A competitive lender makes loans to a pool of borrowers that are identical. After borrowers have received their loans they choose one of two investment projects. Project G pays the borrower a rate of return of r(g) with probability p(g). With probability 1-p(g), the project earns a zero rate of return, the borrower defaults on the loan, and the lender receives back the initial loan amount. Project B pays the borrower a rate of return of r(b) with probability p(b). With probability 1-p(b), the project earns a zero rate of return, the borrower defaults on the loan, and the lender receives back the initial loan amount. We assume that r(g)p(b) and p(g)(1+r(g))>p(b)(1+r(b)).
The lender can’t distinguish between borrower types and so it charges all borrowers the same interest rate r(L). The lender lends an amount L and pays interest r(D)on funds acquired from depositors.
Q1. Which project would the lender prefer that the borrowers undertake?
Project B or Project G
Q2. Explain in words why your answer to the previous question is true.
Q3. Write down an expression for the profit that a borrower expects from Project G and submit an image file depicting your answer.
Q4. Suppose r(g)=0.08, r(b)=0.10, p(g)=0.99, p(b)=0.3, r(D)=0.02, L=100. Find the value for r(L)* such that the borrower is indifferent between projects G and B. Round to three decimal places.
Q5. Either by hand or using a computer, graph the lender’s expected profit function E(π^L) for values of r(L) between 0.00 and 0.10. Make sure axes and r(L)* are clearly labeled.
Q6. Explain in words what is happening to the borrowers’ behavior at the discontinuity in the lender’s profit function that you graphed in the previous question.
Sample Solution
