midterm-answers - MIDTERM, GRADUATE MACRO, ECON 606...

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MIDTERM, GRADUATE MACRO, ECON 606 Jonathan Heathcote March 2nd 2006 Consider the following economy. There are two sectors: an apple sector, and an orange sector. Both sectors use land F and labor n to produce. The amount of land in each sector, F o and F a is f xed: F o = F a = F. The production technology is Cobb-Douglas: Y i = z i F θ i n 1 θ i i = a, o where 0 θ 1 .z i , which determines sector-speci f c productivity, is given by z o =7 0 + ( T 70) z a 0 ( T 70) where T { 60 , 61 , ..., 79 , 80 } is the average temperature in Fahrenheit in the period, and evolves over time according to a f rst-order Markov process de f ned by the transition probability matrix Π . In f ntely-lived consumers have identical preferences over a composite con- sumption good C and leisure l. Suppose, to start with, that labor markets are segmented: half of the population can only work on apple farms, the other half can only work on orange farms. Let superscripts denote the identity of workers / consumers: thus, for example, c o a denotes consumption of apples by workers who work on orange farms. Preferences for workers of type i { a, o } are given by E P t =0 β t u ( C i t ,l i t ) where u ( C i i )= u ( C i )+ v ( l i ) C i = μ ¡ c i a ¢ σ 1 σ + ¡ c i o ¢ σ 1 σ σ σ =1 σ 0 is the elasticity of substitution between apples and oranges. Assume the standard inequality constraints must be respected: c i a ,c i o i 0 , l i 1 . PART 1: Consider a planner who cares equally about apple workers and orange workers. 1
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1. Write down a recursive formulation of the planner’s problem. Take care to choose appropriate state variable(s). V ( T )= max n o ,n a ,c a a ,c o a ,c a o ,c o o 1 2 u μ ³ ( c o a ) σ 1 σ +( c o o ) σ 1 σ ´ σ σ =1 + v (1 n o ) ¸ + 1 2 u μ ³ ( c o a ) σ 1 σ c o o ) σ 1 σ ´ σ σ =1 + v (1 n 0 ) ¸ + βE | T [ V ( T 0 )] subject to c a a + c o a = z a ( T ) F θ ( n a ) (1 θ ) c a o + c o o = z o ( T ) F θ ( n o ) (1 θ ) and the inequality constraints, where z o ( T )=7 0+ ( T 70) z a ( T 0 ( T 70) 2. What are the planner’s f rst order conditions? Assume none of the inequality constraints are binding. Then the FOCs simplify to u c o a = u c a a u c o o = u c a o (1 θ ) 1 2 u c o a z a ( T ) F θ ( n a ) θ = v 0 (1 n a ) (1 θ ) 1 2 u c o o z o ( T ) F θ ( n o ) θ = v 0 (1 n o ) 3. Compare the optimal consumption bundle for apple farm workers to the optimal consumption bundle for orange farm workers.
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midterm-answers - MIDTERM, GRADUATE MACRO, ECON 606...

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