Question

As in Example \(23.2,\) suppose trees are produced by applying one unit of labor 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 instantaneous rate is \(r,\) what is the \(P D V\) of this production process and how should \(t\) be chosen to maximize this \(P D V ?\) a. If the optimal value of \(t\) is denoted by \(t^{*},\) show that the no-pure- profit condition of perfect competition will necessitate that \\[ w=e^{-r f} f\left(t^{*}\right) \\] Can you explain the meaning of this expression? b. \(\mathrm{A}\) 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 \(w e^{\mathrm{re}}\) 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. c. Suppose a landowner has a "balanced" woodlot with one tree of "each" age from 0 to \(t^{*} .\) What is the value of this woodlot? (Hint: It is the sum of the values of all trees in the lot.) d. If the value of the woodlot is \(V\), show that the instantaneous interest on \(V\) (that is, \(r \cdot V\) ) is equal to the "profits" earned at each instant by the landowner, where by profits we mean the difference between the revenue obtained from selling a fully matured tree \(\left[f\left(t^{*}\right)\right]\) and the cost of planting a new one \((w) .\) This result shows there is no pure profit in borrowing to buy a woodlot, because one would have to pay in interest at each instant exactly what would be earned from cutting a fully matured tree.

Short answer

Answer: The market wage rate influences the cost of applying labor in the tree production process, while the instantaneous rate is used for discounting the value of the wood at a given time. To maximize the PDV, we find the first derivative with respect to time and set it to 0, which results in the no-pure-profit condition under perfect competition. The optimal time, denoted as \(t^*\), is chosen such that the tree's price equals the value of the wood contained in it.

Step-by-step solution

Determine the PDV of the production process and optimal t

The PDV of the production process is given by the value of the wood at time \(t\), \(f(t)\), discounted by the factor \(e^{-rt}\), minus the cost of applying the unit of labor at time \(0\), which is the market wage \(w\). This can be written as: PDV = \(e^{-rt} f(t) - w\) To maximize this PDV with respect to \(t\), we need to find the first derivative with respect to \(t\) and set it to 0: \(\frac{d}{dt} (e^{-rt} f(t)) = w\) Now, we solve for the optimal time \(t^*\).

Derive the no-pure-profit condition

From the first derivative equation, we have: \(\frac{d}{dt} (e^{-rt} f(t)) - w = 0\) Rearranging the equation, we get: \(w = e^{-rt^*} f(t^*)\) This equation states the no-pure-profit condition under perfect competition.

Retain the tree until t*

A tree sold before \(t^*\) will not be cut down immediately, as it makes sense for the new owner to let the tree continue to mature until \(t^*\). Thus, the price of a \(u\)-year-old tree will be: \(w e^{r(u - t^*)}\) Note that this price exceeds the value of the wood in the tree \(f(u)\) for every value of \(u\) except \(u = t^*\) when these two values are equal.

Determine the value of a balanced woodlot

The value of a balanced woodlot with trees of each age from 0 to \(t^*\) is the sum of the values of all these trees. The sum can be represented as: \(V = \sum_{u=0}^{t^*} w e^{r(u - t^*)}\)

Show that the instantaneous interest on V is equal to the "profits" earned by the landowner

The instantaneous interest on \(V\) is given by \(rV\). The "profits" earned by the landowner at each instant are the difference between the revenue obtained from selling a fully matured tree, \(f(t^*)\), and the cost of planting a new one, \(w\). We need to show that: \(rV = f(t^*) - w\) If this equation holds, then there is no pure profit in borrowing to buy a woodlot, because one would have to pay in interest at each instant exactly what would be earned from cutting a fully matured tree.

Key concepts

Present Discounted Value (PDV)

Understanding the Present Discounted Value (PDV) is vital in microeconomic theory, especially when analyzing investment decisions over time. PDV is the current value of a future amount of money or stream of cash flows given a specified rate of return, often referred to as the discount rate.

Mathematically, the PDV of receiving a future sum is calculated by multiplying that sum by an exponential factor that incorporates both the time until receipt and the discount rate. The equation is expressed as \( PDV = e^{-rt} f(t) \), where \(f(t)\) is the future value, \(r\) is the discount rate, and \(t\) is the time in the future when the value is received.

For instance, if a tree's wood is worth \(f(t)\) at time \(t\), to determine the optimal time to harvest the tree, one would calculate the PDV for different \(t\) values, seeking to maximize this PDV. This involves factoring in not just the eventual sale price, but also the cost of labor and the time value of money.

No-Pure-Profit Condition

The no-pure-profit condition in microeconomics suggests that under conditions of perfect competition, firms or individuals cannot make economic profits in the long run beyond their opportunity costs. This is because the ease of entry and exit in a perfectly competitive market ensures that profits will attract competition, which will drive prices down until profits are normalized.

In the context of our tree example, if the optimal time to harvest the tree is \(t^*\), the no-pure-profit condition implies that the labor cost (\(w\)) must equal the discounted value of the wood at \(t^*\), or \(w = e^{-rt^*} f(t^*)\). The idea is that the costs of growing the tree (reflecting the labor input) should equate to the present value of the tree’s wood at the optimal harvest time, ensuring that there is no economic profit from growing and selling the tree at that time.

Optimal Harvest Time

Optimal harvest time is the point in time at which the present value of the harvest is maximized. In microeconomic models dealing with renewable resources like trees, determining the optimal harvest time is crucial. This time is denoted as \(t^*\) and represents when the growth in value of the resource (for example, a tree growing and its wood becoming more valuable) is exactly offset by the discount rate, which reflects the opportunity cost of not having harvested earlier.

In mathematical terms, the optimal point \(t^*\) is discovered by setting the derivative of the PDV with respect to \(t\) to zero and solving for \(t\). This results in \(w = e^{-rt^*} f(t^*)\), marrying the value of labor with the PDV of the resource at \(t^*\), where no additional profit can be derived from waiting any longer.

Perfect Competition

Perfect competition is a market structure characterized by a complete absence of rivalry among the individual firms and where all firms sell identical or homogeneous products. In reality, perfect competition is more theoretical than practical; nonetheless, it serves as an essential benchmark for comparing with other market structures.

Properties of perfect competition include a large number of small firms, freedom of entry and exit, perfect information, and no individual buyer or seller having any control over the price. It results in a market equilibrium where goods are produced at the lowest possible cost and where there are no economic profits in the long run, reflecting the no-pure-profit condition. In a perfectly competitive market, prices are dictated by supply and demand, and businesses take these prices as given, acting as 'price takers' rather than 'price makers'.