Imagine a country where each person is either upper class, middle class or lower class, and there is an equal proportion of each type. Regardless of one’s true class, which is known only to the individual (private information), being thought to be upper class (UC) is worth 120; being thought to be middle class (MC) is worth 90; and being thought to be lower class (LC) is worth 0. Members of each class can signal their true class at no cost. Hence, the lowest payoff to an upper class member is 120, to a middle class member is 90, and to a lower class member is 0. Each person in the country has to take one of three tests: an upper class test (UCT), a middle class test (MCT), and a lower class test (LCT). If one passes the upper class test, they are thought of being upper class; if one passes the middle class test, they are thought of being middle class etc. Each person can choose which test to take. The cost of taking the test of one’s true class is 0, and the probability of passing a test of one’s true class is 1. The cost of taking a UCT to a MC person is X, and there is ½ chance of success. If an MC person fails a UCT, he can take the MCT at no cost and pass with certainty, signaling their true class. The cost of taking a UCT to a LC person is 30, and there is 1/10 chance of success. The cost of taking a MCT to a LC person is 20, and the probability of success is p. If a LC person takes either test and fails, then he can take the LCT and pass with certainty.
What should be the values of X and p to get a separating equilibrium where each person takes the test of her/his true class? [Hint: Since the tests are required, participation constraint is not relevant in this case. You only need to check for incentive compatibility constraint for each class—that is, does anyone have an incentive to take a test other than that of their true class?]