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exams9908 - Final Exam Economics 720 Mathematics for...

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Final Exam Economics 720: Mathematics for Economists Fall 1999 This exam has three questions on three pages; before you begin, please check to make sure that your copy has all three questions and all three pages. Each question has fi ve parts. Each part of each question is worth 6 points, for a total of 6 × 5 × 3 = 90 points. 1 The Maximum Principle: Optimal Growth This question asks you to apply the maximum principle to solve a social planner’s problem in continuous time and with an in fi nite horizon. Consider an economy in which output is produced with capital during each period t [0 , ) according to the production function F ( k ( t )) = k ( t ) α , with 1 > α > 0 . Let c ( t ) denote the consumption of a representative consumer, and let δ denote the depreciation rate of capital, where 1 > δ > 0 . Then the capital stock evolves according to k ( t ) α δk ( t ) c ( t ) ˙ k ( t ) for all t [0 , ) . The social planner takes the initial capital stock, k (0) , as given and chooses functions c ( t ) for t [0 , ) and k ( t ) for t (0 , ) to maximize the representative consumer’s utility, given by Z 0 e ρt c ( t ) 1 σ 1 σ ¸ dt with ρ > 0 , σ > 0 , and σ 6 = 1 , subject to the constraint governing the evolution of k ( t ) for all t [0 , ) . a) To solve the social planner’s problem using the maximum principle, begin by setting up the Hamiltonian. b) Write down the fi rst order condition for the static optimization problem that appears in the de fi nition of the Hamiltonian. c) Next, write down the two di ff erential equations that, according to the maximum principle, must be satis fi ed by the problem’s solution. d) Combine your results from parts (b) and (c) above to obtain a system of two di ff erential equations involving c ( t ) and k ( t ) that describe the problem’s solution. 1
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e) Your results from part (d) above should imply that starting from any initial k (0) , the economy will converge to a steady state in which k ( t ) remains constant at some value k and c ( t ) remains constant at some value c . Use your results from part (d) to derive solutions for the steady-state values k and c in terms of the model’s parameters: α , δ , ρ , and σ . 2 Stochastic Dynamic Programming: Saving with a Random Return This problem asks you to use dynamic programming to solve a representative consumer’s problem when savings earn a random rate of return. Let time be discrete and the horizon be in fi nite, so that periods are indexed by t = 0 , 1 , 2 , ... . Let A t denote the consumer’s assets at the beginning of period t . During each period, the consumer divides these assets up into an amount c t to be consumed and an amount s t to be saved. The consumer’s savings earn interest at the gross rate R t +1 , where R t +1 is random, possibly serially correlated, and does not become known until the beginning of period t + 1 . Thus, the consumer must choose s t before knowing the realized value of R t +1 . The consumer takes the initial stock of assets A 0 as given and chooses sequences { c t } t =0 and { A t } t =1 to maximize the expected utility function E 0 X t =0
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