# Price discrimination

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Problem set 3: Price discrimination

Numerical exercise (10 points)
Suppose that IBM has a monopoly position on the printer market. IBM has 100 potential consumers for its LaserPrinter. All consumers have a unit demand (i.e., the buy at most one printer) and have the following willingness to pay:
55 companies are willing to pay 100 € for one LaserPrinter;
45 individuals are willing topay 50 € for one LaserPrinter.
The constant marginal cost of producing one LaserPrinter is equal to 10 €.
(2 points) What is IBM’s optimal price for the LaserPrinter (supposing that a unique price has to be set, i.e., that IBM is not able to price discriminate)? Compute also IBM’s profit and quantity produced at the optimum.

Optimal price p* = 100
Optimal quantity q* = 55
Optimal profit π* =4950
We are looking for a unique price which maximizes the firm’s profit.
Optimal unique price
If - 0 ≤ p ≤ 50 → q = 100 and the maximum profit is π(p=50) = (50 – 10).100 = 4000€
50 < p ≤ 100 → q = 55 and the maximum profit is π(p=100) (100 – 10).55 = 4950€
p > 100 → q = 0 and the maximum profit for any price is zero.

We maximize the profit’sfirm by putting a price equal to 100€. With this price we can pick up all the consumer surplus of the companies but we can’t serve the individuals which have a lower willingness to pay. If we look in term of losses, if we set a price equal to 50, we lose the surplus of the companies which is equal to (100-50)*55 = 2750. On the opposite, if we set a price equal to 100, the individuals are out of themarket and we lose (50-0)*45 = 2250. So we can notice that we lose more in term of profit, when we set a price equal to 50 than when we set a price of 100. So IBM will choose to set a price equal to 100.

Suppose now that IBM wants to introduce a damaged version of its printer, the LaserPrinterE. This printer is almost identical to the original LaserPrinter, except that it includes a piece ofsoftware that reduces the printing speed from 10 pages to 5 pages per minute. Because of the poorer performance of the printer, consumers have a lower willingness to pay. In particular,
companies are willing to pay 60 € for one LaserPrinterE;
individuals are willing to pay 40 € for one LaserPrinterE.
Moreover, because of the extra piece of software, the LaserPrinterE is more expensive toproduce than the LaserPrinter: its constant marginal cost of production is equal to (10 + c) € with c > 0.

(3 points) Find the prices that IBM should set for inducing companies to buy the LaserPrinter and individuals to buy the LaserPrinterE, while maximizing IBM’s total revenues. Which type of price discrimination does IBM implement in such case?

Optimal price for LaserPrinter 80 forLaserPrinterE 40
IBM total revenues 6200 Type of price discrimination Menu pricing

Companies Individuals
LaserPrinter 100 50
LaserPrinterE 60 40
The seller cannot tell companies and individuals apart.
If the price of LaserPrinter (Pl) equals 100 and the price of LaserPrinterE (Pe) equals 40, they have a problem because companies prefer LaserPrinterE as it yields a largersurplus for them:
100 – 100 < 60 – 40
There is a self-selection constraint: Pl – Pe ≤ 100 – 60 → Pl – Pe ≤ 40
If the price of LaserPrinter (Pl) equals 100 and the price of LaserPrinterE (Pe) equals 60 (=100-40), they have a problem because individuals don’t buy anything.
There is a participation constraint Pe ≤ 40
To find the optimum, we combine the two constraints. So, the price ofPrinterLaserE equals 40 and the price of LaserPrinter equals 80 (= 40+40).

The IBM’s total revenues are: 80*55 + 40*45 = 6200.

(2 points) For which values of c does IBM find it profitable to introduce the damaged version of its printer?

Values of c c ≤ 5,5