Now suppose that the firm is contemplating the use of a new abatement (AN) technology, which would generate the following cost functions: TACN = 1000 + 0.125(AN)2 MACN = 0.25(AN) Find the cost savings to the firm of using this new technology when faced with the emission charge.

Consider the following abatement cost functions (TACO andMACO) for a firm using an old abatement (AO) technology:
TACO = 1000 + 0.25(AO)2 MACO = 0.5(AO),
where A isunits of abatement undertaken by the firm, and the cost values are in thousands of dollars.
Further assume that the regulatory authority has set an abatement standard (AST) equal to 40 units for each firm and has proposed an emission charge implemented as a constant per unit tax (t) of $10 (i.e., MT = 10), where Total Tax = t(AST – Ai), and Ai is the existing abatement level.

As in part (a), the abatement level the firm would undertake when faced with the $10 emission charge is as follows:
MT = MACN
0.25(A) = 10, so AN=40 units.
Because of the cost savings associated with the new technology, the firm would abate at a higher level, in this case up to the standard of 40 units. The associated total abatement costs are TACN = 1000 + 0.125(40)2 = $1,200 thousand.
Since the firm avoids any tax payments, its total costs to meet the standard are just the abatement costs. Notice that the use of new technology saves the firm another $100 thousand.