A Review of 'Jumps' in Macroeconomic Models: With Special Reference to the Case when Eigenvalues are Complex-Valued

The dynamic properties of 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, 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 review the Blanchard and Kahn results when eigenvalues are real-valued and then establish that these results also carry through for linear dynamical systems where some of the eigenvalues are complex-valued. We show that the crucial reason why the results continue to hold for complex-valued eigenvalues is because, in order to ensure that the solutions for the endogenous variables are real-valued and thus have an economic interpretation, the coefficients associated with each complex conjugate pair of eigenvalues must also come in complex conjugate pairs. Examples with just one complex conjugate pair of stable eigenvalues and a general n-dimensional model have been presented for both the continuous-time and discrete-time cases.