Question

A plant produces aluminum, facing costs of C(A) = \frac{A^2}{10}, where A is the number of aluminum units produced, and selling them on the competitive market for a price \$ 30. A farmer grows beets next to the aluminum factory, facing costs of C(B) = 5B + \frac{B^2}{100} + \frac{A^2}{20}, where B is the number of beets produced, and selling them for \$ 15 on the competitive market. Because aluminum pollution is harmful to agricultural production, the state government imposes a tax on aluminum production. What is the socially optimal tax on aluminum?

Answer 1

Answer:

$10

Explanation:

We are to account for external costs in production, since we are asked to find optimal tax.

Given:

$C(A) = \frac{A^2}{10}$

We now have:

$C(A) = \frac{A^2}{10} + \frac{A^2}{20}$

A represents number of aluminum units produced, let's find A, since the margnal cost is $30.

Thus,

$30 = \frac{A}{5} + \frac{A}{10}$

$30 = \frac{2A + A}{10}$

$300 = 3A$

$A = \frac{300}{3}$

$A = 100$

Let's equate the private marginal cost with the marginal revenue of each unit in order to achieve this amount of produced units with tax, t.

We have:

$30 - t = \frac{A}{5}$

Substituting 100 for A above, we have:

$30 - t = \frac{100}{5}$

30 - t = 20

t = 30 - 20

t = 10

Therefore, the socially optimal tax on aluminum is $10 per unit