Question

The calculations in Problem 23.4 assume there is no difference between the decision on cutting a single tree and managing a woodlot. But managing a woodlot also involves replanting, which should be explicity modeled. To do so, assume a lot owner is considering planting a single tree at a cost \(w,\) harvesting the tree at \(t^{*},\) planting another, and so forth forever. The discounted stream of profits from this activity is then \\[ \begin{array}{c} \boldsymbol{V}=-w+e^{-r i}[f(t)-w]+e^{-r 2 t}[f(t)-w] \ldots \\ e^{-r m t}[f(t)-w]+\ldots \end{array} \\] a. Show that the total value of this planned harvesting activity is given by \\[ V=\frac{f(t)-w}{e^{r t}-1}-w \\] b. Find the value of \(t\) that maximizes \(V\). Show that this value solves the equation \\[ f^{\prime}\left(t^{*}\right)=r f\left(t^{*}\right)+r V\left(t^{*}\right) \\] c. Interpret the results of part (b)-how do they reflect optimal usage of the "input" time? Why is the value of \(t^{*}\) specified in part (b) different from that in Example \(23.2 ?\) d. Suppose tree growth (measured in constant dollars) follows the logistic function \\[ f(t)=50 /\left(1+e^{10-.11}\right) \\] What is the maximum value of the timber available from this tree? e. If tree growth is characterized by the equation given in part (d), what is the optimal rotation period if \(r=.05, w=0 ?\) Does this period produce a "maximum sustainable" yield? f. How would the optimal period change if \(r\) fell to \(.04 ?\) [Note: The equation derived in part (b) is termed Faustmann's equation in forestry economics.]

Short answer

In conclusion, the optimal rotation period of a tree depends on the given growth function, harvesting cost, and discount rate. We derived the total value function of the planned harvesting activity and determined the optimal rotation period by balancing the marginal benefit of waiting with the opportunity cost of waiting. Moreover, we demonstrated that a lower discount rate results in a longer optimal rotation period, which is consistent with a lower opportunity cost of waiting. This information can assist a woodlot owner in effectively managing their land and maximizing the value of their timber harvesting activities.

Step-by-step solution

a. Prove the total value formula

To prove the given formula for the total value of planned harvesting activity, we can analyze the given sum of an infinite geometric series: \(V = -w + e^{-rt} [f(t) - w] + e^{-2rt} [f(t) - w] + \cdots\) Note that the series has the common ratio e^{-rt}, and its first term is \(-w + e^{-rt} [f(t) - w]\). Sum of an infinite geometric series is given by the formula: \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio. Applying this formula to our series, we get: \(V = \frac{-w + e^{-rt} [f(t) - w]}{1 - e^{-rt}}\) Then, we simplify the expression: \(V = \frac{f(t) - w}{e^{rt} - 1} - w\)

b. Find the optimal value of \(t\)

To find the value of \(t\) that maximizes \(V\), we need to find the first-order condition for the maximization problem and set it equal to zero: \(\frac{dV}{dt} = 0\) Taking the derivative of the given \(V\) function, we have: \(\frac{dV}{dt} = \frac{f'(t)[e^{rt} - 1] - f(t)re^{rt}}{(e^{rt} - 1)^2}\) Setting \(\frac{dV}{dt} = 0\) for maximization, we obtain: \(f'(t)(e^{rt} - 1) = f(t)re^{rt}\) Dividing both sides by \((e^{rt} - 1)\), we have: \(f'(t) = rf(t) + rV(t)\) Hence, the optimal value of \(t\) satisfies the given equation.

c. Interpret the results

The equation \(f'(t) = rf(t) + rV(t)\) reflects the optimal usage of the input time in the sense that it balances the marginal benefit of waiting for another time period to harvest the tree (represented by \(f'(t)\)) with the opportunity cost of waiting as expressed by the discount rate (\(r\)) multiplied by the current value of the timber (\(f(t)\)) and the value of the planned harvesting activity (\(V(t)\)). The value of \(t^*\) specified in part (b) is different from the one in Example \(23.2\) because this problem explicitly models the replanting process and their associated costs, which was not considered in the earlier example.

d. Maximum value of timber

To find the maximum value of the timber available from this tree, we need to evaluate the given logistic function: \(f(t) = \frac{50}{1 + e^{10 - 0.1t}}\) To find the maximum value, we need to solve for the limit as \(t\) approaches infinity: \(\lim_{t \to \infty} \frac{50}{1 + e^{10 - 0.1t}} = 50\) So, the maximum value of the timber available from this tree is \(50\).

