Macroeconomic s explanation needed
Question 1 (20 points) Consider the Mortensen-Pissarides model in continuous time. Labor force is nor- malized to 1. Unemployed workers, with measure u 0 per unit of time, and while a firm is searching for a worker it has to pay a search (or recruiting) cost, pe > 0, per unit of time. Firms that are training their workers do not pay this cost (they are done recruiting). Productive jobs are exogenously destroyed at rate > > 0 (only productive jobs are subject to this shock; matches at the training stage cannot be terminated). All agents discount future at the rate r > 0, and unemployed workers enjoy a benefit z > 0 per unit of time. While at the training stage the worker does not receive an unemployment benefit (a trainee is not unemployed). a) Define the value function of the typical firm for all the possible states of the world it may find itself in. b) Do the same for the typical worker. c) Combine the free entry condition with the expressions you provided in part (a) in order to derive the job creation (JC) curve of this economy. d) Using the same methodology as in the lectures (adjusted to accommodate the differences in the new environment), derive the wage curve (WC) for this economy. 1 Hence, two parties who met at time, say, t are negotiating over an object that will be paid in the future (at time { + 1/a, in expected terms). But, as is always the case, the Nash Bargaining problem is to split the generated surplus as of time t. 2 e) Provide a restriction on parameter values such that a steady state equilibrium pair (w, 0) exists. Is it unique? (No need for a lengthy discussion.) f) What is the effect of a decrease in a on the equilibrium w and 0? Explain analytically and intuitively (but shortly). g) Describe the Beveridge curve (BC) of this economy by looking at the flows of workers in and out of the various states. What effect will the decrease in a (discussed in the previous part) have on equilibrium unemployment?Question 2 (20 points} This question studies the oo-existence of money and credit. Time is discrete with an innite horizon. Each period consists of two subperiods. In the day1 trade is partially bilateral and anonymous as in Kiyotaki and Wright (1991] (call this the KW market} At night trade tales 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 tor buyers is u(q] + U(X}I H, and for sellers it is q + DIX} H, where q is the quantity of the day good produced by the seller and consumed by the buyer, X is oonsmnption of the night good [the nunreraire], and His hours worked in the CM. In the CM, all agents have scores to a technology that turns one unit of work into a unit of good. The functions 11., U satisfy the usual assumptions; I will only spell out the most crucial ones: There exists X\" E [0, on} such that UTX") = 11 and we dene the rst-best quantity traded in the KW market as q" E {q : u'[q'] = l}. The di'erence compared to the baseline model is that there are two types of sellers. Type-0 sellers, with measure rr E [0,1], accept credit. More precisely. in meetings with a type-0 seller (type-[J meetings), no medium of exchange [MOE] is necessary, and the buyer can purchase day good by promising to repay the seller in the arthcoming Chef with numeraire good {this arrangement is called an IO U}. The buyer can premise to repay any amount (no credit limit]1 and her promise is credible (buyers never default]. Type-1 sellers, with measure 1 rr, never accept credit, hence, any purchase of the day good must be paid for on the spot [quid pro gun] with money. All buyers meet a seller in the KW market, so that a is the probability with which a buyer meets a type-0 seller. and 1 or is the probability with which she meets a type-l seller. The rest is standard. Goods are non storable. There exits a storable and reo- ognizable object, at money, that can sem as a MOE in type-l meetings. Money supply is controlled by a monetary authority1 and we consider simple policies of the form Mi+1 = [l + p]Mi, p: b ,3 1. Newsr money is introduced, or withdrawn if p. a: E], via lump-sum transfers to buyers in the CM. Let o denote the unit price of money {in terms of the numeraire}. In KW meetings buyers have all the bargaining power. a} Describe the CM value function of the typical buyer. [Be sure to correctly identify the state variables.) Show that this value funiction is linear in its arg1nnent(s]_ b] Describe the CM value functions of the typical seller of type (i and 1. As in part {a}, show that these value functions are linear. c} Let q,- denote the quantity of good traded in a type-j = 0,1 meeting. Let d denote the amount of numeraire the buyer promises to repay in the forthcoming CM in exchange for on, and let 1' denote the amount of money the buyer pays in exchange for 9'1. Describe the bargaining solution in a typical type-j = I], 1 meeting? d] Describe the objective function of the typical buyer, J(m'], where the prime denotes next periodls choices? e} Describe the equilibrium variables 515,511 as functions of the model's parameters, including the nominal interest rate1 2'. How is Q1 related to the real balanoe; z E om? f] Does a monetary equilibrium (i.e., an equilibrium with z :> [i] always exist? If not, describe the set of parameter values {including the policy parameter i} for which such an equilibrium exists. Finally. dene 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 = Iulqol - as] + {l - U)[u(tn) In]. g} Can you describe the sign of the mun anacr for the various values of o?' Question 3 (20 points} Censider a decentralized real business cycle model. The representative household choose; consumption (c1 and leisure (L = lN. where N is hours worked} to maximize (detrended) lifetime utility: s;s@m+gm\"4 m subject to the detrended household budget constraint: re + 1km = 10W; + (l Welling + rfrh + Fr: ['2] where u, is the degree of capital utilization and is also chosen by the household. in is the real wage, N is hours worked, it is capital, 1-" is the rental price of capital and r are prots from fin-cs. Detrended capital evolves as follows: Thi+1 = (1 51:11:)\"; + it where X,,'Xi_1 = y is the deterministic growth rate of labor augmenting technological change. Assume y = 1. Competitive rms produce output using capital services role, and labor M. The detrended production function is 9': = Aiiutkrlaierli where TFP, denoted by AI, is stochastic and follows an AR{1] procesu in logs. In steady state A = 1. Households own the capital stock, invest and decide on the degree of utilization. Utilization is costly and causes capital to depreciate more quickly, as indicated by the depreciation cost function dlu, where 6(1) = :5 in steady state. Fir-ms choose capital to rent from households, understanding that they will be able to utilize it to a certain degree. You can therefore think of the rm as choosing their demand for labor N, and their demand for capital services ugh. a} Write down the household's problem in recursive form and write down the rm's maximization problem. Derive the household's rst order conditions and the r-m's optimal hiring rules. b} Carefully dene a recursive competitive equilibrium. Talre care to distinguish between the aggregate and individual state variables and explain any market clearing conditions. c] By lineariring the equilibrium mndition governing the optimal degree of utiliza- tion, show that r=s+as+a_ss+:Ieuos+uesl m where E = g and 6' and 6" refer to the rst and second derivatives of the 6 function with respect to of in steady state. As usual1 variables with a hat denote percentage deviations from steady state. (Hint: once you have linearized the equilibrium condition governing the optimal degree of utilization, you can combine this with the linearized production rnctiml 3}, = iii + 1117:; + ah + (l [Ni]. d} Discuss how the addition of variable capital utilisation helps the REC model explain the business cycle facts in the data? If you can, explain why in the limit 5 = so and E = I3 correspond to the cases of no variable capital utilization and full variable capital utilization. e] Briey explain how you would solve this model using a. linearization-based method and how you would produce impulse response functions for the eiIects of a temporary one percent shock to
