@@explain the answers.

1. (10) Briefly discuss the following statements (keep your answers short and concise): (a) Provide an intuitive - but concise - explanation for how the existence and uniqueness of the value function as defined by the Bellman equation associated with the standard growth model was estab lished. In your answer, be sure to identify the metric space used in the analysis. (b) Within the context of the representative agent consumption-based capital asset pricing model, discuss the factors that affect the equilibrium level of the yield on risk-free, one-period bonds. 2. (20) Consider a simple, representative agent RBC model in which output, yr, is produced via a standard Cobb-Douglas production function: where ky denotes beginning-of-period capital, he is labor, and & is an Li.d. technology shock. The depre- ciation rate of capital is 100%. In each period, agents make consumption and labor decisions in order to maximize lifetime expected utility: Within this environment, consider two variations defined by the functional form for U () . (a) In Economy A, agents have preferences given by: U (c,h) = Inc - h; In this economy, do the following i Express the maximization problem as social planner problem and write down the associated Bellman equation. ii. Solve for the equilibrium policy functions describing consumption, investment and labor. (b) In Economy B, agents have preferences given by: U (q, h,) = In (c - -h;) i Express the maximization problem as social planner problem and write down the associated Bellman equation. ii. Solve for the equilibrium policy functions describing consumption, investment and labor. (c) Compare the equilibrium behavior in both economies and provide an explanation for the differences. 3. (20) Consider a variation of the Sidrauski monetary model with a constant population. Specifically, assume that the representative agent's maximize lifetime utility is given by: [gu (a) + V (1) ] where U () and V () are concave, twice-differentiable functions, q denotes consumption and My is money chosen in period t. Each period, agents use beginning of period nominal balances, the revenue from sales of output and a lump-sum monetary transfer to purchase consumption, investment and new money. In contrast to the Sidrauski model, both capital and money are used as inputs into the production process. Letting wr denote output, the production function is given by: where ='(.) 0, =(0) = 1, lim =(M;/P,) = 0. The function f (ki ) has standard properties. The money supply in this economy is growing at the constant rate , > 0 and capital depreciates at the constant rate of 6 0, u"(c) 0, f"(x) #'(Ing + ylna-1), 9>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, # = F(h , m) = kin)-". This technology is owned by firms whose number will be determined in equilibrium. Output can be consumed (q) 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 (r) fully depreciates at the end of a given period, i.e. 5 = 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. (Hint: Derive the Euler equation, and "guess and verify" a policy rule of the form kit1 = ght, where g is an unknown to be determined.) (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 t - co? (d) Express the ADE price of the consumption good in any period T' as a function of parameters of the model and the sequence of capital stock up to period T.Question 4 (20 points) This question considers a variant of the standard New Keynesian model where the government can now purchase a basket of goods Ge, which is completely funded by lump sum taxes. Assume that G, is not productive and does not provide utility. The linearized conditions are given below. In percentage deviations from steady state: & is consumption, to, is the real wage, fu is hours worked, y, is output, d, is real marginal cost and g, is government spending. In deviations from steady state: i, is the nominal interest rate, #, is inflation. " is the output gap (relative to the model with flexible prices): It = 0t - 95. Households (3) it = ou + vin (4) Firms (5) (6) it = BE(1 41) + Ky (7) and y come from household preferences. 1/o is the elasticity of intertemporal substi tution and * is the inverse of the Frisch elasticity. 1. Government spending, gr, follows an AR(1) process It = p9-1 + er (10) where e, is i.i.d. and 0 S p