Solving Non-Linear Models with Saddle-Path Instabilities
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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.
Keywordscomplex-valued eigenvalues computational techniques cyclic convergence macroeconomics monotonic convergence real-valued eigenvalues saddle-path instability
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