:Answer all questions.

QUESTION 2 Let e denote the level of education. There are three types of potential workers: those (type _) with productivity Of , those (type M) with productivity , and those (type #) with productivity ey , with OH > 0w > 0, > 0. For each type i c (L, M, H; the fraction of type i in the population is } . Each potential worker knows her own type, while the potential employer cannot tell the type of any potential worker, although he knows the distribution of types in the population. The employer observes the education level of each potential worker (but not her type) and offers a wage which depends on the applicant's level of education. For every type ic {L, M, /} the cost of acquiring e units of education is . Each worker's utility is given by the difference between the wage she is paid and the cost of education. (a) [Note: for this part do not assume that each worker must be paid a wage equal to her productivity.] Is there an incentive-compatible situation where (1) the employer offers two wages, depending on the education level: wage w to those whose education level is e and wage wy # w to those whose education level is ey # e and refuses to hire anybody with education ec (e , ey ), (2) both types 0, and ey choose education level e', while type , choose education ey ? [Note that you should make no assumptions about whether ey, ey and similarly for wy and w .] If there is such an incentive-compatible situation, please describe it in detail. If your claim is that it does not exist, please prove it. For parts (b) and (c) assume that the employer pays each worker a wage equal to the worker's expected productivity (as computed by the employer, who is risk neutral). (b) Define and describe in detail a pooling equilibrium, that is, a signaling equilibrium where all three types make the same choice of education level, call it a. [Assume that the employer believes that anybody who shows up with education level ex e must be of type L.] (c) Find all the pooling equilibria when 0, = 1, 0M =2, ey =6. Now let us change the situation as follows. There are only two types of potential workers: those with productivity &, and those with productivity Oy, with Oy > 0, >0. The fraction of type Or in the population is equal to the fraction of type Oy . Assume that the cost of education is the same for both types and is given by c(e) = e. Suppose that the utility of worker of type de (0 , , } who is paid wage w and chooses education level e is U(w,e, 0) =0w-e. Assume also that ee [a,b] with 0