Unformatted text preview: k [ sak α − 1 − ( n + δ )] = namely k ∗ 1 = and k ∗ 2 = ± sa n + δ ² − ( 1 α − 1 ) Taking a Frstorder Taylor expansion about point k ∗ , we have f ( k ) = f ( k ∗ ) + f ± ( k ∗ )( k − k ∗ ) where f ± ( k ∗ ) = α sa ( k ∗ ) α − 1 − ( n + δ ) and f ( k ∗ ) = Consider Frst k ∗ = k ∗ 1 = 0, then f ± ( k ∗ 1 ) = lim k → f ± ( k ) = lim k → [ α sak α − 1 − ( n + δ )] = ∞ Next consider k = k ∗ 2 > 0, then f ( k ∗ 2 ) = 0 and f ± ( k ∗ 2 ) = α sa ( k ∗ 2 ) α − 1 − ( n + δ ) = α sa ³ ± sa n + δ ² − ( 1 α − 1 ) ´ α − 1 − ( n + δ ) = α ( n + δ ) − ( n + δ ) = − ( n + δ )(1 − α )...
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 Fall '11
 Dr.Gwartney
 Economics, Derivative, Taylor expansion, x∗, Continuous dynamic systems

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