05 Jun Question Part A. Credit Markets In part A of this problem set, we will t
Question
Part A. Credit Markets
In part A of this problem set, we will think systematically about how the equilibrium interest rate is determined in the credit market. In particular, we will explore how the interest rate, the size of the economic surplus and the distribution of the surplus are affected by asymmetric information in the context of limited liability loans.
In the village of Quahog, people can choose to either farm or work in the factory. If a person works in the factory, she makes $100 with certainty. People in Quahog are either born a SAFE farmer or a RISKY farmer. Half of the people are SAFE farmers and half are RISKY farmers. If either type of person chooses to farm, she will need $300 investment in order to farm. Thus, the opportunity cost of farming is the $100 one could have had earned if she instead worked in the factory. The only difference between SAFE and RISKY farmers is as follows:
• SAFE farmers have a good harvest all the time, they earn revenues of $700 with 100% probability.
• RISKY farmers have a good harvest with 60% probability, in which they earn revenues of $1100 with 60% probability; and they have a bad harvest with 40% probability, in which they earn revenues of $0.
1 PERFECT INFORMATION
Brian is a moneylender who lives in Quahog. His opportunity cost of money is 0.20 (i.e., he would earn 20% if he invested the money in a business instead of lending it to farmers). Brian offers limited liability loans, so a farmer does not have to repay the loan if she has a bad harvest. Since Brian lives in Quahog, he has perfect information about farmers. Specifically, he knows who is a SAFE farmer and who is a RISKY farmer.
1. Let Ysbe the farming income of a SAFE farmer and YRbe the farming income of a RISKY farmer. The farming income for any famer Yis equal to revenues minus all costs (including opportunity cost). Derive expressions for the expected value of farming income E(YS)and E(YR) as functions of the interest rate i. Your expressions should take the form of E(Y) = A+ Bi, where you have to find Aand B.
(a) What are the functions E(YS) and E(YR)
(b) Graph your functions E(YS) and E(YR); place ion the horizontal-axis; title the graph “Figure 1”
2. Let ?be the profit of Brian the moneylender. Derive expressions for the expected value of profit for a loan to a safe farmer E(?S)and a loan to a risky farmer E(?R) as functions of the interest rate i. Your expressions should take the same form of E(?) = A+ Bi, where you have to find Aand B.
(a) What are the functions E(?S) and E(?R)
(b) Graph your functions E(?S) and E(?R) in the same “Figure 1”
3. Using your questions and graph, answer the following questions:
(a) What is the highest interest rate a SAFE farmer would be willing to payfor a loan from Brian?
(b) What is the highest interest rate a RISKY farmer would be willing to payfor a loan from Brian?
(c) What is the lowest interest rate Brian would be willing to chargeon a loan to a SAFE farmer?
(d) What is the lowest interest rate Brian would be willing to chargeon a loan to a RISKY farmer?
4. First, assume that the loan market is perfectly competitive. There are many other lenders who would charge a lower interest than Brian, if Brian is making a profit.
(a) What is the equilibirum interest rate Brian would charge a SAFE farmer?
(b) What is the equilibrium interest rate Brian would charge a RISKY farmer?
(c) What is Brian’s total expected profit E(?) = E(?S)+ E(?R)?
(d) What is total expected income across all types of famers E(Y) = E(YS)+ E(YR)?
5. Now, assume that all the other lenders left Quahog, and now Brian is a monopolistmoneylender.
(a) What is the equilibirum interest rate Brian would charge a SAFE farmer?
(b) What is the equilibrium interest rate Brian would charge a RISKY farmer?
(c) What is Brian’s total expected profit E(?) = E(?S)+ E(?R)?
(d) What is total expected income across all types of famers E(Y) = E(YS)+ E(YR)?
2 ASYMMETRIC INFORMATION
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