Question

Assume that scientific studies provide you with the following information concerning the benefits and costs of sulfur dioxide emissions:

Benefits of abating (reducing) emissions:MB= 500 - 20A

Costs of abating emissions:MC= 200 + 5A

whereAis the quantity abated in millions of tons and the benefits and costs are given in dollars per ton.

a. What is the socially efficient level of emissions abatement?

b. What are the marginal benefit and marginal cost of abatement at the socially efficient level of abatement?

c. What happens to net social benefits (benefits minuscosts) if you abate one million more tons than the efficient level? One million fewer?

d. Why is it socially efficient to set marginal benefits equal to marginal costs rather than abating until total benefits equal total costs?

Short answer

1. The socially efficient level for abatement is 12 million tons.

2. The marginal benefit and marginal cost at the socially efficient level of reduction would be 260.

3. In both cases, if one more million ton or fewer is abated from the efficient level, the social net benefit would fall.

4. It is socially efficient to set marginal benefit equal to marginal cost rather than total benefits to total cost because we want to maximize the additional benefits given the additional cost.

Step-by-step solution

Determination of the socially efficient level 

The socially efficient level of abatement: marginal cost = marginal benefit

$$\begin{gathered} \text{500 - 20A = 200 + 5A} \\ \text{500 - 200 = 20A + 5A} \\ \text{300 = 25A} \\ \text{A =}\frac{\text{300}}{\text{25}} \\ \text{= 12} \end{gathered}$$

Therefore, the socially efficient level of abatement would be 12 million tons.

Marginal cost and benefit at a socially efficient level

The value of A is equal to 12. Therefore by substituting it in the marginal benefit and marginal cost equation, the marginal cost and marginal benefit be:

$$\begin{gathered} \text{MB = 500 - 20}\left(\text{12}\right) \\ \text{= 260} \\ \text{MC = 200 - 5}\left(\text{12}\right) \\ \text{= 260} \end{gathered}$$

Therefore, both will be 260 at a socially efficient level.

The social net benefit would fall

The graph below depicts Marginal Benefit along the x-axis and Marginal Cost along the Y-axis.

Net social benefit is the area under the marginal benefit curve - the area under the marginal cost curve. At socially efficient level this area is = a+b+c+d, or

$0.5\left(500-200\right)\left(12\right)=\$1800$

If you abate one million tons more, then it will be = a+b+c+d-e, or

$$\begin{gathered} 1800-0.5\left(265-240\right)\left(1\right) \\ =1800-12.5 \\ =\$1787.5 \end{gathered}$$

If you abate one million fewer, then it will be = a+b, or

$$\begin{gathered} 0.5\left(500-280\right)\left(11\right)+\left(280-255\right)\left(11\right)+0.5\left(255-200\right)\left(11\right)\backslash \mathrm{hfill} \\ =\$1787.5 \end{gathered}$$

In each case, Net Social Benefit will fall by 1800-1787.5=$12.5 million

Maximizing benefits

It is socially efficient to set the marginal cost equal to marginal benefits because equating the last unit of reduction's cost and benefits will help determine the optimal quantity. However, if we use total cost and total benefits, the net benefits will be zero since the sum of all costs will simplify become equal to some of the benefits, which would compensate for the difference in units.