Problem
Assume that borrowers are not very poor, in that they have some collateral, but they are nevertheless suffering from financial exclusion. Now consider the case of two borrowers with different levels of collateral. Borrower 1 has w = $20 as collateral, Borrower 2 has no collateral at all. Borrower 1 is as productive as Borrower 2, so if they can undertake an investment project that requires $100, both can produce the same gross return y = $190 with certainty, assuming both borrowers put in sufficient levels of effort, which cost c = $30 to each of them. If the borrowers do not work hard enough, the probability of success falls to 0.5. The gross cost of capital lent is K = $140 per each $100 loan.
a. Show that if the lender can observe the effort made by each borrower, it is socially efficient to lend money to both borrowers.
b. If, on the other hand, the borrowers' behavior cannot be observed, then, show that only the one with collateral can borrow.
c. Comment on the main lesson drawn from this exercise with respect to the use of collateral to facilitate credit access.