Problem
Consider an economy with three types of risk-neutral entrepreneurs. If a type 1 entrepreneur invests $200 she gets a gross return of $400 with certainty. If a type 2 entrepreneur invests $100 she gets $200 with certainty. And finally, if a type 3 entrepreneur invests $100 she gets $300 with probability 0.75 and 0 with probability 0.25. A risk neutral, competitive lender is considering extending loans to these entrepreneurs. This bank can determine if potential borrowers are of type 1 (henceforth, high-type borrowers), but it can't distinguish between entrepreneurs of type 2 and 3 (henceforth, low-type borrowers), but it does know that half of the low-type borrowers are of type 2, and the other half are of type 3. All borrowers, on the other hand, can recognize each others' types. Under these conditions, the bank decides to extend individual loans to high-type borrowers and group loans with joint liability to low-type borrowers. As a result, a low-type borrower may have to repay for a defaulting peer. The cost of lending to high-types is $20, while the cost of lending to low-types is $30 (because the bank has to put in additional time and effort to ensure that groups are formed and to enforce debt repayments). Assume that the borrowers are protected by limited liability.
a. If the bank only aims to break even, calculate the interest rates charged to high types and to low types. Compare the two rates.
b. Now suppose that the bank holds a pool of loan contracts. It lends to three high-type borrowers and to four low-type pairs with the compositions: (2,2), (3,3), (2,3) and (3,2). Assume that one borrower in group two succeeds, while type 3 agents fail in pairs three and four. Compute the repayment rates that the bank will receive separately for the high and low types loans. Compare the two rates, and explain your answer.