Your help will be highly appreciated

Detailed explanation

Question 1 (20 points) Consider the Mortensen-Pissarides model in discrete time. The labor force is normalized at 1. Let u denote the unemployment rate. There is a large number of firms who can enter the market and search for a worker. Firms who engage in search have to pay a fixed cost & per period until they find a worker. If in any given period a measure v of vacant firms search for workers, then the total number of matches created in the economy is given by m(u, v) = - uv Each vacant firm has one job opening. Within each match, the firm and the worker bargain (a la Nash) for the wage, w, with n denoting the bargaining power of the worker. If they agree, they move on to production, which will deliver output equal to p per period. All agents discount future at rate B E (0, 1). At the end of every period (after production has taken place), existing matches get destroyed with probability 6.' So far this is just the standard model (in discrete time). We now make two as- sumptions that depart from the baseline model. First, the unemployment benefit, z, does not represent utility of leisure or value of home production, as we conveniently assumed in class. Here, z 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 T (per period) on every matched firm. Thus, the government chooses both z and 7, and must do so in a way so that the budget constraint is satisfied at any t. The second assumption concerns the duration of unemployment benefits. In partic- ular, we will assume that workers are eligible for unemployment benefits only for one period." (This assumption would be quite realistic for the US, if we were to assume that a period of the model corresponds to 6 months.) a) Describe the Beveridge curve (BC) of this economy in steady state, i.e., express u as a function of the market tightness 0 = v/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 1 A worker who was part of a match that got hit by the destruction shock at the end of period , will be unemployed for sure in f + 1 and will try to find a job again in (the beginning of ) period # + 2. 2 Consider again the worker described in footnote 1, i.e., a worker who was part of a match that got hit by the destruction shock at the end of period t. This worker will be unemployed for sure in t + 1 and will receive z. Starting in period + + 2, she will try to find a new job. If she is successful, she will move (immediately) into production. If she is unsuccessful, she will remain unemployed for another period, and, importantly, during that period she will not be eligible for unemployment benefits. This process will continue until the worker finds employment. 2 in the pool of unemployment have been unemployed for different periods of time. This is especially relevant in our question, where unemployment benefit eligibility depends on the duration of unemployment. Describe the measure of workers who have been unemployed for i periods, i = {1, 2, 3, ...}. Verify that your result is correct by adding up the various unemployment durations. (They should add up to the steady state u!) c) Describe the value function for a vacant firm (V) and a firm that has filled its vacancy (J). d) Describe the value function of a typical worker in the various states. e) Exploiting the usual free entry argument, derive the job creation (JC) condition. f) Describe the wage curve (WC) in this economy. g) What is the relationship between 7 and z, u so that the government's budget constraint is satisfied in every period? Use this condition in order to get rid of 7 in the WC and JC expressions you derived earlier. h) Plot the JC curve in the (w, 0) space. Does it have the standard shape? i) Plot the WC in the (w, 0) space. Does it have the standard shape? j) Shortly discuss the existence and uniqueness of a steady state equilibrium.Question 2 [9 points} This question studies the co-existence 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 Kiyctalci and Wright [1939] {call this the KW market}. At night trade takes place in a 1Walrasian 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 uriq) +U(X]| H, and for sellers it is q+ U[X] H, where q is the quantity of the day good produmd 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 Chi, all agents have access to a technology that turns one unit of work into a unit of good. The functions a, U satisfy the usual assumptions; I will only spell out the most crucial ones: There exists X\" E (Um-e] such that UTX") = 1, and we dene the rst-best quantity traded in the KW market as q\" E {q : u''] = 1}. We will assume that there are two types of money, 1111 and 1113." There are also two type; of sellers. For reasons that we will leave out of the model, Type-1 sellers, with measure 9 6 [0,1], do not recognize mg, thus, they accept only the local currency m1. Typo- sellers, with measure 1 a", recognise and, hence, accept mg, as well as \"11. Hence, 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 o 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 evolves according to M.- ;+1 = {l +i.i,-]M.-,t. New money [of both types] is introduced, or withdrawn if p.- 4:\". ll, 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 (m1, my}. b] Desribe the ms of trade in each type of KW meeting. c] Describe the objective finiction of the typical buyer, J{m'1,m'g]. d] For any given (#1, pg}, p,- 3 ,31, for all 1', describe the steady-state equilibrium, summarized by the variables {9-1, I33, 21, 22}, where q,- is the amount of special good traded in a KW meeting with a seller of type i = {1, 2}, and z.- deuotes the equilibrium ' L'Ine possible interpretation is that this is a model of a Latin American economy, and m] is the local currency {e.g., the peso} while 1119 ii the US dollar. Gfoourae, this 'e a very simple modeL an one should not talus this suggestion too literally. real balances of money i = 1,2. {Hints For now, all you need to do is provide 4 equations, the solution to which yields the equilibrium values for our 4 variables.) e] I now ask you to characterize the equilibrium in more detail. To that end, let us assume that the Kw utility function it quadIatic, i.e., ufq} = {1 + m 5,3, which implies q" = 1:. For the various possible combinations of {smog}, ,u, 3 ,3 1, provide closed-5mm solutions for (ghq-g}. Does currency '2 circulate in this economy [i.e., is 22 L\": CI] in every equilibrium? If not, can you provide a condition on policy parameters that would guarantee 132 3 ? Question 3 (20 points) Consider the social planner's problem for a real business cycle model. The house- hold makes consumption (C) and leisure (1 - N, where N is hours worked) decisions to maximize lifetime utility: Eo Bru (Ct, 1 - N.) (1) 1=0 Specific functional forms will be given below. Output is produced using capital K and labor N (2) Z: is a TFP shock and is governed by a discrete state Markov chain. Capital evolves: Kit1 = (1 -6) K+ + h (3) but assume full depreciation so o = 1. There is no trend growth. Finally, First suppose that the utility function u is as follows: In (C - ME (4) a) Write down the recursive formulation of planner's problem and derive the first order conditions. b) Using guess and verify, find the policy functions for investment, consumption and hours worked (Hint: first 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: In Co - 2 (5) c) Repeat parts (a) and (b) using these new preferences. d) Compare the business cycle properties implied by these two models and explain how and why a TFP shock might affect output, consumption, investment and hours worked. Some RBC modelers prefer preferences used in parts (a/b) to those in part (c), why might this be the case? e) If 0 1, 0