Question

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.

Short answer

• The output produced by the firm is 25 units.

• The firm will earn a profit of $800.

• The producer surplus of the firm will be $1250.

Step-by-step solution

Calculating output level of the firm

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.

Calculating the profit of the firm

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.

Determination of producer surplus

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.