Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

Definition 1. Spatial interdependence.

Two forest stands are substitutes, independents or complements in the
ALEP sense when an increase in the age of the exogenous forest stand
decreases, leaves unchanged, or increases the marginal valuation of the
focal private forest stand, respectively.⁵

As one can see from equation [3], this definition is equivalent to the following
mathematical formulation:

What happens to the private rotation age T^J when the age τ of the adjacent stand
changes? Given that the second-order condition Ω_tt < 0 holds, the first-order
condition Ω _τ = 0 defines implicitly the privately optimal rotation age in terms of the
age τ of the adjacent stand, i.e. T^j = T^j (τ,..). Substituting this for T in Ω_τ = 0
gives an identity. Its partial differentiation with respect to τ gives
T_τ^j = (-Ω_tt )^-1Ω_ττ, so that the sign Tτ = sign Ω_ττ, whereΩ_ττ = F_τ (T,τ). Using
Definition 1 yields the following

Proposition 1 For a single rotation a change in the age of the exogenous forest stand
will decrease, have no effect or increase, private rotation age when the
exogenous forest stand is an ALEP substitute, independent or complement to
the private forest stand, respectively.

This result is qualitatively similar to that obtained from the two-period models with
uneven-aged forest management (see Koskela and Ollikainen 1999) and has a natural
interpretation. If forest stands are substitutes, the higher age of the adjacent forest
allows the private forest owner to make his own harvest sooner and to enjoy the
amenities from the adjacent stand for a longer period. For complements the opposite

⁵ See Samuelson (1974) and further discussions in Chipman (1977), Kannai (1980) and
Weber (2000).

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