By Prof. Dr. Günter Gabisch, Dr. Hans-Walter Lorenz (auth.)
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Extra info for Business Cycle Theory: A Survey of Methods and Concepts
2 _ c(1 + ,8)>. 8). The c(l +,8) ± V(C(1 + ,8))2 - 4c,8 2 . 11). Further inspection shows that the signs of c and ,8 do not garantuee that the roots of the characteristic equation are real. First, consider the case (c(1 + ,8))2 ~ 4c,8 4 Compare Allen (1963), pp. 187 If. 13) such that the roots are real. Suppose firstly that the roots AI,2 are not equal. e. the root with the largest absolute value) the system will either be explosive or damped. 9) reveals that the motion will be damped (explosive) if the dominant root is smaller (greater) than unity.
216 f. 6 See Allen (1963), p. 188. 1. The Basic Samuelson Model ~----------------- Monotonic Damping Monotonic Explosion 41 y Fig. e. the Euclidian distance between the origin and the (a, b)- resp. (a , - b)-locus in the plane, and 0 as the angle between the distance line and the a-axis (cf. Fig. 2). ~ can be written 7 8 Dernburg/Dernburg (1969), p. 144£. DeMoivre's theorem states that (r( cos 0 + i sin 0)) n = rn (cos nO + i sin nO). g. Goldberg (1958), p. 139. Chapter 3 42 b (a,b) O ~~---------L--------------~----~a (a, - b) The Modulus of Complex Roots Fig.
Although this is only a minor qualitative change, the additional presentation of the Hicks model can be justified because it uncovers the sensitivity of linear dynamic models to the exact structural specification of the model and because it requires a slightly different technique in analyzing the parameter regimes. 8Yt - 2 = Co +r . 8 = O. 3) with the roots >. 8) ~ V4fi resp . 8 ~ 1 - c. 2. 7) the roots are real for resp. 2 - (c+,8) .. 4). The extremum of this parabolic function ! 9) => Amin = (c + 1')/2> o.