Center Manifold, Stability, and Bifurcations in Continuous Time Macroeconometric Systems

This paper is a follow-on to our earlier paper, "Bifurcations in Continuous-Time Macroeconomic Systems." In this paper, we determine the stability properties of the UK continuous time macroeconometric model on its bifurcation boundaries and we test the null hypothesis that the model's parameters are inside the model's unstable region. We then attach to the model Bergstrom's recommended stabilization policy rule. We find that for most settings of his policy rule's parameters, the model's stable region becomes smaller. Hence the policy may be counter- productive. To the degree that "stabilization policy" is interesting, its intent must be to stabilize a system that is not stable. This is bifurcation. Hence stabilization policy only can be understood as bifurcation selection. We find that selection of a policy that will produce successful bifurcation from the model's unstable region to its stable region is much more difficult than previously believe. If in fact the economy is unstable without policy, the selection of successful stabilization policy is an exercise in a very difficult area of complex dynamics. Since Berstrom is one of the originators of the Bergstrom-Wymer-Nowman model, Bergstrom's choice of stabilization policy cannot be viewed as uninformed. We do find that a policy produced from optimal control theory is capable of bifurcating the system successfully from its unstable region to the stable region. But the form of the resulting policy rule is far too complicated to be of practical use. In addition, such a policy, even if implemented, cannot reasonably be viewed as robust to model specification error, since issues regarding the Lucas critique and time inconsistency (Kydland and Prescott) suggest the need for caution in any assumption of robustness to specification error, when such experiments condition upon a structural model.