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Question 3 {0 posits} Consider the standard growth model in discrete time. There is a large number of identical households normalized to 1. Each household maximises lite-time discounted utility \"Home - Z: aisle}. 3 e {at}. t-l] Each household has an initial capital in; is D at time ii. 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 CR5. This technology is owned by rms [whose measure does not really matter because of the CBS assumption}. Output can be consumed its} or invested {ii}. Households own the capital [so they make the investment decision}, and they rent it out to rms. Let 5 E [{l. 1} denote the depreciation rate of capital. Households own the rms. i.e.. they are claimants to the rms1 prots. but these prots will be acre in equilibrium. The inaction 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 Her simplicity] throws the tax revenues into the ocean. The government can implement one oi the iollwoing two alternative taxation systems. let us call them System A and System B. System A is a proportional tax. 1' E [0.1]. on agents' capital income. In other words. it' the government implements System A, it collects a fraction r of all the income that agents earn by renting out their capital to rms. System B is a proportional tax. 1' E [{l. 'i]. on agents' investmnnt. In other words. it' the government implements System B. it collects a fraction r of all the resources that agents choose to allocate into investment. a} 1Write 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 mider the two specications. Also. notice that I am not asking you to dene a RUE in detail; just state the representative agent's problem within a RUE environment. b] Describe the steady state equilibrium capital stock under taxation System it. For any given 1' E 113.1]. Denote this object by Kr]. c] Describe the steady state eqmlibrium capital steel: under taxation System E. For anyr given 1' E [0.1]. Denote this object by REIT}. d] Assume that FIR. N] - K" Nl'\". a E (0.1}. Provide closed term solutions For the terms KARLKHTJ. described in parts [b].[c}.' e] Plot the terms ELK}. calculated in part (d). against 1' E [0.1] and in the same graph. Discmei shortly. " Herc. the firms Enos a static problem. Iam not asking you to explicitly spell it out. but they problem is critical tor the determination of the various prices. J\"I'lintt Here. it is more onuwuient to ouch. directly with F. in" do not work with the auxiliary function I that we introduced in the lectures. t'] Describe the government's total tax revenue in steady state under System H. Ti]. Plot T3 as a function of the tax rate r [this is the so-called Latter curve}. Discuss the shape [i.e.. the monotoniCity} of the Lamar curve for the various values of o and r. 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 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 > 0 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 wm 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 u;, and let the total measure of unemployed workers be u = u1 + 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 > > 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 u 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. 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