In Chapter 22, we briefly discussed the signaling role of education—i.e. the fact that part of the reason many people get more education is not to learn more but rather to signal high productivity to potential employers (in hopes of getting a higher wage offer). We return to this in part B of this exercise in the context of incomplete information game (built on concepts from Section B of the chapter) but first consider the lack of a role or signaling in a complete information game. Throughout, suppose that there are two types of workers—type 1 workers with low productivity and type 2 workers with high productivity, with a fraction δ of all workers being type 2 and a fraction (1−δ) being type 1. Both types can earn education by expending effort, but it costs type 1 workers e to get education level e > 0 while it costs type 2 workers only e/2. An employer gets profit (2−w) if she hires a type 2worker at wage w and (1−w) if she hires a type 1worker at wage w. (Employers get zero profit if they do not hire a worker). We then assume that the worker decides in stage 1 how much education to get; then, in stage 2, he approaches two competing employers who decide simultaneously how much of a wage w to offer; and finally, in stage 3, he decides which wage offer to accept.
A: Suppose first that worker productivity is directly observable by employers; i.e. firms can tell who is a type 1 and who is a type 2 worker by just looking at them.
(a) Solving this game backwards, what strategy will the worker employ in stage 3 when choosing between wage offers?
(b) Given that firms know what will happen in stage 3, what wage will they offer to each of the two types in the simultaneous move game of stage 2 (assuming that they best respond to one another)?
(c) Note that we have assumed that worker productivity is not influenced by the level of education e chosen by a worker in stage 1. Is there any way that the level of e can then have any impact on the wage offers that a worker gets in equilibrium?
(d) Would the wages offered by the two employers be any different if the employers moved in sequence— with employer 2 being able to observe the wage offer from employer 1 before the worker chooses an offer?
(e) What level of e will the two worker types then get in any sub game perfect equilibrium?
(f) True or False: If education does not contribute to worker productivity and firms can directly observe the productivity level of job applicants, workers will not expend effort to get education, at least not for the purpose of getting good wage offer.
B: Now suppose that employers cannot tell the productivity level of workers directly—all they know is the fraction δ of workers that have high productivity and the education level e of job applicants.
(a) Will workers behave any differently in stage 3 than they did in part A of the exercise?
(b) Suppose that there is a separating equilibrium in which type 2 workers get education  that differs from the education level type 1 workers get—and thus firms can identify the productivity level of job applicants by observing their education level. What level of education must type 1 workers be getting in such a separating equilibrium?
(c) What wages will the competing firms offer to the two types of workers? State their complete strategies and the beliefs that support these.
(d) Given your answers so far, what values could take in this separating equilibrium? Assuming falls in this range specify the separating perfect Bayesian Nash equilibrium — including the strategies used by workers and employers as well as the full beliefs necessary to support the equilibrium.
(e) Next, suppose instead that the equilibrium is a pooling equilibrium—i.e. an equilibrium in which all workers get the same level of education and firms therefore cannot infer anything about the productivity of a job applicant. Will the strategy in stage 3 be any different than it has been?
(f) Assuming that every job applicant is type 2 with probability δ and type 1 with probability (1−δ), what wage offers will firms make in stage 2?
(g)What levels of education could in fact occur in such a perfect Bayesian pooling equilibrium? Assuming e falls in this range specify the pooling perfect Bayesian Nash equilibrium — including the strategies used by workers and employers as well as the full beliefs necessary to support the equilibrium.
(h) Could there be an education level that high productivity workers get in a separating equilibrium and that all workers get in a pooling equilibrium?
(i) What happens to the pooling wage relative to the highest possible wage in a separating equilibrium as δ approaches 1? Does this make sense?