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Comparing different approaches for solving optimizing models with significant nonlinearities

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  • Stemp, Peter J.
  • Herbert, Ric D.

Abstract

The property of saddle-path instability often arises in economic models derived from optimizing behavior by individual agents. In the case when underlying functional forms are nonlinear, it is likely that the stable and unstable arms defining the saddle-path dynamics will also have nonlinear properties. While closed-form analytic solutions can always be derived for linearized deterministic versions of these models, it will be necessary to use numerical techniques to derive the dynamic properties of calibrated versions of the associated nonlinear models. There are a range of different approaches by which it is possible to solve the dynamics of nonlinear models with the saddle-path property. Alternative solution methods using numerical techniques are discussed. In all cases, the success of the solution can be assessed by evaluating whether or not the chosen solution gives a time-path for each variable that goes from the chosen initial condition to (a small neighborhood of) the steady-state. Unfortunately, however, there is generally no way of evaluating whether intermediate points on the path between the initial condition and the steady-state are close approximations to the “true” solution. In order to evaluate the “goodness-of-fit” along the entire dynamic path, it is necessary to have a closed-form solution of the entire solution path. This can only be achieved in special cases. For one special case, we demonstrate how it is possible to construct a model with complex-valued eigenvalues with large imaginary parts and hence significant cycles. Such a model can be employed as a benchmark to compare the properties of solutions derived using a range of solution algorithms.

Suggested Citation

  • Stemp, Peter J. & Herbert, Ric D., 2008. "Comparing different approaches for solving optimizing models with significant nonlinearities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(2), pages 357-366.
  • Handle: RePEc:eee:matcom:v:78:y:2008:i:2:p:357-366
    DOI: 10.1016/j.matcom.2008.01.009
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    References listed on IDEAS

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    1. Peter J. Stemp, 2005. "Finding an Example of an Optimising Agent with Cyclical Behaviour," Computing in Economics and Finance 2005 4, Society for Computational Economics.
    2. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    3. Stephen J. Turnovsky, 2000. "Methods of Macroeconomic Dynamics, 2nd Edition," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262201232, April.
    4. Herbert, Ric D. & Stemp, Peter J. & Griffiths, William E., 2005. "Assessing two common approaches for solving models with saddle-path instabilities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(1), pages 78-89.
    5. Peter Stemp & Ric Herbert, 2006. "Solving Non-Linear Models with Saddle-Path Instabilities," Computational Economics, Springer;Society for Computational Economics, vol. 28(2), pages 211-231, September.
    6. Blanchard, Olivier Jean & Kahn, Charles M, 1980. "The Solution of Linear Difference Models under Rational Expectations," Econometrica, Econometric Society, vol. 48(5), pages 1305-1311, July.
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    1. Herbert, Ric D. & Stemp, Peter J., 2009. "Solving a non-linear model: The importance of model specification for deriving a suitable solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(9), pages 2847-2855.

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