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Unformatted text preview: Macro Prelim 2010 Each question should be answered with a clear and concise explanation. Yesno questions should be answered with a supportive argument or counterexample. Please write legibly. 1) Consider the following continuoustime dynastic model. Each household has a population L ( t ) = exp( nt ) for a constant n and chooses consumption per household member C ( t ) to maximize , )) ( ( ) ( ) exp( ∫ ∞ = dt t C u t L t U ρ where γ γ = 1 1 1 ) ( C C u , and the constants ρ > 0 and γ > 1. The household earns labor income W ( t ) L ( t ) and is initially endowed with capital K (0). Capital K ( t ) earns the instantaneous return r ( t ). Income can either be consumed or saved as capital. a) Write down the instantaneous budget constraint for the household’s problem. Define . ' ) ' ( exp ) ( = ∫ t dt t r t R b) Interpret the condition (*) . ) ( ) ( lim ≥ ∞ → t R t K t The household’s problem is to maximize U for a given K (0) subject to the instantaneous budget constraint and the condition (*). c) Write down the Hamiltonian for the household’s problem. d) Write down Hamilton’s equations for the household’s problem. e) Solve for the optimal path of consumption. Now we assume the economy has a production sector with a constant returns to scale technology )), ( ( ) ( ) ( )) ( ) ( ), ( ( ) ( t k f t L t A t L t A t K F t Y = = where A ( t ) = exp( gt ) for some constant g and ....
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 Spring '10
 Feigenbaum
 Macroeconomics, Optimization, Trigraph, Bellman, max T

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