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 on the lookout for a free meal. Assume that the staff of the charity cannot observe the true type of those coming for free meals. Both a homeless person and a college student receive a payoff of 10 for a free meal. But they differ in their opportunity costs of time. Standing in line for t minutes yields a cost of t/6 to a homeless person and a cost of t/4 to a college student. (a) What is the range of wait time t that will achieve separation of types?
(b) After a while, the charity finds that it can successfully identify and turn away college students half of the time. In particular,
For half of the time, college students who wait in line are turned away with no free meal, and further, incur an additional cost of 8 for embarrassment on top of their time cost
For the other half of the time, students are not identified, which means they receive free meals and only bear the cost of time. With this partial identification of college students, what is the new range of wait time t that will achieve separation of types?