""solve the attached questions.

Question 2 (50 points) This question considers the macroeconomic effects of a collapse in consumer demand in the New Keynesian model There are a continuum of identical households. The representative household makes consumption (C') and labor supply (N) decisions to maximize lifetime expected utility: Nite (1) subject to their budget constraint: Ci + B, = unN, + (1 + 4-1) , (2) where w, is the real wage, N, is hours worked, B, are real bond holdings at the end of period t, i,, is the nominal interest rate paid between t - 1 and t, P, is the price of the final consumption good and D, are real profits from firms that are distributed lump sum. As usual, 0 0 and o > 0. Z, is a household preference shock, which is a way of generating shocks to demand. The production side of the model is the standard New Keynesian environment. Monopolistically competitive intermediate goods firms produce an intermediate good using labor. Intermediate goods firms face a probability that they cannot adjust their price each period (the Calvo pricing mechanism). Intermediate goods are aggregated into a final (homogeneous) consumption good by final goods firms. The production side of the economy, when aggregated and linearized, can be described by the following set of linearized equilibrium conditions (the production function, the optimal hiring condition for labor and the dynamic evolution of prices): (3) ( 4 ) *, = BE,(1,+1) + Amc, (5) The resource constraint is: (6) Monetary policy follows a simple Taylor Rule: i, = d.#, (7) The (linearized) preference shock follows an AR(1) process: & = pa-, + er (8 ) es is i.i.d. In percentage deviations from steady state: me, is real marginal cost, &, is consumption, in, is the real wage, i is hours worked, y, is output. In deviations from 3 steady state: i, is the nominal interest rate, it, is inflation. A is a function of model parameters, including the degree of price stickiness." Assume that o, > 1, 0 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.5. (20) Consider the standard Mortensen-Pissarides model in continuous time. Labor force is normalized to 1. Unemployed workers, with measure u 5 1, search for jobs, and firms with vacancies, with measure v. search for unemployed workers. The matching technology is given by m(u, v) = u"yl-". It is convenient to define the market tightness 0 = v/u. A large measure of firms decide whether to enter the labor market with exactly one vacancy. When a firm meets an unemployed worker a job is formed. The output of a job is p per unit of time. However, while the vacancy is unfilled, firms have to pay a search or recruiting cost equal to pe per unit of time. In an active match (job), the firm pays the worker a wage w per unit of time, which is determined through Nash bargaining when the two parties first match. Let S represent the worker's bargaining power. The destruction rate of existing jobs is exogenous and given by the Poisson rate A. Once a shock arrives, the firm closes the job down. Subsequently, the worker goes back to the pool of unemployment, and the firm exits the labor market. Unemployed workers get a benefit of = > 0 per unit of time. Throughout this question focus on steady state equilibria and let the discount rate of agents be given by r. So far this is just the model we described in class. What is new here is that the unemployment benefit z does not stand for the utility of leisure or the value of home production, as is conveniently assumed in the baseline model. Here, = is a payment (in terms of the numeraire good) that the government delivers to the unemployed. Clearly, the government must tax someone in order to raise funds for the unemployment benefits, and we assume that it raises these funds by levying a lump-sum tax equal to T (per unit of time) on every employed worker. Hence, the government chooses both z and T, and must do so in a way that keeps the budget constraint satisfied at all times. (a) Describe the Beveridge curve (BC) of this economy. (b) Describe the value functions for a vacant firm (V) and a firm with a filled job (J). (c) What condition does / satisfy in equilibrium? Use your answer, together with your findings in part (b), to derive the job creation (JC) condition. (d) Describe the value functions for an unemployed (U) and an employed (W) worker. (e) Describe the wage curve (WC) in this economy. This will be a function of the usual terms and the new term T. (f) What is the relationship between T and z, u so that the government's budget constraint is satisfied at all times? Use this condition in order to get rid of 7 in the WC. (g) In the baseline model, it was easy to characterize the equilibrium values of (0, w), since equations JC and WC contained exclusively these two variables. To do this here, we need a little more work. since at least one of these equations also contains the third endogenous variable, namely, u. Can you get rid of u and replace it with a term that contains o (and other parameters)? Hint: Which equilibrium condition gives you u as a function of 0? (h) Plot the JC and WC curves in the w. 6 space. Do they have the standard shape? Use your graph to discuss existence and uniqueness of equilibrium. What is the intuition behind your findings? (i) Assume that the government increases z. What is the effect of this policy change on equilibrium unemployment? 6. (10) Consider a standard growth model in discrete time. Throughout this question you can focus on the Social Planner's problem (vs the more complicated model with competitive markets). At t = 0 there is a large number of identical agents normalized to 1. The population grows at rate n per period, i.e., N = (1 + n)'. The representative agent's preferences are described by The initial capital stock in this economy is Ko, and each agent can devote one unit of productive time (in each period) to work. Final output is produced using capital and labor, and production is characterized by the so-called labor-augmenting technology: Y = F(Ki, N,(1+ g)'), where F is a CRS production function. Capital depreciates at rate o e (0, 1). The Social Planner wishes to maximize per-capita life-time discounted utility. (a) Describe the resource constraint of the Planner's problem. Hint: It will be useful express all the variables into "growth-adjusted per-capita variables", as we did in class. (b) Characterize the optimal solution to the Planner's problem (i.e. derive the Euler equation). (c) What happens to per-capita consumption in the long run? What happens to total consumption in the long run? (For full credit I expect you to derive the results carefully, and relate them to your work in part (b). However, partial credit will be given to correct, intuitive answers)