You are the Dean of the Faculty at St. Anford University. You hire assistant professors for a probationary period of 7 years, after which they come up for tenure and are either promoted and gain a job for life or turned down, in which case they must find another job elsewhere. Assistant professors come in two types, Good and Brilliant. Each individual assistant professor knows whether he or she is Good or Brilliant, but you cannot directly distinguish between them. You only want to give tenure to Brilliant assistant professors. The payoff from a tenured career at St. Anford is $2 million. Anyone denied tenure will get a faculty position at Boondocks College, with a payoff of $0.5 million. Your faculty can do research and publish the findings. But each publication requires costly effort and time. The monetary equivalent of this cost is $30,000 per publication for a Brilliant assistant professor and $60,000 per publication for a Good one. You must set a minimum number, N, of publications that an assistant professor must produce in order to achieve tenure.(a) Without doing any math, describe, as completely as you can, what would happen in a separating equilibrium to this game.
(b) There are two potential types of pooling outcomes to this game. Without doing any math, describe what they would look like, as completely as you can.
(c) Now please go ahead and do some math. What is the set of possible N that will accomplish your goal of screening the Brilliant professors out from the merely Good ones?