A firm offers two differentiated products, X and Y and faces two types of consumers, types A and B. There are equal numbers of each type of consumers ¡V so, for simplicity, assume there is just one of each type. The valuations of the two types of customers of the two products are summarized in the table below. Assume (for simplicity) that the marginal cost of manufacture of X and Y is identical, constant and equal to zero:
Customers
& Products X-cheapticket Y-VIPticket
Type A 10 15
Type B 12 25
(a)If the firm offers good X for a price of 11 and good Y for a price of 26, which (if any) product will each consumer type buy (if she only wants to buy one)?
(b)If the firm offers good X for a price of 9 and good Y for a price of 24, which (if any) product will each consumer type buy (if she only wants to buy one)?
(c)Calculate the profits earned by the firm on each of the set of prices in parts (a) ¡V (b). Can you propose prices for X and Y to achieve higher profits?
(d)How does the consumer surplus for Type B on good X limit the price that can be set for good Y (to get Type B to select good Y)? Is there a positive or negative relationship between the price on good X and the price that can be charged on good Y? Why (if we want both consumers to purchase a good) does this imply that the price on good X should be set leaving Type A with zero consumer surplus?
(a) To determine which product each consumer type will buy, we need to compare their valuations with the prices of the products.
For Type A:
- Valuation for X = 10
- Valuation for Y = 15
At a price of 11 for X and 26 for Y, Type A will choose to buy X because its price is lower than Type A's valuation for Y.
For Type B:
- Valuation for X = 12
- Valuation for Y = 25
At a price of 11 for X and 26 for Y, Type B will choose to buy Y because its valuation is higher than the prices of both X and Y.
So, Type A will buy X and Type B will buy Y.
(b) Using the same process, we compare the valuations with the new prices.
For Type A:
- Valuation for X = 10
- Valuation for Y = 15
At a price of 9 for X and 24 for Y, Type A will still choose to buy X because its price is lower than Type A's valuation for Y.
For Type B:
- Valuation for X = 12
- Valuation for Y = 25
At a price of 9 for X and 24 for Y, Type B will now also choose to buy X as its valuation for X is higher than the price of X and its valuation for Y.
So, both Type A and Type B will buy X in this case.
(c) To calculate the profits earned by the firm, we need to consider the number of sales and the costs.
At prices (a):
- Type A buys X: Profit from X = price of X - marginal cost of X = 11 - 0 = 11
- Type B buys Y: Profit from Y = price of Y - marginal cost of Y = 26 - 0 = 26
Total profit = Profit from X + Profit from Y = 11 + 26 = 37
At prices (b):
- Type A buys X: Profit from X = price of X - marginal cost of X = 9 - 0 = 9
- Type B buys X: Profit from X = price of X - marginal cost of X = 9 - 0 = 9
Total profit = Profit from X + Profit from X = 9 + 9 = 18
To achieve higher profits, the firm could consider adjusting the prices of X and Y. By analyzing the customers' valuations, the firm can set prices that maximize profit.
(d) The consumer surplus for Type B on good X reduces the amount they are willing to pay for good Y. This is because part of their willingness to pay is already captured by the lower-priced good X.
For example, if Type B's valuation for X is 12 and the price of X is 9, then their consumer surplus for X is 12 - 9 = 3. If the price of Y increases, it reduces the consumer surplus they can gain by choosing Y instead of X.
There is a negative relationship between the price of good X and the price that can be charged for good Y. As the price of X decreases, the consumer surplus for Type B increases, and this reduces the price they are willing to pay for Y.
To ensure that both consumers purchase a good, the price of X should be set to leave Type A with zero consumer surplus. This is because Type A has a lower valuation for both X and Y compared to Type B. By setting a price that captures all of Type A's willingness to pay, the firm can maximize profits.