The asymmetric information problem

In Purdue-ville, people can choose either to farm or work in the factory. If a person works in the factory, she makes $100 with certainty. People in Purdue-ville 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.
Daniel is rich moneylender who lives in Purdue-ville. His opportunity cost of money is 0.20 (i.e., he would earn 20 percent if he invested the money in a business instead of lending it to farmers). Daniel offers limited liability loans, so a farmer does not have to repay the loan if she has a bad harvest. Since Daniel lives in Purdue-ville, he has perfect information about farmers. Specifically, he knows who is a SAFE farmer and who is a RISKY farmer.
1. Let YS be the farming income of a SAFE farmer and YR be the farming income of a RISKY farmer. The farming income for any famer Y is 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 A and B.
a) What are the functions E(YS) and E(YR).
2. Let be the profit of Daniel, our moneylender. Derive expressions for the expected value of profit for a loan to a safe farmer E() and a loan to a risky farmer E() as functions of the interest rate i. Your expressions should take the same form of E() = A + Bi, where you have to find A and B .
a) What are the functions E() and E().
3. Using your answers to questions above, answer the following questions:
a) What is the highest interest rate a SAFE farmer would be willing to pay for a loan from Daniel?
b) What is the highest interest rate a RISKY farmer would be willing to pay for a loan from Daniel?
c) What is the lowest interest rate Daniel would be willing to charge on a loan to a SAFE farmer?
d) What is the lowest interest rate Daniel would be willing to charge on 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 Daniel if Daniel was making a profit.
a) What is the equilibrium interest rate Daniel would charge a SAFE farmer?
b) What is the equilibrium interest rate Daniel would charge a RISKY farmer?
c) What is Daniel’s total expected profit E() = E() + E()?
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 Purdue-ville, and now Daniel is a monopolist moneylender.
a) What is the equilibrium interest rate Daniel would charge a SAFE farmer?
b) What is the equilibrium interest rate Daniel would charge a RISKY farmer?
c) What is Daniel’s total expected profit E() = E() + E()?
d) What is total expected income across all types of famers E(Y) = E(YS) + E(YR)?
6. Now assume that Daniel decides to leave town in search of a better life. But, Javier, another very rich moneylender, moves to Purdue-ville. Javier does not have perfect information. Specifically, he cannot identify who is a SAFE versus a RISKY farmer. (This means that he has to charge a single interest rate to everyone.) He does know, however, that half of the farmers are SAFE and half are RISKY. Continue to assume that the credit market is perfectly competitive and that loans are limited liability.
a. What type of asymmetric information problem does Javier face?
b. What is the equilibrium interest rate under this assumption of asymmetric information?
c. Which types of entrepreneurs will end up borrowing? (SAFE, RISKY, BOTH, or NONE)

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