The Problem Of Backward Dynamics In Economics

This paper discusses the problem known in economics as ``backward dynamics''. In economic models of ``rational'' intertemporal behavior, agents' decisions at any given moment in time depend on their expectations of future values of some variables and, in the deterministic variants of those models, expectations are assumed to be always fulfilled. If to any given current value of the state variable x there corresponds only one expected value, from the models in question one can derive sequences of x going forward in time from any fixed initial conditions.Unfortunately, this condition is not always verified in economic models, nor is there any a priori reason for assuming it. When the condition does not hold (i.e., the functional relation between x(t) and x(t+1) is one-to-many), it is still possible to define the ``backward dynamics'' of the system, in the form of sequences moving from ``now'' to the indefinitely remote past, along which agents' expectations are fulfilled. Since in real life agents are concerned with forward dynamics and forward prediction, the problem arises whether the investigation of backward-moving dynamical models can be used to understand the general properties of the forward, ``true'' dynamics associated with them. We study this problem by means of a mathematical technique known as ``natural extension of non-invertible maps'' that allows us to establish certain correspondences between backward dynamics of a non-invertible map and forward dynamics of a related, invertible map acting on an appropriately defined space of sequences. The results thus obtained are applied to an economic example derived from the overlapping generations model. In particular, we study the properties of (forward-in-time) attracting sets and thereby re-consider certain known results concerning stability of stationary states and sunspots equilibria.