Question

Two firms can control emissions at the following marginal costs: \(M C_{1}=\) \(\mathrm{S} 200 q_{1}, M C_{2}=\$ 100 q_{2},\) where \(q_{1}\) and \(q_{2}\) are, respectively, the amount of emissions reduced by the first and second firms. Assume that with no control at all, each firm would be emitting 20 units of emissions or a total of 40 units for both firms. a. Compute the cost-effective allocation of control responsibility if a total reduction of 21 units of emissions is necessary. b. Compute the cost-effective allocation of control responsibility if the ambient standard is \(27 \mathrm{ppm}\), and the transfer coefficients that translate a unit of emissions into a ppm concentration at the receptor are, respectively, \(a_{1}=2.0\) and \(a_{2}=1.0\)

Short answer

Answer: In scenario b, the cost-effective allocation of emissions control responsibility is approximately \(q_1 = 8.83\) and \(q_2 = 17.67\).

Step-by-step solution

Write the Marginal Cost Equations

We are given the marginal cost equations for both firms: \(MC_1 = 200 q_1\) \(MC_2 = 100 q_2\)

Apply the principle of equal marginal costs

For cost-effective allocation, both firms should have equal marginal control costs. Therefore: \(MC_1 = MC_2\)

Calculate Emissions Reduction Allocations for Scenarios a and b

a. In this scenario, we need to find the allocation for a total reduction of 21 units of emissions: First, set the marginal costs equal to each other to find the relationship between \(q_1\) and \(q_2\): \(200 q_1 = 100 q_2 \Rightarrow q_1 = \frac{1}{2}q_2\) The total emissions reduction required is 21 units, so using the relationship above: \(q_1 + q_2 = 21\) Now, substitute \(q_1 = \frac{1}{2}q_2\) into the equation: \(\frac{1}{2}q_2 + q_2 = 21\) Solve for \(q_2\): \(\frac{3}{2}q_2 = 21 \Rightarrow q_2 = 14\) Now find \(q_1\) using the relationship between \(q_1\) and \(q_2\): \(q_1 = \frac{1}{2}q_2 \Rightarrow q_1 = 7\) So, the cost-effective allocation in scenario a is \(q_1 = 7\) and \(q_2 = 14\). b. In this scenario, we need to find the allocation considering an ambient standard of 27 ppm and given transfer coefficients: First, find the total emissions reduction required in ppm using the transfer coefficients: \(20a_1 + 20a_2 = 40a_1 = 80 \Rightarrow\) ambient concentration without control \(27\) ppm is the required standard, so we need to reduce the concentration by \(80 - 27 = 53\) ppm. Now express the total emissions reduction required in terms of \(a_1\) and \(a_2\): \(q_1a_1 + q_2a_2 = 53\) Then substitute the transfer coefficients \(a_1 = 2.0\) and \(a_2 = 1.0\): \(2q_1 + q_2 = 53\) We know from Step 2 that \(q_1 = \frac{1}{2}q_2\), so substitute this into the equation: \(2(\frac{1}{2}q_2) + q_2 = 53\) Solve for \(q_2\): \(q_2 = \frac{53}{3} \approx 17.67\) Now find \(q_1\) using the relationship between \(q_1\) and \(q_2\): \(q_1 = \frac{1}{2}q_2 \approx \frac{1}{2}(17.67) \approx 8.83\) So, the cost-effective allocation in scenario b is approximately \(q_1 = 8.83\) and \(q_2 = 17.67\).

Key concepts

Marginal Cost of Emissions Reduction

When we discuss the marginal cost of emissions reduction, we refer to the cost of eliminating an additional unit of pollutants from the environment. It is crucial for companies and regulators alike to understand these costs, as they are the deciding factor in determining how much each polluter should reduce their emissions to meet environmental standards in the most economical way.

Let's consider an educational exercise where two firms can control emissions at different marginal costs. Firm 1 has a marginal cost defined by the equation \( MC_1 = 200q_1 \) and Firm 2 by \( MC_2 = 100q_2 \), where \( q_1 \) and \( q_2 \) represent the amount of emissions each firm reduces. The objective is to find a balance where both firms equalize their marginal costs to achieve overall cost-effectiveness in reducing pollution. Otherwise stated, we want to optimize the pollution reduction process so each additional unit of reduction costs the same for both firms, making the entire system efficient.

Emission Reduction Allocation

Emission reduction allocation involves distributing the task of pollution reduction between multiple entities so that overall environmental goals are met in the most cost-effective manner possible. In our exercise example, we apply the principle that cost-effective allocation occurs when marginal costs are equalized across firms.

To solve for the allocation, we consider the total reduction required and use the relationship determined by equalizing the firms' marginal costs. In practice, this means if we need to reduce pollution by a certain amount, we calculate how much each firm contributes to this goal by considering their respective costs of reducing emissions. By doing this strategically, environmental standards can be achieved without placing undue financial burdens on any single firm.

Ambient Environmental Standards

Ambient environmental standards are the levels of pollutants that can be present in the environment without causing harm to human health or ecosystems. These standards are usually set by regulatory agencies and serve as a benchmark for pollution control measures.

In our example, an ambient standard was set at 27 parts per million (ppm). This means that the concentration of emissions must be reduced to this level to meet environmental safety guidelines. Firms must adjust their emissions reductions to ensure that the collective ambient concentration of their pollutants does not exceed this standard. It's an imperative measure that dictates the minimum acceptable environmental quality for public health and the ecosystem's integrity.

Transfer Coefficients

Transfer coefficients play an integral role in understanding how emissions from different sources affect ambient environmental concentrations. They are factors that translate the amount of emissions released into their resultant concentration levels at certain locations or receptors.

In the given problem, transfer coefficients given for firms were \( a_1 = 2.0 \) and \( a_2 = 1.0 \), implying that emissions from Firm 1 have twice the impact on ambient concentration levels than those from Firm 2. These coefficients are used to calculate the total emissions reduction needed in ppm to meet the specified standard. The precise application of transfer coefficients helps in scientifically allocating emission reduction responsibilities, ensuring each firm reduces the right amount of emissions relative to their impact on the environment.