Money and Banking
Spring 2024
Exercise Set 2
Due Thursday, 2/29
It is allowed to work together on problem sets. However, each student must submit an individual solution sheet on Courseworks. In case of collaboration, the names of students you worked with should be stated on top of the first page of the solution sheet.
This problem set consists of four questions. You can obtain a maximum of 60 points.
Question 1 [20 points]
Consider the following bond market. It is common knowledge that the bond pays either $40 or $100 at t=1. Both states are equally likely.
Agent S owns the bond. Since he needs cash at t=0 he is willing to sell the bond at a discount of Δ = $4. It means if the true value of the bond is $40 agent S is willing to sell it for at least $36. If the true value of the bond is $100 agent S is willing to sell it for at least $96.
Agent B is interested in buying the bond. If the true value of the bond is $40 (or $100) agent B is willing to buy it for at most $40 (or $100).
The seller (agent S) proposes a price to sell the bond. Both agents are risk neutral and maximize expected utility.
a) What price does the seller offer in equilibrium? Is there trade? [3 Points]
Now suppose agent B is a sophisticated buyer. After seeing the price offer of the seller, the buyer can try to learn something about the bond. If agent B spends $8 on information acquisition, he can find out the true value of the bond.
b) Suppose the seller offers the expected payoff of the bond and thus the price p=$70. What is the best response of the buyer? Should the seller offer this price? [4 Points]
c) Suppose the seller offers a price p=$66. What is the best response of the buyer? Should the seller offer this price? [4 Points]
d) Suppose the seller offers a price p=$96. What is the best response of the buyer? [3 Points]
e) Is there any price at which agent S is willing to sell and agent B is willing to buy? Please provide a complete proof. [6 Points]
Question 2 [16 points]
Now suppose there is a rating agency in the above bond market. The rating agency evaluates the company that issued the bond. The buyer can acquire information at cost of $8.
Suppose the rating agency announces an AA rating, i.e. there is a 10% probability that the payoff of the bond is $40.
a) Is there trade in equilibrium? If so, what is the price the seller is offering? [8 Points]
Suppose the rating agency announces a BB rating, i.e. there is a 90% probability that the payoff of the bond is $40.
b) Is there trade in equilibrium? If so, what is the price the seller is offering? [4 Points]
c) How does the rating agency create liquidity in the bond market? Please provide an intuitive explanation. [4 Points]
Question 3 [12 points]
Consider the following economy with three dates (t=0, 1, 2). A firm needs to raise I=$100 to finance a project at t=0. At t=2, the project generates $400 if it is a success or pays off nothing if it is a failure. Both cases are equal likely. At t=1, the firm learns about the outcome of the project.
There is an early consumer who has w=$100 at t=0 and the utility function
uE = cE0 +1.2. min[cE1,100] + max[cE1 −100,0]+ cE2 .
There is a late consumer who has W=$320 at t=1 and the utility function uL = cL0 + cL1 + cL2 .
Suppose the firm issues equity at t=0 to raise $100. As a listed company, at t=1 the firm reports whether the project is a failure or success. If the early consumer buys equity at t=0, he sells his equity holding to the late consumer at t=1 after the announcement of the firm.
a) What equity contract does the firm offer the early consumer so as to get $100? What is the fraction of equity the firm is selling? [6 Points]
b) What is the price of the equity at t=1? What is the expected utility of the early consumer at t=0? [4 Points]
c) What is the expected profit of the firm at t=0? [2 Points]
Question 4 [12 points]
Now suppose there is bank in the above economy. The bank provides loans to firms and keeps information secret. Consumers do not observe whether the project is a failure or success. At t=2 the bank is liquidated and consumers who have deposits with the bank get back what the bank owns them at t=2.
The bank offers the early consumer the following demand deposit contract. If the early consumer deposits $100 at t=0, he can withdraw $100 at t=1.
The bank offers the late consumer the following contract. If he deposits $100 at t=1, he gets back the loan that the firm repays to the bank since the bank is liquidated at t=2.
a) What loan contract does the firm offer the bank to get $100? Does the early consumer
deposit $100 at t=0 and does the late consumer deposit $100 at t=1? [10 Points]
b) What is the expected profit of the firm? [2 Points]