(22 marks total) Consider the following modification of the endogenous interest-rate spread

model from Section 2.1 of LN6. Suppose that, rather than there only being two possible

income realizations in the second period (i.e., y2 and 0), there are three: y2 = yH with

probability pH, y2 = yL with probability pL, and y2 = 0 with probability p0 = 1 pH pL,

where yH > yL > 0. All other elements of the model are the same as in LN6, including the

fact that the HH is required to repay as much of the loan in the second period as it feasibly can.

(a) (6 marks) For a given interest rate rB, let GH(Q) denote the amount the HH would

actually repay the bank at t = 2 for an initial loan size Q 0, assuming their secondperiod

income was yH. Similarly, let GL(Q), and G0(Q) denote the actual repayment

amounts assuming the HH's second-period income was yL and 0, respectively. Write down mathematical expressions for GH(Q), GL(Q), and G0(Q). Draw these three functions on the same graph (one with Q on the horizontal axis), being sure to carefully label any important points on the axes.

(b) (8 marks) Assume henceforth that the HH faces the "optimistic" NBL = yH/(1+rB),

i.e., the HH cannot borrow more than it could feasibly repay if it were to end up in the

"high" income state yH in the second period. Given the interest rate rS that the bank

must pay on deposits, for a loan of size Q > 0 let rB(Q) denote the interest rate at which

the bank's expected profit at t = 2 is exactly zero. Obtain a mathematical expression

for the function rB(Q) (NOTE: We're not solving for the spread here, just the interest

rate on loans itself). (HINT: You may find it useful to separately consider a number of

different ranges of Q.)

(c) (5 marks) In your answer to (a), you should have found that one or more of your functions

has a "kink" (a sudden change in the slope) at one or more values of Q. Importantly, the

value(s) of Q at which there is a kink should depend on the relevant interest rate rB(Q)

that's in effect at that value. Using your answer to (b), for each kink, solve explicitly for

the value rB(Q) at that kink, and then use the result to determine explicitly the value

of Q at the kink as a function of exogenous parameters only. (HINT: For each kink, you should have an expression for Q at that kink that depends on rB. Replace rB with rB(Q)

in that expression, then substitute it into the function rB(Q) you found in (b). Finally,

solve for rB(Q), and then plug the result into your original expression for Q to get the

value of Q at the kink.)

(d) (3 marks) Given your answers to (b) and (c), draw the function r(Q) in a graph with Q

on the horizontal axis, being sure to label all important points (including, among things,

any kink values) on the axes.