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.