section describes one very simple, but intuitive stylized example where the clear-cut tax and
lump sum tax are implemented.
In the numerical example, a specific form for the externality function, F(PC,T) is assumed.
The function below shows the instantaneous value of the amenity for any PC and t:
F(t,PC)=γ(1-PC2)b(∞)+γPC2b(t)
(23)
where γ is the dollar value per volume unit of the amenity. The first group of terms on the right
hand side of the equation can be interpreted as the amenity value of the trees that have been left
uncut and have reached peak volume. The second group of terms is the amenity value of the
growing trees that have been replanted since the previous harvest. Squaring the harvest fraction
puts more weight on the value of the fully-grown trees. This approximates the case where the
concern is soil erosion. As the commercial use portion of the forest regrows, the new trees will
develop root systems and eventually aid in the reduction of erosion. However, since trees are
typically harvested long before they are full grown, it is the uncut trees with fully developed root
systems that will produce the most erosion control.
For simplicity, p is assumed to be net of all costs.
Under these assumptions, the first order conditions for a social optimum become:
1 ∂G(T,PC)
G(T) ∂T
1-e
-rT
1 γrPC
pb(T)1-e-rt
T
∫(b(t)-b(T))e-rtdt
0
(24)
21
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