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The inverse demand function that a monopoly faces is p = 10Q - 0.5. The firm's cost curve is C(Q) = 5Q. What is the profit-maximizing price and quantity? Please show steps.

User LWC
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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.

User Ethanabrooks
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