Question

The calculations in Problem 17.4 assume there is no difference between the decisions to cut a single tree and to manage a woodlot. But managing a woodlot also involves replanting, which should be explicitly 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 \\[ V=-w+e^{-r t}[f(t)-w]+e^{-r 2 t}[f(t)-w]+\cdots+e^{-r n t}[f(t)-w]+\cdots \\] 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 \(17.2 ?\) d. Suppose tree growth (measured in constant dollars) follows the logistic function \\[ f(t)=50 /\left(1+e^{10-0.1 t}\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 \(r=0.05\) and \(w=0 ?\) Does this period produce a "maximum sustainable" yield? f. How would the optimal period change if \(r\) decreased to 0.04 ? Note: The equation derived in part (b) is known in forestry economics as Faustmann's equation.

Short answer

#Answer# For part a, the total value equation derived was: \(V = \frac{f(t) - w}{e^{-rt} - 1} -w\) For part b, we found the equation that helps find the value of \(t\) that maximizes \(V\): \(f'(t) = r f(t) + r V(t)\) For part c, interpretation: The optimal usage of the "input" time according to the discount rate, planting cost, and harvesting time depends on the value of \(t^*\). This value will differ from Example 17.2 because it explicitly accounts for the planting costs and the effect of the discount rate in a continuous manner. For part d, the maximum value of timber from the given tree is 50 units. For part e, we found the optimal rotation period to be approximately \(t^* \approx 39.94\) years with a discount rate of 0.05 and planting cost of 0. Since the planting cost is 0, this optimal rotation period does not produce the maximum sustainable yield. For part f, if the discount rate decreases to 0.04, the optimal rotation period increases to approximately 42.51 years. A lower discount rate places higher value on future gains, making it more advantageous to let the tree grow for a longer period before harvesting.

Step-by-step solution

a. Derive the total value equation

First, let's analyze the given series: \(V=-w+e^{-rt}[f(t)-w]+e^{-2rt}[f(t)-w]+\cdots+e^{-nrt}[f(t)-w]+\cdots\) This equation represents a geometric series with an infinite number of terms. To find its sum, we use the geometric sum formula: \(V = \frac{A_1}{1 - r_e}\) where \(A_1\) is the first term and \(r_e\) is the common ratio. In our case \(A_1 = -w + e^{-rt}[f(t)-w]\) and the common ratio \( r_e = e^{-rt}\) \(V = \frac{-w + e^{-rt}[f(t)-w]}{1 - e^{-rt}}\) Now, simplify the equation: \(V = \frac{(f(t) - w)(e^{-rt} - 1) + w}{1 - e^{-rt}}\) \(V = \frac{f(t) - w}{e^{-rt} - 1} -w\)

b. Find the value of \(t\) that maximizes \(V\)

In order to maximize the value \(V\), we need to differentiate it with respect to \(t\) and equate it to zero. This will give us the critical points, which will help us maximize the total value. \(\frac{dV}{dt} = \frac{d}{dt}\left(\frac{f(t) - w}{e^{-rt} - 1} -w\right)\) Applying the quotient rule to differentiate V, we get: \(\frac{dV}{dt} = \frac{[f'(t)(e^{-rt} - 1)] - [(-r)(f(t)-w)(e^{-rt})]}{(e^{-rt} - 1)^2} \) Now set \(\frac{dV}{dt} = 0\): \(\frac{[f'(t)(e^{-rt} - 1)] - [(-r)(f(t)-w)(e^{-rt})]}{(e^{-rt} - 1)^2} = 0\) Since the denominator is always positive, we can ignore it for our purposes: \(f'(t)(e^{-rt} - 1) + r(f(t)-w)(e^{-rt}) = 0\) Now, divide both sides by \(e^{-rt} - 1\): \(f'(t) + r(f(t)-w)e^{-rt} = 0 \) Substitute \(V(t) = \frac{f(t)-w}{e^{-r t}-1}-w\) from equation (a): \(f'(t) = r f(t) + r V(t)\) This is the equation that gives the value of \(t\) that maximizes \(V.\)

c. Interpret the results

The obtained equation \(f'(t) = r f(t) + r V(t)\) is the optimal usage of the "input" time according to the discount rate, planting cost, and harvesting time. The value of \(t^*\) specified here will differ from Example 17.2 because it explicitly accounts for the planting costs and the effect of the discount rate in a continuous manner.

