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.Problem 53. Which of these statements about time inconsistency are true? (a) Time inconsistency is minimized when individual actors are in some manner bound to follow through with a decision before they actually face making that decision. (b) Time inconsistency is minimized when individual actors have the maximum possible discretion to choose the best response to situations as they arise. (c) According to time inconsistency, whatever policy is optimal in one period is optimal in all periods. (d) According to time inconsistency, a policy that is optimal in the first period may no longer be optimal in the next period. (e) An alcoholic man drives by a liquor store on his way from work every day. If he decides to simply use his force of will to restrain his desire to enter the liquor store every time he drives by it, then he has solved his time inconsistency problem. (f) An alcoholic man drives by a liquor store on his way from work every day. If he decides to pick a different route home from work - along which there are no liquor stores - then he has solved his time inconsistency problem.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.PART A - MULTIPLE CHOICE QUESTIONS 1. Real GDP is nominal GDP adjusted for: A) double counting. B) changes in prices. C) population. D) imports. 2. What do a rubbernecking traffic jam and the paradox of thrift have in common? A) In both cases, individual behavior has large negative consequences for the whole of society. B) In both cases, seemingly bad behavior ends up harming everyone. C) In both cases, seemingly careless behavior leads to good times for all. D) In both cases, government intervention can only make matters worse. 3. Every year more and more purchases are made with credit cards on the Internet. Given this trend, all else equal, we would expect: A) the money demand curve to shift outward. B) the money demand curve to shift inward. C) a downward movement along a fixed money demand curve. D) an upward movement along a fixed money demand curve. 4. The course packet and the class lecture contrasted historical growth in real GDP per capita in the US compared to Argentina to. A) slower: Argentina does not have such severely cold winter weather B) faster; Argentina has a tropical climate with poor soil and tropical diseases C) slower; Argentina encouraged land ownership by new immigrants but the US "robber barons" owned all the land in the US D) faster; the US encouraged land ownership by new immigrants but in Argentina Spanish colonists had large land holdings 5. Sam, who is 55 years old and has been a steelworker for 30 years, is unemployed because the steel plant in his town closed and moved to Mexico. Sam is experiencing: A) cyclical unemployment. B) permanent unemployment. C) frictional unemployment. D) structural unemployment. 6. Planned investment spending is: A) actual investment in a period B) investment spending minus depreciation in a period. C) investment spending that businesses plan to undertake during a period. D) always equal to saving. 7. The marginal propensity to consume is: A) increasing if the marginal propensity to save is increasing. B) the proportion of total disposable income that the average family consumes. C) the change in consumer spending divided by the change in aggregate disposable income. D) the change in consumer spending minus the change in aggregate disposable income. 8. As a result of a decrease in the value of the dollar in relation to other currencies, American imports decrease and exports increase. Consequently, there is a(n); A) increase in short-run aggregate supply. B) decrease in the quantity of aggregate output supplied in the short run. C) increase in aggregate demand. D) decrease in the quantity of aggregate output demanded. 9. The money demand curve is: A) downward-sloping because the opportunitycost of holding money is inversely related to the interest rate. B) downward sloping because the opportunity cost of holding money rises as the Interest rate rises. C) downward-sloping because the opportunity cost of holding money rises as the interest rate falls. D) upward-sloping because the opportunity cost of holding money rises with the interest rate. 10. If technology advances, then: A) more output can be obtained from the same inputs. B) more inputs are needed to produce the same output. C) less output can be obtained from the same inputs. D) less output can be produced even with more inputs