Final answer:
To determine the profit-maximizing price and quantity, calculate marginal revenue and marginal cost from the demand and total cost functions, set them equal, and then verify with the second-order condition to ensure profit maximization; finally, find the firm's profit by subtracting total cost from total revenue.
Step-by-step explanation:
To find the profit-maximizing price and quantity for the given company, we must first determine the revenue function by multiplying the demand equation by the price (p), then differentiate this revenue function to find marginal revenue (MR). We must then differentiate the total cost (TC) equation to find the marginal cost (MC). Setting MR equal to MC gives us the quantity where profit is maximized. We verify that the second-order condition (SOC) for a maximum is satisfied by ensuring that the second derivative of profit with respect to quantity is negative.
To find the profit-maximizing levels, here are the steps broken down:
- Determine the demand equation: q = 500 - 5p.
- Calculate the total revenue (TR) function: TR = p × q.
- Determine the marginal revenue by differentiating TR with respect to q.
- Find the total cost (TC) function: TC = 500 + 16q + q².
- Determine the marginal cost by differentiating TC with respect to q.
- Set MR equals MC to find the profit-maximizing quantity (q*).
- Plug q* into the demand equation to find the profit-maximizing price (p*).
- Calculate the firm's profit by subtracting total cost from total revenue at the profit-maximizing quantity and price.
- Verify the SOC for a maximum by checking the second derivative of the profit function is negative at q*.
For a detailed calculation, additional information would be needed, but following these steps will result in finding the profit-maximizing price, quantity, and the firm's profit.