_τ (s, τ)e ^-^rs ds ⇔

F- (T ,τ )(1 - e -^rT )

> ^T∫Fτ(s,τ)e^-^rsds

F (T τ) r

⇔ ^τ------------<-----—. Hence, WT_τ > 0 so that TH > 0.

T -rT ^Tτ ^τ

∫1-e

F_τ (s,τ)e^-^rsds