please answer these questions

1. (22 marks total) Consider the following modication of the endogenous intm-mtrate spread model from Section 2.1 of LNB. Suppose that, rather than there only being two pomihle income realizations in the second period (i.e., fig and 0}, there are three: y2 = 3m with probability pH, 3;; = y; with probability PL, and ya = 0 with probability pa = 1 pH pL, where ya; 2: yr, > 0. All other elements of the model are the same as in LNG, including the fact that the HE is required to repay as much of the loan in the second period as it feasibly can. (a) (I6 marks) For a given interest rate r3, let GH(Q} denote the amount the HE would actuallyrepaythebankatt=2foraninitialloansizeQ230,assumingtheirsecond period income was ya. Similarly, let GL(Q], and Gn(Q] denote the actual repayment amounts assuming the HJ-I's secondperiod income was yr. and 0, respectively. Write down mathematical expressions for GH{Q), GAO}, and Gu(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 HE faces the \"optimistic\" N'BL IV = y}; {(1 +113), i.e., the RH cannot borrow more than it could feasibly repay ifit were to end up in the \"high\" income state ya in the second period. Given the interest rate r3 that the bank must pay on deposits, for a loan of size Q ) {1 let r3{Q) denote the interest rate at which the bank's expected prot at t = 2 is exactly zero. Obtain a mathematical expression for the function avg-(Q) (NOTE: We're not solving for the spread here, just the interest rate on loans itself). (HINT: You may nd 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 hasa \"lo'nk'\" (asuddenchangeintheslope}at oneormorevalueson. Importantly, the value(s] of Q at which there is a kink should depend on the relevant interest rate \"3(0) that's in eect at that value. Using your answer to (b), for each kink, solve explicitly for the value r3(Q) at that kink.J and then use the result to determine explicitly the value of Q at the lcink as a frmction of exogenous parameters only. (HINT: For each kink1 you