Solving a non-linear model: The importance of model specification for deriving a suitable solution

In this paper, we consider a macroeconomic model with alternative linear and non-linear specifications. One version of the model, expressed in levels, is highly non-linear and has at least two steady-state equilibria. One of these equilibria has an economically meaningful interpretation, while the other does not have a sensible economic interpretation. A second version of the model, expressed in logarithms, is linear and has a unique steady-state equilibrium, which corresponds to the economically meaningful equilibrium of the non-linear version of the model. The dynamic solution of each model version has a combination of stable and unstable eigenvalues so that any dynamic solution requires the calculation of appropriate “jumps” in endogenous variables. Attempts to solve these models, using forward-shooting and reverse-shooting algorithms, show that the forward-shooting algorithm chooses the “wrong” solution for the non-linear model, but the “right” solution for the linear model. The reverse-shooting algorithm chooses the “right” solution in both cases. We demonstrate how this result is driven by particular properties of the two versions of the model.