Imagine there are two types of potential borrowers in a village, those with pi=0.7 and those with pi=0.9. As in the model discussed in class, these borrowers succeed with probability pi and fail with probability (1?pi). If they fail, they get no payoff and can pay nothing to the bank (there is no collateral). If they succeed, they pay back (1+r)L to the bank. Assume L = $100. Assume the expected gross payoff (i.e. without accounting for payment to the bank) to borrowing is $200 for all borrowers. Assume all potential borrowers have a non-borrowing option they can choose instead of borrowing: working for a subsistence wage of $70. First consider individual (not group) lending.

a. Write down the expected payment made to the bank by a borrower with pi=0.9. Write down the expected payment made to the bank by a borrower with pi=0.7. Leave both answers in terms of r.

b. Assume the bank can observe the individuals' risk-types, because of credit history for example, and can charge different borrowers different interest rates. If the bank must earn an expected return of 10% on its loans, what interest rate would it charge a borrower with pi=0.9? with pi=0.7?
Justify your answer.

c. For both borrowers, what is their expected net payoff to borrowing if the bank charges the interest rates of part b.? Which types will borrow (comparing the payoff to borrowing to the outside payoff)? Justify your answer.

d. Now assume the bank cannot observe the individuals' risk-types and must charge everyone the same interest rate. Assume the interest rate is the one charged to the risky type (with pi = 0.7) in part b. (If you did not get part b., assume an interest rate of r = 57%.) Write down the expected payment made to the bank by a borrower with pi=0.9. Write down the expected payment made to the bank by a borrower with pi=0.7.

e. For both borrowers, what is their expected net payoff to borrowing? Which types will borrow (comparing the payoff to borrowing to the outside payoff)? Justify your answer.

f. In words, when does the adverse selection problem appear, and why? Now consider group lending. Borrowers form pairs homogeneously. If a borrower succeeds, he pays (1+r)L. In addition, if he succeeds and his partner fails, he makes an extra payment of qL. Assume projects succeed or fail independently of each other. Now assume r = 30% and q = 90%, that is, 90% of the partner's loan principal must be paid by a borrower who succeeds and whose partner fails.

g. Find the effective interest rate for a borrower with pi=0.9. Find the effective interest rate for a borrower with pi=0.7. For each, how does this interest rate compare with that of parts b. and d.?

h. For both borrowers, what is their expected net payoff to borrowing? [Hint: expected repayment to the bank is just pi(1+ i r ~ )L, where i r ~ is the effective interest rate for borrower i.] Which types will borrow (comparing the payoff to borrowing to the outside payoff)? Justify your answer.

i. In words, briefly explain how group lending can overcome the adverse selection problem