Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

14

[11] SW = W * (τ,..) + ( n -1) E (τ, TH (τ...),...) + Vg + nEg (τ, TH (τ,...),...),
144444424444443 14444244443
private forests                                        public forests

where

T ^H (τ,...)
[12a] E(τ,T^H(τ,...)...)=(1-e^-rTH(τ...))^-1 ∫F(s,τ)e^-rsds,
0

[12b] V^g =(1-e^-rτ)^-1[pg(τ)e^-rτ -c], and

τ

[12c] E^g(τ,T^H(τ,...)...)=(1-e^-rτ)^-1∫F^g(T^H(τ,...),x)e^-rxdx,
0

in which g(τ) is the growth of the public stand as a function of its age with the usual
properties (see page 6) and F^g describes the valuation of amenities from the public
stand.

The first-order condition for the socially optimal public rotation age can be expressed
in a general form as

[13] SW = Wg + ( n -1) E_τ + V_τ^g + nEg +{ ( n -1) Et + nET } TH = 0.

The first-order condition describes various channels through which changes in public
rotation will affect social welfare. The first four terms characterize the direct effects
of the change in public stand τ on the welfare of the forest owner and recreationalists,
on public timber revenues, and on the welfare of citizens, respectively. The last two
terms characterize the indirect social welfare effects via the amenity valuation of
private and public stands, which result from the response of the private rotation age to
public harvesting. The detailed expressions for the partial derivatives in equation [13]
are developed in Appendix 5.

We assume that the second-order condition holds and start the analysis of optimal
public harvesting by considering cases 1-3 outlined in the previous section, beginning
with the simplest case 1 and then progressing to the more complex ones, 2 and 3.

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