A Review of 'Jumps' in Macroeconomic Models: With Special Reference to the Case when Eigenvalues are Complex-Valued
AbstractThe 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.
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Bibliographic InfoPaper provided by The University of Melbourne in its series Department of Economics - Working Papers Series with number 920.
Length: 28 pages
Date of creation: 2004
Date of revision:
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More information through EDIRC
Macroeconomics; Dynamical Systems; Complex-valued Eigenvalues; Saddlepath Instability; Jump Variables.;
Find related papers by JEL classification:
- E17 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Forecasting and Simulation: Models and Applications
- E60 - Macroeconomics and Monetary Economics - - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook - - - General
- F41 - International Economics - - Macroeconomic Aspects of International Trade and Finance - - - Open Economy Macroeconomics
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- Dornbusch, Rudiger, 1976. "Expectations and Exchange Rate Dynamics," Journal of Political Economy, University of Chicago Press, vol. 84(6), pages 1161-76, December.
- Stephen J. Turnovsky, 2000. "Methods of Macroeconomic Dynamics, 2nd Edition," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262201232, December.
- Blanchard, Olivier Jean & Kahn, Charles M, 1980. "The Solution of Linear Difference Models under Rational Expectations," Econometrica, Econometric Society, vol. 48(5), pages 1305-11, July.
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