suppose trees are produced by applying 1 unit of labor at time 0. The value of thewood contained in a tree is given at any time t by f (t ). If the market wage rate is w and the real interestrate is r, what is the PDV of this production process, and how should tbe 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 = e^-rtf (t *).

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 we^ruand that this price will exceed the value of the wood in the tree [ f(u)] for every value of u except u = 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 ⋅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*0] 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