‘JUMPS’ IN MACROECONOMIC MODELS: AN EXAMPLE WHEN EIGENVALUES ARE COMPLEX‐VALUED*

Abstract

The dynamic properties of continuous‐time macroeconomic models are typically characterised by having a combination of stable and unstable eigenvalues. In a seminal paper, Blanchard and Kahn showed that, for linear models, in order to ensure a unique solution, the number of discontinuous or ‘jump’ variables must equal the number of unstable eigenvalues in the economy. Assuming no zero eigenvalues and that all eigenvalues are distinct, this also means that the number of predetermined variables, otherwise referred to as continuous or non‐ ‘jump’ variables, must equal the number of stable eigenvalues. In this paper, we investigate the application of the Blanchard and Kahn results and establish that these results also carry through for linear dynamical systems where some of the eigenvalues are complex‐valued. An example with just one complex conjugate pair of stable eigenvalues is presented. The Appendix contains a general n‐dimensional model.