In: Economics

# Model of tree growth, support with graph. (a) Discuss the biological decision rule to harvest the...

Model of tree growth, support with graph.
(a) Discuss the biological decision rule to harvest the forest. [3 points]
(b) Show the optimal time to harvest from a profit maximization perspec- tive. Is it the same as in a)? Why yes or why not? [10 points]
(c) Discuss how the discount rate will affect the harvest decision. [3 points]

## Solutions

##### Expert Solution

a) Biological decision rule to harvest the forest

Forest harvesting decisions are affected by the timber resource both as output and capital good. Harvest decision includes how much timber to harvest, the frequency of harvest and replantation decision after harvesting. Besides, standing trees are regarded as capital asset. Tree growth is generally measured on a volume basis.

V = a + bt + ct2 ; where V is the volume and t = age and a, b and c are parameters

The volume starts to grow slowly with the age of the forest increases initially. After a specific period of time, the volume grows rapidly and eventually reaches to the maximum. Optimal time to harvest in biological terms is the point, where the mean annual increment (MAI) is equal to zero as shown in the figure below with tree growth model.

b) The optimal time of harvesting from the perspective of profit maximization is the age that maximizes the present value (PV) of the net benefits obtained from the timber.

Benefits are calculated using potential timber volume when growth rate and price of the timber is given. Marginal growth is represented by the annual incremental growth. Planting costs are not discounted and calculated immediate basis, however, harvesting costs are discounted as they will be paid in future.

Net benefit is thus calculated by the difference between the value of the timber at the harvest age and the PV of costs.

The year of harvesting (t) is to choose in such a way that the net present value (NPV) can be maximized.

NPV = [PV(t) – ChV(t)]ert – Cp      ;

where p: price per unit of harvested volume

V(t): volume of timber harvested at the age t

Ch: per unit cost of harvesting timber

Cp: fixed cost of planning

[PV(t) – ChV(t) = stampage value

It is not exactly the same as (a). As in the biological decision making, decision is taken based on MAI, which is average annual increment of volume of single tree at specific point of time. However, in the later perspective, decision is taken based on net present value and rate of return. Optimal harvest time is determined when rate of return of waiting for trees to grow one more year equal with market rate of return.

c) Discount rate affects the harvest decision as the harvesting costs are discounted, which is proportional to the amount of timber harvested. When discount rate is high, harvesting period decreases.

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