please guide on this questions thank you

Question 4 (20 points) Consider the planner's problem for a real business cycle model with inelastic labor supply (essentially the stochastic growth model) and no trend growth. Preferences are given by: In C (1) Output is produced using capital K Y1 = A.K (2) where K, is the capital stock at the start of period t, and A, is a TFP shock and is governed by a discrete state Markov chain. There are adjustment costs to capital, which evolves according to the following production function: (3) When 6 - 0, this becomes the simple model we saw in class with full depreciation (i.e. where K141 = It). The resource constraint is Y = G + h a) Write down the recursive formulation of planner's problem. Use two constraints: the typical resource constraint and the capital production function. Denote the Lagrange multiplier on the resource constraint as ), and the one on the capital production constraint as Age. b) Derive the first order conditions. Provide an intuitive explanation of q. c) Using guess and verify, solve the model and find the policy functions for K141, It and C: (Hint: start by guessing that investment and consumption are a constant share of output). d) Explain why we can interpret o as a parameter that affects the degree of capital adjustment costs in this model. Briefly discuss how, and why, the responses of consump- tion and investment to TFP shocks vary with 6. Why do some macroeconomic modelers prefer to include capital adjustment costs in their models?our assumptions, sellers will not carry any money as they enter the KW market). c) Describe the objective function of the typical buyer, J(m), where the hat denotes next period's choice." d) Describe the equilibrium variables go, q1 as functions of the model's parameters, including the nominal interest rate, i. How is q related to the real balances = = om?" To simplify the analysis hereafter, assume that u is quadratic: u(9) = -4 + (1+7)9. y > 0. Notice that this specification implies q' = y, u'(q)-1 = y-q, and u(q") -q' = 7. e) Given this utility specification find closed-form solutions for go, 91. f) For any y, i, with i > ().7 Finally, define the welfare function of this economy as the measure of the various KW market meetings times the net surplus generated in each meeting, i.e., W = ou(go) - 90] + (1 - o) [u(gi ) - 91]. g) Since o is the fraction of type-0 sellers, who accept credit, and since in type-( meetings the buyer is never liquidity constrained, intuition suggests that welfare should increase in o. It turns out that this intuition is wrong! Show that, for o e [0, o], we have OW/do B -1. New money is introduced, or withdrawn if u 0 per unit of time, i.e., p does not depend on the worker's type and p > 2, for all i. Also, while a firm is searching for a worker it has to pay a search (or recruiting) cost, pe > 0, per unit of time. All jobs are exogenously destroyed at rate > > 0 (again, independently of the worker's type). All agents discount future at the rate r > 0. Throughout this question focus on steady state equilibria. a) Based only on your knowledge of the environment (i.e., without analyzing the model), explain whether the following statement is true or false: "In this economy, all workers will be paid the same wage, since they are equally productive". b) Write down the value function of a firm with an unfilled vacancy (V) and the value function of a firm matched with a worker of type i (J.). c) Write down the value function of a worker of type i while unemployed (U.) and while employed (W.). d) Combine the free entry condition (i.e., V = 0) with the expressions for J, provided earlier in order to derive the job creation (JC) curve for this economy." e) Using the same methodoly as in the lectures (adjusted to accommodate the differ- ences in the new environemnt), derive the wage curve (WC) for this economy. f) Combine the JC curve and the WC curve determined in the previous parts in order to provide an equation that (implicitly) determines the equilibrium 0. g) Describe the equilibrium wage for a worker of type i (my;) as a fuention of the model's parameters and the equilibrium @ (which was implicitly determined in part f). h) Is w increasing or decreasing in z? What happens to the distribution of equilibrium wages as the distribution of workers' types becomes more dispersed? Hint: The right-hand side (RHS) of the JC curve should look identical to the one seen in class. On the left-hand side (LHS), instead of just w, you should have an expression that represents the average wage that a firm capecis to pay. "Hint: There will be two differences compared to the WC seen in class: 1) On the LHS, instead of just w, you should have the average wage that a firm expects to pay (the same as in part d); 2) On the RHS, instead of just z, you should have the average unemployment benefit, i.e., = 2i-1 7izi-Question 1 (10 points) Consider the standard deterministic 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({cho) = >B'(Inc + ylne-1). >0, 1=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, y = F(k, n) = kin-". This technology is owned by firms whose number will be determined in equilibrium. Output can be consumed (c) or invested (i,). 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 (x,) 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." 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 f -+ oo