Question

Suppose there are two firms: Firm A and Firm B. These firms are each emitting 75 tons of pollution. Firm A faces marginal abatement cost MACA = 3A and Firm B faces marginal abatement cost MACB = 9A where A is tons of pollution abatement. The government’s control authority wishes the firms to reduce their total emissions by 60 tons using a Cap and Trade system and will initially auction off the permits

1a How many allowances will the control authority auction off?

1b Assuming that demand equals supply, what price would be paid for those allowances?

1c How many allowances would Firm B be expected to buy at the auction?

1d If the control authority decided to use an emissions tax rather than cap and trade, what tax rate would achieve the 60-unit reduction cost-effectively?

Answer 1

A).

Consider the given problem here there are two firms and their “MAC” are given in the above question. Now, each firm is emitting “75 tons of pollution”. So, the total pollution is “75*2=150 tons”. Now, the government control authority wishes to reduce the total emission by “60 tons”, => total pollution reduce to “150-60=90 tons”.

So, the control authority auction off “90 tons” allowances.

B).

Now, the marginal abatement cost of both the firms are given by, “MACA=3A” for firm-A and “MACB = 9B” for firm-B. Now, let’s assume “Pa” and “Pb” are the level of pollution by “firm A” and “firm B” respectively, => Pa+Pb=90. So, the total demand for the pollution for each firm is given by.

=> MACA = 3*A = 3*(75-Pa), => MACA= 3*(75-Pa), => Pa = 75 - MACA/3, be the demand for firm-A. Similarly, for the “firm-B” it is given by, => Pb = 75 - MACB/9.

Now, at the equilibrium “total demand” must be equal to “total supply”.

=> Pa + Pb = 90, => 75 – MACA/3 = 75 – MACB/9, => 150 = MACA/3 + MACB/9.

=> 150 = MAC/3 + MAC/9, => 4*MAC=150*9, => MAC=135.

So, the price paid for those allowances is “$135 per ton”.

C).

The demand for “firm-B” is given by “Pb = 75 – MACB/9”, => for “MAC=135”.

=> Pb = 75 – 135/9 = 75 – 15 = 60 tons. So, the firm-b want to purchase “60 tons” of pollution at the auction.

D).

Now, the marginal abatement cost for each firm are given by, “MACA=3A, where A = level of pollution abated by A” and “MACB = 9B, where B=level of pollution abated by B”. So, “A+B=60” and at the optimum “MACA=MACB=t”, where “t=tax per unit of pollution”.

=> MACA=t, => 3A=t, => A=t/3 and MACB=t, =>9B=t, => B=t/9. So, by “A+B=60”.

=> t/3 + t/9 = 60, => t = 135 per ton.

So, the emission tax is “$135 per ton”.