Necessary and sufficient conditions for the existence of a locally stable message process

When an equilibrium message correspondence is specified for each agent in an economy, there is an infinite number of sets of equations which can be used to represent these correspondences, each of which defines a different dynamic message process for the economy. By working only with C¹ message processes, this paper first explains the relationship between all of the message processes which determine the same equilibrium message correspondences for the agents. Local stability will then be studied; the main goal is to describe when a set of equilibrium message correspondences can be achieved over time by a locally stable message process. By analyzing derivatives, conditions are found under which this both can and cannot be done. A procedure is outlined which can be used to construct a locally stable message process from the equations which define the given equilibrium correspondences when certain regularity conditions are satisfied, and several examples are analyzed.