A group of 13 consumers are trying to decide whether to connect to a new communications network.Consumer 1 is of type 1, consumer 2 of type 2, consumer 3 of type 3, and so on.Each consumer's willingness to pay to belong to the network is proportional to the number of consumers who belong.Where k is the number of consumers who belong, the willingness to pay of a type n consumer is 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)$47
B)$50
C)$43
D)$42
E)$45

Question

Quizplus · Accepted Answer

The willingness to pay for a type n consumer is k*n, where k is the number of consumers in the network. For 9 consumers to connect, we calculate the willingness to pay for the 9th consumer (the one with the lowest willingness to pay among the 9), which is 9*9 = 81. However, to find the highest price at which all 9 could connect and at least break even, we need to consider the consumer with the lowest willingness to pay who would still join, which is the 9th consumer. The total willingness to pay for the first 9 consumers is 1*9 + 2*9 + 3*9 + ... + 9*9 = 9(1+2+3+...+9) = 9*(9*10/2) = 405. To ensure all 9 consumers at least break even, the price must be at or below the average willingness to pay per consumer, which is 405/9 = 45. Therefore, the highest price at which all 9 consumers could connect and at least break even is $45.

A Group of 13 Consumers Are Trying to Decide Whether