##Macroeconomics.
Question 5 (20 points) Consider the following three period economy, with time denoted by t = 0, 1, 2. The economy is populated by a continuum of measure 1 of individuals, each endowed with one unit of a storable consumption good. At t = 0 , individuals have two options with regards to how they can invest their endowment. They can either stuff it in their mattress, where it gets a gross return equal to 1 (i.e., 1 + r = 1), or they can invest it in a long-term project that yields a gross return R > 1 in period two. For example, an individual that invests an amount / will receive RI in period two, and has 1 - I stuffed under the mattress. In t = 1, individuals have the option of liquidating the long-term project at a penalty. If they liquidate, they only receive a return L - 1 (per unit invested) in period 1, rather than the return R in period 2. At time t = 1, a fraction * = 1/2 of the individuals receive a liquidity shock. These individuals are "impatient" and only value consumption in period one. The fraction 1 - " individuals that do not receive a liquidity shock are "patient" and only value consumption in period two. At time t = 0, all individuals have the same chance of being hit by the liquidity shock. Assume that individuals do not discount the future, so that their ex-ante expected utility is given by U = Tu(CI) + (1 - #)u(c2). where c and c, are the consumption in period 1 and 2, respectively, and u(c) = 1, with o > 0. a) Assume there are no financial markets available, so that individuals must simply invest on their own. Given that an individual has invested an amount / at time t = 0, what will be the optimal levels of consumption, C1, C2, if: (i) the individual receives a liquidity shock (i.e. is impatient); (ii) the individual does not receive a liquidity shock (i.c. is patient). Let c and & denote the consumption of an impatient individual in period 1 and of a patient individual in period 2, respectively. b) What is the optimal level of investment when individuals have to invest on their own? Denote this level by /. Hint: Show that there exists L, LE [0, 1] such that if L 2 L, the optimal level of investment is equal to 1, and if L - L, the optimal level of investment is zero. c) Suppose that when types are realized in period 1, this information is publicly observable. Suppose there exists a social planner that individuals entrust all of their endowment to at time 0. The social planner will pay impatient individuals cj in period 1 and patient individuals c; in period 2 (and zero otherwise). Solving the social planner's 9 problem, what is cf and c;? How much does the social planner invest? That is, what is I*? d) Assume that L = 1. Show that I* oz. In other words, show that the planner invests less than the individuals but it makes them face more consumption risk. e) Assume that L = 1 and o > 0. Now suppose an agent's type is private infor- mation, and the social planner can only offer a contract contingent on an individual's announcement of her type at time 1. Furthermore, at time 1, she meets each agent only once, with the meeting order randomly determined. If individuals report honestly, can the social planner achieve the same allocation as in question c)? Is it optimal for an individual to report honestly when everyone else does? f) Suppose that L = 1 and o w are claiming to the planner that they are impatient. Is it optimal for these agents to lie as well? Given your answer to this question, are runs possible when o 1. Preference shocks follow an AR(1) process (in percentage deviations from steady state) (13) In percentage deviations from steady state: & is the preference shock, o, is real marginal cost, & is consumption, w, is the real wage, n, is hours worked, g, is output. In deviations from steady state: 4 is the nominal interest rate, #, is inflation. a) Using the equilibrium conditions above, show that this model can be represented by the standard 3 equations (14) 8 # 1 = BE,(#HI) + Ky (15) (16) Where the natural real rate of interest is: F - (1- P) (17) and where &, follows the process in equation 13. (Hint: you may find it useful to start by showing that the natural rate of output is constant in this model). b) Using the method of undetermined coefficients, find the response of the output gap and inflation to an exogenous decrease in 8, 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 &: c) Interpret your results. In particular, explain how, and why, preference shocks affect the output gap and inflation. Briefly comment on how a decrease in & relates to typical recessions we see in the data. d) Instead of following the Taylor Rule above, policy is now set optimally. Derive the optimal monetary policy rule under discretionary policy. (Hint: As in class, assume that the loss function has quadratic terms for the output gap and inflation, with a relative weight d on the output gap. For simplicity, assume the steady state is efficient). What is the optimal path for the output gap and inflation in response to preference shocks underQuestion 6 (10 points) This question is about fiscal policy in the baseline real business cycle model. You do not need to derive anything and keep your answers clear and concise. a) TFP shocks can explain the positive correlation of GDP, consumption and invest- ment in the data. Government consumption shocks cannot explain these facts. Is this statement true or false? Explain. (Note: in this model, government consumption shocks are defined as the purchase of consumption goods by the government. This spending is not productive and does not enter the household's utility function.) b) The government wants to stimulate private consumption and GDP. They propose a temporary, debt-financed, tax cut. In the RBC model, will a tax cut have the desired effect on the economy? Start by considering lump sum taxes, and then discuss how this result might change for other types of taxes.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 U(taleo) = ) #'(Ingtyla-1), 9>0, 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, #1 = F(M , 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 (r,) fully depreciates at the end of a given period, i.e. 6 = 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 1 = gly, 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
