Solve all the questions with step by step explanation

3. Transitional dynamics in the Ramsey-Cass-Koopmans model. Suppose the planner seeks to maximize the intertemporal utility function Advanced Macroeconomics: Problem Set #1 subject to the sequence of resource constraints Ci + Ki+1 = F(Ki, L) + (1 -6)K, 0 0. The production function has the Cobb-Douglas form Y = F(K, L) = AKOLI-a, 0 0 and the labor force L > 0 are constant. Let o = G/L, k = Ki/L, It = Yi/ L etc denote consumption, capital, output etc in per worker units. Suppose that the period utility function is strictly increasing and strictly concave. (a) Derive optimality conditions that characterize the solution to the planner's problem. Give intuition for those optimality conditions. Explain how these optimality conditions pin down the dynamics of c, and he- (b) Solve for the steady state values c', ", y" in terms of the parameters. How do these steady state values depend on the level of A? (c) Suppose the economy is initially in the steady state you found in (b). Then suddenly there is a permanent increase in productivity from A to A' > A. Use a phase diagram to explain both the short-run and long-run dynamics of q and , in response to this increase in productivity. Does q, increase or decrease? Explain. Now consider the specific utility function u(c) = log(c). (b) Log-linearize the planner's optimality conditions around the steady-state. Guess that in log- deviations capital satisfies and that consumption satisfies Use the method of undetermined coefficients to determine wu and vo in terms of model pa- rameters. How if at all do these depend on the level of A? Now consider the specific numerical values o = 0.3, 3 = 1/1.05, 6 = 0.05 and A = 1. (c) Calculate the values of wi and ver. Suppose the economy is at steady state when suddenly at * = 0 there is a 5% permanent increase in the level of productivity from A = 1 to A' = 1.05. Calculate the transitional dynamics of the economy as it adjusts to its new long run values. In particular, calculate and plot the time-paths of capital, output, and consumption until they have converged to their new steady state levels. (e) How if at all would your answers to parts (b) through (d) change if o was lower, say a = 0.5? Or higher, say o = 2? Give intuition for your answers.6. (10 points) Using first the aggregate supply and demand framework, and then the expectations-augmented Phillips Curve, show the effects of an initially unexpected, once-and-for-all increase in a. The level of the money supply (using AD-AS), and b. The rate of growth of the money supply (using PhC). In both cases, starting from the long-run equilibrium shown in the diagrams, show where the economy goes in the diagram, in both the short run and in the long run, clearly labeling both the location of the economy and any shifts of curves. In addition, in the spaces beneath the diagrams write a short paragraph describing what happens and why. Be sure to explain the role of expectations in the adjustment process in both cases.1. Solow model in continuous time. Consider the Solow model in continuous time with pro- duction function y = /(k) satisfying the usual properties, constant savings rate s, depreciation rate 6, productivity growth g and employment growth n. (a) Use the implicit function theorem to show how an increase in s affects the steady state val- ues k*, y', c'. Does this change in s increase or decrease long run output and consumption per worker? Explain. Now consider the special case of a Cobb-Douglas production function f(k) = ke. (b) Derive expressions for the lasticities of capital and output with respect to the savings rate d log k* dlogy* dlog s dlog s How do these depend on the curvature of the production function a? Explain. (c) Derive an exact solution for the time path k() of capital per effective worker. Now consider the specific numerical values a = 0.3, s = 0.2, 6 = 0.05, g = 0.02, n = 0.03. (d) Calculate and plot the time paths of k(t), y(t), c(t) starting from the initial condition k(0) = k*/2. How long is the half-life of convergence? (e) Now suppose that we are in steady state k(0) = k* when the savings rate suddenly increases to s = 0.22. Calculate and plot the time paths of k(), y(t), c(t) in response to this change. Explain both the short-run and long-run dynamics of k(t), y(t), c(t). What if instead the savings rate had increased to s = 0.30? Do you think these are large or small effects on output? Explain.Answer all questions, on these sheets in the spaces or blanks provided. In questions where it is appropriate, show your work, if you want partial credit for an incorrect answer. Point values of the questions are shown; there are a total of 85 points possible. 1. (10 points) In the long-run, closed-economy model of Mankiw's Chapter 3, compare the effects on GDP, Y, and on the real interest rate, r, of the policies listed below. That is, consider the model whose components are: Production Function: Y = F(K , I) (1) Wage: W = MPL = F, (K.[) (2 ) Consumption: C = C( Y - T) (3) Investment: I = 1(r) (4) Goods Market Equilibrium: Y =C+/+G (5 ) with endogenous variables Y, W, C, /, and r and exogenous variables K , I , T , G and implicit shift parameters for each of the functions. (Assume, as is explicit above but may seem odd below, that the capital stock, K , is not, in the time horizon of the model, changed by investment, I.) Now determine the effects on Y and r of the following four policies: Policy 1: Government increases its purchases, G , by $1 m, spending this on environmental cleanup. That is AG =1 and AK = AZ =AT =0. Policy 2: Government decreases taxes, 7 , by $1 m. That is AT =-1 and AK = AL =AG =0. Policy 3: Government offers a tax credit to firms, causing them to increase their level of investment, /, by $1 m for any given level of the interest rate. (Remember, this investment does not change the level of the capital stock, K .) That is A/ = 1 and AK = AL =AT = AG =0. Policy 4: Government spends $1 m directly increasing the capital stock, K , but having done so, continues with its levels of purchases and taxes unchanged. That is AK =1 and AZ =AT =AG =0