Shilnikov Chaos, Low Interest Rates, and New Keynesian Macroeconomics

Barnett, William and Bella, Giobanni and Ghosh, Taniya and Mattana, Paolo and Venturi, Beatrice (2020): Shilnikov Chaos, Low Interest Rates, and New Keynesian Macroeconomics.

Abstract

The paper shows that in a New Keynesian (NK) model, an active interest rate feedback monetary policy, when combined with a Ricardian passive fiscal policy, à la Leeper-Woodford, may induce the onset of a Shilnikov chaotic attractor in the region of the parameter space where uniqueness of the equilibrium prevails locally. Implications, ranging from long-term unpredictability to global indeterminacy, are discussed in the paper. We find that throughout the attractor, the economy lingers in particular regions, within which the emerging aperiodic dynamics tend to evolve for a long time around lower-than-targeted inflation and nominal interest rates. This can be interpreted as a liquidity trap phenomenon, produced by the existence of a chaotic attractor, and not by the influence of an unintended steady state or the Central Bank's intentional choice of a steady state nominal interest rate at its lower bound. In addition, our finding of Shilnikov chaos can provide an alternative explanation for the controversial “loanable funds” over-saving theory, which seeks to explain why interest rates and, to a lesser extent inflation rates, have declined to current low levels, such that the real rate of interest is below the marginal product of capital. Paradoxically, an active interest rate feedback policy can cause nominal interest rates, inflation rates, and real interest rates unintentionally to drift downwards within a Shilnikov attractor set. Policy options to eliminate or control the chaotic dynamics are developed.

• Item Type: MPRA Paper
• Language: English
• Keywords: Shilnikov chaos criterion, global indeterminacy, long-term un-predictability, liquidity trap
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium; E - Macroeconomics and Monetary Economics > E1 - General Aggregative Models > E12 - Keynes ; Keynesian ; Post-Keynesian; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E52 - Monetary Policy; E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook; E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook > E63 - Comparative or Joint Analysis of Fiscal and Monetary Policy ; Stabilization ; Treasury Policy

References

Afraimovich, V. S., S. V. Gonchenko, L. M. Lerman, A. L. Shilnikov, and D. V. Turaev (2014), “Scientific Heritage of L. P. Shilnikov,” Regular and Chaotic Dynamics, 19(4), pp. 435-460.

Algaba, A., Freire, E. Gamer, E., and Rodriguez-Luis, A.J. (1998). “Analysis of Hopf and Takens-Bogdanov Bifurcations in a Modified van der Pol-Duffing Oscillator.” Nonlinear Dynamics, 16, pp. 369-404.

Ascari, G., Florio, A. and Gobbi, A. (2017). “Controlling Inflation with Switching Monetary and Fiscal Policies: Expectations, Fiscal Guidance and Timid Regime Changes.” Bank of Finland Research Discussion Papers 9/2017.

Barnett, W. A. And Chen, P. (1988a), “The Aggregation Theoretic Monetary Aggregates Are Chaotic and Have Strange Attractors: an Econometric Application of Mathematical Chaos.” In W. A. Barnett, E. Berndt, and H. White (eds.), Dynamic Econometric Modeling, Proceedings of the 3rd International Symposium on Economic Theory and Econometrics, Cambridge U. Press, pp. 199-246. Reprinted in W. A. Barnett and J. M. Binner, Functional Structure and Approximation in Econometrics, Elsevier, Amsterdam, 2004, chapter 22.

Barnett, W. A. and Chen, P. (1988b), “Deterministic Chaos and Fractal Attractors as Tools for Nonparametric Dynamical Econometric Inference, Journal of Mathematical Modeling, 10(4), pp. 275-296.

Barnett, W. A. and Chen, G. (2015), "Bifurcation of Macroeconometric Models and Robustness of Dynamical Inferences." Foundations and Trends in Econometrics, 8(1-2), pp. 1-144.

