Optimal Private and Public Harvesting under Spatial and Temporal Interdependence



28


Decreasing Temporal Dependence: FτT0 TτH0

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

FT0

F (T,τ)e-rsdsFτ     (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, WT0 so that TτH0 .


ii) Assume that F0


Wττ < 0 T F (Tτ)

Fτ(s,τ)e-rsds
0


,

>      - ,T .

1 - e -,


FT0


F (T,τ)e-rsdsFτ     (s,τ)e -rs ds


Fτ (T ,τ )(1 - e -rT )

,


< TFτ (s,τ)e-rsds
0


Fτ (T ,τ)    >    ,

rT . Hence, WTτ0 so that TτH0 . 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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