Final answer:
In a perfectly competitive market, the firm should produce where price equals marginal cost, which is at 24 units, to maximize profits. By subtracting the total cost from the total revenue at this production level, we find that the firm's short-run profits are $1082.
Step-by-step explanation:
Short-Run Profit Maximization in Perfect Competition
In a perfectly competitive market, firms are price takers and can sell their products at the market price, which in this case is $110 per unit. For profit maximization, the firm should produce at a level where the price (P) equals the marginal cost (MC), as that will also be where the marginal revenue (MR) equals the MC; P = MR = MC. Given the MC function MC(Q) = 14 + 4Q, we set P equal to MC to find the optimal quantity:
$110 = 14 + 4Q
4Q = $110 - 14
4Q = $96
Q = 24 units
To calculate the firm's short-run profits, we find total revenue (TR) by multiplying the number of units sold by the price, which gives us TR = P × Q = $110 × 24. Next, we find total cost (TC) using the total cost function C(Q) = 70 + 14Q + 2Q² at Q = 24 to get the costs associated with producing 24 units. Finally, we subtract the total cost from total revenue to determine the profit. The firm's short run profits are calculated as follows:
Profits = TR - TC
Profits = ($110 × 24) - (70 + 14× 24 + 2× 24²)
Profits = $2640 - (70 + 336 + 1152)
Profits = $2640 - $1558
Profits = $1082