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 \(\mathrm{q}_{1}, \mathrm{q}_{2},\) and \(\mathrm{q}_{3}\) 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 \(\$ 2.5 q_{3}\) how much of each method should you use to achieve the removal costeffectively? 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 exclusive use of method 3 isn't cost-effective when marginal costs are increasing because the cost for each subsequent fish removed using method 3 is higher than the previous one. As a result, the total cost of using only method 3 grows too rapidly, and a combination of methods would be more cost-effective.

Step-by-step solution

Equations

We're given the following equations: - Total fish removed: \(q_1 + q_2 + q_3 = 10\) - Marginal costs: \(10q_1\), \(5q_2\), and \(2.5q_3\) The goal is to minimize the total cost, which is given by the following equation: Total Cost (C) = \(10q_1 + 5q_2 + 2.5q_3\) ##Step 2: Minimize the total cost##

Cost Minimization

To achieve the cost-effective goal, we need to minimize the total cost while satisfying the constraint that 10 fish are removed. Using the constraint equation, we can substitute one variable in terms of the others and plug it into the total cost equation. Let's solve for \(q_1\) in terms of \(q_2\) and \(q_3\) from the constraint equation: \(q_1 = 10 - q_2 - q_3\) Now substitute this into the total cost equation: \(C = 10(10 - q_2 - q_3) + 5q_2 + 2.5q_3\) Simplify the equation: \(C = 100 - 10q_2 - 10q_3 + 5q_2 + 2.5q_3 = 100 - 5q_2 - 7.5q_3\) To minimize the cost, we should use more of the method with lower marginal costs. In this case, method 3 has the lowest marginal cost, so we should try to use more of it along with another method to reach our goal. ##Step 3: Explain why exclusive use of method 3 isn't cost-effective##

Ineffectiveness of Method 3

The exclusive use of method 3 would involve removing all 10 fish using only method 3. However, the marginal costs in this situation are increasing, which means the cost for each subsequent fish removed using method 3 is higher than the previous one. As a result, the total cost of using only method 3 is growing too rapidly, and a combination of methods would be more cost-effective. ##Step 4: Find the cost-effective outcome when marginal costs are constant##

Constant Marginal Costs

In this case, the marginal costs are \(MC_1 = 10\), \(MC_2 = 5\), \(MC_3 = 2.5\) Since marginal costs are constant, it is optimal to use the method with the lowest marginal cost. In this case, the method with the lowest marginal cost is method 3. To find the cost-effective outcome, we must meet the constraint equation \(q_1 + q_2 + q_3 = 10\). Since method 3 has the lowest cost with a constant marginal cost, we should use method 3 to remove all 10 fish. Therefore, the cost-effective outcome is \(q_1 = 0\), \(q_2 = 0\), and \(q_3 = 10\).

Key concepts

Marginal Cost Analysis

Marginal cost analysis is a critical tool in environmental economics, particularly when managing limited resources or addressing pollution and invasive species issues. It involves examining the additional cost of producing one more unit of a good or service. In the context of invasive species removal from a lake, marginal costs represent the increase in total cost associated with the removal of each additional fish.

In our exercise, the cost of removing an additional fish via three different methods is represented by the marginal costs \(\$ 10 q_{1}\), \(\$ 5 q_{2}\), and \(\$ 2.5 q_{3}\). To determine the most cost-effective strategy, one would need to minimize the total expense of removing the ten invasive fish, taking into account the escalating marginal costs of each method. Thus, the solution hinges on balancing the different marginal costs to achieve the lowest possible total cost, while also removing the targeted number of fish.

Marginal cost analysis not only aids in making budget-friendly decisions but also underpins the 'polluter pays' principle, where the cost of pollution is internalized by those responsible, compelling them to consider the incremental costs of their actions.

Cost-Effective Resource Management

Cost-effective resource management requires identifying strategies that accomplish environmental goals at the lowest total cost. It's an optimization problem that ensures resources are not over-utilized or wasted. In the exercise, we are tasked with removing invasive species while minimizing costs, which is a typical real-world scenario for environmental managers.

The economic principle highlighted here dictates that among the removal methods, we should utilize more of the method with the lower marginal cost to ensure cost-effectiveness. However, on closer examination, if the marginal costs increase with each additional unit, using only the cheapest method (\(\$ 2.5 q_{3}\)) may not remain cost-effective as quantities increase. Rendering the task an optimization one that requires a strategic mix of methods, rather than a singular approach.

In real-world applications, this optimization extends beyond invasive species control and into areas such as waste management, water conservation, and air quality improvement, highlighting the importance of finding a balance between environmental sustainability and economic feasibility.

Economics of Invasive Species Control

The economics of invasive species control is complex and involves weighing the costs and benefits of different control methods. Controlling invasive species is crucial as they can have significant negative impacts on native ecosystems, biodiversity, and the economy. However, the control methods themselves need to be economically viable.

In the example provided, we see that the constant marginal costs change the problem's dynamics. When costs remain static, it logically follows that we should commit entirely to the method with the lowest cost (\(\$ 2.5\) per fish), since, in this scenario, the cost would not increase with each additional fish removed. This illustrates a simplified scenario where cost-effectiveness is straightforward to determine.

In reality, the economic management of invasive species often deals with budget constraints and must factor in not only the direct costs of removal but also long-term ecological benefits and the prevention of future expenses related to the species' spread. Thus, interdisciplinary approaches that involve both ecological and economic expertise are essential for devising strategies that are both effective and financially sensible.