## 2. Optimal Rotation (Part II)

The Maximum sustained yield (MSY) or the mean annual increment (MAI) is used by biologists to determine the optimal harvest age for timber. The allowable annual cut is usually based on the MSY concept. This objective function is simple and includes no monetary variables. For example:

Maximize Maximum sustained yield=volume at harvest age/harvest age or

Maximize MSY=f(T)/T

Optimum condition (the cutting rule) for this maximization problem can be solved by computing the first derivative of the objective function with respect to T and setting the derivative as equal to zero. The solution can be presented as follows (by allowing a meaningful interpretation of the cutting rule).

Cutting rule is the current annual increment at harvest age meaning annual increment at harvest age

or

f'(T)=f(T)/T (CAI=MAI)

**Interpretation:**Clear cut the forest, when the current annual increment (CAI) is equal to the mean annual increment (MAI) (Figure E10 illustrates this geometric interpretation).

**Figure E10: The optimum rotation (T**

^{MAI}) maximizes the mean annual increment= MAI (Numbers are illustrative)NOTE: The slope of the (red) straight line depicting the mean annual increment at age TMAI. It is equal to the slope of (green) f(T),( i.e., f'(T) (current annual increment), at age TMAI).

It is also noteworthy that this cutting rule is totally independent of prices, costs and discount rates (i.e., there is no economic reasoning at all). Nowadays, forestry is guided by this (or some similar rule) that is often justified to guarantee sustainability instead of economic efficiency. Note: This cutting rule is only desirable from the timber buyer's viewpoint as it increases timber supply.

In the forest rent approach the aim is to maximize the average annual net income (i.e., the "Forest Rent"). Thus,

Max Forest rent = [cutting revenue at harvest age - planting costs]/harvest age or

Max FR = [S(T)-w]/T

where S(T)=pf(T) is the potential clear-cutting revenue (cutting value) at age T, and p is unit stumpage price and w is planting cost.

The cutting rule is the marginal change in cutting value at harvest age=[cutting value at harvest age - planting costs]/harvest age or

S'(T)=[S(T)-w]/T

**Interpretation:**By cutting the forest when the rate of change is in its cutting value with respect to time, this can result in net income over rotation. For geometric interpretation of the cutting rule, planting costs are in Figure E11 measured in timber, e.g. 1 000 € divided by unit stumpage price of 1 m3 timber. Importantly, following this cutting criterion leads always to longer rotations than MSY criterion.

**Figure E11: Optimum rotation (T ^{FR}) maximizing forest rent (Numbers are illustrative)**