,,answer all of them plz..

Question 3 (20 points) Consider the following decentralized business cycle model. The representative household makes consumption (C) decisions to maximize lifetime expected utility: Bo _ B' (In Ci) (1) subject to their budget constraint: (2) where w is the real wage, N is hours worked, K is capital, I is investment, * is the rental price of capital and II are profits from firms. As usual, 0 k, and hence a planner would be able to make all agents better off by reducing the capital stock in all periods. f) Suppose now that the agents are allowed to trade a useless, non-reproducible asset in fixed unit supply, which trades at the price p. We call this asset a "bubble" Argue that if p. > 0 and t+1 > 0 the agent must be indifferent between holding capital and the bubble asset, and derive the associated arbitrage condition. g) Show that if (14) holds, there exists a steady state equilibrium with p = pss > 0.Question 1 (10 points) Consider an pure exchange economy that consists of two islands, i = {1,2}. Each island has a large population of infinitely-lived, identical agents, normalized to the unit. There is a unique consumption good, say, coconuts, which is not storable across periods. Agents' preferences are given by u(c) = > A'In(c), Vi, 1=0 where of is consumption in period / for the typical agent in island i and B e (0, 1). The total endowment of coconuts in the economy in period { is given by the sequence {e, heo. such that er > 0 for all t. But due to weather conditions in this economy, we have e, = et, if t is even, and e, =0, if t is odd. (Naturally, e? = e -e;.) Agents cannot do anything to boost the production of coconuts, but they can trade coconuts, so that the consumption of the typical agent in island i, in period , is not necessarily equal to the production of coconuts on that island in that period. Assume that shipping coconuts across islands is costless. a) Define an Arrow Debreu equilibrium (ADE) and a sequential markets equilibrium (SME) for this economy. b) Assuming that e = 2 - e-', and using any method you like, characterize the ADE prices, {pro- Are prices increasing or decreasing in t, and why?Question 2 (20 points) This question studies the co-existence of money and credit. Time is discrete with an infinite horizon. Each period consists of two subperiods. In the day, trade is partially bilateral and anonymous as in Kiyotaki and Wright (1991) (call this the KW market). At night trade takes place in a Walrasian or centralized market (call this the CM). There are two types of agents, buyers and sellers, and the measure of both is normalized to 1. The per period utility for buyers is u(q) + U(X) - H, and for sellers it is -q + U(X) - H, where q is the quantity of the day good produced by the seller and consumed by the buyer, X is consumption of the night good (the numeraire), and H is hours worked in the CM. In the CM, all agents have access to a technology that turns one unit of work into a unit of good. The functions u, U satisfy the usual assumptions; I will only spell out the most crucial one al ones: There exists X* E (0, co) such that U'(X*) = 1, and we define the first-best quantity traded in the KW market as (' = (q : (q') =1}. The difference compared to the baseline model is that there are two types of sellers. Type-0 sellers, with measure o E [0, 1), accept credit. More precisely, in meetings with a type-0 seller (type-0 meetings), no medium of exchange (MOE) is necessary, and the buyer can purchase day good by promising to repay the seller in the forthcoming CM with numeraire good (this arrangement is called an IOU). The buyer can promise to repay any amount (no credit limit), and her promise is credible (buyers never default). Type-1 sellers, with measure 1 - o, never accept credit, hence, any purchase of the day good must be paid for on the spot (quid pro quo) with money. All buyers meet a seller in the KW market, so that o is the probability with which a buyer meets a type-0 seller, and 1 - o is the probability with which she meets a type-1 seller. The rest is standard. Goods are non storable. There exits a storable and rec- ognizable object, fiat money, that can serve as a MOE in type-1 meetings. Money supply is controlled by a monetary authority, and we consider simple policies of the form Mi+1 = (1 + p)M, p> 8 - 1. New money is introduced, or withdrawn if p 0) always exist? If not, describe the set of parameter values (including the policy parameter ?) for which such an equilibrium exists. 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 = ofu(go) - 90] + (1 - o)[u(m) - q]- g) Can you describe the sign of the term OW/do for the various values of a