Question

ques - Explain graphically and intuitively why Wicksell rotation is inefficient and why it is longer than Faustmann rotation. Be detailed in whatever you write.

Answer 1

The Wicksellian rotation refers to the age at which a stand of even-aged trees will be harvested one time only. When timber price and regeneration costs are constant, the Wicksellian rotation length "aw" is constant, determined implicitly as the solution to the following equation, as in :

Va(aw)/V(aw)=r, Eq. (1)

where aw is the Wicksellian rotation length, r is the discount rate, and V(a) and Va(a) are, respectively, the growth volume function and its first derivative.
When timber prices evolve at a constant rate μ, the rotation length remains constant and equal to the implicit solution of the equation:

Va(aw)/V(aw)=δ, Eq. (2)

which is the same as Eq. (1), where r is replaced by δ=r−μδ to account for a price increase resulting in higher rotation lengths.
When timber prices p are constant and it is possible to plant the land and harvest the stand an unlimited number of times according to the Faustmann’s setting, the rotation length aF, referred to as the static Faustmann’s rotation length, is constant from one rotation to the next. This is determined implicitly by the following Faustmann’s rule as, for instance :
(Va(aF)/V(aF)) −(D/p)= r/(1−e^(−raF) ), Eq. (3)
Where, D is the constant regeneration cost.
The static Faustmann’s rotation depends on the timber price level as long as the regeneration cost D is positive. In this situation, one known implication is that a one-time rise in the timber price level implies a decrease in the Faustmann’s rotation length. An increase in timber revenues, therefore, makes longer rotations less attractive. Note that when the regeneration cost is equal to zero or accounted for in the timber price, the Faustmann’s rotation length is independent of the price level as it is determined implicitly as the solution to:

Va(aF)/V(aF)=r/(1−e^(−raF)), Eq.(4)

When timber prices evolve at a constant rate μ and regeneration costs are absent or accounted for in timber prices, the rotation length remains constant and equal to afaf, the implicit solution of the equation:

Va(af)/V(af) =δ/(1−e^(−δaf)) , Eq. (5)

where δ=r−μ. It can be shown that δ(1−e^(−δaf)) is decreasing in δ for a given age a. Therefore, Eq. (5) admits a higher solution for the rotation length when δ decreases (or μ increases). Hence, the rotation length increases when the price rate of change μ increases. This result holds when regeneration costs are positive.
Some confusion exists in the literature with respect to timber price impact on forest rotation length. To clear up this confusion, it is important to distinguish between a one-time static increase in the price change rate and its impact on the rotation length on the one hand, and the continuous increase of timber prices in time and their impact on successive rotation lengths on the other. For instance, when timber prices increase exponentially at a constant rate in the presence of constant regeneration costs, succeeding rotation lengths continuously decrease over time. Under these conditions, successive rotation lengths converge after some rotations to a certain length that can be higher or lower than the static Faustmann’s rotation length aF. However, when timber prices are rising or stochastic and a conversion of the site to an alternative use is possible, changes over time in the optimal rotation lengths remain uncertain.

The three graphs below shows the revenue generated,yield produced and rate of return vs the the length of rotation.