Question

Two firms (firm 1 and firm 2) can control pollution with the following marginal abatement costs:

MAC1=100q1

MAC2=200q2

where q1 and q2 are the amount of pollution reduced by the first and second firms. Assume that with no controls at all, each firm would be emitting 100 units of pollution or 200 units for both firms.

Suppose that the government sets up an emissions trading market mechanism to achieve a pollution goal of 100 units of total pollution. The government would issue 100 emissions permits. Each permit represents the right to emit one ton of emissions. The government then freely allocates 50 permits to each firm. In the emissions trading market, who will buy permits? How many permits the buyer will buy?

Answer 1

Marginal Abatement cost of second is double that of first, for every positive units of pollution.

When there was no restriction each of them produced 100 tons of pollution. Therefore if somehow any one of them completely stops polluting, and the other gains all the permit then the it would produce the same number of pollutants as it proudced with restriction i.e., 100 tons .

Now Initially, one and two both have 50 permits which means each of them can produce no more than 50 pollutant. Which means for 51st unit they would have to incurr marginal cost of abatement . MAC1 < MAC2. for 50th unit, MAC1 = 5000 and for 51st unit MAC2 = 10,200. Now this gap in MAC gives an incentive to second to offer first any amount greater than 5000 and lesser than 10,200 to trade permit. If first trades it's permit to second , say at 10,000 , then it will not only cover it's MAC for 50th ton of pollutant but also leaves it with surplus 5000. Whereas second will only have to incur the permit cost of 10,000 instead of MAC2 = 10,200 for 51st unit . Hence it will gain a surplus of 200 by trading.

This will continue happening because MAC of one is lesser than two and for every additional pollutant , say for nth unit ,MAC2 for 50+nth output > MAC 50-nth output.

Assuming that they have traded 49 permit ,first has sold and second has bought , let's check whether the explanation provided above holds for 50th permit. For 100th unit of pollutant by second firm ,MAC2 = 20,000 . We have to keep in mind that 49 permits has already been sold by first, that is first is left with only one permit, the first permit saves the cost of the q1= one ton of pollutant which is MAC1 = 100 . Second can offer first any amount greater than 100 and lesser than 20,000. Say for example 500 , if second offers first 500, there isn't any reason why first will not sell it's permit at this . 500 is enough to cover MAC1 for q=1 plus it also leaves a surplus of 400 for it. And second will not have to incurr MAC2 = 20,000 after buying permit of worth 500. Hence this trade is feasible for every permit that one posses.

All of the permit will be sold by first to second .