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Macro1_PS4sol - Macroeconomics 1 Master APE 2010-2011 PS4...

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Unformatted text preview: Macroeconomics 1. Master APE. 2010-2011. PS4 Prof. Xavier Ragot / T.A : Eric Monnet Solutions 1 Bellman equations Let's revisit an in nite consumption-savings problem in which household are facing an exogenous income stream y t (instead of a xed initial endowment), that is known to the household with certainty. The income stream can vary overtime. Households solves the following problem : Max c t ,s t +1 = ∞ X t =0 β t u ( c t ) subject to : c t + s t +1 = (1 + r t ) s t + y t , ∀ t with consumption ( c t ≥ ), saving ( s given), and the interest rate at time t r t > . Assume the utility function is of CRRA form, u ( c t ) = c 1- σ t- 1 1- σ , with σ > . For more details on the answers of this exercise, interpretations of Euler equation, and Envelope theorem, have a look at the notes/slides of your third class on con- sumption and at the attached handout on dynamic programming 1- Formulate this problem using Bellman equation and derive the rst-order nec- essary conditions. Provide details of your use of the Envelope Theorem. Interpretessary conditions....
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