A firm produces a product in a competitive industry and has a total cost function C = 50 + 4q + 2q2 and a marginal cost function MC = 4 + 4q. At the given market price of $20, the firm is producing 5 units of output. Is the firm maximizing its profit? What quantity of output should the firm produce in the long run?

The product's price is $20, and the output produced is 5 units.

The total revenue (R) is calculated in the following manner:

$$\begin{gathered} R(\$)=p\times q \\ =20\times 5 \\ =100 \end{gathered}$$

The total cost function is, C = 50 +4q+2q². Putting the value of q=5,

$$\begin{gathered} C\left(\$\right)=50+4q+2q^2 \\ =50+4\left(5\right)+2{\left(5\right)}^2 \\ =50+20+50 \\ =120 \end{gathered}$$

For a firm to maximize profit, the value of marginal cost mustn't exceed the value of marginal revenue. But the value of total cost (c) exceeds the value of total revenue (R).

Therefore, the firm is not maximizing profit.

The output at which the firm is maximizing its profit is the output that the firm should produce in the long run.The output at which the marginal cost equals the marginal revenue is the profit-maximizing output. The optimal output for long-run is calculated below:

$$\begin{gathered} TR=p\times q \\ =20q \\ MR=\frac{d\left(TR\right)}{dq} \\ =\frac{d\left(20q\right)}{dq} \\ =20 \end{gathered}$$

Equating MC = MR

4 + 4q = 20

4q = 16

q = 4

The firm should produce 4 units of output.