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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((cho) = [B'u(a), BE (0, 1). 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 (c) or invested (it). 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 u'(c) > 0, u"(c) 0, f"(z) 0 per unit of time, and while a firm is searching for a worker it has to pay a search (or recruiting) cost, pe > 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 2 > 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 J (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), I (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, 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 #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?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 )" K1- 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 n), 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 = 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 k on the horizontal axis. Let &*denote the balanced growth path level of & and compare k'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 k 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.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 B' Ince; BE (0, 1) where e denotes consumption. Output is produced via a standard Cobb-Douglas production function: where & 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 Li.d. 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 A while the Lagrange multiplier on the capital production constraint is given by the product of Age where q 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 q. (c) Define and solve for the recursive competitive equilibrium in this economy. (Note: This is simplified by first using output (y) 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