Answer the following questions:

Question 1 Consider the sequence of mction {n,I {ani dened by a+1(1} = max {HEX -I'l +ualx'l} :r'E[tl,Jr] witlt lllix] = t], Under which conditions on u and ,8 do we know that x] = linings\Question 4 In the first problem-set of the course (Rendahl) we considered the following "income-fluctuation problem" max Eo _ B'ucci(v')) (1) 1=0 subject to c( ) + a+ (') = (1+ra(* )+> (2) where y, is the individual's stochastic endowment in period , and y' = (yo, VI, . . ., M) is the corresponding history. Assume B (1 + r) = 1. a. Let yi+1 = pyr + er, where a, is a zero mean random shock. What is the Bellman equation corresponding to (1)-(2) above? (Feel free to use the expectations operator instead of summing/integrating over possible events.) b. In the problem-set, we showed that if u (c) = ac - abc, the solution took the form C = It,(1 +r ) + E. > (1 tris Vits] (3) $=0 That is, consumption in period / equals the annuity value of total assets plus "perma- nent income" (a term coined by Milton Friedman). Find the consumption policy function that satisfies your Bellman equation in (a), and show that your (recursive) solution coincides with (3) (Hint: Use the (recur- sive) Euler equation to derive the policy rule - the value function is very difficult to recover.). c. What is the marginal propensity to consume, ay, 24? How does the MPC change with the parameter p? Interpret.Question 5 Consider the following sequence problem max (4) subject to dita = (ltra, + wor, 1=0, 1,... (5) a. Assume that n = r(x), wy = m(x,), and that *41 = (1 - p)i + pyr. What is the Bellman equation corresponding to the above sequence problem? b. Now assume for simplicity that r, = r, (a constant), but that my = aw, + fun-2. What is the relevant (the smallest sufficient) state vector? c. What is your answer to (b) if wret = Zi by' un-? d. How does your answer to (c) change if N = co? Note that in all problems associated with Question 5. there is an implicit assumption that there are sufficient givens in (4)-(5) in order to accurately predict future prices. E.g. in (i) it is implicitly assumed that xo is given, and so on. Question 6 Consider the following simple RBC model with capital adjustment costs in which output is a function of beginning of period capital and an i. i.d. stochastic technology shock: M = Ek, ; a