Choice between the Discrete and Continuous Models in Economic Applications and Its Implications

When building a dynamic model we express an impact of past on future state variables. We usually know if the change should be positive or negative and have an idea about a magnitude of the change. But if we express the change with either a difference or a derivative often depends on such factors, as is ease of evaluation of discrete models. We show that this choice may have important stability implications. We choose a more general approach and consider the choice not only between discrete and continuous models but among discrete models with various lengths of the step (considering a continuous model be a discrete model with infinitely small step). We derive a closed form formula that says if the equilibrium point in a system with a particular step length is stable or not and some implications about the stability of the same system with different step lengths. As a special case we get a relationship between stability of a discrete model and the corresponding continuous model where a difference was substituted by a derivative.