Macro...solve all the following
1.5 Options 1. A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two- month European call option with a strike price of $49? 2. A stock price is currently $80. It is known that at the end of four months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four- month European put option with a strike price of $80? 3. Today's price of three traded call options on BackBay.com, all expiring in one month, are as follows: Strike Price Option Price $50 60 $3 70 You are considering buying a "butterfly spread" consisting of the fol- lowing positions: . Buy 1 call at strike price of $50 . Sell (write) 2 calls at strike price of $60 . Buy 1 call at strike price of $70. (a) Plot the payoff of your total position for different values of the stock price on the maturity date. (b) What is the dollar investment required to establish the spread? (c) For what stock prices on the maturity date will you be making an overall profit? 4. You are given the following prices: Security Maturity (years) Strike Price ($) JEK stock 94 Put on JEK stock 50 3 Put on JEK stock 60 Call on JEK 50 Call on JEK 60 Tbill (FV=100) 1 91 What is the price of the two call options?7. Suppose that the Federal Reserve acts to increase the money supply. a. In the aggregate demand/aggregate supply diagram, will this monetary policy action work initially to shift the aggregate demand curve, the short-run aggregate supply curve, or the long-run aggregate supply curve? (Note: focusing for now on just the short-run effects of the change in policy, only one of these curves will shift.) b. In which direction will the curve you mentioned above shift: to the left or to the right? When the curve you mentioned above shifts, what will the short-run effect on the economywide level of prices be: with it rise, fall, or stay the same? d. When the curve you mentioned above shifts, what will the short-run effect on real GDP be: will it rise, fall, or stay the same? e. When the curve you mentioned above shifts, what will the short-run effect on unemployment be: will it rise, fall, or stay the same? 8. This last question builds directly on the previous one. Suppose that after observing the short-run effects of the increase in the money supply, the Federal Reserve decides not to reverse that policy action and, instead, leaves the money supply at its new, higher level permanently. a. Given that the Federal Reserve does not reverse its initial policy action, how will the economy move from the short-run equilibrium you described in question 7, above, to a new long-run equilibrium: through a shift in the aggregate demand curve, through a shift in the short-run aggregate supply curve, or through a shift in the long-run aggregate supply curve? (Note: focusing now just on the transition from the short run to the long run, only one of these curves will shift.) b. In which direction will the curve you mentioned above shift: to the left or to the right? C. Compared to its level in the initial long-run equilibrium, will the economywide level of prices in the new long-run equilibrium be higher, lower, or the same? d. Compared to its level in the initial long-run equilibrium, will real GDP in the new long-run equilibrium be higher, lower, or the same?Question 6 (20 points) Consider the following OLG economy. Time is discrete and goes from t = 0 to infinity. Agents live for two periods, and there is a constant measure 1 of individuals in each generation. Young agents born in t have preferences over consumption streams of a single good that are ordered by u(d) +u(c41 ) where u(c) = 17, and where of is the consumption of an agent born at i in time t. Assume that o > 0 and that when o = 1, u(c) = Inc. Each young agent born at t 2 0 has identical preferences and endowment pattern (wi, w2), where wj is the endowment when young and we is the endowment when old. Assume 0 0 and with probability 1 -* the price will be zero. What happens after the price is zero? How does the price p change with a? e) Assume o = 1 and w2 = 0. Show that there are equilibria in which in some periods agents are optimistic (so that a, is high) and the price of money is high, and some periods in which they are pessimistic (so that a, is low) and the price of money is low. Argue that this model produces non-fundamental fluctuation in asset prices.b) Using the method of undetermined coefficients, find the response of the output gap and inflation to an exogenous increase in g, when prices are sticky and monetary policy follows the Taylor Rule above. To do this, guess that the solution for each variable is a linear function of the shock ge: Ut = Ay91 (Hint: follow the steps used in the problem set. Start by substituting the guesses, the monetary policy rule and the AR(1) process for 9, into the dynamic equations (13) and (14).) c) How, and why, does the response of GDP differ from the model with flexible prices? Do positive government spending shocks increase inflation? Provide economic intuition and (if you can) discuss the solution you found in part (b). d) In principle, could monetary policy fully stabilize the output gap and inflation after a government spending shock? (Hint: think about how the 3-equation setup above looks like the cases we studied in class)? Would there be any additional benefit from conducting optimal monetary policy under commitment? 