Economics Dynamics Problems 83

Economics Dynamics Problems 83 - k [ sak 1 ( n + )] =...

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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 Frst-order approximation takes the form f ( x ) = f ( x ) + f ± ( x )( x x ) + R 2 ( x , x ) If the initial point x 0 is sufFciently close to x , then R 2 ( x , x ) ² 0. ±urthermore, if we choose x as being a Fxed point, then f ( x ) = 0. Hence we can approximate f ( x ) about a Fxed 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 Fxed points obtained from solving
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Unformatted text preview: k [ sak 1 ( n + )] = namely k 1 = and k 2 = sa n + ( 1 1 ) Taking a Frst-order Taylor expansion about point k , we have f ( k ) = f ( k ) + f ( k )( k k ) where f ( k ) = sa ( k ) 1 ( n + ) and f ( k ) = Consider Frst k = k 1 = 0, then f ( k 1 ) = lim k f ( k ) = lim k [ sak 1 ( n + )] = Next consider k = k 2 > 0, then f ( k 2 ) = 0 and f ( k 2 ) = sa ( k 2 ) 1 ( n + ) = sa sa n + ( 1 1 ) 1 ( n + ) = ( n + ) ( n + ) = ( n + )(1 )...
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