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Question 3 (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 (1989) (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 # 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 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 : u'(q') = 1}. The only difference comapared 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, but there exits a storable and recog- nizable object, fiat money, that can serve as a MOE in type-1 meetings. Money sup- ply is controlled by a monetary authority, and we consider simple policies of the form Miti = (1 + p)Me, p > 8 - 1. New money is introduced, or withdrawn if p 0. Notice that this specification implies q' = y, "(q)-1 = y-q, and u(9") -q' = 7. e) Given this utility specification find closed-form solutions for go, 91- () For any y, i, with i 0).' 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(q1) - q]. g) Since a is the fraction of type-0 sellers, who accept credit, and since in type-0 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, a], we have Ow/80 1. Government spending follows an AR(1) process 9 = p91-1 +er (12) a) Show that the natural level of output can be written as Explain the mechanism through which an increase in government spending leads to an 7 increase in output in this model with flexible prices (similar to the RBC model). (Hint: start by combining equations 5, 6, 7 and 10 and note that, under flexible prices (1 =0). For the rest of this question note that this model can be simplified to the familiar 3 equations: (13) At = BE(# 1 ) + Ky (14) (15) Plus the process for government spending 9 = p91-1 + et (16) and a definition of the natural real rate of interest * = =(1-r)(1 -p)g (17) b) Using the method of undetermined coefficients, find the response of the output gap and inflation to an exogenous increase in g, when prices are sticky and monetary policy follows the Taylor Rule above. To do this, guess that the solution for each variable is a linear function of the shock gf: i = Ag.Question 6 (10 points) This question is about the standard decentralized real business cycle model. You do not need to derive anything for this question and keep your answers clear and concise. a) Briefly explain the mechanisms through which TFP shocks affect output, con- sumption, hours worked and investment in the standard RBC model. How well does the model replicate the business cycle facts seen in the data? How would adding habits in consumption affect the dynamics of consumption and investment? b) Suppose you want to solve the model using computational methods. Explain one approach, the advantages of this method and the steps you would need to take.1. Consider a pure exchange economy with two consumers (4 and B) and two goods (X and 1). The utility functions for the two consumers are represented by: U.(X,,Y.)= X, +Y, and U,(X,,Y, )= min(X,, Y,) where X, and Y are the amounts of the two goods consumed by person i. Suppose that the initial endowments owned by the two consumers are X, =80 X, = 20 Y, = 20 Y, = 30 a. Is the initial allocation of goods Pareto efficient? Explain. b. Derive the contract curve for this economy. C. What would the competitive equilibrium look like in this economy? d. Is the equilibrium calculated above Pareto efficient? Explain. e. Describe all initial allocations that lead to this same final allocation in the competitive equilibrium. f. Describe all initial allocations that lead to this same competitive price ratio as that obtained with the allocation provided at the start of this