Correlations and Variances over the Ordinary Business Cycles

The choice between the two normal forms of cycle equations is irrelevant in Chapter 8 (because of their symmetry) — we shall here use the first of them, (3.29), since it applies in Chapter 9 too. The Solution of the linear approximation (3.46)-(3.47) can be written in the form 8.1$$s - {s^*}\mathop = \limits^{Lin} C{e^{ut}} + \overline C {e^{\overline u t}},C = \left( {\frac{1}{2} + \frac{{ia}}{{4\omega }}} \right)\left[ {s(0) - {s^*}} \right]$$, 8.2$$x - {b^*}\mathop = \limits^{Lin} D{e^{ut}} + \overline D {e^{\overline u t}},D = \left[ {\frac{{ia({\alpha ^2} + 4{\omega ^2})}}{{8(1 - {s^*})\omega }}} \right]\left[ {s(0) - {s^*}} \right]$$.