Question

In a perfectly competitive market, a firm's Total Revenue and Total Cost functions are: \[ TR = 120Q \] \[ TC = 200 + 40Q + Q^2 \] The profit-maximizing output level is:

• A. 20 units
• B. 30 units
• C. 40 units
• D. 50 units

Hint:

For Perfect Competition: \[ MR = AR = Price \] Profit maximization occurs at: \[ MR = MC \] with a rising MC curve.

Answer 1

The Correct Option is C

Concept: A competitive firm maximizes profit at the output level where: \[ MR = MC \] and \[ MC \text{ is rising} \] Since firms in perfect competition are price takers: \[ MR = AR = Price \]

Step 1: Find Marginal Revenue. Given: \[ TR=120Q \] Differentiate with respect to \(Q\): \[ MR=\frac{d(TR)}{dQ} \] \[ MR=120 \]

Step 2: Find Marginal Cost. Given: \[ TC=200+40Q+Q^2 \] Differentiate: \[ MC=\frac{d(TC)}{dQ} \] \[ MC=40+2Q \]

Step 3: Apply profit maximization condition. \[ MR=MC \] \[ 120=40+2Q \] \[ 80=2Q \] \[ Q=40 \]

Step 4: Verify second condition. \[ MC=40+2Q \] Since coefficient of \(Q\) is positive, MC is rising. Therefore the condition for profit maximization is satisfied. Hence, \[ \boxed{Q=40} \] Thus option (C) is correct.