Question

Suppose there are 10 firms that each have the profit function π= 24q- 3q², where q is the quantity of pollution that a firm generates.

a. Assuming that the firms are profit maximizers, what tax rate on pollution would you need to set in order to reduce pollution by a total of 15 units (compared to no tax).

Answer 1

Given

Let us calculate the amount of q at which profit is maximized. Put

We can see that profit is maximized when q=4 units.

Now suppose t tax is imposed on per unit pollution. Now, profit is changed to

Let us calculate the amount of q at which profit is maximized. Put

We want to reduce 15 units of pollution. There are 10 firms.

Needed pollution reduction per firm=15/10=1.5 unit

So, allowable pollution limit per firm=Initial pollution-Needed pollution reduction=4-1.5=2.50 units

Put q=2.50

Tax should be 9 per unit of pollution.