{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Economics Dynamics Problems 243

# Economics Dynamics Problems 243 - Discrete systems of...

This preview shows page 1. Sign up to view the full content.

Discrete systems of equations 227 or 2 . 85078 v r 1 v r 2 = 0 v r 1 + 0 . 35078 v r 2 = 0 Let v r 1 = 1 then v r 2 = − 2 . 8508. The second eigenvector is found from ( A 0 . 5 I ) v s = 0 i.e. 0 . 35078 1 1 2 . 85078 v s 1 v s 2 = 0 0 or 0 . 35078 v s 1 v s 2 = 0 v s 1 + 2 . 85078 v s 2 = 0 Let v s 2 = 1 then v s 1 = − 2 . 8508. Hence v r = 1 2 . 8508 , v s = 2 . 8508 1 One eigenvector represents the stable arm while the other represents the unstable arm. But which, then, represents the stable arm? To establish this, convert the system to its canonical form, with z t + 1 = V 1 u t + 1 . Now take a point on the first eigenvector, i.e., point (1 , 2 . 8508), then z 0 = V 1 u 0 = 1 2 . 8508 2 . 8508 1 1 1 2 . 8508 = 1 0 Hence, z 1 t = 2 t (1) z 2 t = ( 0 . 5) t (0) = 0 Therefore z 1 t → +∞ as t → ∞ . So the eigenvector v r must represent the unstable arm. Now take a point on the eigenvector v s , i.e., the point ( 2 . 8508 , 1), then z 0 = V 1 u 0 = 1 2 . 8508 2 . 8508 1 1 2 . 8508 1 = 0 1 Hence, z 1 t = 2 t (0) = 0 z 2 t = ( 0 . 5) t (1)
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online