Backward dynamics, inverse limits and global sunspots

In economic theory, there exist many important problems whose mathematical formulation is a function relating the current value of a certain state variable to its (accurately forecast) future value. The most widely studied class of such problems is the Overlapping generations (OLG) models in their innumerable variations are probably the best known example of such problems. A difficulty arises when, for certain specifications of ``fundamentals"", the map relating future to present values of the state variable is many—to—one and therefore the dynamics defined by the iterates of the map is backward in time. Since in real life agents are concerned about the future not the past, one would like to use the investigation of backward-moving dynamical models to understand the general properties of the forward dynamics associated with them. The purpose of this paper is to provide a complete characterization of this problem by means of certain concepts and methods known in the mathematical literature as ``inverse limits"". After a concise introduction to these ideas relatively little known in economics, the paper provides a systematic application to a basic OLG model. In this context, we also provide an analysis of global sunspots (i.e., sunspots that are not necessarily located near a stationary state or a periodic orbit), arising in OLG models characterized by backward dynamics.