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homework8 2009 solutions_revised

homework8 2009 solutions_revised - ECN/APEC 6000/7230 Fall...

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ECN/APEC 6000/7230 Fall 2009 Homework 8 Solutions 1) The capital map is , ) 1 )( 1 ( ) ( ) ( ) ( 1 1 g n k w k s k k t t t t + + = + + where ( 29 . 1 ) ( ' 1 1 1 ) ( / 1 1 θ θ ρ δ - - + - + + = k f k s The slope of the capital map is , )) ( 1 )( 1 )( 1 ( ) ( ) ( ' ' 1 1 1 + + + - + + - = t s t t t t t k g n k s k f k dk dk η (*) where η s ( k ) is the elasticity of s with respect to k : . ln ) ( ln ) ( k d k s d k s = η ( 29 ). ( ' ' 1 ) ( ' 1 1 1 ) ( ' 1 1 1 ) ( ' / 1 2 / 1 1 k f k f k f k s θ θ θ ρ δ θ θ ρ δ - - - + - + - + - + + - = This can be simplified to ). ( ' ' ) ( 1 ) ( ' 1 1 ) ( ' 2 / 1 k f k s k f k s θ ρ δ θ θ - + - + - = Thus ). ( ' ' ) ( 1 ) ( ' 1 1 ) ( / 1 k f k ks k f k s θ ρ δ θ θ η - + - + - = Since f ′′ ( k ) < 0, we have η s ( k ) ≤ 0 if θ ≤ 1, in which case (*) implies that dk t +1 / dk t > 0. Suppose now that θ > 1 and f ( k ) = k α . Then
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ECN/APEC 6000/7230 Fall 2009 1 ) ( ' - = α α k k f ) ( ' ) 1 ( ) 1 ( ) ( ' ' 1 k f k k kf - = - = - α α α α . Let us define ( 29 θ θ ρ δ φ / 1 1 1 ) ( ' 1 ) ( - - + - + = k f k Then we have . ) ( 1 ) ( ' ) 1 ( 1 ) ( ' 1 1 ) ( / 1 k k f k f k s φ α ρ δ θ θ η θ + - + - + - = - [ ] .
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