Economics fr.

3. (20) Consider a standard Solow growth model that is augmented with labor migration. As is typical, the aggregate production function is given by Y = (AL)" Klo where Y is output, A is effectiveness of labor, L is labor, and K is capital. Also, as is typical, the law of motion for aggregate physical capital is given by K = sY -5k where K = dk/dt, s is the savings rate and 6 is the depreciation rate. The effectiveness of labor grows at the constant rate of g : A = gA. In addition to population growth (given by the rate "), the country experiences migration M so that L = (nL + M). Migrants bring no physical capital and assume that the migration rate is positively related to the capital per worker. In particular, assume that the migration rate m = M/L is given by m = blog (1 + k) where b > 0 and & = K/ (AL) is the capital per effective units of labor. Given this, do the following: (a) Derive the expression for & in this economy. Compare this to the expression in the standard Solow model. (b) As in the standard Solow model, analyze graphically the behavior of the economy in a graph with & on the horizontal axis. Let &*denote the balanced growth path level of & and compare &*in the economy with migration to that in the standard Solow model. (c) Analyze using a phase diagram the stability properties of the balanced growth path for the case with migration. (d) Linearize the & function around the balanced growth path, define the speed of convergence of the economy to its steady-state and calculate it. Compare the speed of convergence in this economy to that in the standard Solow model.4. (20) Consider the Mortensen-Pissarides model in continuous time. Labor force is normalized to 1. Un- employed workers, with measure a $ 1, and firms with one vacancy each and total measure o 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 ( = w/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 @ e (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 " > 0, and unemployed workers enjoy a benefit > > 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 functions of the typical firm at one of the three possible states: V (with an open vacancy), M (matched but still at the stage of training), and / (matched at the stage of production). Describe the steady state expressions for these value functions. (b) Define the value functions of the typical worker at one of the three possible states: U (unemployed), T ( matched but still at the stage of training), and W (matched at the stage of production). Describe the steady state expressions for these value functions. (c) Combine the free entry condition (i.e., V = 0) with the expressions that you provided for V, M, J in order to derive the job creation curve of the economy. (d) Using the same methodology as in the lecture (adjusted only to accommodate the differences in the new environment), derive the wage curve for this economy. (e) Provide a restriction on parameter values such that a steady state equilibrium pair (w, () 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 #7 Explain intuitively (but shortly). (g) Describe the Beveridge curve 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 unemployment?5. (20) Consider the standard growth model in discrete time. There is a large number of identical households normalized to 1. Each household wants to maximize life-time discounted utility U(also) = Bu(a), BE (0, 1). 1=0 Each household has an initial capital ko 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 CRS production function F. This technology is owned by firms (whose measure does not really matter because of the CRS assumption). Output can be consumed (q) or invested (4). Households own the capital (so they make the investment decision), and they rent it out to firms. Let & 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 is twice continuously differentiable and bounded, with a'(c) > 0, u"(c) 0, f"(r) 0 such that U'(X*) = 1. Goods are non storable, but there exits a storable and recognizable object, called fiat money, that can serve as a means of payment. The supply of money, controlled by the monetary authority, follows the process Mitt = (1 + #)My, and new money is introduced via lump-sum transfers to buyers in the CM. So far, this is just a description of the model we saw in class. What is different here is that only a fraction of buyers turn out to have a desire to consume the DM good in the current period; let us refer to these buyers as C-types (for consumption) and to the remaining 1- a buyers as N-types (for no-consumption). The shock that determines each buyer's type in every period is did A buyer learns her type after all CM trade has concluded but before the DM opens. To make things interesting we will assume that between the CM and the DM there is a third market, where C-types and N-types can meet and trade "liquidity", i.e., money. Let us refer to this market as the loan market (LM). The LM is a bilateral market for loans, where N-types, who may carry some money that they do not need, meet C-types, who may need additional liquidity. A CRS matching function f (x, 1 - x) brings the two types together. Importantly, the LM is not anonymous, so that agents can make credible (and enforceable) promises. Hence, when an N-type and a C-type meet, they mutually benefit from a contract specifying that the N-type will give i units of money to the C-type right away, and the C-type will repay d (for debt) units of the numeraire good in the forthcoming CM. After the LM trades have concluded (for the agents who matched with someone), C-types proceed to the DM, where they use money to purchase goods from sellers. Assume that all C-type buyers match with a seller. Notice that I have not said anything about the splitting of the various surpluses (i.e., bargaining), because this information will not be necessary for what I am asking here. Let W( ) be the CM value function of a buyer, and V(.) the DM value function of a C-type buyer (since only these buyers visit the DM). Also, let ?'(.) be the LM value function of a type-i buyer, be {C, N). Your task in this question is to describe these value functions. I am not asking you to analyze them. I recommend that you draw a graph summarizing the timing of the model. (a) Describe the function W(.), and show that it is linear in all its arguments/state variables (what these arguments are, however, is for you to determine). (b) Let (q. p) be the quantity of good and the units of money exchanged in a typical DM meeting. Let (1, d) be the size of the loan (in dollars) and the promised repayment (in terms of the numeraire) specified in a typical LM meeting. What variables do the terms q, p, I, d depend on? Hint: Provide quick answers of the form "q is a function of the money holdings of the (C-type) buyer". (c) Describe the function V(.), where, again, determining the state variables is your task. (d) Describe the functions ?'(m), ie {C, N), for a buyer who enters the LM with m units of money. Hint: Recall that some buyers (of either type) will match in the LM and some will not, and the outcome of the matching process will critically affect a buyer's continuation value. Make sure that this is reflected in the expression you provide.1. (10) Briefly discuss the following statements (keep your answers short and concise): (a) The consumption-based capital asset pricing model is inconsistent with high volatility of stock prices. (b) In standard real business cycle models, the MPK is highly procyclical. This implies that interest rates (i.e. real) will be as well. 2. (20) An economy is populated by identical, infinitely-lived agents (there is no population growth) that maximize the present discounted value of lifetime utility given by s'Inca BE (0,1) 1= 0 where c denotes consumption. Output is produced via a standard Cobb-Douglas production function: where ky denotes the beginning of period capital stock. (Implicitly it is assumed that labor supply is inelastically supplied by households to firms and that the labor input has been normalized to unity. Hence the labor market is ignored in this analysis.) In addition to consumption, households choose investment. This produces next period's capital stock using the following production function: where 6, denotes a stochastic depreciation factor. It is assumed that this shock, the only source of uncertainty in the economy, is an did. random variable. Given this environment, do the following: (a) Solve for the recursive competitive equilibrium by solving the associated social planner's problem. In setting up the dynamic programming problem, use two constraints: the typical resource constraint and the capital production constraint. Denote the Lagrange multiplier on the budget constraint as At while the Lagrange multiplier on the capital production constraint is given by the product of Aqr where of is the shadow price of capital (in terms of consumption). (b) Derive the associated Euler equations for the social planner problem. Give an intuitive explanation for the determination of qf. (c) Define and solve for the recursive competitive equilibrium in this economy. (Note: This is simplified by first using output (#) as a state variable and then employing the guess and verify solution method.) (d) Give an intuitive explanation for the behavior of consumption and investment in this economy