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Economics Dynamics Problems 140

# Economics Dynamics Problems 140 - − r s rs = 1 − b v v...

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124 Economic Dynamics Example 3.17 Determine the path of income for the equations C t = 50 + 0 . 75 Y t 1 I t = 4( Y t 1 Y t 2 ) G = 100 The equilibrium is readily found to be Y = 600, which is the particular solution. The complementary solution is found by solving the quadratic x 2 (19 / 4) x + 4 = 0 i.e. r = 3 . 6559 and s = 1 . 0941 Since r and s are real and distinct, then the solution is Y t = c 1 (3 . 6559) t + c 2 (1 . 0941) t + 600 and c 1 and c 2 can be obtained if we know Y 0 and Y 1 . Of more interest is the fact that the model can give rise to a whole variety of paths for Y t depending on the various parameter values for b and v . It is to this issue that we now turn. From the roots of the characteristic equation given above we have three possible outcomes: (i) real distinct roots ( b + v ) 2 > 4 v (ii) real equal roots ( b + v ) 2 = 4 v (iii) complex roots ( b + v ) 2 < 4 v In determining the implications of these possible outcomes we use the two prop- erties of roots r + s = b + v rs = v It also follows using these two results that (1 r )(1 s ) = 1
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Unformatted text preview: − ( r + s ) + rs = 1 − ( b + v ) + v = 1 − b and since 0 < b < 1, then 0 < (1 − r )(1 − s ) < 1. With both roots real and distinct, the general solution is Y t = c 1 r t + c 2 s t + Y ∗ where r is the larger of the two roots. The path of Y t is determined by the largest root, r > s . Since b > 0 and v > 0, then rs = v > 0 and so the roots must have the same sign. Furthermore, since r + s = b + v > 0, then both r and s must be positive. The path of income cannot oscillate. However, it will be damped if the largest root lies between zero and unity. Thus, a damped path occurs if 0 < s < r < 1, which arises if 0 < b < 1 and v < 1. Similarly, the path is explosive if the largest root exceeds unity, i.e., if r > s > 1, which implies 0 < b < 1 and rs = v > 1....
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