\( [ \pm 2 \) You are the manager of a forest products company with land recently planted \( (t=0) \) with a fast-growing species of pine. The merchantable volume of timber at instant \( t \geq 0 \) is given by \( Q(t)= \) \( a t+\beta t^{2}-\gamma t^{3} \), where \( \alpha=10, \beta=1 \), and \( \gamma=0.01 \).
(a) What is the maximum volume, and when does it occur? What rotation length maximizes mean annual increment \( [Q(T) / T] \), and what is the associated volume?
(b) If the net price per unit volume is \( p=1 \) and the discount rate is \( \delta=0.05 \), what is the optimal single rotation \( T_{S} \), volume at harvest \( Q\left(T_{S}\right) \), and present value \( \pi_{S}\left(T_{S}\right) \) ?
(c) If the cost of replanting is \( c=150 \), what is the optimal Faustmann rotation \( T^{*} \), volume at the Faustmann rotation \( Q\left(T^{*}\right) \), and present value \( \pi\left(T^{*}\right) \) ?
(d) If the price increases to \( p=2 \), what are the new values for \( T_{S} \) and \( T^{*} \) ? Do the new values make sense relative to their values when \( p=1 \) ?
