Problem: Consider the two-period bankruptcy model with utility u(c) = ln(c), discount factor , savings interest rate i, default penalty , known first period income y₁, and unknown second period income y₂.

(a)Given debt b₁> 0, what is the minimum income y(b₁) at which the borrower repays instead of defaults?

(b) Supposey₂ takes on anyvaluebetween 0 and 1 with equal probability,i.e. prob(y₂ y) = yfor all y [0, 1].What is the effective interest rate function r_b(b₁)?

(c) Suppose = 0.96, i = 2%, y₁ = 0.1, and = 0.1. Using the "Bankruptcy Example" Excel file on Dropbox, what is the borrower's preferred choice of b1, the effective interest rate r_b(b1), and the bankruptcy rate prob(y₂ y(b1))?

(d) What happens to borrowing, interest rates, and the bankruptcy rate if the penalty increases to = 0.2? What about when = 2? What about when = 3?

(e) What do the results from (c) - (d) say about the relationship between default penalties and the quantity of borrowing, interest rates, and the bankruptcy rate?