Question 1 {EH points] Consider the MortensenPissarides model in discrete time. The labor force is normalized at 1. Let 1.: denote the unemployment rate. There is a large number of rms who can enter the market and search for a worker. Firms who engage in search have to pay a fixed cost l: per period until they nd a worker. If in any given period a measure \"U of vacant rms search for workers, then the total number of matches created in the economy is given by no u. + v . Each vacant rm has one job opening. Within each match, the rm and the worker bargain [a la Nash} for the wage, to, with r} denoting the bargaining power of the worker. If they agree, they move on to production, which will deliver output equal to 1: per period. ll agents discount future at rate .3 IE {I}, I]. At the end of every period {after production has taken place}, existing matches get destroyed with probability 5.1 \"Mimi-'1' = So Ear this is just the standard model [in discrete time]. We now make two as- sumptions that depart from the baseline model. First, the unemployment benet, 3, does not represent utility of leisure or value of home production, as we conveniently assumed in class. Here, 2 is a payment made by the government and, naturally, this payment needs to be funded somehow. We assume that the government raises these funds by imposing a lump-sum {flat} tax 7' {per period} on every matched rm. Thus, the government chooses both 2 and T, and must do so in a way so that the budget constraint is satised at any i. The second assumption concerns the duration of unemployment benets. In partic ular, we will assume that workers are eligible for unemployment benets only for one period.a {This assumption would be quite realistic for the US, if we were to assume that a period of the model corresponds to 5 months.} a} Describe the Beveridge curve {EC} of this economy in steady state, i_e_, express a as a function of the market tightness H E \"cf-u. b} This model predicts that a certain level of unemployment will persist even in the steady state. What is perhaps a little more subtle is that workers who are currently in the pool of unemployment have been unemployed for different periods of time. This is especially relevant in our question, where unemployment benet eligibility depends on the duration of unemployment. Describe the measure of workers who have been unemployed for 1'. periods, 1;: {1,213, ...}.3 Verify that your result is correct by adding up the various unemployment durations. {They should add up to the steady state a!) c] Describe the value function for a vacant rm {1"} and a rm that has lled its vacancy {J}. d} Describe the value Function of a typical worker in the various states. e] Exploiting the usual Free entry argumentJ derive the job creation {JG} condition. f] Describe the wage curve [WC] in this economy. g} Wl'lat is the relationship between T and 2,1: so that the government's budget constraint is satised in every period? Use this condition in order to get rid of r in the WC and JG expressions you derived earlier. h} Plot the JC curve in the {1913] space. Does it have the standard shape? i] Plot the WC in the {1913] space. Does it have the standard shape? j} Shortly discuss the existence and uniqueness of a steady state equilibrium. Question 2 {EH points] This question studies the oo-eicistenoe of different currencies. Time is discrete with an innite horizon. Each period consists of two subperiods. In the day, trade is bilateral and anonymous as in Kiyotaki and Wright [1939} {call this the KW market]. At night trade takes place in a Walrasian or centralised market [call this the CM}. There are two types of agents, buyers and sellers, and the measure of both is normalized to l. The per period utility for buyers is uh} + DIX} 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 1:, U satisfy the usual assumptions; I will only spell out the most crucial ones: There exists X' E {[1, oo] such that U'{X'} = I, and we dene the firstbest quantity traded in the KW market as If E {g : it"[q'] = 1}. We 1will assume that there are two types of money, 1711 and 1112.4 There are also two types of sellers. For reasons that we will leave out of the model, Typel sellers, with measure r.:Ir E [13,1], do not recognize mag, thus, they accept only the local currency as]. Type2 sellers, with measure I or, recognize and, hence, accept mg, as well as m]. I-Ienoe, local currency has a liquidity advantage over the foreign one, since it is recognised by all sellers. All buyers meet a seller in the KW market, so that or is the probability with which a buyer meets a type-l seller, and l or is the probability with which she meets a type-2 seller. In any type of meeting, buyers have all the bargaining power. The rest is standard. Goods are non storable. The supply of each money is con- trolled by an individual authority, and evolve: according to Mil+1 = {I +Fi}Mi,i~ New money {of both types] is introduced, or withdravm if p,- 4: El, via lump-sum transfers to buyers in the CM. Throughout this question focus on steady states. a} Describe the value function of a buyer and a seller who enter the Walrasian market with arbitrary money holdings {1111,1112}. b} Desribe the terms of trade in each type of KW meeting. c] Describe the objective function of the typical buyer, J{m',, mg}. d} For any given {#1, {.52}, pr.- 2 ,8 I, for all i, describe the steadystate equilibrium, summarized by the variables {111,92, 31,313}, where g.- is the amount of special good traded in a KW meeting with a seller of type i = {1, 2}, and z.- denotes the equilibrium Question 3 {29 points] Consider the social plannerls problem for a real business cycle model. The house- hold makes consumption {C} and leisure [1 N, where N is hours worked} decisions to maximise lifetime utility: EuZEIIChl-Nd [1} [:0 Specic functional forms will be given below. Output is produced using capital K and labor N v, = Zfo'\" E2} .3; is a TFP shock and is governed by a discrete state Markov chain. Capital evolves: KH1= {lE]H}+I [3} but assume full depreciation so I5 = 1. There is no trend growth. Finally1 Y: = Ci + I: First suppose that the utility Function 1.: is as follows: 2 In (C: %) [4} a} Write down the recursive formulation of plannerls problem and derive the rst order conditions. b} Using guess and verify1 nd the policy functions for investment, consumption and hours worked (Hint: rst consider the equilibrium condition for hours worked and guess that investment is a constant share of output}. Now suppose the utility function is given by: 2 Inc. % [5} c] Repeat parts [a] and {b} using these new prerenoes. d} Compare the business cycle properties implied by these two models and explain how and why a TFP' shock might affect output1 consumption, investment and hours worked. Some REC modelers prefer preferences used in parts {afb} to those in part