The business manager of Bunyan Forests, Inc. is trying to decide when they shoul

Question

The business manager of Bunyan Forests, Inc. is trying to decide when they should cut down their trees. Given the growth patterns of their trees and market conditions, the market value of the lumber they will get if they let their trees reach the age of t years and then cut them down and sell the lumber is given by the function W(t)=e^2t-001t^2 The company has an opportunity cost of funds of 5% per year (the interest rate on their cash funds in the bank). If the manager is only interested in the trees as an investment, how long should he let the trees grow? At what age do the trees have the greatest market value?

Explanation / Answer

The growth rate of the market value of the trees will be greater than 5% until the trees reach 75 years of age, i.e. when the growth rate of the forest equals the interest rate. Because before that, the forest is earning a higher rate of return than money in the bank, and after that period it is earning less than money in the bank. Thus, the optimal time to cut a forest is when its growth rate just equals the interest rate.

Given: W(t) = e0.20t – 0.001t2

Following the elementary calculus, we know

If W(t) = eg(t), then W’(t)/W(t) = g’(t)

Therefore,

W’(t)/W(t) = g’(t) = 0.2 – 0.002t

Equating this with interest rate gives

0.2 – 0.002t = 0.05

0.002t = 0.15

t = 75 years

Thus, if he is only interested in the trees as an investment, then he should let the trees grow for 75 years.

Similarly, the greatest market value for the trees can be achieved at an age of 100 years.

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