8 Question 6 (10 points) This question is about the standard decentralized real business cycle model. You do not need to derive anything for this question and keep your answers clear and concise. a) Briefly explain the mechanisms through which TFP shocks affect output, con- sumption, hours worked and investment in the standard RBC model. How well does the model replicate the business cycle facts seen in the data? How would adding habits in consumption affect the dynamics of consumption and investment? b) Suppose you want to solve the model using computational methods. Explain one approach, the advantages of this method and the steps you would need to take.4. (20) Consider a version of the continuous time Mortensen-Pissarides labor market model with match- specific productivity. Labor force is normalized to 1, and there is a large measure of firms that can enter the market and search for workers. A firm can enter the labor market with exactly one vacancy, and the total measure of vacancies v will be determined endogenously by free entry. A matching function, m = m(u, v), brings together unemployed workers and vacant firms; m is increasing in both arguments and exhibits CRS. As is standard, let 0 = v/u denote the market tightness. Unlike the baseline model, here not all meetings result in a match. When a firm and an unemployed worker first meet, they draw a match-specific productivity a from a cdf F(r), with support in the set (0, 1]. The random draws of a are did across matches and time. Upon observing the specific realization of x, the firm and worker decide whether they will form a productive match, which can produce a units of the numeraire good per unit of time. Alternatively (if the realization of a is too low), they may decide that it is not worth forming a match. In this case, the worker stays unemployed, and the firm stays in the large pool of firms that can potentially search for workers. If a match is indeed formed, the two parties negotiate over the wage, which will be contingent on the match-specific productivity, z. In the negotiation process, let P E (0, 1) denote the worker's bargaining power. Crucially, a productive match keeps its idiosyncratic productivity r for as long as it is active/alive. Existing productive matches are terminated at an exogenous Poisson rate, > > 0. To close the model, we will make a few more standard assumptions. While a firm is searching for a worker it has to pay a recruiting cost, c > 0, per unit of time. All agents discount future at the rate r > 0, and all unemployed workers enjoy a benefit z > 0 per unit of time. We further impose that z > R. (e) For any r 2 R, describe the wage curve (WC) equation, w(r). The right-hand side should contain only parameters and endogenous variables (not value functions). (f) Use your findings in part (e) to provide a formula for J(x), r 2 R, that does not contain the term w ( I). Since here we have 4 equilibrium objects (u, 0, w(x), R), we also need 4 equilibrium conditions. We already have the BC and the WC. The remaining two are just the job creation (JC) and job destruction (JD) conditions. In what follows, to simplify the analysis, feel free to assume that z is distributed uniformly in [0, 1]. g. Derive the JC condition for this economy, by substituting your formula for J(r), from part (f), into the equilibrium condition that you reported in part (d). h. Derive the JD condition for this economy. (Hint: Evaluate your formula for J(r), from part (f), at the value where jobs are "destructed" (or not worth keeping), just like we did in the standard model with endogenous destruction in class.) i. Summarize the 4 steady state equilibrium conditions, and discuss in as much detail as you can the existence and uniqueness of equilibrium. 5. (20) Consider an economy that consists of two islands, i = {1, 2). Each island has a large population of infinitely-lived, identical agents, normalized to the unit. There is a unique consumption good, say, coconuts, which is not storable across periods. Although within each island agents have identical prefer- ences over consumption, across islands there is a difference: Agents in island 2 are more patient. More precisely, the lifetime utility for the typical agent in island & is given by Vi(Glo) = >B; In(c) 1= 0 where B, E (0, 1), for all i, and B2 > 81. Due to weather conditions in this economy, island 1 has a production of e > 0 units of coconuts in even periods and zero otherwise, and island 2 has a production of e units of coconuts in odd periods and zero otherwise. Agents cannot do anything to boost this production, but they can trade coconuts, so that the consumption of the typical agent in island , in period t, is not necessarily equal to the production of coconuts on that island in that period (which may very well be zero). Assume that shipping coconuts across islands is costless. (a) Describe the Arrow-Debreu equilibrium (ADE) allocations in this economy using Negishi's method. (b) Describe the ADE prices in this economy. (c) Plot the equilibrium allocation for the typical agent in island , i.e., {G)12,, i = {1, 2}, against t. Is there any period t in which & = ? If yes, please provide a closed form solution for that value of t
