A Polluter Has Total Abatement Cost Of Tac 10a2 Where A Is T

1 A Polluter Has Total Abatement Cost Of Tac 10a2 Where A Is The

1. A polluter has total abatement cost of TAC = 10A^2, where A is the amount it abates relative to uncontrolled emissions of 200 pounds. A performance standard requires it to abate As = 100. The polluter faces a fine of $20,000 if any non-compliance is detected. (a) Initially, suppose any noncompliance is detected with certainty. Will the polluter comply with the performance standard? What A does the polluter choose? Explain with numbers. (b) Now suppose the probability of detection increases with the amount of the violation, where the violation is the difference between the standard and actual abatement, i.e., v = As - A. If the probability of detection p(v) = 0.02v, what is the expected marginal cost to the polluter of violating the standard? Does the polluter comply? What level of A does the polluter choose? (c) Now suppose the detection returns to being certain. How much will the polluter violate the standard with a fine of $200 per ton of violation? (d) With the $200 per ton fine, what is the violation with 50% probability of detection?

Paper For Above instruction

The scenario described involves analyzing the behavior of a polluter under different regulatory and detection conditions, assessing compliance decisions based on economic incentives and costs. Understanding the problem requires grasping concepts of abatement costs, fines, detection probabilities, and optimal compliance strategies, which are essential in environmental economics and regulation policy analysis.

Initially, consider the case where non-compliance is detected with certainty. The problem states that the total abatement cost (TAC) is given by TAC = 10A^2, where A reflects the abatement amount relative to an uncontrolled emission level of 200 pounds. The standard mandated is to abate at least As = 100 pounds, translating into actual abatement A ≥ 100. The penalty for violations is a fixed $20,000 fine if non-compliance is detected.

Under guaranteed detection, the polluter faces two options: comply or evade. If the polluter chooses not to comply, the violation v (difference between the required and actual abatement) becomes critical. Since the policy mandates A ≥ 100, any less than this constitutes non-compliance, incurring the penalty. The first step is to evaluate whether the polluter finds it profitable to comply. The marginal abatement cost (MAC) derived from TAC = 10A^2, is calculated by differentiating TAC with respect to A:

MAC = d(TAC)/dA = 20A.

To find the optimal A, where the marginal abatement cost equals the marginal benefit (which in this context is the avoided fine), set the MAC equal to the penalty, leading to:

20A = 20,000, so A = 1,000 pounds.

This suggests that if the polluter's ideal abatement A exceeds 100, it would likely choose to comply, given the high cost of non-compliance and certain detection. Since 1,000 is much larger than 100, compliance is more economically favorable, because evading compliance would involve a large violation, incurring the penalty, which outweighs the marginal abatement costs at A=100.

In the second scenario, the probability of detection depends on the violation size, modeled as p(v) = 0.02v, with v = As - A. The expected marginal cost (EMC) of violating the standard now depends on the detection probability and the penalty.

The expected cost of non-compliance per unit of violation is:

EMC = p(v) * penalty per unit of violation.

Given the penalty is $20,000 per violation, and the violation v measured in pounds, the expected marginal cost is:

EMC(v) = 0.02v * 20,000 = 400v.

This indicates that for each additional pound of violation, the expected cost increases by $400, influencing the polluter's decision. The polluter aims to minimize total costs, which include abatement costs and expected violation costs, leading to an optimal violation level where marginal abatement cost equals expected marginal violation cost:

20A = 400(As - A).

The fixed standard As = 100, so the equation becomes:

20A = 400(100 - A).

Solving for A:

20A + 400A = 40,000;

420A = 40,000;

A ≈ 95.24 pounds.

The optimal abatement A is approximately 95.24 pounds, just below the standard, indicating that the increased detection probability encourages near-compliance. The actual violation v is about 4.76 pounds (100 - 95.24). Consequently, the polluter chooses an abatement level close to but slightly below the standard to balance abatement costs and expected violation penalties.

In the final case, where detection certainty resumes, the violator's optimal choice is straightforward. With a fine of $200 per ton (which equals $0.2 per pound, assuming 1 ton = 2000 pounds), the total fine for violation v pounds is $0.2 * v. To find the violation level that maximizes net benefits, the total expected cost of violation (fine) is calculated, and the optimal violation occurs where the marginal abatement cost equals the marginal penalty:

20A = 0.2 v, where A is the actual abatement and v = 100 - A.

Since A ≤ 100 (the standard), the violation v = 100 - A. The cost of violating depends on v, so substituting gives:

20A = 0.2 (100 - A).

Expanding:

20A = 20 - 0.2A;

20A + 0.2A = 20;

20.2A = 20;

A ≈ 0.99 pounds.

This implies that under certain detection, the polluter violates negligibly, close to zero, because the small violation incurs a penalty that outweighs the marginal abatement cost.

When the probability of detection is 50%, expected costs are halved, affecting optimal violation levels. The marginal expected cost of violation becomes:

EMC = 0.5 * 0.2 v = 0.1 v.

Setting equal to marginal abatement cost:

20A = 0.1 (100 - A),

20A + 0.1A = 10,

20.1A = 10,

A ≈ 0.5 pounds.

The violation level increases slightly compared to certain detection, illustrating how detection probability influences compliance behavior. Overall, these models reveal that increasing detection probability decreases violations, but the impact diminishes as the probability approaches certainty. Conversely, higher fines incentivize compliance by increasing the marginal cost of violations.

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