4. (20 points) Consider the following sequence problem, = MAX Bu(a) (7) 120 1=0 s.t. t-1 WE= (10) i=0 Oo, Wo given (11) (a) What conditions do you need on 6 and y in order to ensure that the sequence {w0 90 Nzo is bounded?b) Use the logic of Theorem 1 and derive the Bellman equation associated with the sequence problem. There is no need for a formal proof, but make sure you state the logical steps clearly (i.e. "prove" it verbally).(c) What is aggregate (or average) consumption? Does it equal aggregate resources?3. (20 points) Consider the following economy. A continuum of individuals are born in period 0 with no resources. With idiosyncratic probability p, an indi- vidual will be employed in period 1, and with the complementary probability (1 - p) she will be unemployed. However, the agents have the opportunity of writing contracts with each other, promising the payment of some resources contingent on which state occurs. As they consume nothing in period 0, each individual's problem is given by max {pula) + (1 -p)u(ce)} s.t. 0 = pib1 + p2b2 C1 = bit w C2 = by where ci and c2 denotes consumption at the employed and the unemployed state, respectively. by denotes the quantity purchased of the asset (or contract) which pays bj units of the consumption good if the agent turns out to be employed. Similarly, by denotes the quantity purchased of the asset which pays by units of the consumption good if the agent turns out to be unemployed. (a) Is this a complete- or an incomplete markets economy?(b) What is the market clearing condition? And what are the market clearing prices