Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

-⅛E yields (1 - e~^rT )∫ sF(s, τ)e^-^rsds = TE - E, so that
0

—E = - Te ~^rT (1 - e ~^rT ) ^-¹E - TE + E = Ed, where d = (1 + r - rT - rT ( e^rT -1)^-¹ ) =
dr

(1 + r)(e^rτ -1) -rTe^rτ

rT 1 '

e^r -1
numerator and the

To apply the L’Hopital’s rule for d, we differentiate its

denominator with respect to rT and get

(1 + r)e^rT -e^rTd = -------—

e^rT

- rTe ^rT (1 + r) - 1

------- which gives that lim d =--------= r > 0 . Hence, we
rT→0 1

have - (E +rE) = = E[1 +r -rT - rT(e ^rT -1)^-¹]< 0. Now the overall term

rT rT

B1 =-(1 - )( pf (T ) + V ) - (1 + r - rT - -_r- ) E < 0, so that T₁H

e^r -1 e^r -1

<0. Q.E.D.

Appendix 3. Proof of Lemma 2:

Note first that we can write

A.3.1. F(T,τ) -rE =

∫F(s,τ)e^-^rsds

F (T ,τ )

∫F(s,τ)e^-^rsds

₍₁_-_e-rT₎

If F_T > (≤) 0, then ∫ F(T,τ)e ^-^rs ds > (≤)∫F(s,τ)e^-^rsds ⇔

FT"(1 -e-^rT)>(≤)∫F(s,τ)e-^rsds ⇔ T ^f^(T∙^τ) >(≤) ^r-_rT)

⁰ ∫ F(s, τ)e^-^rsds ⁽^e⁾

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

Appendix 4. Proof of the Theorem: the sign of T_τ^H = the sign of F_τ_T

Recall from the text that the cross-derivative of equation [9] can be written as