Barnett, W.A., Duzhak, E. A. (2008), “Non-robust dynamic inferences from macroeconometric models: Bifurcation Stratification of Confidence Regions.” Physica A, 387(15), pp. 3817-3825.

Barnett W.A. and Duzhak, E. A. (2010), “Empirical Assessment of Bifurcation Regions within New Keynesian Models,” Economic Theory, 45, 99-128.

Barnett, W. A., and Duzhak, E. A. (2019), “Structural Stability of the Generalized Taylor Rule.” Macroeconomic Dynamics, 23 (4), June, pp. 1664-1678.

Barnett, W. A. and Ghosh, T. (2013), “Bifurcation Analysis of an Endogenous Growth Model.” Journal of Economic Asymmetries, 10, pp. 53-64.

Barnett, W. A. and Ghosh, T. (2014), “Stability Analysis of Uzawa-Lucas Endogenous Growth Model.” Econ. Theory Bull., 2, pp. 33-44.

Barnett, W. A., Gallant, A. R., Hinich, M. J., Jungeilges, J., Kaplan, D., and Jensen, M. J. (1997). “A Single-Blind Controlled Competition between Tests for Nonlinearity and Chaos. Journal of Econometrics, 82, pp. 157-192.

Bella G., Mattana P., and Venturi B. (2017), “Shilnikov chaos in the Lucas model of endogenous growth.” Journal of Economic Theory, 172, pp. 451-477.

Belongia, M. (1996), “Measurement Matters: Recent Results from Monetary Economics Reexamined,” Journal of Political Economy 104, pp. 1065-1083.

Belongia, M., and Ireland, P. (2014), “The Barnett Critique after Three Decades: A New Keynesian Analysis.” Journal of Econometrics, 183(1), pp. 5-21.

Belongia, M., and Ireland, P. (2017), “Circumventing the Zero Lower Bound with Monetary Policy Rules Based on Money.” Journal of Macroeconomics, 54 (Part A), pp. 42-58.

Belongia, M., Ireland, P. (2018), “Targeting Constant Money Growth at the Zero Lower Bound.” International Journal of Central Banking, March, pp. 159-204.

Benhabib, J., Schmitt-Grohé, S., and Uribe, M. (2001a), "Monetary Policy and Multiple Equilibria." American Economic Review, 91(1), pp. 167-186.

Benhabib, J., Schmitt-Grohé, S., and Uribe, M. (2001b), "The Perils of Taylor Rules." Journal of Economic Theory, 96(1-2), pp. 40-69.

Benhabib, J., Schmitt-Grohé, S., and Uribe, M. (2002), “On Taylor Rules and Monetary Policy: Chaotic Interest-Rate Rules.” AEA Papers and Proceedings, 92(2), pp. 72-78.

Bernanke, Ben S. (2020), “The New Tools of Monetary Policy, American Economic Association Presidential Address, January 4, https://www.brookings.edu/wp-content/uploads/2019/12/Bernanke_ASSA_lecture.pdf, with summary on the Brookings Institution web site at https://www.brookings.edu/wp-content/uploads/2019/12/Bernanke_ASSA_lecture.pdf.

Bhattarai, S., Lee, J. W., and Park, W. Y. (2012), “Monetary-Fiscal Policy Interactions and Indeterminacy in Postwar US Data.” American Economic Review: Papers & Proceedings 2012, 102(3), pp. 173–178.

Blanchard, O. J. and Fischer, S. (1989), Lectures on Macroeconomics, MIT Press, Cambridge, MA.

Bofinger, P. and Ries, M. (2017), “Excess Saving and Low Interest Rates: Theory and Empirical Evidence,” CEPR Discussion paper 12111.

Bullard, J. (2010), “Seven Faces of ‘The Peril’.” The Federal Reserve Bank of St. Louis Review, September/October 2010, 92(5), pp. 339-52.

Carlstrom, C. T. and Fuerst, T. S. (2001), “Timing and real indeterminacy in monetary models.” Journal of Monetary Economics, 47, 285–298.

