Question

Economics

Two firms are ordered by the federal government to reduce
their pollution levels Firm A’s marginal costs associated with pollution
reduction is MC=20+4Q and firm B’s MC=10+8Q The marginal benefit of pollution
reduction is MB=400-4Q

Compare the social efficiency of three possible outcomes:
require all firms to reduce pollution by the same amount; charge a common tax
per unit of pollution; or require all firms to reduce pollution by the same
amount, but allow pollution permits to be bought and sold.

Answer 1

Total surplus is equal to total benefit minus total cost. The total benefit of pollution reduction will be the same in all three cases, because they all generate the same quantity of pollution reduction, Q◦ = 57.5. The total benefit is equal to the area under the marginal benefit curve, MB = 400 − 4QT , from QT = 0 to QT = 57.5, which is 16387.5.

1. The same total reduction in pollution could be achieved by requiring each firm to reduce pollution by 28.75 units. The total cost to firm A of producing 28.75 units of pollution reduction is the area under firm A’s marginal cost curve, MCA = 20 + 4QA, from QA = 0 to QA = 28.75, which is 2228.13. The total cost to firm B of producing 28.75 units of pollution reduction is the area under firm B’s marginal cost curve, MCB = 10 + 8QB, from QB = 0 to QB = 28.75, which is 3593.75. Total surplus is thus 16387.5 - 2228.13 - 3593.75 = 10565.6. We can tell that this would be less efficient than the social optimum, since it would be less costly for firm A to reduce pollution by more and for firm B to reduce pollution by less (since MCA < MCB at 28.75 units).

2. A common tax could be used to achieve the social optimum. The marginal benefit of pollution reduction at the optimum is MB(57.5) = 400 − 4(57.5) = 170. In other words, the marginal external cost of pollution (not pollution reduction) is equal to 170 at the optimum. Thus, the Pigouvian tax on pollution (or, equivalently, the good whose production causes pollution) is 170 per unit. Setting a tax on pollution (not pollution reduction) of 170 would lead firm A (respectively, B) to reduce pollution to the point where MCA = 170 (respectively MCB = 170). In other words, each firm reduces pollution until the marginal cost of doing so exceeds the marginal benefit, which is the value of avoiding the tax. Solving gives QA = 37.5 and QB = 20. The total cost to firm A of producing 37.5 units of pollution reduction is the area under firm A’s marginal cost curve from QA = 0 to QA = 37.5, which is 3562.5. The total cost to firm B of producing 20 units of pollution reduction is the area under firm B’s marginal cost curve from QB = 0 to QB = 20, which is 1800. It follows that total surplus in this case is 16387.5 - 3562.5 - 2800 = 11025.

3. Requiring both firms to reduce pollution by 27.25 units but allowing them to trade pollution permits can also be used to achieve the social optimum. The value to firm B of being able to produce 1 more unit of pollution (i.e., MCB) is higher than the cost to firm A of reducing pollution by one unit (i.e., MCB) when QB = 27.25 = QA, so both can gain by trading a unit of pollution permits. This continues to be true as long as QA < 37.5 and QB > 20, so they will trade permits until QA = 37.5 and QB = 20