Question

The marginal control cost curves for two air pollutant sources affecting a single receptor are \(M C_{1}=\$ 0.3 q_{1}\) and \(M C_{2}=\$ 0.5 q_{2},\) where \(q_{1}\) and \(q_{2}\) are controlled emissions. Their respective transfer coefficients are \(a_{1}=1.5\) and \(a_{2}=1.0 .\) With no control they would emit 20 units of emission apiece. The ambient standard is 12 ppm. a. If an ambient permit system were established, how many permits would be issued and what price would prevail? b. How much would each source spend on permits if they were auctioned off? How much would each source ultimately spend on permits if each source were initially given, free of charge, half of the permits?

Short answer

Answer: Under an ambient permit system, 5 permits would be issued to source 1 and 3 permits to source 2, with a prevailing price of $1.5. If permits were auctioned, source 1 would spend $7.5 and source 2 would spend $4.5. If half the permits were given for free, source 1 would spend $1.5 on permits, and source 2 would receive $1.5 from selling one of its permits.

Step-by-step solution

1. Define the given variables and equations

Let \(q_1\) and \(q_2\) represent the controlled emissions for the two sources, and \(MC_1\) and \(MC_2\) represent their respective marginal control costs. Given equations: \(MC_1 = 0.3q_1\) \(MC_2 = 0.5q_2\) Let \(a_1\) and \(a_2\) represent their respective transfer coefficients. Given values: \(a_1 = 1.5\) \(a_2 = 1.0\) Initial emissions without control: \(E_1 = E_2 = 20\) Ambient standard is \(12\) ppm.

2. Establish the system of equations for the ambient permit allocation

To reach the ambient standard, the total controlled emissions must be equal to the difference between the initial emissions and the ambient standard divided by the summation of the transfer coefficients of the sources. \((E_1 - q_1)\times a_1 + (E_2 - q_2) \times a_2 = 12\) Plug in the values: \((20 - q_1)(1.5) + (20 - q_2)(1.0) = 12\)

3. Find the optimal point in terms of cost for both sources

In an ambient permit system, sources will minimize their control costs. The cost minimization condition is when the marginal control costs are equal: \(MC_1 = 0.3q_1 = 0.5q_2 = MC_2\)

4. Solve for permits and prices for each source

Using the cost minimization condition (\(MC_1 = 0.5q_2\)), we can solve for \(q_2\): \(q_2 = 0.6q_1\) Now, plug the equation for \(q_2\) into the equation we derived in step 2 and solve for \(q_1\): \((20-q_1)(1.5) + (20 - 0.6q_1)(1.0) = 12\) Solving this equation, we find: \(q_1 = 5\) Now find the value of \(q_2\): \(q_2 = 0.6 * 5 = 3\) Now, find the permit price from the marginal control cost equation: Price = \(MC_1 = 0.3q_1 = 0.3(5) = \$1.5\) So, if an ambient permit system were established, \(5\) permits would be issued to source \(1\) and \(3\) permits to source \(2\), with a prevailing price of \$1.5.

5. Calculate spending under auction and free permit allocation

a. If permits were auctioned off: Source \(1\) would spend: \(5\) permits * \(1.5 = \)7.5 Source \(2\) would spend: \(3\) permits * \(1.5 = \)4.5 b. If each source were initially given half the permits (total \(8\), so \(4\) each) for free: Source \(1\) still needs \(1\) more permit: \(1 * 1.5 = \)1.5 Source \(2\) has one extra permit that it can sell: \(1 * 1.5 = \)-1.5 (negative sign indicates it actually receives money from selling its permit) So, with half the permits given out for free, Source \(1\) would ultimately spend \(1.5\) on permits, and Source \(2\) would receive \(1.5\) from selling one of its permits.

Key concepts

Understanding Marginal Control Cost

Marginal control cost represents the cost of reducing emissions by one additional unit. It is an essential economic concept in environmental policy analysis, helping to determine the most efficient allocation of resources to control pollution. In our textbook exercise, marginal control costs for two pollutant sources were given by the equations MC_1 = \(0.3 q_1 and MC_2 = \)0.5 q_2.

When it comes to minimizing pollution control costs across multiple sources, an aim is to equalize marginal control costs. The rationale is that equal marginal costs ensure that no source can reduce pollution at a lower cost than another, optimizing the use of financial resources. This is why the exercise seeks an equilibrium where MC_1 = MC_2 to find the optimal quantity of emissions reductions for each source.

Practical Implications

In real-world applications, businesses might calculate their marginal control costs to determine at which point it would be cheaper to purchase emissions permits rather than further reduce their own emissions. Economists and policymakers use these costs to design efficient environmental regulations, like cap-and-trade systems.

The Role of Transfer Coefficients

Transfer coefficients are numerical values used to measure how pollutants from a particular source affect a receptor site. They can represent how many units of pollution from a source will reach a given location. In our exercise problem, we had a_1 = 1.5 and a_2 = 1.0 as the transfer coefficients for the two sources.

Knowing these coefficients allows for the calculation of the actual impact of emissions on the ambient air quality standard, which in this case was set at 12 ppm. By understanding the different effects each source has on the receptor, an effective ambient permit system can be designed to achieve the standard in the most cost-effective manner.

Application in Environmental Policy

Transfer coefficients are paramount in creating spatially differentiated environmental regulations. This allows policymakers to target specific areas with higher pollution impacts more aggressively and to apply less stringent measures to areas with lower transfer coefficients, tailoring measures to the region's specific needs.

Emission Standards and Permit Systems

Emission standards are regulatory requirements that set specific limits on the amount of pollutants that can be emitted into the environment from various sources. These standards are crucial to protect public health and the environment. The ambient standard mentioned in our problem, 12 ppm, indicates the maximum allowable concentration of a pollutant in the ambient air.

Emission standards can be enforced using a permit system. An example is the ambient permit system highlighted in the exercise. This system would issue permits that allow a certain level of emissions, in line with the overall goal of keeping ambient pollution levels below a specific threshold.

Trading and Auctions

Permits can be distributed in different ways. They might be auctioned off, which potentially generates revenue for the government and ensures that companies who value the permits most highly (because controlling emissions is more costly for them) will buy them. Alternatively, permits might be distributed for free, which can mitigate the financial impact on businesses but doesn't provide revenue to the government. Our exercise explored both scenarios, giving students insight into the economic dynamics at play in environmental regulation.