Backward dynamics in economics. The inverse limit approach

Abstract

We discuss the problem known in economics as ‘backward dynamics’. In models of perfect foresight, intertemporal equilibrium, rational agents’ decisions at any given moment in time depend on (accurately anticipated) future values of some variables. For certain kinds of structural (e.g., utility) functions, those models determine forward equilibrium sequences only implicitly. Then, the problem arises whether the investigation of backward-moving dynamics can be used to understand the forward dynamics associated with them. We study this problem by means of a mathematical technique known as ‘inverse limits theory (ILT)’ that allows us to establish a correspondence between backward dynamics of a non-invertible map and forward dynamics of a related, invertible map acting on an appropriately defined space of sequences. We apply ILT to certain implicit difference equations occurring in economics, and suggest precise criteria for identifying typical orbits forward in time.