@ detailed solutions..

Question 1 (20 points) Consider the Mortensen-Pissarides model in continuous time. Labor force is nor- malized to 1. Unemployed workers, with measure a $ 1, and firms with one vacancy each and total measure v search for each other, and v is determined endogenously by free entry. A CRS matching function, m(u, v), brings together unemployed workers and vacant firms; m is increasing in both arguments. As is standard, let 0 = v/u denote the market tightness and q(0) = m/v the arrival rate of unemployed workers to the typical firm. What is different here compared to the baseline model is that a "match" and a "productive job" are not equivalent by default. When a worker and a vacant firm meet, the firm must train the worker before she can start producing. A formed match turns into a productive job at a stochastic rate, a E (0, too), so that 1/a can be thought of as the average time necessary for the training to be completed. Assume that the firm and the worker determine the wage level when they first meet (i.e., even before training starts), through Nash bargaining, with 3 6 (0, 1) representing the worker's power. However, the wage upon which they have agreed will only be paid to the worker when she starts producing." To close the model, we will make a few more standard assumptions. The output of a productive job is p > 0 per unit of time, and while a firm is searching for a worker it has to pay a search (or recruiting) cost, pc > 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. Hence, two parties who met at time, say, t are negotiating over an object that will be paid in the future (at time t + 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 #? 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 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?The (linearized) preference shock follows an AR(1) process: & = pa-1 + er (8) er is i.i.d. In percentage deviations from steady state: me, is real marginal cost, & is consumption, w, is the real wage, f is hours worked, y, is output. In deviations from 3 steady state: i, is the nominal interest rate, #, is inflation. A is a function of model parameters, including the degree of price stickiness." Assume that o. > 1, 0 > 0. To close the model, we will make a few more standard assumptions. While a firm is searching for a worker it has to pay a recruiting cost, c > 0, per unit of time. All agents discount future at the rate r > 0, and all unemployed workers enjoy a benefit > > 0 per unit of time. We further impose that = 31- Due to weather conditions in this economy, island 1 has a production of e > 0 units of coconuts in even periods and zero otherwise, and island 2 has a production of e units of coconuts in odd periods and zero otherwise. Agents cannot do anything to boost this production, but they can trade coconuts, so that the consumption of the typical agent in island , in period t, is not necessarily equal to the production of coconuts on that island in that period (which may very well be zero). Assume that shipping coconuts across islands is costless. (a) Describe the Arrow-Debreu equilibrium (ADE) allocations in this economy using Negishi's method. (b) Describe the ADE prices in this economy. (c) Plot the equilibrium allocation for the typical agent in island , i.e., (C!)fen, i = {1, 2), against t. Is there any period t in which & = 4? If yes, please provide a closed form solution for that value of t