Carlstrom, C. T., and Fuerst, T. S. (2003), “Backward-Looking Interest-Rate Rules, Interest-Rate Smoothing, and Macroeconomic Instability.” Journal of Money, Credit and Banking, 35(6), pp. 1413-1423.

Champneys, A. (2010), “To the Memory of L. P. Shilnikov.” The Dynamical Systems Web, https://dsweb.siam.org/The-Magazine/All-Issues/to-the-memory-of-lp-shilnikov-1.

Chen, B., and Zhou, T. (2011), “Shilnikov homoclinic orbits in two classes of 3D autonomous nonlinear systems.” International Journal of Modern Physics B, 25(20), pp. 2697-2712.

Christiano, L and Takahashi, Y. (2018), “Discouraging Deviant Behavior in Monetary Economics,” NBER Working Paper 24949, August.

Cochrane, J. H. (2011), “Determinacy and Identification with Taylor Rules,” Journal of Political Economy, vol. 119, no. 3, June, pp. 565-615.

Coibion, O. and Gorodnichenko, Y. (2011), “Monetary Policy, Trend Inflation, and the Great Moderation: An Alternative Interpretation.” American Economic Review, 101, 341-70.

Davig, T., and Leeper, E. M. (2011), “Monetary–fiscal policy interactions and fiscal stimulus.” European Economic Review, 55 (2), pp. 211-227.

Edge, R. M., and Rudd, J. B. (2007), “Taxation and the Taylor principle.” Journal of Monetary Economics, 54, pp. 2554-2567.

Farmer, J.D., Ott E., Yorke, J.A. (1983), “The dimension of chaotic attractors.” Physica D: Nonlinear Phenomena, 7, pp. 153-180.

Feenstra, R. C. (1986), “Functional Equivalence between Liquidity Costs and the Utility of Money,” Journal of Monetary Economics 17, pp. 271-291.

Freire, E. Gamero, E. Rodriguez-Luis, A.J., Algaba, A. (2002), “A note on the triple-zero linear degeneracy: normal forms, dynamical and bifurcation behaviors of an unfolding.” International Journal of Bifurcation and Chaos, 12, pp. 2799-2820.

Galí, J., López‐Salido, J. D., and Vallés, J. (2004), “Rule‐of‐Thumb Consumers and the Design of Interest Rate Rules.” Journal of Money Credit and Banking, 36, pp. 739-63.

Geweke, J. (1992), “Inference and Prediction in the Presence of Uncertainty and Determinism.” Comment on L. M. Berliner, “Statistics, Probability, and Chaos,” and S. Chatterjee and M. Yilmaz, “Chaos, Fractals, and Statistics”), Statistical Science 7, pp. 94-101.

Geweke, J. and Durham, G. (2019). “Sequentially Adaptive Bayesian Learning for Inference and Optimization,” Journal of Econometrics 2010: 4-25.

Geweke, J., Barnett, W. A., and Shell, K. (1989), Economic Complexity: Chaos, Sunspots, Bubbles, and Nonlinearity, Cambridge University Press.

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

Hanke, S. (2019), “Trade Wars: Facts and Fallacies.” Forbes magazine, October 15, 3:59 pm.

Hanke, S. and Li, E. (2019), “The Strange and Futile World of Trade Wars,” Journal of Applied Corporate Finance, Fall.

Kaas, L. (1998), “Stabilizing chaos in a dynamic macroeconomic model.” Journal of Economic Behavior & Organization, 33, pp. 313-332.

Kiley, M. T. (2007), “Is Moderate-to-High Inflation Inherently Unstable?” International Journal of Central Banking, 3(2), pp. 173-201.

Kumhof, M., Nunes, R., and Yakadina, I. (2010), “Simple monetary rules under fiscal dominance.” J Money Credit Bank, 42(1), pp. 63-92.

Kuznetsov, N. V. (2016), “The Lyapunov Dimension and Its Estimation Via the Leonov Method,” Physics Letters 380(25-26), June, pp. 2142-2149.

