There are three types of workers in the world, I, II, and III; each type has probability 1/3. The output of a type I worker is 100, the output of a type II is 80, and the output of a type III is 40. Employers observe only the worker's level of education e, not her type t, and pay a wage w equal to the expected output given the beliefs about the worker's type (I, II or III) induced by her education. The worker's utility is the wage she gets minus the cost of the education. The cost of education for a type I is 5 per year, the cost to a type II is 10 per year, and the cost to a type III is 20 per year. The worker can choose her years of education freely from [0, ) that is, she's not constrained to integer amounts. (a) Find a pooling PBE in pure strategies. Specify beliefs. (b) Find the least years of education for each type for which there is a pure fully separating PBE (one in which each type gets a different number of years of education). If no fully separating equilibria exist, show this