91Ó°ÊÓ

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.

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

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 (Ï€) is calculated by subtracting the total cost (C) from total revenue (R).

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

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.