Final answer:
The minimum price for a perfectly competitive firm to earn a profit is just above the average cost at the quantity where marginal revenue equals marginal cost. The price should be at the minimum point of the average cost curve.
Step-by-step explanation:
The student has asked what the minimum price (p) for a perfectly competitive firm should be to earn a profit, given the total cost function tc = 100 + 4q2. The answer lies within understanding the relationship between market price, average cost (AC), and profit maximization in a perfectly competitive market. For the firm to earn profit, the market price must exceed the average cost at the profit-maximizing level of output.
To determine the minimum price, we need to calculate the firm's average cost at the quantity where marginal revenue (MR) equals marginal cost (MC), which is the profit-maximizing condition. Since this firm's marginal cost function is derived from the total cost function, we would first need to find the MC and set it equal to MR (price for a perfectly competitive firm) to find the quantity that maximizes profit.
Once the quantity is known, we can calculate the average cost (AC) by dividing the total cost (from the cost function) by the quantity (q). The minimum price for the firm to earn a profit is just above this average cost at the profit-maximizing quantity. Moreover, this price would be at the minimum point of the AC curve, where the firm earns zero profits; anything higher than this price would result in profits.