The problem states that a farmer is a price taker in soybeans. The total cost (TC) function is given by $TC = 0.1q^2 + 2q + 100$, and the marginal cost (MC) function is given by $MC = 0.2q + 2$. The farmer has to purchase a license for $50 per period to stay in business. The question asks for the new marginal cost function after considering the license fee.

1. Problem Description

The problem states that a farmer is a price taker in soybeans. The total cost (TC) function is given by $TC = 0.1q^2 + 2q + 100$, and the marginal cost (MC) function is given by $MC = 0.2q + 2$. The farmer has to purchase a license for $50 per period to stay in business. The question asks for the new marginal cost function after considering the license fee.

2. Solution Steps

The license fee is a fixed cost. Fixed costs do not affect marginal cost. Marginal cost is the derivative of the total cost function with respect to quantity, $q$.

$$MC = \frac{dTC}{dq}$$

The total cost function is $TC = 0.1q^2 + 2q + 100$. When the farmer purchases the license, the total cost function becomes $TC = 0.1q^2 + 2q + 100 + 50 = 0.1q^2 + 2q + 150$.

Now, we can calculate the marginal cost by taking the derivative of the new total cost function with respect to $q$.

$$MC = \frac{d}{dq}(0.1q^2 + 2q + 150) = 0.2q + 2$$

Since the license fee is a fixed cost, it does not change the marginal cost. Therefore, the marginal cost remains the same, i.e., $MC = 0.2q + 2$.

3. Final Answer

still MC = .2q + 2