Question 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 $1 at T-1. An agent has $1 and is risk averse and can be of two types. With probability 0.2 an agent is a type-1 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 Hi = 2 - 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 H2 = 2 –- C1 +C2 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=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,, d2) where d, is the amount and agent can withdraw at T=1 and de is the amount be can withdraw at T-2

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Thank you for your answering part a to c already. Please answer part d & part e ONLY.
Question 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 $1 at T-1.
An agent has $1 and is risk averse and can be of two types. With probability 0.2 an agent is a
type-1 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
Hi = 2 --
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
Hz = 2 –
C1 +C2
where C, 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=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 (d1, d2) where d, is the amount and agent can
withdraw at T=1 and d, is the amount he can withdraw at T-2.
b)
Suppose d1-1.2. What is the amount dz that the bank can offer an agent who withdraws at
T=2? What is the expected utility of an agent?
c)
Suppose d2-3.6. 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, dz)= (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?
What is the maximum number of withdrawals at T=1 such that a type-2 consumer has no
e)
incentive to withdraw at T=1.
Transcribed Image Text:Question 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 $1 at T-1. An agent has $1 and is risk averse and can be of two types. With probability 0.2 an agent is a type-1 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 Hi = 2 -- 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 Hz = 2 – C1 +C2 where C, 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=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 (d1, d2) where d, is the amount and agent can withdraw at T=1 and d, is the amount he can withdraw at T-2. b) Suppose d1-1.2. What is the amount dz that the bank can offer an agent who withdraws at T=2? What is the expected utility of an agent? c) Suppose d2-3.6. 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, dz)= (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? What is the maximum number of withdrawals at T=1 such that a type-2 consumer has no e) incentive to withdraw at T=1.
Expert Solution
Step 1

Utility function of type 1 :  U = 2 - 1/c1

Utility function of type  U = 2 - 1/ (c1+ c2 )

d1 & d2 are the return values that the bank provides in the period t = 1 & 2  respectively .

d1 = 1.4 

d2 = 3.6 

Now ,

For type 2 customer to withdraw at T= 1 time period :

Utility of type 2 customer liquidating from bank  in T = 1  > Utility of type 2 customer liquidating in T = 2 

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