Suppose that a competitive firm has a total cost function C(q) = 450 + 15q + 2q² and a marginal cost function MC(q) = 15 + 4q. If the market price is P = $115 per unit, find the level of output produced by the firm. Find the level of profit and the level of producer surplus.

The output at which the firm will maximize its profit will be the optimal output level for the firm.The profit is maximized when marginal revenue equals the marginal cost.

$$\begin{gathered} MR=\frac{d\left(TR\right)}{dq} \\ =\frac{d\left(p\times q\right)}{dq} \\ =\frac{d\left(115q\right)}{dq} \\ =115 \\ MC=MR \\ 15+4q=115 \\ 4q=100 \\ q=25 \end{gathered}$$

The firm should produce 25 units of the product.

Since the value of q is determined 25, the value of total cost (C) and total revenue (R) is calculated by putting the value of q:

C($) = 450 + 15q + 2q²

= 450 + 15(25) + 2(25)²

=450 + 375 + 2 x 625

= 2075

R($) = p x q

=25 x 115

= 2875

Profit (π) is calculated by subtracting the total cost (C) from total revenue (R).

$$\begin{gathered} \mathrm{\pi }=\mathrm{R}-\mathrm{C} \\ =\$2875-\$2075 \\ =\$800 \end{gathered}$$

The firm will earn a profit of $800 at an output level of 25 units.

Producer surplus is calculated by subtracting variable cost (VC) from revenue (R).The value of variable cost is calculated below:

VC($) = 15q + 2q²

= 15(25) + 2(25)²

= 375 + 1250

= 1625

The value of producer surplus (PS) is determined below:

PS = R - VC

= $2875 - $1625

= $1250

The firm is earning a producer surplus of $1250.