Does a Unique Solution Exist for a Nonlinear Rational Expectation Equation with Zero Lower Bound?

  • Takashi TamuraEmail author
Original Research


To formalize a nonlinear rational expectation model in macroeconomics as a functional equation, we investigate the existence of a unique solution. Assuming that the probability of large exogenous disturbance occurring is sufficiently small, we prove that a unique solution exists. This result provides a rigorous foundation for research on nonlinear rational expectation models as a natural extension of existing linear models.


Nonlinear rational expectation model Functional equation Zero lower bound on interest rates Lyapunov functions 

JEL Classification




This work was supported by the JST PRESTO program. I am grateful to the anonymous referees for their invaluable comments.


  1. Adam, K., & Billi, R. M. (2006). Optimal monetary policy under commitment with a zero bound on nominal interest rates. Journal of Money, Credit, and Banking, 38(7), 1877–1905. Scholar
  2. Adam, K., & Billi, R. M. (2007). Discretionary monetary policy and the zero lower bound on nominal interest rates. Journal of Monetary Economics, 54(3), 728–752. Scholar
  3. Benhabib, J., Schmitt-Grohé, S., & Uribe, M. (2001). The perils of taylor rules. Journal of Economic Theory, 96(1–2), 40–69.CrossRefGoogle Scholar
  4. Blanchard, O., & Kahn, C. (1980). The solution of linear difference models under rational expectations. Econometrica: Journal of the Econometric Society, 48(5), 1305–1311.CrossRefGoogle Scholar
  5. Bullard, J. B. (2010). Seven faces of “the peril”. Review, Federal Reserve Bank of St. Louis, 92, 339–352. Google Scholar
  6. Cochrane, J. (2007). Identification with Taylor rules: A critical review. NBER working paper .Google Scholar
  7. Durrett, R. (2010). Probability: Theory and examples (4th ed., Vol. 31). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  8. Holden, T. D. (2016). Existence, uniqueness and computation of solutions to dynamic models with occasionally binding constraints, pp. 1–74. EconStor Preprints 127430, ZBW - Leibniz Information Centre for Economics.Google Scholar
  9. Horn, R. A., & Johnson, C. R. (2013). Matrix analysis (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  10. Jin, H., & Judd, K. (2002). Perturbation methods for general dynamic stochastic models. unpublished, Stanford University.Google Scholar
  11. Kim, J., Kim, S., Schaumburg, E., & Sims, C. (2008). Calculating and using second-order accurate solutions of discrete time dynamic equilibrium models. Journal of Economic Dynamics and Control, 32(11), 3397–3414.CrossRefGoogle Scholar
  12. Klein, P. (2000). Using the generalized schur form to solve a multivariate linear rational expectations model. Journal of Economic Dynamics and Control, 24(10), 1405–1423.CrossRefGoogle Scholar
  13. Lan, H., & Meyer-Gohde, A. (2013). Solving DSGE models with a nonlinear moving average. Journal of Economic Dynamics and Control, 37(12), 2643–2667.CrossRefGoogle Scholar
  14. McCallum, B. (2009). Inflation determination with taylor rules: Is newkeynesian analysis critically flawed? Journal of Monetary Economics, 56(8), 1101–1108.CrossRefGoogle Scholar
  15. Mccallum, B. T. (2009). Indeterminacy from inflation forecast targeting: Problem or pseudo-problem? FRB Richmond Economic Quarterly, 95(1), 25–51. Google Scholar
  16. Schmitt-Grohe, S., & Uribe, M. (2004). Solving dynamic general equilibrium models using a second-order approximation to the policy function. Journal of Economic Dynamics and Control, 28(4), 755–775.CrossRefGoogle Scholar
  17. Sims, C. (2002). Solving linear rational expectations models. Computational Economics, 20(1), 1–20.CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of ScienceOsaka Prefecture UniversitySakaiJapan

Personalised recommendations