Business Cycle Models: closing the gap between the different approaches

This paper combines the two different business cycle methodologies and shows that in a quite standard discrete two-dimensional endogenous business cycle model fluctations near the interior steady state exhibit properties that resemble those generated by the exogenous approach. In order to generate these fluctuations however we do not require large, frequent and perisistent shocks as is the case in the exogenous models. This is explained by the property that the linearized dynamics near the steady state may have multiple eigenvalues which are close to unity. If these multiple eigenvalues cross the unit circle through -1 or through +1 socalled Bogdanov-Takens bifurcations (either a flip and a Hopf bifurcation or a saddle node and a Hopf bifurcation coincide) occur giving rise to all sorts of complicated dynamic structures. Since near to the Bogdanov-Takens bifurcation parameter value the eigenvalues of the linearized dynamics are close to unity time series generated by these economies posess the socalled unit root property whilst the complicated (chaotic) fluctuations appear in a vicinity of the steady state, as asked by certain groups of economists. Moreover, if one applies, for example, i.i.d. shocks to such an economy (either on the level of the technology or on the level of expectations) this will lead, because of a dominant off-diagonal term in the linearized dynamics, to the amplification of these shocks. To put it in the language of Frisch and Slutsky, there is persistence in the propagation mechanism itself and therefore persistence of the impulse, as in the exogenous models, is not required.