Question

14.Suppose that the marginal damages to society from air pollution are MD = 6+ 2e, where e is the level of air pollution. Suppose also that the marginal cost of reducing the air pollution on the part of firms is MC = 150 - e.

a. Find the optimal level of pollution. Illustrate graphically.

b. Find the net gain to society, assuming firms were initially not controlling emissions at all.

c. What level of tax would achieve the optimal level of pollution?

Answer 1

a) The marginal damages to society from e level of air pollution are : MD = 6 + 2e

The marginal cost of reducing air pollution are : MC = 150 - e

If we want to find the socially optimum level of pollution, then we need to find that level of pollution at which marginal damage to the society due to air pollution is exactly equal to the marginal cost of reducing air pollution , i.e.,

MD = MC or, 6 + 2e = 150 - e

or, 3e = 150 - 6

or, 3e = 144

or, e = 144 / 3

or, e = 48

So, the optimum level of pollution : e_opt^* = 48

The reason behind equating the MC and MD of air pollution is that firms pollute air through the production activities undertaken by them while society as a whole suffers from the damaging impacts of air pollution aggravated by production activities of firms. Unfortunately, rational profit maximizing firms do not take into consideration the damage cost they impose upon the society while making economic decisions about their operations such that the negative pollution impact remains unaccounted for and thus treated as an externality issue arising out of production activities. Now when we want to decide the socially optimum level of air pollution that may be caused by firms, we would take into account the damage cost of such activities as well as the marginal pollution abatement cost of firms, i.e., how much cost the firms would need to incur in order to marginally reduce air pollution caused by them. So, the idea here is not only to take into account of the damage cost but also to account for the economic viability of reducing pollution by the firms. If the marginal cost that firms need to incur to abate air pollution is too high(or, opportunity cost of polluting air is too high) then that must be weighed against the marginal damage that level of pollution is inflicting on the society and then firms may decide not to abate pollution if the abatement cost outweighs social damage cost. For that purpose, we need to compare MD and MC of pollution and find optimal level of pollution by equating the cost as that level of pollution would be the most desirable both from the point of view of social welfare and economic feasibility.

b) To find out the net gain to the society, one needs to visualize the problem graphically as follows :

To find the net gain of the society assuming firms were initially not controlling emissions, we take the initial pollution abatement of firms as 0 thus MC of pollution abatement must be 0 as well. So,

MC = 150 - e = 0 or, e = 150

Thus, initial level of pollution is 150 when the firms were not controlling emissions. At e = 150 , marginal damage to the society is : MD = 6 + 2 150 = 6 + 300 = 306 .

Now, later suppose the firms control pollution to reach the optimum level of air pollution i.e., e_opt^* = 48 . At optimum level of pollution : MC = 150 - 48 = 102 ; MD = 6 + 2 48 = 102 ; MC = MD = 102

Now the net gain from controlling pollution is given by the E^*AB in the above diagram. As pollution level is marginally reduced, the extra amount of Marginal Damage cost over and above the Marginal abatement cost of firms gives the Marginal gain or benefit of the society from reducing pollution. In this case, as pollution is reduced from 150 to 48, the total gain is given by adding marginal gains from all those units which is precisely what the triangle E^*AB denotes.

To find the area of the desired triangle , we follow the formula :

Area of E^*AB = (1 / 2) length of E^*C length of AB

= (1 / 2) (150 - 48) 306

= (1 / 2) 102 306

= 51 306

= 15606

Therefore, the net gain to the society is given by 15606 units.

c) The level of tax that would achieve the optimum level of pollution is given by MC of reducing pollution at the optimal pollution level. So, t_opt = MC_opt = 150 -48 = 102 .

The reason for charging this level of tax is that if the tax charged on pollution units is less than the MC of pollution abatement at the optimal pollution level (MC_opt = 102), then firms would find it profitable to pay the pollution tax instead of investing in pollution abatement. The higher than optimal societal damage will continue due to high level of pollution and optimum pollution cannot be reached as firms do not take enough initiative to control pollution. Charging a higher tax rate than MC_opt would then result in firms abating more than optimal pollution level, which might be socially desirable but not economically viable for the society as marginal damage is exceeded by marginal abatement cost of pollution for any level of pollution less than e_opt^* = 48. However, if a tax rate is charged which is equal to MC_opt = 102, then firms would find it profitable to abate pollution to reach the exact optimal level of pollution. Investing in reducing more pollution than the optimal pollution level will however not be profitable for firms as MC of pollution abatement would then exceed pollution tax rate and they would instead continue to pollute until optimal pollution is reached.