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 pro cess, 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.

c.

d.

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).