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 Hy = 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 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 d2-3.6. What is the amount dl that the bank can offer an agent who withdraws at T=1? What is the expected utility of an agent?
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 Hy = 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 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 d2-3.6. What is the amount dl that the bank can offer an agent who withdraws at T=1? What is the expected utility of an agent?
Managerial Economics: A Problem Solving Approach
5th Edition
ISBN:9781337106665
Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Chapter17: Making Decisions With Uncertainty
Section: Chapter Questions
Problem 4MC
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Given:
time periods - 0,1,2
If invests $1 in T=0, it becomes $4 in T=2 but in T=1 liquidated at $1
For type 1 probability, p1=0.2
For type 2 probability, p2=0.8
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