# SOLVING ECONOMIC MODELS WITH SADDLE-PATH INSTABILITIES

*by Peter J. Stemp*

*Institution:*Department of Economics, University of Melbourne

*Date: Fri 17th September 2004*

*Time: 3:15 PM*

*Location: Th. 2, Grd Floor, ICT Building (111 Barry St, Carlton)*

*Abstract*: Dynamic economic models are typically derived from assumed underlying optimising behaviour on the part of consumers and firms. The simplest dynamic models of optimising agents (consumers or producers) are characterised by the property of saddle-path instability (one stable and one unstable eigenvalue) and are solved by allowing the optimising agent to choose a 'jump' or control variable (such as consumption or investment) so that the dynamic solution of the model follows the stable arm of the

saddle to a stable steady-state equilibrium. Larger models are typically characterised by a number of stable and unstable eigenvalues with the model being solved through choosing the initial conditions of a number of 'jump' variables (equal to the number of unstable eigenvalues), which

allow the dynamic solution of the model to proceed to equilibrium along a stable manifold. As a consequence, in order to solve these models, it is necessary to derive the stable arm or stable manifold of the optimal solution. In the case of linear (or linearised) models this can be achieved using standard matrix techniques. However, in the case of non-linear models, it is necessary to calibrate the model, and derive the stable path or stable manifold using iterative search techniques. Two

approaches that commonly underly these iterative search techniques are

forward-shooting and reverse-shooting. In this seminar, we investigate the relative success of these approaches under two specific scenarios. Firstly, we investigate success as the complexity of the model, and hence the number of stable and unstable eigenvalues, increases. Secondly, we

attempt to gain an insight into the likely success of the algorithms in solving highly non-linear models by investigating success when the underlying model is linear but has complex-valued eigenvalues and so

exhibits cyclical behaviour.

(Joint Work with Ric D. Herbert)

*For More Information:* Emma Lockwood: 8344 1617, emmal@ms.unimelb.edu.au