kindly help
Exercises 2|]? Exercise 117.2 {Moderatel For this problem, assume that there are only two types of potential borrowers: Safe (who comprise a: of le population} and Risky {who comprise the remaining 1 o of the popula- tion]. Banks cannot tell the difference between them.r and with probability o. a borrower is safe and probability 1 t]: a borrower is risky. Safe borrowers have access to safe projects. which pay off s5 if they succeed and [I if they fail. Safe projects succeed with probability p 3. Risky borrowers have access to risky projects, which pay off Tr if they succeed and zero if they fail. Risky projects succeed with probability pH. Risky and safe projects have the same expected payoff: PEN s = PHTTH, but the probability of success is lower for risky projects. so Fa at P5, and the payoff from succeeding is greater, so \"In; :3- g. Both risky and safe projects have public failure,r that is, there is no need to audit agents who claim that their project failed. To finanoe Hie projects borrowers need a unit of capital from a bank. The bank in turn announces a repayment amount a in die event that the borrower's project does not fail. If the project fails. borrowers owe noming [they declare bankruptcy). If the project succeeds, borrowers consume their output minus 2. if the project fails. borrowers consume zero. Assume that borrowers are risk neutral so that their utility function is just their expected consumption. There is a risk-free interest rate of 5' that banks must pay to their depositors (ius iey have to realize at least 1 + r in expected value on their loan to meet their deposit liability]. 1. Write down a bank's balanoe sheet (in terms of z, 5', p5, and PH] assuming that. with probability a: the borrower is safe and wii probability 1 o the borrower is risky. 2. Assume that banks compete by offering the lowest value of a that gives them non- negative prots in expectation. Determine the equilibrium interest rate s*{r, ct} as a function of the interest rate r and the proportion of safe agents ct. 3. Find the expected utility of a safe agent who borrows, VSIII'), as a function of the interest rate 5' when a is given by a*{r, t1]. Repeat for a risky agent. 4. Agents stop borrowing if the expected utility of being a borrower falls below zero. Show that if a safe agent decides to borrow; a risky agents will too. Find the critical interest rate r* at which safe agents stop borrowing. At interest rates greater than or equal to this critical value. 5' 3 r* all safe agents leave the pool. so or = I]. What happens to the equilibrium payment a? Exercise 117.3 {Moderatel Consider the model of costly audits again. Now suppose that intermediaries gain access to a technology which allows them to extract more from each borrower [that is. for each value of announced repayment a and audit cost 1*. suppose ids, y} shifts up). What happens to Exercise 17.3 (Moderate) Consider the model of costly audits again. Now suppose that intermediaries gain access to a technology which allows them to extract more from each borrower (that is, for each value of announced repayment a and audit cost y, suppose (r, 7) shifts up). What happens to 208 Financial Intermediation the demand schedule of capital? What happens to the supply schedule of capital? What happens to the equilibrium interest rate? What happens to equilibrium economy-wide output? Are agents made better off or worse off? Exercise 17.4 (Moderate) Yale University costs 1 dollars to attend. After graduation, Yalies (that is, graduates of Yale) either land good jobs paying w or no job at all, paying nothing. The probability of landing the good job is a where a is hidden effort exerted by the Yalie. Yalies are born with wealth a 2 0, and those Yalies born with wealth a 1 to finance the loans. Student borrowers who get the good job must repay Yale University some amount a out of their wages w. Student borrowers who do not land the good job pay nothing. All students have preferences over lifetime expected consumption E(c) and private labor effort * of: V(E(c), T) = E(c) - 4X Assume 0 1. Show that her optimal effort ,* is a. 2. Now consider poor Yalies, with a 0. The larger dis, the nicer the central banker (that is, the more the central banker cares about the unemployed. Assume that there is a Phillips curve of the form in equation (19.6). Answer the following questions: 1. Assume that inflationary expectations are fixed at 7. Find the optimal inflation rate choice of the government, To(). 2. For fixed inflationary expectations, find the corresponding choice of unemployment rate, uo($). 3. Now assume that the private sector is aware of the government's maximization prob- lem and knows perfectly. Find the inflation rate m at which expectations are met. What is the associated unemployment rate, u1
