Question

a. Suppose you want to remove ten fish of an exotic species that have illegally been introduced to a lake. You have three possible removal methods. Assume that \(q_{1}, q_{2},\) and \(q_{3}\) are, respectively, the amount of fish removed by each method that you choose to use so that the goal will be accomplished by any combination of methods such that \(\mathrm{q}_{1}+\mathrm{q}_{2}+\mathrm{q}_{3}=10 .\) If the marginal costs of each removal method are, respectively, \(\$ 10 q_{1}, \$ 5 q_{2},\) and \(\$ 2.5 \mathrm{q}_{3},\) how much of each method should you use to achieve the removal cost-effectively? b. Why isn't an exclusive use of method 3 cost effective? c. Suppose that the three marginal costs were constant (not increasing as in the previous case) such that \(M C_{1}=\$ 10, M C_{2}=\$ 5,\) and \(M C_{3}=\$ 2.5\) What is the cost-effective outcome in that case?

Short answer

Answer: The cost-effective combination for removing ten exotic fish is to use method 3 exclusively, with \(q_1 = 0\), \(q_2 = 0\), and \(q_3 = 10\).

Step-by-step solution

Part a: Determine the Cost-Effective Fish Removal Combination

To determine the optimal combination of fish removal methods to achieve the removal cost-effectively, we have to minimize the total cost while ensuring all 10 fish are removed from the lake. First, we set up the following equations to fulfill our goal and considering marginal costs of each removal method: 1. \(q_1 + q_2 + q_3 = 10\) 2. \(10q_1 = MC_1\) 3. \(5q_2 = MC_2\) 4. \(2.5q_3 = MC_3\) To minimize the total cost TC, we have to find the combination of \(\textit{q’s}\) that results in the lowest total marginal cost. The total cost function can be expressed as: 5. TC = \(MC_1 + MC_2 + MC_3 = 10q_1 + 5q_2 + 2.5q_3\) To minimize this, we can find the partial derivatives of TC with respect to \(q_1\), \(q_2\), and \(q_3\) and set them equal to zero: \(\frac{\partial TC}{\partial q_1} = 10, \frac{\partial TC}{\partial q_2} = 5, \frac{\partial TC}{\partial q_3} = 2.5\) Since \(\frac{\partial TC}{\partial q_1}, \frac{\partial TC}{\partial q_2}, \frac{\partial TC}{\partial q_3}\) are all constants, we conclude that the only way to minimize the total cost is to use the lowest cost method, which is method 3. Solving for \(q_3\) in equation 1: \(q_3 = 10 - q_1 - q_2\) Since we decided to use only method 3, we have \(q_1 = 0\) and \(q_2 = 0\). Thus: \(q_3 = 10\) The cost-effective combination will be using method 3 only, so \(q_1 = 0\), \(q_2 = 0\), and \(q_3 = 10\).

Part b: Discussing the Exclusive Use of Method 3

While using only method 3 seems to be the cost-effective solution in part a, this may not necessarily be true under certain circumstances. If method 3 has limitations, such as only being efficient at removing a certain number of fish or having a diminishing returns in its effectiveness, exclusively using method 3 might not always be the most cost-effective. Additionally, if any negative externalities are associated with method 3, such as causing environmental harm, the overall cost could be higher than anticipated.

Part c: Constant Marginal Costs Scenario

In this scenario, the marginal costs are given as constants: \(MC_1=\$ 10, MC_2=\$ 5,\) and \(MC_3=\$ 2.5.\) Our goal is still to remove 10 fish with a combination of methods: \(q_1 + q_2 + q_3 = 10\) Since the marginal costs are constant, we should prioritize using method 3, which has the lowest cost per fish removed. We can continue using this method until we reach the desired total: \(q_3 = 10\) as it is the method with the lowest marginal cost. To be cost-effective in this scenario, the optimal solution would involve using method 3 exclusively. Therefore, \(q_1 = 0\), \(q_2 = 0\), and \(q_3 = 10\).

Key concepts

Environmental Economics

Environmental economics is a crucial area of study that deals with how economic activities impact the environment, and conversely, how environmental considerations can guide resource allocation decisions. In the case of removing invasive species from a lake, the practice falls under the realm of environmental economics as it involves the cost and impact of different methods of resource use—or in this case, the resource being the ecosystem of the lake itself.

One foundational principle of environmental economics is the concept of externalities, which are costs or benefits that affect a third party not involved in the transaction. In our exercise, if method 3, for example, causes harm to other species in the lake or to the water quality, this would be considered a negative externality. Environmental economics also considers the value of biodiversity and ecosystem services that might be affected by economic decisions regarding resource use. Balancing the costs of removal methods with ecological preservation is key. Effective resource allocation accounts not only for the cost of removal but also for long-term environmental sustainability.

In summary, environmental economics guides us to make choices that achieve our goals cost-effectively while also weighing in on the environmental impact of these actions. Recognizing that eliminating the invasive fish has inherent ecological value, the cost of removal needs to incorporate both the direct costs and potential hidden environmental costs associated with each method.

Marginal Cost Analysis

Marginal cost analysis is an economic principle that plays a pivotal role in effectively allocating resources. It is based on the marginal concept of economics, which evaluates the additional or incremental cost or benefit of producing one more unit of a good or service. In our exercise, the marginal cost of each fish removal method is given and varies with the quantity chosen for each method, represented by the expressions \(10q_1 \) for method 1, \(5q_2 \) for method 2, and \(2.5q_3 \) for method 3.

Marginal cost analysis helps identify the least expensive way to achieve a particular outcome—in this case, the removal of ten exotic fish from a lake. By comparing the marginal costs, one can determine the most cost-effective strategy. The marginal cost analysis suggests that to minimize costs, one should allocate resources starting with the method that has the lowest marginal cost, and only move to the higher-cost methods if necessary.

However, merely selecting the method with the lowest marginal cost might not always be cost-effective, as other factors, including the effectiveness of each method and possible externalities, need to be considered. This is precisely why method 3 is questioned in the exercise, prompting further analysis beyond the face-value cost calculation. In sustainable economic practices, the marginal cost analysis is a tool for guiding the allocation of resources, but it's crucial to consider its broader implications as well.

Optimization of Resource Use

Optimization of resource use is the process of allocating resources in the most efficient way to achieve a specific objective, while minimizing waste and cost. It involves making the best possible use of available resources, given the constraints and the goals that need to be met.

In the context of our fish removal scenario, optimization of resource use implies choosing a mix of the three removal methods such that all ten invasive fish are removed at the least possible cost. To do this, we initially consider the marginal costs of each method and select the one that offers the lowest cost per unit of outcome (removal of fish).

Optimization exercises often involve setting up an objective function—in this case, minimizing total cost—and subject to constraints, which here is the requirement that \(q_1 + q_2 + q_3 = 10\). Given that the cost function is linear, we deduced that using only the method with the lowest marginal cost (method 3) would be optimal. But, this result assumes there are no diminishing returns or capacity constraints on method 3.

Optimization isn't just about the costs on paper. It requires holistic thinking to assess all contributing factors. In the real world, constraints such as method capacity, effectiveness, regulatory, and environmental constraints could all alter the 'optimal' decision. Ultimately, optimization isn't just an exercise in arithmetic; it requires critical thinking and a comprehensive evaluation of all variables involved.