Consider the Following Abatement Cost Functions (TAC_O and MAC_O) for a Firm

Consider the following abatement cost functions (TAC_O and MAC_O) for a firm using an old abatement (A_O) technology:
TAC_O = 1000 + 0.25(A_O)²
MAC_O = 0.5(A_O),

where A is units 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 (A_ST) equal to 40 units for each firm and has proposed an emission charge implemented as a constant per unit tax (t) of $10 , where Total Tax = t(A_ST - A_i), and A_i is the existing abatement level.

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

b. Now suppose that the firm is contemplating the use of a new abatement (A_N) technology, which would generate the following cost functions:
TAC_N = 1000 + 0.125(A_N)²
MAC_N = 0.25(A_N)

Find the cost savings to the firm of using this new technology when faced with the emission charge.