Industry A is a monopoly. Firm 1, the monopolist, produces a single good. The demand curve for the industry is P = 180 – Q1, where Q1 is the quantity of the good produced by Firm 1. So the total revenue function for firm 1 is TR = P*Q1 = 180Q1 - Q1^2.

Firm 1 has a marginal cost of $60 and no fixed cost. So the total cost function for firm 1 is TC = 60*Q1.

What is the profit-maximizing price for Firm 1?

Hint: we first construct MR (marginal revenue function in quantity Q1) and MC (marginal cost function in quantity Q1). We then choose Q1* such that MR = MC. Finally, we plug Q1* back into the demand curve to recover the profit-maximizing price for firm 1.

Question

Industry A is a monopoly. Firm 1, the monopolist, produces a single good. The demand curve for the industry is P = 180 – Q1, where Q1 is the quantity of the good produced by Firm 1. So the total revenue function for firm 1 is TR = P*Q1 = 180Q1 - Q1^2.

Firm 1 has a marginal cost of $60 and no fixed cost. So the total cost function for firm 1 is TC = 60*Q1.

What is the profit-maximizing price for Firm 1?

Hint: we first construct MR (marginal revenue function in quantity Q1) and MC (marginal cost function in quantity Q1). We then choose Q1* such that MR = MC. Finally, we plug Q1* back into the demand curve to recover the profit-maximizing price for firm 1.

BlackTom AI · Accepted Answer

For a linear demand P = a − bQ, marginal revenue is MR = a − 2bQ; here MR = 180 − 2Q1.

1) Relevant knowledge points and concepts
- Profit maximization for a monopoly occurs where MR = MC. 
- MR is the derivative of total revenue TR with respect to quantity Q1. 
- MC is the derivative of total cost TC with respect to Q1. 
- Once the profit-maximizing quantity Q1* is found, plug it into the demand curve to get the monopoly price P*.

2) Complete reasoning process step by step
- Given TR = 180Q1 − Q1^2, compute MR = d(TR)/dQ1.
- Given TC = 60Q1, compute MC = d(TC)/dQ1.
- Set MR = MC and solve for Q1*.
- Substitute Q1* into P = 180 − Q1 to get the profit-maximizing price.

3) Calculations with intermediate steps
- MR = d(180Q1 − Q1^2)/dQ1 = 180 − 2Q1.
- MC = d(60Q1)/dQ1 = 60.
- Set MR = MC: 180 − 2Q1 = 60.
- Rearranging: 2Q1 = 180 − 60 = 120.
- So Q1* = 120 / 2 = 60.
- Price: P* = 180 − Q1* = 180 − 60 = 120.
(As a check, profit = TR − TC = (180·60 − 60^2) − (60·60) = (10800 − 3600) − 3600 = 3600.)

4) Why the correct answer is correct
MR = MC identifies the output where the firm’s incremental revenue equals its incremental cost. Solving MR = 180 − 2Q1 = 60 gives the monopoly output Q1* = 60; the demand curve then yields the monopoly markup price P* = 120. This is the price that maximizes profit given the cost and demand functions.

5) Why the other options are incorrect
- 20 and 30: These prices would correspond to much larger quantities (P = 20 → Q = 160; P = 30 → Q = 150) where MR < MC, not profit-maximizing.
- 60: This is the marginal cost, not the profit-maximizing price. A monopoly charges a price above MC.
- "None of the other choices": incorrect because 120 is attainable and correct.

6) Conclusion
The profit-maximizing price for Firm 1 is 120, which matches the provided answer.

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