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macro_s08 - Ph.D Preliminary Examination MACROECONOMIC...

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Ph.D. Preliminary Examination MACROECONOMIC THEORY Majors : Answer two (2) questions in Part I and two (2) questions in Part II. Points for each question in Part I are twenty (20), and points for each question in Part II are thirty (30). Minors: Answer THREE (3) of the five questions in Part I and ONE (1) question in Part II OR answer one (1) question in Part I and two (2) questions in Part II. Points for each question in Part I are twenty (20), and points for each question in Part II are forty (40). PLEASE MAKE YOUR ANSWERS NEAT AND CONCISE Make whatever assumptions you need to answer the questions. BE SURE TO STATE THEM CLEARLY.
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Macroeconomic Theory Spring 2008 Part I Short Questions 2
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Macroeconomic Theory Spring 2008 Question I.1 Define V ( h, k ) = sup { c t ,x kt ,x ht ,k t ,h t ,n t } t =0 β t log( c t ) s.t. c t + x ht + x kt Ak α t ( n t h t ) 1 α k t +1 = k t (1 δ k ) + x kt h t +1 = h t (1 δ h ) + x ht 0 n t 1 ( h 0 , k 0 ) = ( h, k ) given Consider some ( h, k ) IR 2 ++ . Show that there exists a function B : IR + IR + such that V ( λh, λk ) = B ( λ ) + V ( h, k ) for all λ > 0. 3
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Macroeconomic Theory Spring 2008 Question I.2 Optimal Consumption/Asset Allocation Consider a household that lives for a finite number of periods, T . Wealth, W t , at the beginning of the period is allocated towards current consumption, c t , purchases of a risky asset x t , and a risk-free asset, y t , according to c t + x t + y t W t . Wealth evolves according to W t +1 = Rx t + R f y t where R denotes the (gross) return to the risky asset and R f denotes the gross return to the risk-free asset. Assume that R is i.i.d over time and is drawn from a distribution F ( R ). The household’s preferences are given by T t =0 β t u ( c t ) where β is the discount function and u ( c ) = c 1 σ 1 σ . The household maximizes expected utility. (a) Set up the household’s problem as a dynamic program. (b) Financial planners recommend that households’ portfolios should become less risky as they age. Evaluate this recommendation. Hint : Guess at the functional form of the value function. Derive the household’s optimal asset allocation given this guess. 4
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Macroeconomic Theory Spring 2008 Question I.3 Search and Bargaining Imagine a firm has a worker and it can produce one unit of output per period for 10 periods (there is a zero interest rate). If the firm does not operate today, it loses its license and is out. The worker is risk neutral, can either take the job or go home and grow .1 of output this period and apply for a government resarch grant that yields .5 per period for 5 years but leaves her incapacitated for anything else than for growing thinks at home but that she gets for sure. (a) What is the minimum wage that the worker would accept. (b) Pose a wage that is the bargaining solution with the worker having twice the weight than the firm.
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