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-Cheaptickets Y-VIPtickets
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 at the given prices, we compare their valuations for each product. For Type A, the valuation for X is 10 and for Y is 15. Since the price of X is 11, which is higher than Type A's valuation, Type A will not buy X. However, the price of Y is 26, which is higher than Type A's valuation as well, so Type A will not buy Y either. Type A will not buy any product.
For Type B, the valuation for X is 12 and for Y is 25. The price of X is 11, which is lower than Type B's valuation, so Type B will buy X. However, the price of Y is 26, which is higher than Type B's valuation, so Type B will not buy Y.
Therefore, at these prices, Type A will not buy any product and Type B will buy X.
(b) Using the same method, we compare the valuations for each product with the given prices.
For Type A, the price of X is 9, which is lower than their valuation of 10. Thus, Type A will buy X. The price of Y is 24, which is lower than their valuation of 15. Type A will still prefer X over Y, so Type A will buy X.
For Type B, the price of X is 9, which is lower than their valuation of 12. Type B will buy X. The price of Y is 24, which is lower than their valuation of 25. Thus, Type B will buy Y.
Therefore, at these prices, both Type A and Type B will buy X, and Type B will also buy Y.
(c) To calculate the profits earned by the firm, we need to consider the costs as well. Since the marginal cost of manufacture is assumed to be zero, the firm's profit is equal to the revenue earned from selling the products.
At a price of 11 for X and 26 for Y (from part a), the firm sells one unit of X to Type B, earning a revenue of 11. The profit earned by the firm is 11.
At a price of 9 for X and 24 for Y (from part b), the firm sells one unit of X to Type A and one unit of Y to Type B. The revenue earned from X is 9, and the revenue earned from Y is 24. The total profit earned by the firm is 9 + 24 = 33.
To achieve higher profits, the firm can adjust the prices of X and Y. One possible strategy is to increase the price of X to capture more of Type B's valuation, while keeping the price of Y lower to attract more buyers. The firm can experiment with different price combinations to find the optimal pricing strategy that maximizes its profits.
(d) The consumer surplus for Type B on good X limits the price that can be set for good Y. Consumer surplus is the difference between a consumer's valuation and the price they actually pay. In this case, Type B's valuation for X is 12, but they only pay a price of 9. Therefore, their consumer surplus is 12 - 9 = 3.
If the price of Y is set too high, Type B will prefer to buy X instead, as their valuation for X is higher. This is because they are getting a positive consumer surplus on X. To get Type B to select Y instead, the price of Y must be set lower than or equal to the consumer surplus they get from X. In this case, it should be less than or equal to 3.
There is a negative relationship between the price of X and the price that can be charged on Y. As the price of X increases, the consumer surplus for Type B on X decreases. This reduces the value they place on X relative to Y, allowing for a higher price to be set on Y in order to get Type B to select it.
To ensure that Type A buys X, the price on X should be set in a way that eliminates Type A's consumer surplus. This means the price of X should be set equal to or higher than their valuation, in this case higher than 10, so that they don't have any surplus left after the purchase. By setting the price to eliminate consumer surplus for Type A, the firm can maximize its profits.
(a) If the firm offers good X for a price of 11 and good Y for a price of 26, we can compare the valuations of each consumer type for the two products:
- Type A: Valuation for X = 10, Valuation for Y = 15. Since the price of X is lower than their valuation, Type A will buy X.
- Type B: Valuation for X = 12, Valuation for Y = 25. Since the price of X is lower than their valuation, Type B will buy X.
(b) If the firm offers good X for a price of 9 and good Y for a price of 24, the calculations remain the same:
- Type A: Valuation for X = 10, Valuation for Y = 15. Since the price of X is equal to their valuation, Type A can choose either X or Y. Let's assume they choose X.
- Type B: Valuation for X = 12, Valuation for Y = 25. Since the price of X is lower than their valuation, Type B will buy X.
(c) To calculate the profits earned by the firm on each set of prices, we need to determine the quantity demanded for each product by each consumer type:
- Set of prices in (a):
- Quantity of X demanded by Type A = 1
- Quantity of X demanded by Type B = 1
- Quantity of Y demanded by Type A = 0
- Quantity of Y demanded by Type B = 0
Total profit = (Price of X * Quantity of X) + (Price of Y * Quantity of Y)
= (11 * 1) + (26 * 0)
= 11
- Set of prices in (b):
- Quantity of X demanded by Type A = 1
- Quantity of X demanded by Type B = 1
- Quantity of Y demanded by Type A = 0
- Quantity of Y demanded by Type B = 0
Total profit = (Price of X * Quantity of X) + (Price of Y * Quantity of Y)
= (9 * 1) + (24 * 0)
= 9
To achieve higher profits, the firm could potentially adjust the prices of X and Y. They could increase the price of X to capture additional consumer surplus from Type A and increase the price of Y to capture additional consumer surplus from Type B.
(d) The consumer surplus for Type B on good X limits the price that can be set for good Y. Type B is willing to pay up to 12 for good X, so if the price of X exceeds 12, Type B will choose Y instead. This implies a negative relationship between the price of X and the price that can be charged for Y - as the price of X increases, the price that can be charged for Y decreases in order to make it more attractive to Type B.
To ensure that Type A has zero consumer surplus, the price of X should be set equal to or slightly higher than their valuation. This maximizes the profit from Type A while making it more likely for Type B to choose Y, which has a higher valuation for them.