017 016.. eco aug j

6. (10) Consider a standard growth model in discrete time. Throughout this question you can focus on the Social Planners 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., Nt = (1 + n) t . The representative agents preferences are described by U(fctg 1 t=0) = X1 t=0 t c 1 t 1 : The initial capital stock in this economy is K0, 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: Yt = F(Kt ; Nt(1 + g) t ); where It will be useful express all the variables into (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 answer

Consider the following decentralized real business cycle model with no trend growth. There is a continuum of households and the representative household's preferences are given by: ln (ct hCt1) 1 2 n 2 t (1) Where c is household consumption and C is aggregate consumption (which the household takes as given). The household is infinitely lived and maximizes utility subject to their budget constraint. The household budget constraint can be written as: ct + kt+1 (1 )kt = wtnt + r k t kt + t (2) where w is the real wage, n is hours worked, k is capital, r k is the rental price of capital and are profits from firms. The representative firm produces output using capital k and labor nt : yt = Atk t n 1 t (3) Total factor productivity A follows a Markov chain over the set A = {a1, .....aN } with transition probabilities given by pij . The aggregate resource constraint is Yt = Ct + It where upper case letters denote aggregate variables. a) Write down the household's problem in recursive form and write down the firm's maximization problem. Derive the household's first order conditions and the firm's optimal hiring rules. b) Carefully define a recursive competitive equilibrium. Take care to distinguish between the aggregate and individual state variables and explain any market clearing conditions. c) Linearize the consumption Euler equation you found in part (a) around the deterministic steady state. d) With reference to your answer in part (c) (if you can), discuss how a TFP shock affects consumption in this model. Would the dynamics of consumption be different if consumption preferences were given by ln (ct hct1)? (Hint: note the second term is now household consumption at t 1, not aggregate consumption. You also do not need to derive anything for this question). e) What is the labor supply elasticity in this model? Given this value, how well will the model match the data? Explain.

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 rms with vacancies, with measure v, search for unemployed workers. The matching technology is given by m(u; v) = u av 1a . It is convenient to dene the market tightness v=u. A large measure of rms decide whether to enter the labor market with exactly one vacancy. When a rm meets an unemployed worker a job is formed. The output of a job is p per unit of time. However, while the vacancy is unlled, rms have to pay a search or recruiting cost equal to pc per unit of time. In an active match (job), the rm pays the worker a wage w per unit of time, which is determined through Nash bargaining when the two parties rst match. Let represent the workers bargaining power. The destruction rate of existing jobs is exogenous and given by the Poisson rate . Once a shock arrives, the rm closes the job down. Subsequently, the worker goes back to the pool of unemployment, and the rm exits the labor market. Unemployed workers get a benet of z > 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 benet z 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 benets, 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 satised at all times. (a) Describe the Beveridge curve (BC) of this economy. (b) Describe the value functions for a vacant rm (V ) and a rm with a lled job (J). (c) What condition does J satisfy in equilibrium? Use your answer, together with your ndings 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 governments budget constraint is satised at all times? Use this condition in order to get rid of T in the WC. (g) In the baseline model, it was easy to characterize the equilibrium values of (; 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 (and other parameters)? Hint: Which equilibrium condition gives you u as a function of ? (h) Plot the JC and WC curves in the w; space. Do they have the standard shape? Use your graph to discuss existence and uniqueness of equilibrium. What is the intuition behind your ndings? (i) Assume that the government increases z. What is the eect of this policy change on equilibrium unemployment?