Eleven consumers are trying to decide whether to connect to a new communications network.Consumer 1 is of type 1, consumer 2 is of type 2, consumer 3 is of type 3, and so on.Where k is the number of consumers connected to the network (including oneself), a consumer of type n has a willingness to pay to belong to this network equal to k times n.What is the highest price at which 9 consumers could all connect to the network and either make a profit or at least break even?
A)$30
B)$9
C)$18
D)$24
E)$27

Question

Quizplus · Accepted Answer

For 9 consumers to connect and at least break even, the price must be equal to or less than the willingness to pay of the consumer with the lowest willingness to pay among the 9. This would be consumer 9, as the willingness to pay increases with the consumer type number. The willingness to pay for consumer 9 when 9 consumers are connected is 9 times 9, which equals 81. However, the question asks for the highest price at which 9 consumers could connect, not the willingness to pay of the 9th consumer. To find this price, we need to consider the consumer with the lowest willingness to pay who would still choose to connect, which would actually be consumer 1 when 9 consumers are connected. Consumer 1's willingness to pay would be 1 times 9 = 9. However, my explanation mistakenly focused on the wrong aspect of the problem, leading to an incorrect conclusion. The correct approach is to find the consumer with the highest type among those who are willing to connect and use their willingness to pay to determine the maximum price. For 9 consumers, the 9th consumer's willingness to pay sets the maximum price they are all willing to pay or break even on, which is 9 times 9 = 81. However, to find the highest price at which exactly 9 consumers could connect, we need to consider the marginal consumer, who is the 9th consumer in this case. The correct calculation involves finding the price at which the consumer with the lowest willingness to pay among the 9 (which would actually be the 9th consumer, not the 1st, in terms of willingness to pay, but the calculation involves ensuring all 9 are willing to pay the price) is willing to pay. The error in the initial explanation was in the interpretation of how to apply the willingness to pay to determine the maximum price. The correct answer involves calculating the willingness to pay of the marginal consumer willing to join the network at a given price, ensuring that this price allows for 9 consumers to connect without any of them incurring a loss. This involves a more nuanced understanding of the economic principles at play, including the concept of marginal utility and the role of the marginal consumer in determining market outcomes.

Eleven Consumers Are Trying to Decide Whether to Connect to a New