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2. Consider a perfectly competitive credit market in which a commercial bank makes loans to two types of borrowers: A and B. Both types of borrowers take the loan to start some new business. Type A borrower runs a low risk business that generates a gross return of 2 when succeeded with probability 0.8, and a gross return of 0 when failed with probability 0.2. Type B borrower runs a high risk business that generates a gross return of 15 when succeeded with probability 0.1, and a gross return of 0 when failed with probability 0.9. Both types of borrowers repay their loans if the business succeeds, in which case the bank receives loan amount plus interests, and default (without getting punished) on the loan if the business fails, in which case the bank receives nothing back. Finally, the probability that bank gets either type of borrower is 0.5. Please answer all parts of the question.

a. Suppose the bank has perfect information on the borrower type, i.e., the bank is able to distinguish two types of borrower, what would be the gross interest rate R (one plus interest rate r) that the bank charges on each type of borrower? Assume the opportunity cost of banks funds is the (gross) risk-free rate, which is equal to 1.

b. Suppose there is information asymmetry between borrowers and the bank, i.e., the bank is unable to distinguish two types of borrowers, what would be the interest rate r that the bank charges to all borrowers?

c. Continue with information asymmetry case, given the uniform interest rate that you just calculated, would each type of borrower still want to borrow in the first place? Show your calculation.

d. What would be the banks expected profit/loss per dollar of loan, based on your answers in part 3?

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