explained welll

Exercise 9.6 (Moderate) In the last step, you will simulate business cycles in the model economy. All you need to know is the capital ki at the beginning of time and the random shocks er. As a starting capi- tal, use ki = 3.7. You can generate the random shocks with the random number generator in your spreadsheet. In Excel, just type "=RAND()", and you will get a uniformly distributed random variable between zero and one. Generate 50 such random numbers, and use your formulas for a and i and the equation for capital in the next period, keel = (1-6) k + is, to simulate the economy. Plot consumption and investment (on a single graph). How does the volatility of the two series compare? Plot a graph of GDP, that is, consumption plus in- vestment. How do the business cycles you see compare with the ones you found in the real world? You don't need to compute the length of each cycle, but try to make some concrete comparisons. 84 Business Cycles Exercise 9.7 (Easy) Read the following article: Plosser, Charles. 1989. "Understanding Real Business Cycles". Journal of Economic Perspectives 3(3): 51-78. Flosser is one of the pioneers of real business cycle theory. What you have done in the previous exercises is very similar to what Flosser does in his article. His economy is a little more realistic, and he gets his shocks from the real world, instead of having the computer draw random numbers, but the basic idea is the same. Describe the real business cycle research program in no more than two paragraphs. What question is the theory trying to answer? What is the approach to answering the question? Exercise 9.8 (Moderate) What does Plosser's model imply for government policy? Specifically, can the government influence the economy, and is government intervention called for?Exercises Exercise 13.1 (Hard) Consider an economy with infinitely many agents, each of whom is very very small. An agent i has preferences over consumption o and labor effort & of: u(de)=d -ye. The preference parameter y' is distributed uniformly on the interval [0, 1]. So the fraction of agents with preference parameters y less than some number z is just z, for 0 s a $ 1. For example, exactly half of the population have values of y less than or equal to 0.5. Agents may only choose whether or not to work, not how many hours to work. If an agent chooses to work, she supplies exactly one unit of labor effort to the common backyard technology transforming labor effort into output as y' = 6. If an agent chooses not work, her labor effort is zero, she produces nothing and consumes nothing. All agents have the same backyard technology. The government levies a flat income tax at a rate 0 *s. Both risky and safe projects have public failure, that is, there is no need to audit agents who claim that their project failed. To finance the projects borrowers need a unit of capital from a bank. The bank in turn announces a repayment amount a in the event that the borrower's project does not fail. If the project fails, borrowers owe nothing (they declare bankruptcy). If the project succeeds, borrowers consume their output minus a, 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 r that banks must pay to their depositors (thus they have to realize at least 1 + r in expected value on their loan to meet their deposit liability). 1. Write down a bank's balance sheet (in terms of z, r, ps, and pa) assuming that, with probability a the borrower is safe and with probability 1 - a the borrower is risky. 2. Assume that banks compete by offering the lowest value of a that gives them non- negative profits in expectation. Determine the equilibrium interest rate a" (r, a) as a function of the interest rate r and the proportion of safe agents a. 3. Find the expected utility of a safe agent who borrows, Vs(r), as a function of the interest rate r when a is given by a*(r, a). 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, r 2 r* all safe agents leave the pool, so a = 0. What happens to the equilibrium payment z?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 .(2, 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 z 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) - -7 Assume 0 1. Show that her optimal effort a* is a. 2. Now consider poor Yalies, with a 0. The larger of is, 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 . Find the optimal inflation rate choice of the government, no(@). 2. For fixed inflationary expectations, find the corresponding choice of unemployment rate, uo( d). 3. Now assume that the private sector is aware of the government's maximization prob- lem and knows o perfectly. Find the inflation rate mi at which expectations are met. What is the associated unemployment rate, u1?Exercise 19.4 (Moderate) For this exercise, we will consider what happens when the government and the private sector repeatedly interact. Unemployment in period tu, inflation a, and inflationary ex- Exercises 241 pectations of are related by the simple Phillips curve: w =u; ty(n! - m), for all t = 0, 1, ... , 00. The parameter y is fixed over time. The government knows about the Phillips curve, but the private sector does not. The government has preferences over unemployment and inflation in period t of: V(1, " ) = -12 -x, for allt = 0, 1, ... , 00. The private sector sets inflationary expectations based on last period's inflation. This is known as adaptive expectations. As a result, of is given by: *1 = m-1, for all t = 1, 2, . .. , 00. Assume that * = 0, that is, the private sector begins by believing that inflation will be zero. Answer the following questions: 1. Assume that the government takes as given expectations in a period , and picks the inflation rate at which gives it the highest payoff in period t. Find the government's choice rule *; (";) 2. If the government sets inflation * = *"(";), how do expectations evolve over time? Thus right down a law of motion for inflation, *(*-1). 3. What do the trajectories of inflation and unemployment look like over time? Are they rising or falling? Do they settle down? If so, where? 4. How would your answer have been different if, instead of the initial expected infla- tion being zero, it had been some very large number instead? 5. Now assume that the Phillips curve is augmented with a mean zero shock term, z, so: Assume that the government knows the value of a, and reacts appropriately. Now what happens? Exercise 19.5 (Easy) To answer this exercise, you need to answer Exercise 19.4 above. Imagine that the pri- vate sector has adaptive expectations about the government's inflationary policy over Time but that part of expected inflation is the government's announced inflation target. This an nounced inflation target is merely an announcement and has nothing to do with realty. I " is the announced target for period-t inflation, expectations satisfy: of = 601-1 + (1 -8)x,, for allt = 1, 2, ... , 00. Here 0