Macroeconomics Exam Review 62

Macroeconomics Exam Review 62 - Solutions for Foundations...

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2.17 Assume x is optimal, so that 𝑈 ( x ) 𝑈 ( x ) for every x Γ( 𝑥 0 ) This implies (using (2.39)) 𝑓 ( 𝑥 0 , 𝑥 1 ) + 𝛽𝑈 ( x ∗′ ) 𝑓 ( 𝑥 0 , 𝑥 1 ) + 𝛽𝑈 ( x ) where x = ( 𝑥 1 , 𝑥 2 , . . . ) is the continuation of the plan x starting at 𝑥 1 and x ∗′ = ( 𝑥 1 , 𝑥 2 , . . . ) is the continuation of the plan x . In particular, this is true for every plan x Γ( 𝑥 0 ) with 𝑥 1 = 𝑥 1 and therefore 𝑓 ( 𝑥 0 , 𝑥 1 ) + 𝛽𝑈 ( x ∗′ ) 𝑓 ( 𝑥 0 , 𝑥 1 ) + 𝛽𝑈 ( x ) for every x Γ( 𝑥 1 ) which implies that 𝑈 ( x ∗′ ) 𝑈 ( x ) for every x Γ( 𝑥 1 ) That is, x ∗′ is optimal starting at 𝑥 1 and therefore 𝑈 ( x ∗′ ) = 𝑣 ( 𝑥 1 ) (Exercise 2.15). Consequently 𝑣 ( 𝑥 0 ) = 𝑈 ( x ) = 𝑓 ( 𝑥 0 , 𝑥 1 ) + 𝛽𝑈 ( x ∗′ ) = 𝑓 ( 𝑥 0 , 𝑥 1 ) + 𝛽𝑣 ( 𝑥 1 ) This verifies (2.13) for 𝑡 = 0. A similar argument verifies (2.13) for any period 𝑡 .
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