Question

A monopolist faces some consumers (called group-H consumers) with an inverse demand function for each consumer given by p = 80 − q. The firm’s total cost function is given by: C(q) = 20q (so that MC = 20 for all q).

First suppose that the firm only uses linear pricing (i.e., charges only a unit price p).

1. Find the price maximizing the firm’s profits.

2. What are the corresponding profit and surplus per consumer?

Now suppose that the firm can use a two-part tariff (pH, FH), with pH the unit price and FH the fixed part.

3. First suppose the firm sets its unit price pH equal to the profit-maximizing linear price found in part (1) above. What fixed fee FH will the firm set at this price, and why?

4. Bearing in mind the opportunity to set a fixed fee FH, what unit price pH will the firm set? What fixed fee FH will it charge in view of this unit price?

5. Compute the firm’s profit per consumer and compare with the one found in the linear pricing case. Comment.

Answer 1

1) Total revenue = Pq = (80-q)q = 80q - q²

Marginal revenue (MR) can be found out by differentiating total revenue with respect to q

MR = 80 - 2q

A monopolist equate MR = MC in order to maimize the profits.

80-2q = 20

2q=60

q = 30

Put q = 30 in the demand equation in order to find out the price

p = 80 - q = 50

price maximizing the firm’s profits = $50

b) Profits = total revenue - total cost = 80q - q² -20q = 60q-q² = 60*30 - 30² = 900

Consumer surplus = area above the price line and below the demand curve = area highlighted in green in figure 1 = 1/2 * (80-50) *30 = 450

Figure 1

c) the firm sets its unit price pH equal to the profit-maximizing linear price = $50

Fixed fee FH will the firm set at this price = consumer surplus = $450

This is so becuase if monoploist charges the consumer surplus, the consumer will still buy the good and the monopolist will be able to earn the maximum profits available. If the monoplist charges a sum higher than the consumer surplus then the consumer will not buy the commodity.

d) The monopolist will equate P = MC and then charge the entire consumer surplus as the fixed fee.

P = MC

80-q = 20

q = 60

Fixed fee = consumer surplus = area above the price line and below the demand curve = area highlighted in red in figure 2 = 1/2 * (80-20) *60 = 1800

The unit price that is charges = MC = 20

Figure 2

e) Now, a consumer comes and pay 1800 as fixed fee and consumes 60 units at $20 each.

Profits = Total revenue - total costs = 1800 + 60*20 - 20*60 = 1800

Profit is higher by 900 as compared to linear pricing

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