Economics Dynamics Problems 83

Economics Dynamics Problems 83 - k sak α − 1 − n δ =...

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Continuous dynamic systems 67 If f is continuously differentiable in an open interval containing x = x , then we approximate f using the Taylor expansion f ( x ) = f ( x ) + f ( x )( x x ) + f ( x )( x x ) 2! + . . . + f n ( x )( x x ) n ! + R n ( x , x ) where R n ( x , x ) is the remainder. In particular, a first-order approximation takes the form f ( x ) = f ( x ) + f ( x )( x x ) + R 2 ( x , x ) If the initial point x 0 is sufficiently close to x , then R 2 ( x , x ) 0. Furthermore, if we choose x as being a fixed point, then f ( x ) = 0. Hence we can approximate f ( x ) about a fixed point x with f ( x ) = f ( x )( x x ) (2.30) Example 2.21 Although we could solve the Solow growth model explicitly if the production function was a Cobb–Douglas by using a transformation suggested by Bernoulli, it provides a good example of a typical nonlinear differential equation problem. Our equation is ˙ k = f ( k ) = sak α ( n + δ ) k This function has two fixed points obtained from solving
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Unformatted text preview: k [ sak α − 1 − ( n + δ )] = namely k ∗ 1 = and k ∗ 2 = ± sa n + δ ² − ( 1 α − 1 ) Taking a Frst-order 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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