Question

A local charity has been given a grant to serve free meals to the homeless in its community, but it is worried that its program might be exploited by nearby college students who are always interested in a free meal. both a homeless person and a college student receive a payoff of 10 for a free meal. the cost of standing in line for the meal is m2 400 for a homeless person and m2 250 for a college student, where m is the number of minutes spent waiting in line. assume that the staff of the charity cannot observe the true type of people coming for free meals.

a. what is the minimum wait time, m, that will achieve separation of types?

Answer 1

To prevent college students from exploiting a free meal program, the charity should set the minimum wait time high enough so that the net benefit for waiting becomes negative for students but remains positive for the homeless. The ideal minimum wait time, m, is calculated to be 0.2 minutes. However, this time would be rounded up to 1 minute in practice to achieve separation.

Step-by-step explanation:

The question is based on designing a mechanism that differentiates between homeless individuals and college students when both groups could receive the same benefit from a free meal. This mechanism relies on the concept of opportunity costs and assumes that time has different values for these two groups. We are asked to find the minimum wait time, m, that will separate the homeless (who value the meal more than their time) from the college students (who value their time more and might not be willing to wait as long for a free meal).

To achieve type separation, we must set the waiting cost high enough to deter college students but not so high as to deter homeless individuals. The cost of waiting in line is represented by m^2 times the respective cost factors, 400 for a homeless person and 250 for a college student. The benefit of the meal is 10 for both groups. Separation occurs when the net payoff of waiting for a meal is greater for the homeless than for college students. The net benefit for homeless persons is
`10 - m^2` × 400, and for college students, it is
`10 - m^2` × 250.

We find the minimum m by solving for the situation where the net benefit equals zero for college students, which sets the threshold above which they will not participate:

`10 - 250m^2 = 0`

`250m^2 = 10`

`m^2 = 10/250`

`m = √(0.04)`m
m = 0.2

However, since the waiting time cannot realistically be 0.2 minutes, the charity might round up to the next full minute to ensure proper separation.

Answer 2

Homeless:
10 – t^2 /320 > 0
10 > t^2 /320
3200 > t^2
t < 40√2 = 56.6
College students:
10 – t^2 /160 > 0
10 > t^2 /160
1600 > t^2
t < 40
Wait time would have to be at least 40 minutes to prevent college students from waiting.