Ken runs a barber shop. Given the popularity and location of the restaurant, he has a monopoly position in the market. The market demand curve is given by Q = 120 – 2P. Ken has a total cost of TC = Q2. If he charges the same price to all customers, what are Ken’s profit-maximising price PM and quantity QM?

Question

BlackTom AI · Accepted Answer

First, restate the scenario: Ken has a monopoly with market demand Q = 120 − 2P, so the inverse demand is P = 60 − Q/2. The total cost is TC = Q^2, so marginal cost is MC = d(TC)/dQ = 2Q.

Now, evaluate the profit-maximizing condition for a monopolist. In general, a monopolist chooses output where MR = MC. For a linear demand curve, the marginal revenue MR lies below the price and can be derived from the inverse demand: P = 60 − Q/2. The revenue is R = P·Q = (60 − Q/2)Q = 60Q − (1/2)Q^2. Differentiating, MR = dR/dQ = 60 − Q.

Set MR equal to MC to find the profit-maximizing quantity:
MR = MC ⇒ 60 − Q = 2Q.
Solving gives 60 = 3Q, so Q = 20.

To find the corresponding price, substitute Q = 20 into the inverse demand: P = 60 − (20)/2 = 60 − 10 = 50.

Sanity check: The total revenue at Q = 20 is TR = P·Q = 50·20 = 1000, and total cost is TC = Q^2 = 400. Profit is TR − TC = 600, which is feasible given the cost structure.

Option analysis:
- If there were alternative options claiming different Q or P, they would fail to satisfy MR = MC or the inverse demand relationship. For instance, a higher Q would imply MR < MC, reducing profit, while a lower Q would yield MR > MC but for a different P that would not align with the demand schedule. The derived Q = 20 and P = 50 are consistent with both the MR = MC condition and the market demand curve.
- If an option suggested a price or quantity that does not satisfy the inverse demand P = 60 − Q/2, it would be inconsistent with the given demand. Similarly, any option that implies MC ≠ 2Q at the chosen Q would violate the marginal cost relationship from TC = Q^2.

In summary, the profit-maximizing output is determined where MR equals MC, and the corresponding price is obtained from the demand curve. The calculations above show the steps to obtain Q = 20 and P = 50, and any alternative values would be inconsistent with either the marginal revenue/ marginal cost condition or the demand function.

类似问题

Suppose a profit maximizing monopolist faces the following demand curve: Q = 90 – 2P, and sets a uniform price where the elasticity of demand is - 2. What must be the MC of this firm at the profit-maximizing level of output?

Below is the reaction function diagram for an oligopoly. There are two firms each has the marginal cost of 4 and the demand function is P=20-2Q.What are the monopoly profits in this industry?

Below is the reaction function diagram for an oligopoly. There are two firms each has the marginal cost of 4 and the demand function is P=20-2Q.What is the monopoly price in this industry?

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