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 4 consumers could all connect to the network and either make a profit or at least break even?
A)$24
B)$64
C)$40
D)$28
E)$32

Question

Quizplus · Accepted Answer

For 4 consumers to connect and at least break even, we need to find the consumer with the lowest willingness to pay among the 4, because this will determine the highest price at which all 4 can afford to connect. The 4th consumer (type 4) will have the lowest willingness to pay among the 4, with a willingness to pay of 4 times 4, or $16. However, to find the highest price at which 4 consumers could connect, we should consider the consumer with the highest willingness to pay among the first 4, which is consumer 4. Their willingness to pay is 4 * 4 = $16, but this calculation is incorrect for determining the highest price at which 4 consumers could connect. The correct approach is to consider the total willingness to pay for the 4 consumers with the highest willingness to pay, which are consumers 8, 9, 10, and 11, since their willingness to pay increases with their type number. Consumer 8's willingness to pay is 8 * 4 = $32, consumer 9's is 9 * 4 = $36, consumer 10's is 10 * 4 = $40, and consumer 11's is 11 * 4 = $44. The correct answer should be based on the willingness to pay of the consumer who is just willing to pay enough to join, which in the case of 4 consumers connecting, would actually be the 4th highest willing consumer, not necessarily the 4th in sequence. However, the premise of finding the "highest price at which 4 consumers could all connect" based on the willingness to pay of the 4th consumer in sequence was flawed. The correct approach involves calculating the willingness to pay based on the scenario given, which was not accurately represented in the initial explanation. The highest price at which 4 consumers could connect would actually depend on the total network effect and the individual willingness to pay, which is not correctly captured by simply multiplying the type number by 4. The question seems to imply looking for a price point where the network is attractive enough for 4 specific consumers to join, considering their individual willingness to pay as a function of the network size and their type. Given the error in the initial reasoning, the correct answer should involve a more nuanced calculation considering the network effects and the specific types of consumers who are connecting, but the provided options do not directly correlate with this corrected understanding. The mistake was in the interpretation of how to calculate the highest price for 4 consumers connecting, which should consider the dynamics of network effects and the incremental willingness to pay as more consumers join, not just a simple multiplication of the consumer type by the number of connected consumers.

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