Multiple Equilibria, Points of Indifference, and Thresholds

This chapter addresses the interesting and important topics of multiplicity and history-dependence of optimal solutions. Multiplicity means that for given initial states there exist multiple optimal solutions; thus the decision-maker is indifferent about which to chose. This explains why such initial states are called points of indifference. In contrast, history-dependence occurs when the optimal solution depends on the problem's temporal history.

The existence of multiple equilibria has been long-recognized in physics, and more recently has provided an important enrichment of economic control models. In policy-making, it may be crucial to recognize whether or not a given problem has multiple stable optimal equilibria and, if so, to locate the thresholds separating the basins of attraction surrounding these different equilibria. At such a threshold different optimal courses of actions are separated. Thus, in general, starting at a threshold, a rational economic agent is indifferent between moving toward one or the other equilibrium. Small movements away from the threshold can destroy the indifference and motivate a unique optimal course of action. Among the economic consequences of such an unstable threshold is history-dependence (also denoted as path-dependence): the optimal long-run stationary solution toward which an optimally controlled system converges can depend on the initial conditions.

In the following section we present general results on discounted, autonomous problems, which are applied to a typical problem in the subsequent section. Here multiplicity and history-dependence are explained by means of a simple one-state model that already exhibits many of the interesting features connected to indifference points. In the next section we introduce a definition of multiplicity and history-dependence for optimal control problems with an arbitrary number of states.