C62 - Existence and Stability Conditions of EquilibriumReturn

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.

We analyze a discrete version of a simple Kaldor model. As is typical for Kaldor model we consider an S shaped investment function. This leads to either a one or three equilibria of the model. For simplicity reasons we do not consider an S shape saving function as assumed in the original Kaldor paper. This does not affect any analytical conclusions as for presentation of dynamic properties nonlinear investment function is sufficient. Our aim is to study changes in the model dynamics under varying parameters. We study transition between one and two equilibria setup and also each of the set up separately.