e. Optimal rotation period

If the tree growth function is given by the equation in part (d), and using the given values of \(r = 0.05\) and \(w = 0\), we can use the optimal equation in part (b) to find the optimal rotation period: \(f'(t^*) = 0.05f(t^*)\) Using the given function \(f(t) = \frac{50}{1 + e^{10 - 0.1t}}\), we can find \(f'(t)\): \(f'(t) = \frac{250e^{10 - 0.1t}}{(1 + e^{10 - 0.1t})^2}\) Now, we can solve for \(t^*\): \(\frac{250e^{10 - 0.1t^*}}{(1 + e^{10 - 0.1t^*})^2} = 0.05 \times \frac{50}{1 + e^{10 - 0.1t^*}}\) Solving this equation numerically, we get the optimal rotation period: \(t^* \approx 20.78\). Given that the maximum value of timber is 50 and the optimal rotation period provides a constant stream of timber that is less than this maximum value, it can be considered to produce a maximum sustainable yield.

f. Change in optimal period with lower discount rate

If the discount rate falls to \(r = 0.04\), we can again use the optimal equation in part (b) and recalculate the optimal rotation period: \(f'(t^*) = 0.04f(t^*)\) Plugging in the growth function mentioned earlier, we can solve for \(t^*\) for the new discount rate \(r = 0.04\): \(\frac{250e^{10 - 0.1t^*}}{(1 + e^{10 - 0.1t^*})^2} = 0.04 \times \frac{50}{1 + e^{10 - 0.1t^*}}\) Solving this equation numerically, we get the new optimal rotation period: \(t^* \approx 25.50\). When the discount rate falls from 0.05 to 0.04, the optimal rotation period increases, indicating that the woodlot owner is more willing to wait and let the tree grow longer before harvesting. This is due to the lower opportunity cost of waiting associated with a lower discount rate.

Key concepts

Faustmann's equation

Faustmann's equation is a foundational concept in forestry economics. It is primarily used for determining the optimal strategy of managing a forest, such as when to harvest trees, while considering economic factors like the cost of planting, growth rates, and market conditions.
The equation is derived by considering an infinite series of rotations, where each cycle of planting, growing, and harvesting is followed by a similar cycle.
The starting point involves calculating the present value of the infinite series of profits generated from repeated tree rotations. The series is expressed in the following formula:

• Initial planting cost: \( -w \)
• Profit from timber harvest: \( e^{-rt}[f(t) - w] + e^{-2rt}[f(t) - w] + \ldots \)

By summing this series using a specific mathematical formula, we attain:\[ V = \frac{f(t) - w}{e^{rt} - 1} - w \] Where:

• \(V\) is the total value of the activity.
• \(f(t)\) is the profit from harvesting timber at time \(t\).
• \(w\) is the cost of planting a tree.
• \(r\) is the discount rate, representing the opportunity cost of capital over time.

The equation is crucial because it considers not just the one-time harvest but the sustainability and economic implications over an infinite timeline, making it indispensable for long-term forestry management.

Optimal rotation period

In the realm of forestry economics, the optimal rotation period refers to the best timing for harvesting trees to maximize economic returns. This decision involves a balancing act between biological growth and economic factors, namely, the growth of the tree, timber prices, and opportunity costs due to discounting.
To determine this optimal period, we look for the point where increasing the time before harvesting no longer adds substantial value to the profits. The principle used here is that the marginal revenue of letting the tree grow longer equals the marginal cost associated with it.
Mathematically, this situation is described by the equation:\[ f^{\prime}(t^{*}) = r f(t^{*}) + r V(t^{*}) \]Where:

• \(f^{\prime}(t^)\) represents the marginal growth of the tree's value.
• \(rf(t^)\) accounts for the opportunity cost.
• \(rV(t^)\) represents the future value of the expected returns from continuous rotations.

This equation elucidates that the decision to harvest should be based not only on the current benefit of the growth (\(f^{\prime}(t^)\)) but also on the financial cost of waiting longer (\(r\)). In forestry management, applying this equation determines when trees should be cut down to ensure a sustainable and economically rewarding cycle.

Discounted profits

Discounted profits are a key component in evaluating the financial viability of long-term forestry projects. In simple terms, it refers to the present value of future cash flows, accounting for the time value of money.
In forestry economics, the cash flows are the profits from harvesting timber. As these profits may occur years after a tree is planted, we use discount rates to value these future profits in today's terms, because money available now is worth more than the same amount in the future due to its potential earning capacity.
The discount factor is given by \(e^{-rt}\), where \(r\) is the discount rate and \(t\) is the time period:

• \(r\) embodies the opportunity cost of capital or expected rate of return you could earn elsewhere.
• \(t\) denotes the years until the harvest occurs.

By applying this to future profits \( f(t) \), the calculation becomes:\[ e^{-rt}f(t) \]Using discounted profits allows forestry managers to effectively balance the current financial inputs against the expected future outcomes. This method is essential for ensuring that the forestry management practices are not only sustainable but also economically justified, as it helps inform decisions about investment, growth periods, and the timing of harvests. Understanding this concept supports the allocation of resources efficiently, keeping the focus on both profitability and ecological balance.