Question

A monopoly is considering selling several units of a homogeneous product as a single package. Analysts at your firm have determined that a typical consumer’s demand for the product is Q^d = 60 − 0.25P, and the marginal cost of production is $80.

a. Determine the optimal number of units to put in a package.
 

 units
 

b. How much should the firm charge for this package?

Answer 1

To determine the optimal number of units to put in a package and the price the firm should charge, we need to find the point where the firm maximizes its profit. The profit is maximized when marginal revenue (MR) equals marginal cost (MC). Here's how we can approach it:

Given: Demand equation: Qd = 60 - 0.25P Marginal cost (MC) = $80

Step 1: Find the marginal revenue (MR) equation: Since the firm is a monopoly, it faces the entire market demand curve for its product. The marginal revenue is calculated as the derivative of the total revenue (TR) with respect to quantity (Q): TR = P * Q MR = d(TR)/dQ

To find MR, we differentiate the total revenue equation: TR = P * Q MR = d(P * Q) / dQ = P * dQ/dQ + Q * dP/dQ (applying the product rule) = P + Q * dP/dQ

Since Qd = 60 - 0.25P, we can find dP/dQ by differentiating Qd with respect to Q: dP/dQ = -0.25

Therefore, MR = P + Q * (-0.25) = P - 0.25Q

Step 2: Set MR equal to MC to find the optimal quantity: P - 0.25Q = MC P - 0.25Q = $80 (since MC = $80)

Solving for P, we have: P = $80 + 0.25Q

Step 3: Substitute P back into the demand equation to find the optimal quantity (Q): Qd = 60 - 0.25P Qd = 60 - 0.25($80 + 0.25Q) (substituting P = $80 + 0.25Q)

Solving for Q, we have: Qd = 60 - 0.25($80) - 0.0625Q Qd + 0.0625Q = 60 - 0.25($80) (1 + 0.0625)Q = 60 - 0.25($80) 1.0625Q = 60 - 20 1.0625Q = 40 Q = 40 / 1.0625 Q ≈ 37.65

Step 4: Determine the optimal number of units in the package: Since the demand equation represents the demand for a single unit, the optimal number of units in the package will be rounded to the nearest whole number: Optimal number of units in the package = 38 units

Step 5: Calculate the price the firm should charge for the package: Using the price equation we derived earlier: P = $80 + 0.25Q P = $80 + 0.25(38) P = $80 + 9.5 P ≈ $89.50

Therefore, the firm should put 38 units in the package and charge approximately $89.50 for this package to maximize its profit.