17.4 As in Example 17.3, suppose trees are produced by applying 1 unit of labor at time 0. The value of the wood contained in a tree is given at any time t by f(t). If the market wage rate is w and the real interest rate is r, what is the PDV of this production process, and how should t be chosen to maximize this PDV?

a. If the optimal value of t is denoted by t

$

, show that the ‘‘no pure profit’’ condition of perfect competition will necessitate that w ¼ e

!rtfðt

$

Þ:

Can you explain the meaning of this expression?

b. A tree sold before t

$ will not be cut down immediately. Rather, it still will make sense for the new owner to let the tree continue to mature until t

$

. Show that the price of a u-year-old tree will be weru and that this price will exceed the value of the wood in the tree [ f (u)] for every value of u except u ¼ t

$ (when these two values are equal).

c. Suppose a landowner has a ‘‘balanced’’ woodlot with one tree of ‘‘each’’ age from 0 to t

$

. What is the value of this woodlot?

Hint: It is the sum of the values of all trees in the lot.

d. If the value of the woodlot is V, show that the instantaneous interest on V (that is, r Æ V) is equal to the ‘‘profits’’ earned at each instant by the landowner, where by profits we mean the difference between the revenue obtained from selling a fully matured tree [ f (t

$

)] and the cost of planting a new one (w). This result shows there is no pure profit in borrowing to buy a woodlot, because one would have to pay in interest at each instant exactly what would be earned from cutting a fully matured tree.