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1. An exchange economy has two dates t = 0, 1 and two states of nature s = 1, 2 which will be revealed at date 1. Unlike the model in class, agents in this economy do have endowments, consume and trade in goods at t = 0. Use s = 0 to denote the date-event pair corresponding to date 0. There is one physical commodity, and two consumers i = 1, 2 whose endowments wis are as follows: 10 = 2, w11 = 4, w12 = 3, w20 = 4, w21 = 2, w2 = 3. Both share the von-Neumann-Moregenstern utility log co + logo, where a denotes date t consumption. Consumer 1 believes s = 1 will occur with probability }, while consumer 2 believes s = 1 will occur with probability 2. At date 0, there is a spot commodity (i.e., for delivery at s = 0) market, besides two assets & = 1, 2 whose date-1 returns fax are given by mn = 1, m12 = 2, 121 = 0, r22 = 1. So consumers divide their date 0 wealth between consumption at t = 0 and purchasing assets that yield returns at $ = 1. At date 1, agents realize their asset returns and trade in spot commodity markets after the state is revealed. (a) Derive the entire set of er ante Pareto optimal allocations in this economy. Are these allocations ex post Pareto optimal as well?1. True or False? Briefly explain (or qualify) your answers. (a) The present value of a share of common stock is an increasing function of the future growth rate of earnings per share. 2. True or false? Briefly explain (or qualify) your answers. (a) In principle, the market price of a share of stock equals the discounted value of the stream of future earnings per share. 3. True, false or "it depends" (give a brief explanation): (a) Present value is good to value only traded assets since the discount rate comes from returns on traded assets. (b) Growth stocks should either have growing dividends or earnings. 4. True, false or "it depends" (give a brief explanation): Managers should maximize the firm's current market value, but only when maximization does not create unacceptable risks for shareholders. 5. True, false or "it depends" (give a brief explanation): The DDM (Dividend Discount Model or the DCF Valuation model) works only for firms with a dividend history. 6. True or false. Briefly explain your answer in each case. (a) Growth stocks usually have growing dividends. 7. Question 1 (40 points) True, false or "it depends"? Briefly explain or qualify your answer. (a) A company that has not made a profit since its IPO (initial public offering) cannot possibly be a growth stock. 8. True, false (give a brief explanation): (a) Firms with higher than average plow back ratios are growth companies. (b) Small stocks earn higher average returns because their returns are more volatile. 9. True or false (give a brief explanation): The price of a stock equals the present value of expected future earnings per share. 10. True or false (give a brief explanation): Within an industry, holding business risk and financial leverage constant, differences in firms' P/Es depend only on differences in rates of asset growth. Fall 2008 Page 28 of 66 11. MetaTrend Corp. earns a book rate of return (ROE) of 12%. It reinvests one-half its earnings and pays out the other half as cash dividends. The nominal cost of capital is 12%. (a) Given this ROE and dividend payout ratio, what is the growth rate of Meta- Trends earnings and dividends? (b) Assume this growth rate is expected to continue in perpetuity. What is the present value of MetaTrend shares? Assume that book value per share is $10. (c) What does your answer to (b) assume about the timing of dividend payments? Explain briefly. (d) Your calculation in (b) assumes a nominal cost of capital and a nominal growth rate. Restate the cost of capital and growth rate in real (inflation-adjusted) terms and recompute the present value of MetaTrend shares. Show that the present value does not change. (e) Suppose Meta Trend decides to pay out all its earnings as cash dividends. There- fore it does not grow. What is the change, if any, in MetaTrends stock price? Why?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.1. (10) Briefly discuss the following statements (keep your answers short and concise): (a) Provide an intuitive - but concise - explanation for how the existence and uniqueness of the value function as defined by the Bellman equation associated with the standard growth model was estab lished. In your answer, be sure to identify the metric space used in the analysis. (b) Within the context of the representative agent consumption-based capital asset pricing model, discuss the factors that affect the equilibrium level of the yield on risk-free, one-period bonds. 2. (20) Consider a simple, representative agent RBC model in which output, y, is produced via a standard Cobb-Douglas production function: where ke denotes beginning-of-period capital, he is labor, and & is an i.i.d.technology shock. The depre- ciation rate of capital is 100%. In each period, agents make consumption and labor decisions in order to maximize lifetime expected utility: ED [ BU ( C , he ) Within this environment, consider two variations defined by the functional form for U (.). (a) In Economy A, agents have preferences given by: U (c, hi) = Inc - ;hi In this economy, do the following i. Express the maximization problem as social planner problem and write down the associated Bellman equation. ii. Solve for the equilibrium policy functions describing consumption, investment and labor. (b) In Economy B, agents have preferences given by: U (c, he) = In (c - -h; ) i. Express the maximization problem as social planner problem and write down the associated Bellman equation. ii. Solve for the equilibrium policy functions describing consumption, investment and labor. (c) Compare the equilibrium behavior in both economies and provide an explanation for the differences. 3. (20) Consider a variation of the Sidrauski monetary model with a constant population. Specifically, assume that the representative agent's maximize lifetime utility is given by: Est [U (a) + V (#4)] where U () and V () are concave, twice-differentiable functions, q denotes consumption and Me is money chosen in period t. Each period, agents use beginning of period nominal balances, the revenue from sales of output and a lump-sum monetary transfer to purchase consumption, investment and new money. In contrast to the Sidrauski model, both capital and money are used as inputs into the production process. Letting y denote output, the production function is given by: 3t = 1 - = (7) f ( kil) where =' (.) 0, = (0) = 1, Me/P-+00 lim = (M:/ P) = 0. The function f (kt) has standard properties. The money supply in this economy is growing at the constant rate / > 0 and capital depreciates at the constant rate of 6 > 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.6. (10) Consider the standard growth model in discrete time. There is a large number of identical households (normalized to 1). Each household wants to maximize life-time discounted utility V((a)20) = >B'(Ing + yla-1), 920, that is, households preferences are characterized by "habit persistence". Each household has an initial capital stock ro at time 0, and one unit of productive time in each period, that can be devoted to work. Final output is produced using capital and labor services, Mt = F(ht, m) = kin)-. This technology is owned by firms whose number will be determined in equilibrium. Output can be consumed (@) or invested (4). We assume that households own the capital stock (so they make the investment decision) and rent out capital services to the firms. We also assume that the capital stock (If) fully depreciates at the end of a given period, i.e. 5 = 1. Finally, it is assumed that households own the firms, i.e. they are claimants to the firms' profits. (a) In this economy, why is it a good idea to describe the AD equilibrium capital stock allocation by solving the (easier) Social Planner's Problem? (b) Fully characterize (i.e. find a closed form solution for) the equilibrium allocation of the capital stock. (Hint: Derive the Euler equation, and "guess and verify" a policy rule of the form kit = ght, where g is an unknown to be determined.) (c) What is the capital stock equal to as t - co? What is the ADE value of the rental rate of capital and the rental rate of labor as t - co? (d) Express the ADE price of the consumption good in any period T as a function of parameters of the model and the sequence of capital stock up to period T