The worker has private information about her level of ability. With probability p she is a high-ability type (H) and with probability 1- p she is a low-ability type (L). After observing her own type, the worker decides whether to obtain a costly education (E) or not (N); think of E as getting a degree. The firm observes the worker's education but the firm does not observe the worker's quality type. The firm then decides whether to employ the worker in an important managerial job (M) or in a much less important job (C). The payoffs are represented in the above extensive form. Is there a separating perfect Bayesian equilibrium in which the high- ability type worker obtains an education and the low-ability type does not? If yes, fully describe such an equilibrium. If no, prove there is not such an equilibrium.
