Question 4. There are two types of worker, high productivity and low productivity. A worker knows what their type is. There is one potential employer who only knows that a given worker is of high type with probability p and low type with probability 1 p. The employer is considering signing a onetime contract with the worker. From the employers perspective. signing a high type worker is worth $32,000 and signing a low type worker is worth $12,000. A high type worker's opportunity cost of entering into such a contract is $25,000 (that's what they would earn it they don't sign a contract employer}. Similarly. a low type workers opportunity cost is $3,000. Assume that the employer is acting in a competitive market [just like in the problem sets where there were two competing employers}. Given this assumption, you can conclude that the equi- librium wage offer is the expected productivityfworth of the worker according to the employer's beliefs. Note: a worker can always opt out of signing a contract at the offered wage and save them- selves the opportunity ccst [the payoff from not signing is zero). 1. For what values of p is there an equilibrium where a worker ends up being signed by the employer regardless of their type? Now suppose that the worker has the ability to take a certification test. There is a $1,000 fee and a $1 ,000 deposit to take the test. If a worker takes it and passes. they are refunded the deposit Ant 1: .-'. tr} and receive a certificate. However, if they fail, they lose the $1,000 (note that they don't get the $1 .000 fee back). The employer gets to observe whether or not the worker is certified. A high type of worker passes with certainty, but a low type worker passes with 50% probability. 2. Suppose the employer thinks the worker will take the test regardless of their type (in other words, suppose the employer thinks the worker will follow a pooling strategy). Calculate the employer's beliefs when they observe that the worker is certified. That is, give the probability that the employer thinks the worker is of high type upon seeing that the worker is certified. Similarly. what are the employers beliefs upon seeing an uncertified worker? [Hints Since the employer can observe whether a worker is certified and since the employer conjectures that the worker wiii take the test regardiess of their type, the empioyer conciudes that a worker who is not certified has taiied the test. 1 . Still assuming that the employer thinks the worker follows a pooling strategy of taking the test regardless of type, what is the expected productivity of the worker, from the employer's perspective, if the worker is certified? Similarly. what is the expected productivity if they are not certified? . Given that we have assumed a competitive labor market, meaning that the employer will offer as a wage the expected productivity, will the high ability worker take the certification test? Does their decision depend on p? . Does the low type worker take the test? Does their decision depend on p? . For what values of p would the worker take the certication test regardless of their type? Does this form a pooling equilibrium? Which worker types are signed by the employer? . Give an explanation of how and why the answers to 6. and 1 . are different. At