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Question 2 (20 points) Consider the following extension of the Mortensen-Pissarides model in continuous time. Labor force is normalized to 1, but there are two types of workers, Type 1, with measure I E (0, 1), and Type 2, with measure 1 - #. The two types of workers have different productivities: when a type 1 worker is matched with a firm, she can produce p > O units of the numeraire good per unit of time, but when a type 2 worker is matched with a firm, she cannot produce anything (the type 2 is a total lemon). Although firms would clearly prefer to match with Type 1 workers, they can only observe the worker's type after they have matched. This implies that the matching process is "unbiased", i.e., when a firm matches with a worker, the probability that this worker is of Type 1, depends only on the relative measure of Type 1 workers in the pool of unemployed.' On the flip side, this assumption means that the arrival rate of jobs to a worker does not depend on her type (since firms cannot discriminate, even though they would like to). Once a match has been formed, the worker's type is immediately revealed. If the worker is of type 1, the two parties negotiate over the wage as in the baseline model (with B E (0, 1) denoting the worker bargaining power) and production starts right away. If the worker is type 2, clearly, there is no need for any negotiation, since there is no production and no surplus to split. In this case, by law, the firm must pay the worker a fixed wage um per unit of time (think of it as the minimum wage), until it can prove that the worker is a lemon. The firm will eventually be able to prove this in a court of law, but the court decision takes a random amount of time. Specifically, the decision of the court arrives at a Poisson rate a > 0. When the decision is made, the firm can (finally) fire the unproductive worker and stop paying her the amount wm- Let the measure of unemployed workers of Type i be up, and let the total measure of unemployed workers be u = uj + u2. There is a very large measure of (identical) firms that can enter the market and search for workers. A firm can enter the labor market with exactly one vacancy, and the total measure of vacancies v will be determined endogenously by free entry. A CRS matching function brings together unemployed workers and vacant firms, and, due to the "unbiased" matching technology assumed here, the total number of matches depends only on v, u, i.e., m = m(u, v), and it is increasing in both arguments. As is standard, let 0 = v/u denote the market tightness. 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 search (or recruiting) cost, pc > 0, per unit of time. Productive jobs are exogenously destroyed at Poisson rate A > 0, and, as already explained, unproductive jobs are terminated (through the legal process) at the rate a > 0. To avoid weird equilibria, assume that a > A. All agents discount future at the rate r > 0, and all unemployed workers enjoy a benefit z > 0 per unit of time. We will impose p > Wm > z. a) Describe the Beveridge curve (the relationship between unemployment, u;, and market tightness, #) for each type of worker.? Suppose there are a unemployed workers out there looking for jobs, and 75% of them are of Type 1. Then, conditional on meeting a worker, the probability that this worker is a Type 1 is 75%. 'Hint: This economy will have two Beveridge curves, one for each type. To find them equate the inflows and outflows out of the pool of unemployment for each type. 3 b) For i = 1, 2, use your findings in part (a) to define the fraction of Type i workers who are unemployed (i.e., the unemployment rate within the Type i population). Denoting this term by y, show that 71 Bu(a), BE (0, 1). Each household has an initial capital ko > 0 at time 0, and one unit of productive time in each period that can be devoted to work. Final output is produced using capital and labor, according to a production function, F, which has the standard properties discussed in class, most notably, it is increasing in both arguments and exhibits CRS. This technology is owned by firms (whose measure does not really matter because of the CRS assumption). Output can be consumed (c) or invested (). Households own the capital (so they make the investment decision), and they rent it out to firms. Let 6 E (0, 1) denote the depreciation rate of capital. Households own the firms, i.e., they are claimants to the firms' profits, but these profits will be zero in equilibrium. The function u also has the usual nice properties, which I will not spell out here since you will not need them explicitly. In this economy there is a government that collects taxes and (for simplicity ) throws the tax revenues into the ocean. The government can implement one of the follwoing two alternative taxation systems, let us call them System A and System B. System A is a proportional tax, 7 6 0, 1], on agents' capital income. In other words, if the government implements System A, it collects a fraction 7 of all the income that agents earn by renting out their capital to firms. System B is a proportional tax, 7 6 [0, 1], on agents' investment. In other words, if the government implements System B, it collects a fraction 7 of all the resources that agents choose to allocate into investment. a) Write down the problem of the household recursively, under both taxation systems. Pay special attention to the budget constraints. These constraints will not be the same under the two specifications. Also, notice that I am not asking you to define a RCE in detail; just state the representative agent's problem within a RCE environment. bj Describe the steady state equilibrium capital stock under taxation System A, for any given + E [0, 1). Denote this object by KA(7). c) Describe the steady state equilibrium capital stock under taxation System B, for any given 7 6 [0, 1). Denote this object by KE(7). d) Assume that F(K, N) = Ko Ni-s, a e (0, 1). Provide closed form solutions for the terms KA(T), KB(T), described in parts (b),(c)." e) Plot the terms KA, KB, calculated in part (d), against + 6 [0, 1] and in the same graph. Discuss shortly. Here, the firms face a static problem. I am not asking you to explicitly spell it out, but this problem is critical for the determination of the various prices. "Hint: Here, it is more convenient to work directly with F, ie., do not work with the auxiliary function f that we introduced in the lectures. 5 f) Describe the governement's total tax revenue in steady state under System B, TB. Plot To as a function of the tax rate 7 (this is the so-called Laffer curve). Discuss the shape (i.e., the monotonicity) of the Laffer curve for the various values of a and T.Question 2 (20 points) Consider an economy that consists of two islands, i = {1, 2}. Each island has a large population of infinitely-lived, identical agents, normalized to the unit. There is a unique consumption good, say, coconuts, which is not storable across periods. Although within each island agents have identical preferences over consumption, across islands there is a difference: Agents in island 2 are more patient. More precisely, the lifetime utility for the typical agent in island i is given by U. ((q).) = >s; In(c). where B E (0, 1), for all i, and &, > 81- Due to weather conditions in this cconomy, 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 i, 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. You can use any method you like, but I strongly recommend that you exploit Negishi's method. b) Describe the ADE prices in this economy. c) Plot the equilibrium allocation for the typical agent in island i, i.e., (C}pe i = {1,2), against t. Is there any period t in which & = @? If yes, please provide a closed form solution for that value of t.*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 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 : (q') = 1}. The difference compared to the baseline model is that there are two types of sellers. Type-0 sellers, with measure o c [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 Mitt = (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 = olu(go) - 90] + (1 - o)[u(q) - q]. g) Can you describe the sign of the term OW/do for the various values of o