## 1. Optimal Rotation (Part I)

Previously, we laid the foundations for the basic logic of investment calculations. Now we further apply these principles to forest management by studying the problem of optimal rotation for even-aged stands.

The problem of optimal rotation means choosing the optimal time for clear-cutting, followed by reforestation and another even-aged stand. In the previous chapter, Table E3 illustrated that the rotation was fixed for 60 years without any justification. Now we will derive the optimum condition for rotation. In economics, these kinds of statements - in general - should be derived from the forest owner's optimization behavior. Therefore, we will start by first specifying the target (or actually three alternative targets): (1) maximum sustained yield, (2) forest rent or (3) land rent  that a forest owner tries to maximize. Then we will compare the optimum conditions (cutting rules) to see how different goals (can quite naturally) imply different optimum cutting behavior. At the same time we will discuss the economic rationale of different cutting rules.

But first, if only to reveal the basic logic, let us simplify. Timber production is the only goal   (non-timber values are ignored). Thinnings are not allowed (cutting revenue from thinnings could be compounded at the end of every rotation), planting and the final clear-cut are the only actions in a stand where an initial state is bare land (year 0) and the unit price for timber (p) is constant. Thus, here we are studying only one decision, the timing of clear-cut, that by which the forest owner can control the profitability of timber production.

Optimizing is based on the knowledge (or expectation) of timber volume development as the stand is aging1 (i.e., growth modeling and predictions).  At first, the volume is increasing very rapidly, but later on the growth slows down to stop totally at the end (Figure E9). Hence, mathematically, the timber volume can be presented as Q(T)=f(T), where Q is volume at age T. The general growth pattern of trees implies that there exists:  age Tmax so that current annual increment (the marginal volume growth) f'(T) > 0, when 0 ≤ T < Tmax and f'(Tmax)=0. In addition there exists age T* so that f''(T) > 0 when T < T* and f''(T) < 0 when T > T*. These properties of the general growth model are presented graphically in Figure E9.

Figure E9: Volume growth as a function of age (Numbers are illustrative)

The debate on the appropriate criteria for determining the optimal forest rotation has been a long-lasting one (e.g. Möhring 2001, Hyytiäinen & Tahvonen 2003, Viitala 2006). According to Nautiyal (1988), the most common criteria adopted in forestry has been the rotation of maximum sustained yield or maximum mean annual increment. He further states that this rotation "has been obviously popular because of the simplicity in its determination and the apparent logical assertation that it results in the production of a maximum average volume per year and thus also uses the site to its fullest extent."

In the next section, we will see next that this is not so. As Möhring (2001) presents, there has been a historical struggle between land rent theory and the forest rent approach.

1. This physical growth development is emphasized here as unit price is kept constant, for simplicity.