If the state enacts the emission charge, find the associated cost savings to the firm, assuming the use of the old technology.

Question

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.

Answer 1

Using the old technology, the total abatement costs of meeting the 40-unit abatement standard are: TACO = 1000 + .25(40)2 = $1,400 thousand.
Faced with the emission charge, the firm would abate up to the point where the tax is equal to its MAC, which is found as follows:
MT = MACO
10= 0.5(AO), so AO= 20
Therefore, the firm’s total abatement costs would be TACO = 1000 + .25(20)2 = $1,100 thousand.
Since the actual level of abatement (AO) is less than the abatement standard of 40 units, the firm also will pay taxon the difference equal to Total Tax = t(AST – AO) = $10(40 – 20) = $200 thousand.
Taken together, the firm’s total costs of meeting the abatement standard are the sum of the TACO and the tax, or $1,100 + $200 = $1,300 thousand.
Notice that the emission charge allows the firm to realize a cost savings of $100 thousand ($1,400 thousand – $1,300 thousand).