Hello tutors am stuck with the following please help
5. (20) Consider the one-sided job search model that we studied in class. There is a continuum of identical workers whose measure is normalized to the unit. Workers maximize expected discounted utility and, for simplicity, let u(y) = y. The term y, takes the value w when the worker is employed at wage w, and = > 0 when the worker is unemployed. Workers who were unemployed in period t - 1 will receive a wage offer in period t. With probability 1/2 this offer will be drawn from a random distribution with CDF given by Fi(w) that has support on the set [0, w]. With probability 1/2 the offer will be drawn from another random distribution with CDF given by Fa(w) that has support on the set [0, w]. Assume that z 0, u"(q) 0 per unit. What is the economic meaning of i? What is the buyer's optimal choice of m?3. {211] In Kydland and Prescott's original REC model, they made the assumption that investment does not produce capital immediately; i.e. the economy exhibited \"time to build\". Consider a representative agent [i.e. no population growth}, non-stochastic version of their economy and amume that agents have prefereners given by: issue +sln(1 at]; {=0 where q is consmnptiou and h; is time spent in work activity. Aggregate output 'm produced using a standard Cobb-Douglas production function: y: = ifhl'\" where y! is output and 1h denotes capital. {Note that there is no technological progress in the economy.) In each period, agents choose consumption, work effort and investment in order to maximize lifetime utility. In this economy, an investment project started at time t does not produce capital until period t + 2. The costs associated with this project are spread out evenly over the two-period horizon. Hence, when an investment project is started at time t, the agent is committing to an equal expenditure in period t+ 1. Let 3.} denote investment expenditures on a project that is nished after 2' periods {1' = 1, 2}. Then total investment expenditures at time t are given by: il=31t+sst ill The law of motion for the capital stock is given by: ki+1=hi1-~'5l+31t {2] Given this environment, do the following: {a} Express the amociated social planner problem for this economy as a dynamic programming problem. Be explicit in identifying the states and control variables in each period [along with the laws of motion for the state variables]. [Note it is easiest to write the law of motion for capital as an additional constraint. Also, note that because of the timehto-build feature, the price of capital will not be equal to 1. Let the shadow price ofcapital be denoted qt.) (b) Derive and interpret the necmry conditions emaciated with an optimum. [c] Solve for the steady-state output-capital ratio, the investment-capital ratio, and the ratio of time spent in work activity to time spent in leisure as a function of the exogenous parameters. Also solve for the steady-state value of j. Interpret this result
