Final answer:
The profit-maximizing price is $5 and the profit-maximizing quantity is 0.55 for the monopoly. However, the monopoly is not earning any profit at this price and quantity.
Step-by-step explanation:
Step 1: The monopolist determines its profit-maximizing level of output by finding the quantity where marginal revenue (MR) equals marginal cost (MC). In this case, the inverse demand function is given as p = 10Q - 0.5, and the monopolist's cost function is C(Q) = 5Q. Therefore, to find the profit-maximizing quantity, we set MR = MC:
10Q - 0.5 = 5
10Q = 5.5
Q = 0.55
Step 2: The monopolist decides what price to charge by drawing a line straight up from the profit-maximizing quantity to the demand curve. Substituting Q = 0.55 into the inverse demand function, we can determine the price (P):
P = 10(0.55) - 0.5
P = 5.5 - 0.5
P = 5
Step 3: The monopolist can calculate total revenue by multiplying the price by the quantity:
Total Revenue = P × Q
Total Revenue = 5 × 0.55
Total Revenue = 2.75
Total cost will be Q multiplied by the average cost of producing Q, which is given as 5Q.
Total Cost = C(Q)
Total Cost = 5 × 0.55
Total Cost = 2.75
Profits will be the total revenue minus the total cost:
Profit = Total Revenue - Total Cost
Profit = 2.75 - 2.75
Profit = 0
Since the profit is zero, the monopolist is not earning any profit.