Find the cost effective allocation of abatement across the two sources that also satisfies the 40 percent aggregate abatement level. Support with calculations.

Question

Suppose that there are only two stationary sources in a given air quality region. The first source has been in existence for several years, while the second source is new. The following functions represent marginal abatement costs (MAC) for each polluting source:

MACEX = 10.0 + 0.7A EX, MACN = 9.2 + 0.5A N,

whereAEX is the percentage abatement level for the existing source, and AN is the percentage abatement level for the new source.

Assume that the aggregate abatement standard (i.e., for the region as a whole) is set at 40 percent and that the two firms’ current abatement levels areAEX = 10 and AN = 30.

Answer 1

To find the cost effective allocation of individual abatement levels, set MACEX = MACNwhile maintaining the 40 percent aggregate abatement level. This is determined by simultaneously solving the following two equations:

MACEX = MACN
AEX + AN = 40
Therefore:
10.0 + 0.7(AEX) = 9.2 + 0.5(AN), where A N = 40 ?AEX
Substituting and solving:
10.0 + 0.7(AEX) = 9.2 + 0.5(40 ?AEX)
10.0 + 0.7(AEX) = 29.2 ? 0.5(AEX)
1.2(AEX) = 19.2
AEX = 16
AN = 40 ?AEX = 40 – 16 = 24.

Notice that when AEX = 16, MACEX = 10.0 +.7(16) = 21.2, and when AN= 24, MACN = 9.2 +.5(24) = 21.2. Since MACEX = MACN= 21.2, the abatement allocation of AEX = 16 and AN= 24 is cost-effective.