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?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
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
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
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education