Problem

Consider an economy where a representative entrepreneur is active for three periods, and assume the following timing of events: At date 0, the bank lends an amount I to the entrepreneur (henceforth: the borrower), and she invests the entire loan in a project. At date 1, the borrower obtains a return y. If the borrower repays R1 to the bank at date 1, she will be able to access a new loan I from the bank with certainty. Otherwise, she will be denied access to a new loan and therefore will not be able to execute the project. At date 2, the borrower faces exactly the same situation: if she invests at date 1, she will obtain a return y with certainty, otherwise she will get nothing. Only if she repays R2, she is able to receive a loan in the second period, invest I again, and obtain a return y at date 3 with certainty. The borrower's discount factor is δ with δ < 1="" and="" δy=""> I, and the borrower's consumption decisions start at date 1. The gross cost of lending I for the bank is K. Assume that the bank sets R1 = R2, that it has a discount factor equal to 1, and that it just wants to break even. Suppose that the bank cannot verify the borrower's returns at date 1 and 2, but it knows that they should be y.

a. What is the minimum gross repayment R* for each I loan at which the banks would be willing to lend?

b. What is the maximum gross interest repayment R** for each I loan at which the borrowers would be willing to repay at date 1 and 2?

c. If I = $100, y = $300, K = $120, and δ = 0.8, is the bank willing to lend to the borrower at date 1 and date 2 and the borrower willing to pay back the bank at these two dates?