Discrete Choice and Complex Dynamics in Deterministic Optimization Problems

This papers shows that complex dynamics arise naturally in deterministic discrete choice problems. In particular, we show that if the objective function of a maximization problem can be written as a function of a sequence of discrete variables, and if the (maximized) value function is strictly increasing in an exogenous variable, then for almost all values of the exogenous variable, any optimal path exhibits aperiodic dynamics. This result is applied to a maximization problem with indivisible durable goods as well as to a Ramsey model with an indivisible consumption good. In each model, it is shown that optimal dynamics are almost always complex. These results are illustrated with various numerical examples.