Question 1 (22 points) Consider an economy with three dates (T=0, 1, 2) and the following investment opportunity. If an agent invests $1 in a project at T=0, the project yields $4 at T=2. The project can be liquidated at T=1 but early liquidation yields $1 at T=1. An agent has $1 and is risk avers and can be of two types. With probability 0.2 an agent is a type-l consumer and with probability 0.8 an agent is a type-2 consumer. If an agent is a typel-consumer, he only values consumption at T=1 and his utility function is 1 u = 2- where cy is the amount consumed at T=1. If an agent is a type-2 consumer, he values consumption at both T=1 and T=2 according to the utility function 1 u, = 2- a+c where ci and c2 are the amounts consumed at T=1 and T=2, respectively. a) What is the expected utility of the agent? [3 Points) 1 Now consider a bank that invests in these projects. There are N=1,000 agents. All agents are identical ex ante in the above sense. Suppose they all deposit $1 each with the bank. The bank offers the following demand deposit contract (d, dz) where di is the amount and agent can withdraw at T=1 and d2 is the amount he can withdraw at T=2. b) Suppose dy=1.2. What is the amount d2 that the bank can offer an agent who withdraws at T=2? What is the expected utility of an agent? (4 Points] c) Suppose d2=3.6. What is the amount d that the bank can offer an agent who withdraws at T=1? What is the expected utility of an agent? [4 Points) Suppose the bank offers (d,d2)= (1.4, 3.6). An agent expects that M=630 other agents will withdraw at T=1. d) What is the best response of the type-2 consumer, i.e. does he has an incentive to run to the bank and withdraw at T=1? [3 Points] e) What is the maximum number of withdrawals at T=1 such that a type-2 consumer has no incentive to withdraw at T=1. [8 Points)