please answer in respective order please

4. (20) Consider a labor search model with continuous time and an infinite horizon. There are some dif- ferences compared to the baseline DMP model we saw in class. First, the measure of firms (as well as that of workers) is fixed, i.e. there is no free entry. In turn, arrival rates will not depend on a matching function; they will be exogenous. What makes the model interesting is that workers who are currently employed also search for new jobs, i.e. this model has "on-the-job" search. At any moment in time each worker is either unemployed (state 0) or employed (state 1). At random time intervals, however, a worker receives a new job offer. Allowing the arrival rate to depend on a worker's state, let A;, i 6 {0, 1}, represent the Poisson arrival rate of offers while a worker is currently in state i. As workers are assumed to randomly search among employers, an offer is simply a realization of a random draw from F(w), the distribution of wage offers across employers with support in [w, w), i.e. F is the cdf. In this question you can treat F as given (I do not ask any questions regarding the behavior of firms; all the questions concern the behavior of workers). All jobs are identical except from the wage associated with them. Employed workers move from lower to higher paying jobs as the opportunity arises, and, of course, unemployed workers move to employment when they agree to accept a specific offer. Also, existing job-worker matches are destroyed at an exogenous rate 6 > 0. This shock is common to all jobs and does not depend on the match-specific wage. Unemployed workers receive utility flow = 2 0 per unit of time. All agents discount future at rate r." (a) Write down formulas for the value function of an unemployed worker, Vo, and for a worker who is employed at wage w, Vi(w). (b) Provide a formula for the derivative of the value function Vi(w) with respect to w, i.e. Vi(we). What is the sign of this expression? (c) Use the result in part (b) to claim that the unemployed worker's optimal behavior involves a "reser- vation strategy", i.e. there exists a unique R such that the unemployed worker accepts a job offer, w, if and only if w 2 R. (d) Given that unemployed workers use a reservation strategy, show that the optimal R solves an equation of the form (1) where a1, 02 are parameters for you to determine. (e) Based on equation (1), derive an expression that links R (implicitly) only to the model's parameters (including the cdf F). Hint: Basically you need to get rid of Vo. Vi in (1), and replace them with the term Vi. You can do this using integration by parts: recall that [F(x)G(2) = [ F(x)G(x)dr + [ F(x)G (x)dr, where you should let G(x) = [Vi(x) - Vol. Once you do this, you can replace V, with the expression that you found in part (b). (f) Let do = A1. Provide a closed form solution for the optimal R. (g) Assuming that the measure of workers is normalized to the unit, describe the steady-state equilibrium rate of unemployment, u, as a function of parameters of the model and the optimal R.3. (20) Consider a standard RBC framework with Cobb-Douglas technology and a 100% capital depreciation rate. Specifically, output is given by: Uncertainty is introduced into the economy via preference shocks which affect the disutility of working. The one-period utility function has the following functional form: U (q, hi) = In 4 - 817 1 where of is consumption and he is labor with 6, the stochastic shock to preferences. It is assumed that is distributed i.i.d. with c.d. f. given by G () and p.d.f. given by g (0). Each period agents choose labor, consumption and capital in order to maximize expected lifetime utility: (a) Given this environment, define a recursive competitive equilibrium for this economy. (b) Solve the model as a Social Planner problem. In particular, determine the functions that define equilibrium consumption, labor and investment. (Hint: First examine the conditions describing the labor market and then use these insights to solve for the goods market equilibrium.) (c) Describe the business cycle properties implied by this model. Which are consistent with observation? Many RBC modelers prefer these preferences to those found in a Hansen-Rogerson type indivisible labor model. Why?1. (10) In his highly acclaimed book, Capital in the Twenty-First Century, Thomas Piketty claims that, if the rate of return on capital exceeds the per-capita growth rate of the economy, then capital's share of income will grow. Is this statement consistent with the Ramsey-Cass-Koopmans optimal growth model as studied in class? 2. (20) An economy is populated by identical, infinitely-lived agents (there is no population growth) that maximize the present discounted value of lifetime utility given by Song + A(1 - me - st)]; BE (0,1), A>0 where q denotes consumption, he is time spent in work activity and s is time spent in acquiring con- sumption goods, i.e. s, denotes shopping time. (Note that the above functional form implies that utility is linear in leisure.) It is assumed that shopping time is an increasing function of consumption but a decreasing function of real balances. That is, holding money reduces shopping time. It is assumed that this function is linear in the level consumption and the log of real balances: $t = 0 - In 25: where M, is money chosen at time t, and P is the nominal price level. At the beginning of the period, agents hold money from the previous period (M-1) and also receive new money from the government which is distributed as a lump sum transfer. Agents produce output (y) using a logarithmic production function with labor as the only input: y = In my. This is sold and the proceeds, along with the beginning- of-period money and monetary transfer, are used to purchase consumption and new money. The money supply is growing at the constant rate / implying Meth = Me(1 + p) where M is used to denote the aggregate money stock. It is assumed that / > 0. Given this environment, do the following: (a) Express the agent's maximization problem as a dynamic programming problem and derive and interpret the associated necessary conditions. (b) Define a steady-state equilibrium in this economy. (c) How do changes in / affect steady-state consumption, labor, leisure, and real balances? In particular, what is the relationship between money growth and utility (in steady-state)? (d) Suppose that one period nominal bonds are introduced into this economy. These bonds cost $1 at time t and return $ (1 + m) in period t + 1. Determine the steady-state nominal interest rate in this economy. (You can work directly from the Euler equation associated with one-period nominal bonds. You do NOT need to write down the dynamic programming problem.) (e) Now suppose that the monetary transfer in this economy is distributed as an interest payment on beginning-of-period money holdings with the interest rate equal to the nominal interest rate determined in part (d). Would this affect the relationship between money growth (#), steady-state real balances and utility as described in part (c)? Explain. (Again: you may work directly from the Euler equation for real balances in this new setting; it is NOT required to write down the new dynamic programming problem.)