Need help with my microeconomics question

Question

A monopoly provides a service and has two types of consumers: Professionals and students. The corresponding demand functions for these groups are as follows; For Professionals: 𝑃p = 100 − 𝑄p For Students: Ps = 50 − 𝑄s Assume that marginal cost of the firm is 20 for professionals and 10 for students. The firm can separate the agents by asking their student cards.

a) Find the profit maximizing prices and quantities for both groups.

b) Assume that this firm can charge a membership fee for its consumers. Find the profit maximizing membership fee, price and quantity for both groups. Is this outcome better than part (a)?

Answer 1

a) Find the profit-maximizing prices and quantities for both groups.

Recall that a profit-maximizing monopolist offering a single product will operate at a quantity that equates its marginal revenue and marginal cost.

This question appears more complicated - there are two consumer types, and two different marginal costs. However, note that:

1. The monopolist can set different prices for the different consumer types;

2. The willingness to pay for one type of consumer is independent of the services provided to the other, i.e., there are no demand-side interactions in the monopolist's problem. For example, the marginal revenue from providing an extra unit to professionals is independent of the amount provided to students;

3. The cost of providing the service to one consumer type is independent of the cost of providing the service to the other type, i.e., there are no supply-based interactions in the monopolist's problem.

As a result of these independence, the monopolist can decompose its problem into two: one for each consumer type and set a price for each consumer type. To find the profit-maximizing price and quantity, we'll equate the use marginal revenue (MR) equals marginal cost (MC) for each group.

Step 1: Demand Functions

- Professionals: P_p = 100 - Q_p

- Students: P_s = 50 - Q_s

Step 2: Revenue and Marginal Revenue for Each Group

Revenue is R = P x Q, and Marginal Revenue (MR) is the derivative of total revenue with respect to quantity Q.

For professionals:

- Revenue: R_p = P_p x Q_p = (100 - Q_p) x Q_p = 100Q_p - Q_p²

- Marginal Revenue: MR_p = dR_p/dQ_p = 100 - 2Q_p

For students:

- Revenue: R_s = P_s x Q_s = (50 - Q_s) x Q_s = 50Q_s - Q_s²

- Marginal Revenue: MR_s = dR_s/dQ_s = 50 - 2Q_s

Step 3: Set MR = MC to Maximize Profit

For professionals:

- Set MR_p = MC_p : 100 - 2Q_p = 20

2Q_p = 80, so Q_p = 40

- Plug Q_p = 40 into the demand equation to get the corresponding price P_p = 100 - 40 = $60/unit

For students:

- Set MR_s = MC_s : 50 - 2Q_s = 10

2Q_s = 40, so Q_s = 20

- Plug Q_s = 20 into the demand equation: P_s = 50 - 20 = $30/unit

Summary for Part a:

- Professionals: Price P_p = 60, Quantity Q_p = 40 units

- Students: Price P_s = 30, Quantity Q_s = 20 units

b) Find the profit-maximizing membership fee, price, and quantity for both groups.

When the firm charges a membership fee, it can set the price equal to the marginal cost (since it's recovering additional profits from the membership fee) and use the membership fee to capture the entire consumer surplus.

Step 1: Set Prices Equal to Marginal Costs

- For professionals, set P_p = MC_p = $20/unit

- For students, set P_s = MC_s = $10/unit

Step 2: Find the Quantities

Using the demand equations, plug the prices into the demand functions:

For professionals:

Q_p = 100 - P_p = 100 - 20 = 80 units

For students:

Q_s = 50 - P_s = 50 - 10 = 40 units

Step 3: Calculate Consumer Surplus

Consumer surplus is the area under the demand curve and above the price, up to the quantity demanded. It's a triangle with height (maximum willingness to pay minus price) and base (quantity).

For professionals:

- Maximum willingness to pay: $100/unit

- Price: $20/unit

- Quantity: 80 units

Consumer surplus for professionals = 0.5 x (100 - 20) x 80 = $3,200

For students:

- Maximum willingness to pay: $50/unit

- Price: $10/unit

- Quantity: 40 units

Consumer surplus for students = 0.5 x (50 - 10) x 40 = $800

Step 4: Set the Membership Fee

The profit-maximizing membership fee is the total consumer surplus for each group.

- Membership fee for professionals: F_p = $3,200

- Membership fee for students: F_s = $800

Summary for Part b:

- Price for professionals: P_p = $20/unit, Quantity Q_p = 80 units, Membership fee F_p = $3,200

- Price for students: P_s = $10/unit, Quantity Q_s = 40 units, Membership fee F_s = $800

Comparison with Part a:

- In part (a), the firm sets higher prices and sells lower quantities.

- In part (b), the firm sets prices equal to marginal cost and charges a membership fee to capture the consumer surplus, leading to higher quantities sold.

- The outcome in part (b) is more efficient (more total surplus captured) and likely more profitable since it maximizes both output and total consumer surplus captured through membership fees.