Question

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 (i.e., MT = 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.

Answer 1

Answer;

a)Now, under old technology the marginal abatement cost is given by, “MAC0 = 0.5*A0”. Now under “abatement standard = 40”, the “total abatement cost” is given by, “TAC0 = 1000 + 0.25*A0^2”.

=> TAC0 = 1000 + 0.25*40^2 = 1000 + 0.25*1600 = 1000 + 400 = 1400.

Now, under “emission charge” of “$10” per unit tax if “Ai < 40. So, the optimum emission will be determined by the following condition.

=> MAC = emission fee, => 0.5*A0 = 10, => A0 = 10*2 = 20 < 40. So, here the optimum emission reduction is less than “40”, => each firm having old technology will reduce “20” units and pay tax for “40-20=20” units.

So, under the old technology the “total cost” is given below.

=> TAC = 1000 + 0.25*20^2 + 20*10 = 1000 + 100 + 200 = 1300.

So, here the cost savings under emission fee is “1400 – 1300 = 100”.

b).

Now, under new technology, the “TAC” under the emission standard “40” is given below.

=> TAC = 1000 + 0.125*An^2 = 1000 + 0.125*40^2 = 1000 + 0.125*1600 = 1000 + 200 = 1200.

Now, if emission fee will be charge of “$10”, then the optimum emission reduction is given by.

=> MACn = 10, => 0.25*An = 10, => An = 40 same as emission standard. So, here the optimum emission reduction is same as the emission standard, => there will be any cost savings under new technology.