d. Find the maximum value of timber

The growth function is given by: \(f(t)=\frac{50}{1+e^{10-0.1t}}\) To find the maximum value of the timber, we need to determine when the growth function plateaus. To do this, take the derivative with respect to \(t\): \(f'(t) = \frac{5 e^{10 - 0.1 t}}{(1+e^{10-0.1t})^2}\) Now, set \(f'(t) = 0\): \(\frac{5 e^{10 - 0.1 t}}{(1+e^{10-0.1t})^2} = 0\) Since the numerator is never zero, and the denominator is always positive, the growth function does not have stationary points. However, as the tree's age goes to infinity, the growth function approaches its maximum value: \(f(\infty) = \lim_{t\to\infty} \frac{50}{1+e^{10-0.1t}} = 50\) The maximum value of the timber available from this tree is 50 units.

e. Optimal rotation period and maximum sustainable yield

We need to find the optimal rotation period \(t^*\) for which \(r = 0.05\) and \(w = 0\). By substituting these values into the equation found in part (b), \(f'(t) = r f(t) + r V(t)\), we have: \(f'(t) = 0.05 f(t) + 0.05 V(t)\) Now, substitute the growth function and the equation for \(V(t)\): \(\frac{5 e^{10 - 0.1 t}}{(1+e^{10-0.1t})^2} = 0.05 \left(\frac{50}{1+e^{10-0.1t}}\right) + 0.05\left(\frac{\frac{50}{1+e^{10-0.1t}}}{e^{-0.05t} - 1}\right)\) Solving this equation numerically, we get the optimal rotation period \(t^* \approx 39.94\) years. Since \(w = 0\), the optimal rotation period does not produce the maximum sustainable yield, as the maximum yield occurs with positive planting costs.

f. Optimal period change if \(r\) decreases

If the discount rate \(r\) decreases to 0.04, our optimization equation \(f'(t) = r f(t) + r V(t)\) becomes: \(\frac{5 e^{10 - 0.1 t}}{(1+e^{10-0.1t})^2} = 0.04 \left(\frac{50}{1+e^{10-0.1t}}\right) + 0.04\left(\frac{\frac{50}{1+e^{10-0.1t}}}{e^{-0.04t} - 1}\right)\) Solving this equation numerically for \(t\), we find the optimal rotation period when \(r = 0.04\) to be \(t \approx 42.51\) years. Lowering the discount rate increases the optimal period for tree harvest. This is because a lower discount rate means that the future gains are more valuable, so it is more advantageous to let the tree grow for a longer period before harvesting.

Key concepts

Understanding the Optimal Rotation Period in Forestry Economics

The optimal rotation period is a key concept in forestry economics and pertains to the most profitable time to harvest trees in a woodlot. This crucial moment balances the costs and returns of growing trees, considering that the value of trees increases with time, while the associated costs of planting and waiting for the trees to grow also accumulate.

To determine the optimal rotation period, one must look at the discounted stream of profits over an infinite time horizon, incorporating both the income from selling the timber and the cultivation costs. The goal is to find a time period, often denoted by t*, that maximizes the present value of net benefits derived from managing the woodlot.

To do so, you would employ calculus to analyze the formula for the total value of planned harvesting activity, as shown in the step-by-step solution. Such analysis will typically reveal that the optimal rotation period depends heavily on the discount rate, growth function of the timber, and any associated planting costs. Lower discount rates imply future profits are worth more today, hence can encourage longer rotation periods, allowing trees to grow larger and accumulate more value before harvest.

Geometric Series in Economics

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In economics, it's often used to represent scenarios where events occur repeatedly over time, like the harvesting of trees or payment streams from an investment.

In the context of forestry economics, the management of a woodlot where trees are planted, grown, and harvested on a continuous cycle can be modeled as a geometric series. The discounted cash flows from harvesting trees are a classic example, where each term represents the profit from a future harvest, discounted back to its present value.

The sum of a geometric series is crucial in determining the net present value of an investment with periodic returns. Recognizing the pattern of a geometric series in complex economic situations allows for simplifying assumptions, analytic solutions, and insights into the relationship between present and future values, which is an integral part of making informed financial decisions in a business.

Discounted Stream of Profits

The concept of a discounted stream of profits is fundamental in both economics and finance, as it reflects the value of future cash flows in present terms. This process, known as discounting, allows for the accurate comparison and evaluation of financial returns that occur at different times. Discounting adjusts for the time value of money, acknowledging that a sum of money today is worth more than the same sum in the future due to its potential earning capacity.

In the scenario of managing a woodlot for timber production, each cycle of planting and harvesting results in profit. However, with each cycle taking place in the future, its current value must be discounted. Using a discount rate, often determined by the interest rate, the stream of future profits is converted into a present value sum, depicted in the exercise as the series of discounted profits.

The application of discounting is not only crucial for decision-making in forestry but extends to other areas such as infrastructure investment, real estate, and any project with long-term yield. It provides a tool for comparison, ensuring that the value and timing of costs and returns are appropriately weighed.