Kuznetzov, Y.A. (1998). Elements of Applied Bifurcation Theory. 2nd edition. New York: Springer-Verlag.

Le Riche, A., Magris, F., and Parent, A. (2017), “Liquidity Trap and stability of Taylor rules.” Mathematical Social Sciences, 88, pp. 16-27.

Leeper, E.M. (1991), “Equilibria under `active' and `passive' monetary and fiscal policies.” Journal of Monetary Economics, 27(1), pp. 129-147.

Lucas, R. E. (2000), “Inflation and welfare.” Econometrica, 68(62), pp. 247-274.

Mendes, V. M., and Mendes D. A. (2006), “Active Interest Rate Rules and the Role of Stabilization Policy R&D Tax Credits.” Working Papers Series 1, ercwp0208, ISCTE-IUL, Business Research Unit (BRU-IUL).

Naimzada, A. K., and Sordi, S. (2018), “On controlling chaos in a discrete-time Walrasian tâtonnement process.” Metroeconomica, 69, pp. 178–194.

Natvik, G. J. (2009), “Government Spending and the Taylor Principle.” Journal of Money, Credit and Banking, 41(1), pp. 57-77.

Ott, E., Grebogi, C, and Yorke, J. A. (1990), “Controlling chaos.” Phys. Rev. Lett., 64, 1196-1199.

Peters, E. E. (1991). Chaos and Order in the Capital Markets. John Wiley & Sons, Inc. Publisher.

Piazza, R. (2016), “Self-fulfilling deflations.” Journal of Economic Dynamics and Control, 73, pp. 18-40.

Poterba, J. M. and Rotemberg, J. J. (1987). “Money in the Utility Function.” In Barnett, W. A. and Singleton, K.J. (eds.), New Approaches to Monetary Economics, Cambridge: Cambridge University Press, pp. 219-240.

Røisland, Ø. (2003), “Capital income taxation, equilibrium determinacy, and the Taylor principle.” Economics Letters, 81(2), pp. 147-153.

Serletis, A., and Rahman, S. (2013), “The Case for Divisia Money Targeting.” Macroeconomic Dynamics, 17, pp. 1638-1658.

Serletis, A. and Xu, L. (2019), “Consumption, Leisure, and Money,” Macroeconomic Dynamics, 23, pp. 1 – 30.

Shang, D., and Han, M. (2005), “The existence of homoclinic orbits to saddle-focus.” Applied Mathematics and Computation, 163, pp. 621-631.

Shilnikov, L.P. (1965), “A case of the existence of a denumerable set of periodic motions.” Sov. Math. Docl., 6, pp. 163-166.

Stokes, H.H. (2016), “Using nonlinear testing procedures to specify the right-hand side of an aggregate production function containing financial variables in the period 1967–2011.” The Journal of Economic Asymmetries, 14, pp. 147-156.

Sveen, T., and Weinke, L. (2007), “Firm-specific capital, nominal rigidities, and the Taylor principle.” Journal of Economic Theory, 136(1), pp. 729-737.

Sveen, T., and Weinke, L. (2005), “New perspectives on capital, sticky prices, and the Taylor principle.” Journal of Economic Theory, Volume 123, Pages 21-39. Tsuzuki, E. (2016), “Fiscal policy lag and equilibrium determinacy in a continuous-time New Keynesian model.” Int Rev Econ, 63, 215-232.

Walsh, C. E. (2010), Monetary Theory and Policy, MIT Press, Cambridge, MA, Third Edition.

Wang, P. and Yip, C. (1992). “Alternative Approaches to Money and Growth,” Journal of Money, Credit, and Banking 24(4), pp. 553-562.

Woodford, M. (2003). Interest and prices: foundations of a theory of monetary policy. Princeton University Press, Princeton.

Xu, L. and Serletis, A. (2016), “Monetary and fiscal policy switching with time-varying volatilities.” Economics Letters, 145, pp. 202-205.