Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

amenity services from a private stand under a single harvest cycle of length T can be
expressed as

T

[2]     v(T,τ) = ∫F(s,τ)e^-rsds ,

0

where F(s,τ) is the flow of amenities from the focal stand of age s when it is
potentially affected by an adjacent exogenous stand of age τ.³

From equation [2] we get the discounted marginal valuation from amenity services as
a function of the age of the private stand by differentiation

[3]     vT(T,τ) =F(T,τ)e^-rT.

It is often assumed that amenity valuation increases with the stand age, i.e.
F_T(T,τ) > 0 (see e.g. Hartman 1976). Depending on the specific amenities the
valuation function F can have other properties as well. We may have F_T(T,τ) < 0,
indicating that a young forest is valued more than an old one; or if only the site-
specific features of the forests count, then F_T(T,τ) = 0 .⁴

The sign of v_Tτ (T,τ) indicates how the discounted marginal valuation of amenity
services from a private forest stand depends on the age of an exogenous stand. To
explore this interdependence more precisely, we define the “static” concept of
Auspitz-Lieben-Edgeworth-Pareto (ALEP) complementarity or substitutability
between forest stands in our framework as follows.

^f '(T) = ^dfTp- for ^f (T)^, while Ax (^x, y) = ¾y) for A(^x, y)^, e^tc∙

³ Swallow and Wear (1993) originally suggested this formulation. Snyder and Bhattacharyya
(1990) have analyzed the situation where consideration is given to the maintenance costs
associated with a flow of non-timber values by assuming that otherwise they would vanish via
a process of decay∙ Abstracting from the maintenance costs and from their assumption of the
decay of non-timber values leads to the same formulation, which is used in this paper∙

⁴ Calish, Fight and Teeguarden (1978) studied several alternative forest non-timber benefits
for Douglas fir and found that they included a variety of increasing and decreasing time paths∙
Swallow, Parks and Wear (1990) extended their analysis by providing functional forms for
various types of non-timber benefits and by presenting numerical simulations∙

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