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 f 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 incon t inuoust imeandw ithanin f 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 .L e t 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) ,a sg iv enand 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 f rst order condition for the static optimization problem that appears in the de f nition of the Hamiltonian. c) Next, write down the two di f erential equations that, according to the maximum principle, must be satis f ed by the problem’s solution. d) Combine your results from parts (b) and (c) above to obtain a system of two di f 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 ) rema insconstantatsomeva lue 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 f nite, so that periods are indexed by t =0 , 1 , 2 ,... .Le t 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 β t u ( c
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This note was uploaded on 02/19/2010 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.

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exams9908 - Final Exam Economics 720: Mathematics for...

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