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# sol1 - Macroeconomics B Solution to problem set 1 The...

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Macroeconomics B Solution to problem set 1 The recursive problem can be written in the following form W ( a t , z t ) = max { c t ,a t +1 } u ( c t ) + βE t W ( a t +1 , z t +1 ) (1) s.t. a t +1 = (1 + r ) a t + y t c t , a t given, lim t →∞ a t (1 + r ) t 0 , where z t is the appropriate state variable (to be determined) characterizing the evolution over time of the income process. Replacing for c t and maximizing with respect to a t +1 we obtain the FOC u ( c t ) = β E t ∂W ( a t +1 , z t +1 ) ∂a t +1 . (2) Shifting the envelope condition ∂W ( a t , z t ) ∂a t = β (1 + r ) E t ∂W ( a t +1 , z t +1 ) ∂a t +1 = (1 + r ) u ( c t ) (3) one period forward and using β (1 + r ) = 1 we obtain the Euler equation u ( c t ) = E t u ( c t +1 ) . (4) Given that u is linear we Euler equation can be rewritten as c t = E t c t +1 (5) which we use in what follows. 1. Suppose labour income follows the stochastic process y t = ¯ y + ε t δε t 1 , (6) with ε t white noise. In choosing the state variables for the income process consider that

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sol1 - Macroeconomics B Solution to problem set 1 The...

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