Simple and Complex Dynamics: A Hidden Parameter

In discrete-time dynamical systems arising from economic problems, the length of the period (henceforth simply "period") separating two successive values of the state variables may play an essential role in determining the type of dynamics followed by the system. In economics, this fact is generally hidden by the habit of implicitly choosing the unit of measure of time so that the period is always equal to one. We abandon this hypothesis and study a one-parameter family of models where the controlling parameter, Delta, denotes the period not necessarily equal to the unit of time and whose member for Delta=1 corresponds to a well-known optimal growth model which, for sufficiently large values of the discount rate, r, generates cyclical or chaotic paths. We find a positive relation between the degree of complexity of paths and the value of both Delta and r but the effect of changes in these two parameters is not symmetrical. In fact, we prove that for any arbitrarily large but finite value of r, there exists a critical value of Delta, not necessarily close to zero, such that along optimal paths the dynamics are simple, i.e., convergence to a stationary state.