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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) = ko. (b) Derive expressions for the lasticities of capital and output with respect to the savings rate d log &* 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(f), y(), 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(), c(t) in response to this change. Explain both the short-run and long-run dynamics of k(t), y(), 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.2. Natural resource depletion in the Solow model. Consider a Solow model where output is given by the CRS production function Y(1) = K(OR(D)(A(OL(1) )1--$, 0 () R(t) = -OR(t) The rest of the model is as standard with constant savings rate s, depreciation rate 6, produc- tivity growth g and employment growth n. (a) Let gy (t) and gu(t) denote the growth rates of output and the capital stock. Derive a formula for gy (t) in terms of gr(t). (b) Let g; and g; denote the growth rates of output and the capital stock along a balanced growth path. Show that along any balanced growth path gp = of. Solve for this growth rate. (c) Does the economy necessarily converge to a balanced growth path? Explain. (d) Now suppose instead that resources R(t) grew in line with population, R(t) = n R(t). Compare the long-run growth rate of the economy with resource depletion from part (b) to the long growth rate of this alternative economy without resource depletion. What would make this gap between the growth rates large? Explain.Economics 101 Summer 2014 Answers to Questions on Consumer Theory 1. Darcie buys two goods: happy-face magnets and cheeseburgers. The price of a magnet is $1, and the price of a cheeseburgers is $2. Each month, Darcie spends all of her income and buys 30 magnets and 5 cheeseburgers. Next month, the price of magnet will fall to $0.50 and the price of a cheeseburger will rise to $5. Assume that Darcie's preferences are similar to that in Figure A.3. on page 150 of the textbook. a) Will Darcie be able to buy 30 magnets and 5 cheeseburgers next month? Yes, she haslito in income and 30 magnets 95 cheeseburgers will cost this amount. b) Will Darcie want to buy 30 magnets and 5 cheeseburgers at these new prices? This is a viable choice but her choice depends on her thatis preferences ( hall c) When the prices change next month, will there be an income effect and a substitution effect at work or just one of them? 4 No =) more d) If at the original prices, Darcie 's marginal utility of another magnet is 40, and the maphil marginal utility of another cheeseburger is 80, is she maximizing her utility? If not, how can she adjust her bundle so that she does maximize her utility?8.1 Exercise 7.2 The performance of a system with one processor and another with two processors will be compared. Let the inter-arrival times of jobs be exponentially distributed with parameter A. We consider, first, the system with one processor. The service time of the jobs is exponentially distributed with a mean of 0.5sec. 1. For an average response time of 2.5sec (the total time for a job in the system) how many jobs per second can be handled? 2. For an increase of A with 10% how much will the response time increase? 3. Calculate the average waiting time, the average number of customers in the server and the utilization of the server. What is the probability of the server being empty? Let us now compare this system with a system of two cheaper processors, each with a mean service time of 1 sec. 1. How many jobs can now be handled per second with a mean response time of 2.5 sec? 2. For an increase of A with 10% how much will the response time increase? 3. Calculate the average waiting time, the average number of customers in the server and the utilization of the server. What is the probability of both of the servers being busy?10 Exercise 6.6 There are K computers in an office, and a angle repairman. Each computer breaks down after an exponentially distributed time with parameter a. The me. per takes an exponentially distribured amount of time with parameter f. Only one computer is being repaired at a time, computers break down independently from the repair process, and repair times and lifetimes of the computers are Independent. 1. What is the probability that i computers are working of time {? 2. What is the average failure nate (be. the average number of computers that fail per time unit]? 1. What is the percentage of time a repairman is busy? 4. What is the percentage of time when all computers are out-of-order (he- ken] ! 5. How many computers should we have if we would like to have N computers to work on oneyet