macro F09 - Ph.D. Preliminary Examination MACROECONOMIC...

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Ph.D. Preliminary Examination MACROECONOMIC THEORY Fall 2009 Majors and Minors : Answer ALL FOUR parts. Please make your answers neat and concise. Make whatever assumptions you need to answer the questions. Be sure to state your assumptions clearly.
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Macroeconomic Theory Fall 2009 Part I Answer the question in Part I. 2
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Macroeconomic Theory Fall 2009 Question I.1: Consumption/Savings with Production Risk Consider an economy with a continuum of consumers/entrepreneurs of measure 1, indexed by i . Each consumer has access to a production technology that uses capital and labor to produce a homogeneous good. This production technology is speciFc to the entrepreneur who owns the technology. Letting k it denote the amount of capital entrepreneur i has at date t , the production function is y it = z it f ( k it ,n it ) where z it is a random variable that is iid over time and across consumers, f is a constant returns to scale function and n it denotes the amount of labor hired by entrepreneur i .The economy has no aggregate uncertainty. Capital fully depreciates after one period. Entrepreneurs can invest their capital only in their own production technology and cannot rent/sell it to other consumers. In addition to capital, entrepreneurs trade among themselves one-period risk-free bonds. Let r t denote the interest rate on bonds purchased at date t. The proFts generated by entrepreneur i are given by π i ( k i ,z i ; w )=max n i z i f ( k i i ) wn i The consumer/entrepreneur’s problem is to maximize life-time utility max c jt ,k +1 ,b +1 E X t =0 β t u ( c ) subject to c + k +1 + b +1 = π i ( k ; w t )+ w t +(1+ r t 1 ) b where, note that the endowment of labor of each entrepreneur is normalized to be 1 . Assume u ( c )= c 1 γ 1 γ with γ> 1 . Let λ denote the initial distribution of assets ( k i 0 ,b i 0 ) across consumers. The market clearing condition for bonds is R b it di =0fora l l t. There is no constraint (other than the natural debt limit) on how much the consumer can borrow using the risk-free bond. We require however k +1 0 . Question I.1 continues on the next page. 3
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Macroeconomic Theory Fall 2009 Question I.1 continued: (a) Write the consumer’s problem recursively. Defne a recursive competitive equilibrium. (b) Characterize the consumer’s decision rules in the stationary steady-state: derive expressions relating the consumption and portFolio decisions to primitives oF the model. Show that these consumption rules aggregate into an expression For aggregate consumption as a Function oF the aggregate capital stock. Use this aggregation result to characterize the steady state level oF the capital stock. (Hint: You may fnd it useFul to let human wealth be defned as the present discounted value oF wages.) (c) Does β (1 + r ) = 1 in this economy?
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macro F09 - Ph.D. Preliminary Examination MACROECONOMIC...

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