13.10 Suppose trees are produced by applying 1 unit of labour 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 maximise this PDV ?

a. If the optimal value of t is denoted by t*, show that the ‘no pure proit’ 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 weru and 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 ‘proits’ earned at each instant by the landowner, where by proits 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 proit 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.