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2017 | OriginalPaper | Buchkapitel

3. From Sunspots to Black Holes: Singular Dynamics in Macroeconomic Models

verfasst von : Paulo B. Brito, Luís F. Costa, Huw D. Dixon

Erschienen in: Sunspots and Non-Linear Dynamics

Verlag: Springer International Publishing

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Abstract

We present conditions for the emergence of singularities in DGE models. We distinguish between slow-fast and impasse singularity types, review geometrical methods to deal with both types of singularity and apply them to DGE dynamics. We find that impasse singularities can generate new types of DGE dynamics, in particular temporary determinacy/indeterminacy. We illustrate the different nature of the two types of singularities and apply our results to two simple models: the Benhabib and Farmer (1994) model and one with a cyclical fiscal policy rule.

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Vorheriges Kapitel Assessing the Local Stability Properties of Discrete Three-Dimensional Dynamical Systems: A Geometrical Approach with Triangles and Planes and an Application with Some Cones

Nächstes Kapitel Sunspot Fluctuations in Two-Sector Models with Variable Income Effects

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We use the notation for derivatives \(f_{x}\equiv \frac{\partial y}{\partial x }\) and \(f_{xy}\equiv \frac{\partial ^{2}y}{\partial x\partial y}\).

For simplicity we exclude the existence of periodic solutions, but the analysis can easily be extended to that case.

Observe that we are now referring to a two-dimensional projection in (K, L) of a three-dimensional system in (K, L, C).

See Kuehn (2015) for a recent textbook presentation.

Major references for continuous-time regular dynamics and bifurcations are Guckenheimer and Holmes (1990) or Kuznetsov (2005).

Recall we are assuming that \(\delta (K,L,\epsilon )\ne 0\).

A singular impasse point is a point in \(\mathcal {S}\) such that \(\nabla \delta = \mathbf 0 \).

Indeed, a one-dimensional manifold.

A zero-dimensional manifold.

For a proof, see Cardin et al. (2012) inter alia.

Benhabib and Farmer (1994, p. 34) already noted this behavior for particular values of the parameters: “As \(\chi \) moves below \(-0.015\) the roots both become real but remain negative until at (approximately) \(\chi =-0.05\) [i.e. \(\epsilon =0\)] one root passes through minus infinity and reemerges as a positive real root”. To compare with our results please note that we introduced a slight change in notation: while the authors set \(\chi \) as non-positive we set \(\chi \) as non-negative.

For another example in a macro model with imperfect competition see Brito et al. (2016).

Considering that we have an infinitely-lived representative household, it would act as if budget was balanced at all moments in time, i.e. Ricardian equivalence holds.

One special case is given by \(\phi =G\) and \(\mu =-1\), corresponding to setting the expenditure level.

We thank an anonymous referee and Nuno Barradas for suggesting this clarification.

Barnett, W. A., & He S. (2010). Existence of singularity bifurcation in an Euler-equations model of the United States economy: Grandmont was right. Economic Modelling, 27(6), 1345–1354.

Benhabib, J., & Farmer, R. (1994). Indeterminacy and increasing returns. Journal of Economic Theory, 63, 19–41.CrossRef

Brito, P., Costa, L., & Dixon, H. (2016). Singular macroeconomic dynamics and temporary indeterminacy. Mimeo.

Cardin, P. T., da Silva, P. R., & Teixeira, M. A. (2012). Implicit differential equations with impasse singularities and singular perturbation problems. Israel Journal of Mathematics, 189, 307–322.CrossRef

Cass, D., & Shell, K. (1983). Do sunspots matter? Journal of Political Economy, 91(2), 193–227.

Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. Journal of Difference Equations and Applications, 31, 53–98.

Grandmont, J.-M. (1985). On endogenous competitive business cycles. Econometrica, 53, 995–1045.CrossRef

Grandmont, J. -M. (1998). Expectations formation and stability of large socioeconomic systems. Econometrica, 66(4), 741–782.

Grandmont, J. -M. (2008). Nonlinear difference equations, bifurcations and chaos: An introduction. Research in Economics, 62(3), 122–177.

Grandmont, J.-M., Pintus, P., & de Vilder, R. (1998). Capital-labor substitution and competitive nonlinear endogenous business cycles. Journal of Economic Theory, 80(1), 14–59.

Guckenheimer, J., & Holmes, P. (1990). Nonlinear oscillations and bifurcations of vector fields (2nd ed.). Springer.

He, Y., & Barnett, W. A. (2006). Singularity bifurcations. Journal of Macroeconomics, 28, 5–22.CrossRef

Kuehn, C. (2015). Multiple time scale dynamics (1st ed.). Applied Mathematical Sciences 191. Springer.

Kuznetsov, Y. A. (2005). Elements of applied bifurcation theory (3rd ed.). Springer.

Llibre, J., Sotomayor, J., & Zhitomirskii, M. (2002). Impasse bifurcations of constrained systems. Fields Institute Communications, 31, 235–256.

Stiglitz, J. (2011). Rethinking macroeconomics: What failed, and how to repair it. Journal of the European Economic Association, 9(4), 591–645.

Zhitomirskii, M. (1993). Local normal forms for constrained systems on 2-manifolds. Boletim da Sociedade Brasileira de Matemática, 24, 211–232.CrossRef

Titel: From Sunspots to Black Holes: Singular Dynamics in Macroeconomic Models
verfasst von: Paulo B. Brito
Luís F. Costa
Huw D. Dixon
Verlag: Springer International Publishing
Buch: Sunspots and Non-Linear Dynamics
Print ISBN: 978-3-319-44074-3

Electronic ISBN: 978-3-319-44076-7

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-44076-7_3