Consider an economy, which is identical to that of exercise 7, except that in this economy there is a moral hazard problem: At date 0, provided the borrower puts in an adequate effort level, she is capable of obtaining a gross return y at date 1 with certainty; the cost of her effort is

e. If the borrower does not put in any effort, she can get y with probability p < 1. With complementary probability, 1 - p, she does not get anything, but in this case e = 0. The bank can set a gross interest rate R* at date 1. If the borrower repays at date 1, the bank extends a new loan, which is identical to that of the previous loan. If granted a new loan, the borrower obtains a gross return y with certainty at date 2.

Assume that the borrower does not have to make any effort in order to obtain a second-period return. Her discount factor is

d. Describe the gross interest rate R* that the bank should set in order to elicit effort from the borrower at date 0 and that will at the same time ensure that the borrower to be willing to repay at date 1.

Data from in Exercise 7

Consider an economy with risk-neutral individuals. And suppose the following timing of events: At date 0, an individual wants to borrow an amount I in order to invest in a project that yields a gross return y with certainty at date 1. The bank cannot verify the return realization on that individual’s project, but it knows that the return should be y. Now suppose that, conditioning on the borrower repaying an amount R, the bank will extend a new loan of size I at date 1. The borrower then invests the entire proceeds from the new loan I, and obtains y at date 2 with certainty. But if the borrower defaults at date 1, that is, if she fails to repay R, the bank does not extend a new loan I at date 1.

Therefore, the borrower can not invest. The lender’s gross cost of lending I is K. Let d denote the borrower’s discount factor. Define the R at which the bank would wish to extend a loan and the borrower would wish to repay.