Suppose there are three types of chip consumers in the world with three different inverse demand functions given by Pa=30-1/2P, Pb=40-1/2P, and Pc=50-1/2P. The marginal cost of the monopoly that produces chips is a constant $20. What size packages should the perfectly price discriminating monopolist make? What price should the price discriminating monopolist charge for each package?
How much profit will the price discriminating monopolist make?
see my post to jennifer (just below this post) for a good hint on how to solve this problem.
Repost if you have questions.
To find the optimal package size and price for a price-discriminating monopolist, we need to follow a few steps.
Step 1: Derive the inverse demand functions
The inverse demand functions for the three types of chip consumers are given as:
Pa = 30 - 1/2P
Pb = 40 - 1/2P
Pc = 50 - 1/2P
Step 2: Calculate the marginal revenue (MR) for each consumer type
The marginal revenue is calculated by taking the derivative of the inverse demand function with respect to quantity.
For consumer type A:
MRa = ∂(Pa * Qa) / ∂Qa = Pa - Qa * ∂Pa / ∂Qa
where Qa is the quantity demanded by consumer type A.
Differentiating Pa = 30 - 1/2P with respect to quantity Qa, we get:
∂Pa / ∂Qa = -1/2
Substituting the values, we get:
MRa = (30 - 1/2P) - Qa * (-1/2)
Similarly, we can calculate MRb and MRc for consumer types B and C, respectively.
Step 3: Find the quantity and price for each consumer type
To maximize profit, the monopolist sets marginal cost (MC) equal to marginal revenue (MR). The monopolist will charge each consumer type a price equal to their respective willingness to pay (given by the inverse demand function) for the quantity demanded.
For consumer type A:
MC = MRa
20 = 30 - 1/2P - Qa * (-1/2)
1/2P = 10 + Qa/2
P = 20 + Qa
Similarly, we can find the prices for consumer types B and C:
Pb = 20 + Qb
Pc = 20 + Qc
Step 4: Calculate the quantity demanded and profit for each consumer type
To determine how much of each package size to produce, substitute the derived price equations back into the inverse demand functions.
For consumer type A:
Pa = 30 - 1/2(20 + Qa)
20 + Qa = 30 - 1/2Qa
3/2Qa = 10
Qa = 20/3
Similarly, we can calculate Qb and Qc for consumer types B and C, respectively.
Step 5: Calculate the profit for each consumer type
Profit can be calculated using the formula:
Profit = (Pa - MC) * Qa + (Pb - MC) * Qb + (Pc - MC) * Qc
Substituting the calculated values, we get:
Profit = (30 - 20) * (20/3) + (40 - 20) * (20/3) + (50 - 20) * (20/3)
By following these steps, you can find the optimal package sizes, prices, and profits for a price-discriminating monopolist.