Macroeconomics Exam Review 227

Macroeconomics Exam Review 227 - 5.5 We apply the reasoning...

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Unformatted text preview: 5.5 We apply the reasoning of Example 5.5 to each component. Formally, for each , let ˆ be the projection of along the ℎ axis ˆ ( ) = ( ∗ 1 , ∗ 2 ,..., ∗ − 1 , , ∗ +1 ,..., ∗ ) ∗ maximizes ˆ ( ) over ℜ + , for which it is necessary that ˆ ( ∗ ) ≤ ∗ ≥ ∗ ˆ ( ∗ ) = 0 Substituting ˆ ( ∗ ) = [ x ∗ ] yields [ x ∗ ] ≤ ∗ ≥ ∗ [ x ∗ ] = 0 5.6 By Taylor’s Theorem (Example 4.33) ( x ∗ + dx ) = ( x ∗ ) + ∇ ( x ∗ ) dx + 1 2 dx ( x ∗ ) dx + ( dx ) ∥ dx ∥ 2 with ( dx ) → as dx → . Given 1. ∇ ( x ∗ ) = and 2. ( x ∗ ) is negative definite and letting...
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