st 017 econ

Consider an pure exchange 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. Agents' preferences are given by ui(c i ) = X t=0 t ln(c i t ), i, where c i t is consumption in period t for the typical agent in island i and (0, 1). The total endowment of coconuts in the economy in period t is given by the sequence {et} t=0, such that et > 0 for all t. But due to weather conditions in this economy, we have e 1 t = et , if t is even, and e 1 t = 0, if t is odd. (Naturally, e 2 t = ete 1 t .) Agents cannot do anything to boost the production of coconuts, 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. Assume that shipping coconuts across islands is costless. a) Define an Arrow Debreu equilibrium (ADE) and a sequential markets equilibrium (SME) for this economy. b) Assuming that et = 2 e t , and using any method you like, characterize the ADE prices, {pt} t=0. Are prices increasing or decreasing in t, and why?

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 (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.1 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 (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. 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 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 > .

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 i , show that 1 < 2. c) Let V, J1, J2 denote the value functions of a firm that is vacant, matched with a Type 1 worker, and matched with a Type 2 worker, respectively. Also, let Ui , Wi , i = 1, 2, denote the value functions of a Type i worker who is unemployed or employed, respectively. Describe these value functions. d) Exploit the free entry condition (i.e., V = 0) in order to provide the job creation (JC) curve for this economy.3 e) Describe the wage curve (W C) for this economy. f) Discuss shortly the existence and uniqueness of equilibrium (no need to go into great detail and formal proofs)

Consider the standard growth model in discrete time. There is a large number of identical households normalized to 1. Each household maximizea life-time discounted utility U({ct} t=0) = X t=0 tu(ct), (0, 1). Each household has an initial capital k0 > 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 (ct) or invested (it). Households own the capital (so they make the investment decision), and they rent it out to firms. Let (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, [0, 1], on agents' capital income. In other words, if the government implements System A, it collects a fraction of all the income that agents earn by renting out their capital to firms. System B is a proportional tax, [0, 1], on agents' investment. In other words, if the government implements System B, it collects a fraction of all the resources that agents choose to allocate into investment. a) Write down the problem of the household recursively, under both taxation systems.4 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. b) Describe the steady state equilibrium capital stock under taxation System A, for any given [0, 1]. Denote this object by K A( ). c) Describe the steady state equilibrium capital stock under taxation System B, for any given [0, 1]. Denote this object by K B( ). d) Assume that F(K, N) = Ka N1a , a (0, 1). Provide closed form solutions for the terms K A( ), K B( ), described in parts (b),(c).5 e) Plot the terms K A, K B, calculated in part (d), against [0, 1] and in the same graph. Discuss shortly. 4 Here, the firms face a static prob