Comparing different approaches for solving optimizing models with significant nonlinearities

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.