Game theory Economics

A firm is looking to hire an employee for a vacant position. Applicants for this position can be of two types: they can be a slacker or they can be a hard-worker. Firm does not observe the applicants type ( a worker's type is their private information). However, applicants may choose to obtain a college degree and the firm observes whether the applicant has a college degree or not. It is mentally costly for the applicant to obtain a college degree (there is clearly a monetary cost of going to college, but we assume those costs are 0.) The cost of attending college depends on the applicant's type: it's more costly for slackers to attend college compared to hard-workers. Applicants' costs are summarized in the following table: Does not have a college degree Has a college degree Stacker 10 135 Hard-worker 30 100 The applicant receives a utility of 150 regardless of their type if the firm offers them the job or 30 if it does not. The applicant's payoff is the utility from the job minus the cost. For example, if a slacker obtains a 2 college degree and they are offered a job they receive a payoff of 150 - 135 = 15. Before the firm meets an applicant, it believes the probability the applicant is a slacker is 0.5 and the probability the applicant is a hard-worker is 0.5. The firm will receive a payoff of 100 if it hires a hard worker, 0 if it hires a slacker, and 55 if it hires no one. (a) (12 points) Does this game have a pooling equilibrium? Explain your answer carefully. You have to specify the strategies of the firm, the strategies of the applicants and the beliefs of the firm when defending your argument. b) (13 points) Does this game have a separating equilibrium? Explain your answer carefully. You have to specify the strategies of the firm, the strategies of the applicants and the beliefs of the firm when defending your argument