= ¯ c note that this result is a consequence of the

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Unformatted text preview: = ¯ c . Note that this result is a consequence of the assumption on the discount rate. If β > 1 1+ r , the consumption would tend to in nity because the agent has always interest to wait and report a part of his consumption. In the other case, the path is normal. Iterating the transition equation to nd the relationship between s t +1 and s , you get : s t +1 (1 + r ) t = (1 + r ) s + t +1 X t =0 y t 1 (1 + r ) t- t +1 X t =0 c t 1 (1 + r ) t If t → ∞ , (1 + r ) s + ∞ X t =0 y t 1 (1 + r ) t = ∞ X t =0 c t 1 (1 + r ) t assuming that lim t →∞ s t +1 (1 + r ) t = 0 This assumption, necessary to avoid an explosive system, is known as the transver- sality condition. It means that the representative agent will consume everything before the in nity... So, going on with the previous assumptions and using geometric series'formula, we get the optimal level of (constant) consumption : ¯ c = (1- β ) " βs + ∞ X t =0 y t β t # 3- With uncertainty, the Euler equation is u ( c t ) = (1 + r ) E t βu ( c t +1 ) . With a quadratic or a linear utility function and the assumption β (1 + r ) = 1 , consumption is then a random walk. 3...
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