3. We consider the principal-agent problems. Now, assume there are n possible outcomes (x1,...,xn). The agent can take one of two actions, a or b, which influence the probability of occurrence of the outcomes. Thus, we let Tia be the probability that Xi is observed if the agent chooses action a, and Tib be the probability that Xi is observed if the agent chooses action b. s(xi) be the payment from the principal to the agent if Xi is observed. Then the expected profit of the principal if the agent chooses the action b, say, is Let si = n (xi si) Tib i=1 As for the agent, let us suppose that he is risk-averse and seeks to maximize Von Neumann-Morgenstern utility function of the payment, and the cost of his action is Ca if a is chosen, and ch if b is chosen. Usia (a) Show the IC condition which has to be satisfied if agent chooses the action b even if the choice of action is not observable for principal. (b) Suppose that if the agent does not participate, he gets utility . Show the IR condition. (c) Suppose there's no asymmetric information problem, and the prin- cipal chooses the action b for an agent. In this case, the agent chooses b. Show the first order condition of the profit maximization problem with respect to si by the principle. 3. We consider the principal-agent problems. Now, assume there are n possible outcomes (x1,...,xn). The agent can take one of two actions, a or b, which influence the probability of occurrence of the outcomes. Thus, we let Tia be the probability that Xi is observed if the agent chooses action a, and Tib be the probability that Xi is observed if the agent chooses action b. s(xi) be the payment from the principal to the agent if Xi is observed. Then the expected profit of the principal if the agent chooses the action b, say, is Let si = n (xi si) Tib i=1 As for the agent, let us suppose that he is risk-averse and seeks to maximize Von Neumann-Morgenstern utility function of the payment, and the cost of his action is Ca if a is chosen, and ch if b is chosen. Usia (a) Show the IC condition which has to be satisfied if agent chooses the action b even if the choice of action is not observable for principal. (b) Suppose that if the agent does not participate, he gets utility . Show the IR condition. (c) Suppose there's no asymmetric information problem, and the prin- cipal chooses the action b for an agent. In this case, the agent chooses b. Show the first order condition of the profit maximization problem with respect to si by the principle