Please help with these two questions! These were taken from Money and Banking problem set 1(Columbia University, Professor Tri V Dang), if that information helps. Explanation/Answers/answers to similar questions will all be greatly appreciated. Thank you!
1. 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 at $1 at T=1.
An agent has $1 and is risk avers and can be of two types. With probability 0.4 an agent is a type-1 consumer and with probability 0.6 an agent is a type-2 consumer.
If an agent is a type1-consumer, he only values consumption at T=1 and his utility function is
u1= 2- (1/c1)
where c 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
where c1 and c2 are the amounts consumed at T=1 and T=2, respectively.
a) what is the expected utility of the agent?
Now consider a bank that invests in these projects. There are N=1000 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 (d1,d2) where d1 is the amount an agent can withdraw at T=1 and d2 is the amount he can withdraw at T=2.
b) suppose d=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?
c) Suppose d2=3.4 What is the amount d1 that the bank can offer an agent who withdraws at T=1? What is the expected utility of an agent?
Suppose the bank offers (d1,d2) = (1.4 , 2.93). An agent expects that M=640 other agents will withdraw at T=1.
d) What is the best response of the type-2 consumer, i.e. does he have an incentive to run to the bank and withdraw at T=1?
e) What is the maximum number of withdrawals at T=1 such that a type-2 consumer has no incentive to withdraw at T=2?
2. Now consider policy responses in the above economy where a bank offers a demand deposit of (d1,d2) = (1.225, 3.4)
a) Suppose the government wants to suspend convertibility of demand deposit into cash if too many agents are withdrawing. When should a suspension kick in (i.e. how many agents are allowed to withdraw at T=1) in order to avoid that a type-2 consumer withdraws at T=1? Is this number unique?
Suppose the government designs a deposit insurance fund and insures an amount of I.
b) In order to avoid a bank run of type-2 consumers, the minimum insurance is I =3.4. Please explain if this is correct.
c) suppose I=1.225. How many type-2 consumers will withdraw at T=2?