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The Economics of Forestry 151 If the trees are allowed to grow another \( T \) years, the volume of nerchantable timber will be \( Q(T)=e^{a-b /(A+T)} \). The unit price of timber, also independent of \( T \), is given by \( P>0 \). You seek the value of \( T \) that will maximize \( \pi(T)=\left(P e^{a-b /(A+T)}-\mathrm{c}+v\right) e^{-j T} \). Suppose that \( a=13, b=185, A=40, P=1.78, \mathrm{c}=1,000 \), \( \delta=0.05 \), and \( v=2,000 \). What is the value of \( T \) that maximizes \( \pi(T) \) ? Plot \( \pi(T) \) for \( T=\varepsilon, 1,2, \ldots, 50, \varepsilon=0.0001 \), to make sure that you have found a global maximum. [4.2 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 \( Q(t)= \) \( \alpha t+\beta t^{2}-\gamma t^{3} \), 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 \( p=1 \) ? E4.3 Suppose that the inventory of old-growth forest yields an amenity value given by \( A_{t}=a \ln \left(X_{t}\right) \), where \( a>0 \) and \( \ln (\bullet) \) is the natural \( { }^{l o g} \) operator. As in Section 4.6, let \( \delta>0 \) denote the discount rate, \( N>0 \) the net revenue from old-growth timber, and \( \pi \) the present value of recently replanted land under the optimal Faustmann rotation. (a) What is the expression defining \( X^{} \), the optimal old-growth inventory to preserve? (b) Suppose that the initial stock of old-growth forest has been normalized to \( X_{0}=1 \) and that you estimate \( a=615, \delta=0.05 \), \( \pi=1,000 \), and \( N=40,000 \). What is the value for \( X^{} \) ?

          The Economics of Forestry
151

If the trees are allowed to grow another \( T \) years, the volume of nerchantable timber will be \( Q(T)=e^{a-b /(A+T)} \). The unit price of timber, also independent of \( T \), is given by \( P>0 \). You seek the value of \( T \) that will maximize \( \pi(T)=\left(P e^{a-b /(A+T)}-\mathrm{c}+v\right) e^{-j T} \). Suppose that \( a=13, b=185, A=40, P=1.78, \mathrm{c}=1,000 \), \( \delta=0.05 \), and \( v=2,000 \). What is the value of \( T \) that maximizes \( \pi(T) \) ? Plot \( \pi(T) \) for \( T=\varepsilon, 1,2, \ldots, 50, \varepsilon=0.0001 \), to make sure that you have found a global maximum.
[4.2 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 \( Q(t)= \) \( \alpha t+\beta t^{2}-\gamma t^{3} \), 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 \( p=1 \) ?

E4.3 Suppose that the inventory of old-growth forest yields an amenity value given by \( A_{t}=a \ln \left(X_{t}\right) \), where \( a>0 \) and \( \ln (\bullet) \) is the natural \( { }^{l o g} \) operator. As in Section 4.6, let \( \delta>0 \) denote the discount rate, \( N>0 \) the net revenue from old-growth timber, and \( \pi \) the present value of recently replanted land under the optimal Faustmann rotation.
(a) What is the expression defining \( X^{*} \), the optimal old-growth inventory to preserve?
(b) Suppose that the initial stock of old-growth forest has been normalized to \( X_{0}=1 \) and that you estimate \( a=615, \delta=0.05 \), \( \pi=1,000 \), and \( N=40,000 \). What is the value for \( X^{*} \) ?

The Economics of Forestry
151

If the trees are allowed to grow another T years, the volume of nerchantable timber will be Q(T)=e^a-b /(A+T). The unit price of timber, also independent of T, is given by P>0. You seek the value of T that will maximize π(T)=(P e^a-b /(A+T)-c+v) e^-j T. Suppose that a=13, b=185, A=40, P=1.78, c=1,000, δ=0.05, and v=2,000. What is the value of T that maximizes π(T) ? Plot π(T) for T=ε, 1,2, …, 50, ε=0.0001, to make sure that you have found a global maximum.
[4.2 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 ≥ 0 is given by Q(t)= α t+β t^2-γ t^3, where α=10, β=1, and γ=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 δ=0.05, what is the optimal single rotation TS, volume at harvest Q(TS), and present value πS(TS) ?
(c) If the cost of replanting is c=150, what is the optimal Faustmann rotation T^*, volume at the Faustmann rotation Q(T^*), and present value π(T^*) ?
(d) If the price increases to p=2, what are the new values for TS and T^* ? Do the new values make sense relative to their values when p=1 ?

E4.3 Suppose that the inventory of old-growth forest yields an amenity value given by At=a ln(Xt), where a>0 and ln (∙) is the natural ^l o g operator. As in Section 4.6, let δ>0 denote the discount rate, N>0 the net revenue from old-growth timber, and π the present value of recently replanted land under the optimal Faustmann rotation.
(a) What is the expression defining X^*, the optimal old-growth inventory to preserve?
(b) Suppose that the initial stock of old-growth forest has been normalized to X0=1 and that you estimate a=615, δ=0.05, π=1,000, and N=40,000. What is the value for X^* ?

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