An entrepreneur has no funds to finance a project that requires an investment I > 0. The project yields R in the case of success and 0 in the case of failure. The borrower and the lenders are risk-neutral, and the borrower is protected by limited liability, that is, the borrower cannot receive a negative payoff in any state of nature. The interest rate in the economy is 0. The borrower can be one of two types. A good borrower has a probability of success p and a bad borrower has a probability of success q with p > q. Assume that pR > I, that is, at least the good project has positive net present value (the expected payoff is greater than the initial investment). We refer to this scenario as the "good borrower is creditworthy." We allow for the possibilities that qR ≥ I OR qR < I, that is, the bad borrower may or may not be creditworthy. A borrower has private information about her type. The probability that a borrower is good is α and bad is 1- α. Assume that the capital market is competitive, that is, a lender demands an expected return on its investment of at least 0. In other words, a lender is willing to make a loan as long as it gets an expected payoff that equals its investment I.
(i) Consider first the "symmetric information" case where the borrower's type is perfectly known to the lender. What are the optimal contracts for the borrowers? Consider both the cases qR ≥ I AND qR < I.
Now consider the asymmetric information case. Assume that the only feasible contracts are those that give the borrower a compensation Rb ≥ 0 in the case of success and 0 in the case of failure. Notice that such contracts necessarily pool (bunch) the two types of borrowers because both receive the same contract. Define m = αp + (1-α)q
(ii) First consider the case where mR < I. Analyze the optimal contracts for different values of α. Is lending possible for all possible values of α? Discuss your answer.
(iii) Now consider the case where mR ≥ I. Derive the optimal contract. Analyze the implications of the contract for the "good" and "bad" borrowers by comparing their payoffs with the payoffs in the first best scenario analyzed in part (i).