$ solve the following questions.

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).4. (20) 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(14)20) = EAU(a), BE (0,1). Each household has an initial capital ko > 0 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, according to a production function, F, which has the standard properties discussed in class, most notably, it is increasing in both arguments and exhibits CRS. This technology is owned by firms (whose measure does not really matter because of the CRS assumption). Output can be consumed ( ) or invested (4). Households own the capital (so they make the investment decision), and they rent it out to firms, Let o e (0,1) denote the depreciation rate of capital. Households own the firms, i.e., they are claimants to the firms' profits, but these profits will be zero in equilibrium. The function u also has the usual nice properties, which I will not spell out here since you will not need them explicitly. In this economy there is a government that collects taxes and (for simplicity) throws the tax revenues into the ocean. The government can implement one of the following two alternative taxation systems, let us call them System A and System B. System A is a proportional tax, 7 6 [0, 1], on agents' capital income. In other words, if the government implements System A, it collects a fraction 7 of all the income that agents earn by renting out their capital to firms. System B is a proportional tax, re [0, 1], on agents' investment. In other words, if the government implements System B. it collects a fraction 7 of all the resources that agents choose to allocate into investment. (a) Write down the problem of the household recursively, under both taxation systems.' Pay special attention to the budget constraints. These constraints will not be the same under the two specifica- tions. Also, notice that I am not asking you to define a RCE in detail; just state the representative agent's problem within a RCE environment. (b) Describe the steady state equilibrium capital stock under taxation System A, for any given 7 e [0, 1]. Denote this object by Ki(7). (c) Describe the steady state equilibrium capital stock under taxation System B, for any given 7 e [0, 1]. Denote this object by K;(T). (d) Assume that F(K, N) = K" NI-", a E (0, 1). Provide closed form solutions for the terms Kj(7), K;(7), described in parts (b),(c). Hint: Here, it is more convenient to work directly with F, i.e., do not work with the auxiliary function f that we introduced in the lectures. (e) Plot the terms KA, Kj, calculated in part (d), against 7 6 [0, 1) and in the same graph. Discuss briefly. (f) Describe the government's total tax revenue in steady state under System B, Tg. Plot To as a function of the tax rate 7 (this is the so-called Laffer curve). Discuss the shape (i.e., the monotonicity) of the Laffer curve for the various values of a and r. Here. the firms face a static problem. I am not asking you to explicitly spell it out, but this problem is critical for the determination of the various prices. 5. (20) Consider the standard Mortensen-Pissarides model in continuous time. Labor force is normalized to 1. Unemployed workers, with measure u - 1, search for jobs, and firms with vacancies, with measure v, search for unemployed workers. The matching technology is given by m(u, v) = up -, It is convenient to define the market tightness o = /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 X. 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 : does not stand for the utility of leisure or the value of home production, as is conveniently assumed in the baseline model. Here, z 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 7, 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 theQuestion 1 (50 points) Consider a pure exchange economy with 2 islands. Each island consists of infinitely- lived identical agents whose measure is normalized to one. There is a single consump- tion good, or fruit, that is non-storable. The representative agent of island i values different consumption streams according to Vi, with o > 0. The total endowment in this economy (i.e., in both islands) in period f is given by a deterministic sequence {}fo, with a > 0 for all t. However, due to the location of the two islands, weather conditions are different affecting the fruit yield on each island. Letting each period t denote a season, we assume that in even periods the fruit yield on island 1 is given by el = aer, a e [0, 1], and in odd periods it is given by el = (1 - a) e. Of course, by definition, the endowment of fruit on island 2 must be given by ef = e - er, in all periods. a) For this economy define an Arrow-Debreu equilibrium (ADE) and a Sequential Markets equilibrium (SME). bj Fully characterize (i.e., find a closed-form solution for) the ADE prices. c) Using any method you like, characterize the SME consumption allocation in as much detail as you can.' In the remaining questions, we will assume that the total endowment in the economy (in both islands) follows the process e = ye, with y > 0. d) Given this new parametric specification, provide a closed-form solution for the SME consumption allocation. e) What happens to the consumption allocation you calculated in part (d) when y = 1 (no growth in the endowment)? What if (y = 1 and) a = 1? () Back to the model with general a, y; can you specify parameter values for which agents on island 1 consume more than agents on island 2 in a typical period ? g) For what parameter values do the ADE prices you calculated in part (b) increase as a function of t? Provide some intuition for your result.Briefly explain how and why a reduction in government spending causes a fall in output in this flexible price model. (Hints: start by combining equations 4, 5, 6 and 8. Also note that the real wage is constant in the flexible price model given the constant marginal product of labor). b) The full sticky price model can be simplified to 3 equations (and equation (10)): ElJet - 01 = (1 - Entit - 190 (1 -D)(1 -p)g.) (11) it = BE(# +1) + ky: (12) (13) Using the method of undetermined coefficients, find the response of the output gap and inflation to an exogenous decrease in g, when prices are sticky and monetary policy follows the Taylor Rule above. Guess that the solution for each variable is a linear function of the shock g. Is the fall in output larger or smaller in this sticky price model (than in part (a))? Explain. c) Instead of following the Taylor Rule above, monetary policy is now set optimally. Derive the optimal monetary policy rule under discretion. Assume the steady state is efficient. (Hint: As in class, assume that the loss function has quadratic terms for the output gap and inflation, with a relative weight of on the output gap). What is the optimal path for output and inflation following the reduction in government spending? d) Suppose the monetary policymaker wants to implement the optimal policy from part (c) using an interest rate rule for 4. The policymaker is considering a rule which sets is equal to 4:"(1 -P)(1 - p)gt, the natural real interest rate in this model. Briefly explain why this will not work. Furthermore, suggest a modification to the proposed rule that would successfully generate the outcomes in part (c). (Hint: you do not need to derive anything. You should be able to answer this from your knowledge of the model) e) Now consider two modifications to the model: (i) government spending provides utility where the utility function is: 7 + xlog Go - 74, (ii) the decrease in govern- ment spending is initially used to lend money to households rather than to cut taxes (although, over time, the government reverses this policy and cuts lump sum taxes later). Discuss how these changes might affect your answers in parts (a) and (b). You do not need to derive anything, just explain the economic intuition