Computational Economics

, Volume 28, Issue 2, pp 211–231 | Cite as

Solving Non-Linear Models with Saddle-Path Instabilities



Economic models derived from optimizing behavior are typically characterized by the properties of non-linearity and saddle-path instability. The typical solution method involves deriving the stable arm of the saddle-path and calculating suitable “jumps” to bring the path of endogenous variables onto this stable arm. The solution for the stable arm can be determined using a range of different approaches. In this paper we examine the extent to which the success of these alternative approaches can be evaluated. Any method of evaluation will be dependent upon the amount of information that is known about a particular model solution. For some deterministic models the only information known with certainty about the path of the model solution are values taken by steady-state solutions; the rest of the path must be approximated in some way based on numerical solutions derived from non-linear ordinary differential equations. In some special cases it is possible to derive a closed-form solution of the entire path. As an example of a model with a closed-form solution, we consider a simple linear model with two stable complex-valued eigenvalues and one unstable real-valued eigenvalue. The model is then employed as a benchmark to compare the properties of model solutions derived using two well-known solution algorithms. Because the model has complex-valued eigenvalues it will have cyclic dynamics and thus problems encountered in solving these dynamics will likely coincide with some of the problems that solution algorithms have in solving non-linear models. Since the entire solution path of the model is known, it is possible to derive deeper insights into the factors that are likely to ensure the success or failure of different solution approaches than would be the case if less information about the solution path was available.


complex-valued eigenvalues computational techniques cyclic convergence macroeconomics monotonic convergence real-valued eigenvalues saddle-path instability 


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  1. Alvarez, F., Monteiro, G. and Turnovsky, S.J. (2004). Habit formation, catching up with the joneses, and economic growth. Journal of Economic Growth, 9, 47–80.CrossRefGoogle Scholar
  2. Anderson, G.S. (1997). Continuous-time Application of the Anderson-Moore (AIM) Algorithm for Imposing the Saddle Point Property in Dynamic Models, Unpublished Manuscript, Board of Governors, Federal Reserve System, Washington, D.C.Google Scholar
  3. Anderson, G. and Moore, G. (1985). A linear algebraic procedure for solving linear perfect foresight models. Economics Letters, 17, 247–252.CrossRefGoogle Scholar
  4. Azariadis, C., Bullard, J. and Ohanian L. (2004). Trend-Reverting fluctuations in the life-cycle model. Journal of Economic Theory, 119, 334–566.CrossRefGoogle Scholar
  5. Blanchard, O.J. (1985). Debt, deficits, and finite horizons. Journal of Political Economy, 81, 637–654.Google Scholar
  6. Blanchard, O.J. and Kahn, C.M. (1980). The solution of linear difference models under rational expectations. Econometrica, 48, 1305–1311.CrossRefGoogle Scholar
  7. Cagan, P. (1956). The Monetary Dynamics of Hyperinflation. In M Friedman (ed.), Studies in the Quantity Theory of Money. University of Chicago Press, Chicago.Google Scholar
  8. Dormand, J.R. and Prince P.J. (1980). A family of embedded Runge-Kutta formulae. Journal of Computing and Applied Mathematics, 6, 19–26.CrossRefGoogle Scholar
  9. Gould, J.P. (1968). Adjustment costs in the theory of investment of the firm. Review of Economic Studies, 35, 47–56.CrossRefGoogle Scholar
  10. Gear, C.W. (1971). Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ, USA.Google Scholar
  11. Hayashi, F. (1982). Tobin's marginal q and average q: A neoclassical interpretation. Econometrica, 50, 213–224.CrossRefGoogle Scholar
  12. Judd, K.L. (1998). Numerical Methods in Economics. MIT Press, Cambridge, Massachusetts.Google Scholar
  13. Kunkel, P. and von dem Hagen, O. (2000). Numerical solution of infinite horizon optimal control problems. Computational Economics, 16(3), 189–205.CrossRefGoogle Scholar
  14. Lagarias, J.C., Reeds, J.A., Wright, M.H. and Wright, P.E. (1998). Convergence properties of the nelder-mead simplex method in low dimensions. SIAM Journal of Optimization, 9(1), 112–147.CrossRefGoogle Scholar
  15. Lucas, R.E. (1967). Adjustment costs and the theory of supply. Journal of Political Economy, 91, 39–69.Google Scholar
  16. Matsuyama, K. (1987). Current account dynamics in a finite horizon model. Journal of International Economics, 23, 299–333.CrossRefGoogle Scholar
  17. MathWorks (2006). website:
  18. Morshed, M. and Turnovsky, S.J. (2004). Intersectoral adjustment costs and real exchange dynamics in a two-sector dependent economy model. Journal of International Economics, 62, 147–177.CrossRefGoogle Scholar
  19. Ramsey, F.P. (1928). A mathematical theory of saving. Economic Journal, 38, 543–553, Reprinted in J.E. Stiglitz and H. Uzawa (eds.), Readings in the Modern Theory of Economic Growth. MIT Press, 1969.CrossRefGoogle Scholar
  20. Shampine, L.F. and Gordon, M.K. (1975). Computer Solution of Ordinary Differential Equations: The Initial Value Problem. W. H. Freeman, San Francisco.Google Scholar
  21. Shampine, L.F. and Reichelt, M.W. (1997). The MATLAB ODE suite. SIAM Journal on Scientific Computing, 18(1).Google Scholar
  22. Sims, C.A. (2002). Solving linear rational expectations models, unpublished manuscript,
  23. Stemp, P.J. (2005). Finding an Example of an Optimizing Agent with Cyclical Behaviour, paper presented to the Society for Computational Economics. 11th International Conference on Computing in Economics and Finance, Washington, D.C., June 23–25.Google Scholar
  24. Stemp, P.J. and Herbert, R.D. (2005). Comparing different approaches for solving optimizing models with significant nonlinearities. In A. Zerger and R.M. Argent (eds.), MODSIM 2005 International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, 1099–1105.Google Scholar
  25. Treadway, A.B. (1969). On rational entrepreneurial behaviour and the demand for investment. Review of Economic Studies, 36, 227–239.CrossRefGoogle Scholar
  26. Turnovsky, S.J. (1997). International Macroeconomic Dynamics. MIT Press, Cambridge, Massachusetts, London, England.Google Scholar
  27. Turnovsky, S.J. (2000). Methods of Macroeconomic Dynamics, Second edition, MIT Press, Cambridge Massachusetts and London England.Google Scholar
  28. Yaari, M.E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies, 32, 137–150.CrossRefGoogle Scholar

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© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of EconomicsThe University of MelbourneMelbourneAustralia
  2. 2.School of Design, Communication and Information TechnologyThe University of NewcastleOurimbahAustralia

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