Question

A monopolist produces a product and sells to two types of buyers: A and B. The inverse demand functions for A and B are given by \( P = 50 - 5Q \) and \( P = 100 - 10Q \), respectively. The monopolist’s cost function is given by \( C = 90 + 20Q \). The profit maximizing output of the monopolist for buyer B is .........

Hint:

To maximize profit, the monopolist sets the marginal revenue equal to marginal cost and solves for the output level.

Answer 1

The monopolist maximizes profit by choosing output levels for each buyer. The profit function is: \[ \pi_B = P_B \cdot Q_B - C_B \] Where:
- \( P_B = 100 - 10Q_B \) (inverse demand function for buyer B),
- \( C_B = 90 + 20Q_B \) (cost function),
- \( Q_B \) is the quantity sold to buyer B.
Step 1: Revenue for buyer B is: \[ R_B = P_B \cdot Q_B = (100 - 10Q_B) \cdot Q_B = 100Q_B - 10Q_B^2 \] Step 2: Profit is: \[ \pi_B = R_B - C_B = 100Q_B - 10Q_B^2 - (90 + 20Q_B) = 100Q_B - 10Q_B^2 - 90 - 20Q_B = -10Q_B^2 + 80Q_B - 90 \] Step 3: To maximize profit, take the first derivative of the profit function with respect to \( Q_B \): \[ \frac{d\pi_B}{dQ_B} = -20Q_B + 80 \] Step 4: Set the derivative equal to zero to find the profit-maximizing quantity: \[ -20Q_B + 80 = 0 \] \[ Q_B = 4 \] Thus, the profit-maximizing output for buyer B is 4 units. Final Answer: \[ \boxed{4} \]