Expectational coordination in simple economic contexts: Concepts and analysis with emphasis on strategic substitutabilities

We consider an economic model that features (1) a continuum of agents and (2) an aggregate state of the world over which agents have an infinitesimal influence. We first review the connections between the "eductive" viewpoint on expectational stability and standard game-theoretical rationalizability concepts. The "eductive" reasoning selects different plausible beliefs that are a priori, and possibly a posteriori, "diverse". Such beliefs are associated with the sets of "Cobweb tâtonnement" outcomes, "Rationalizable States" and "Point-Rationalizable States" (the latter two being shown to be convex). In the case where our model displays strategic complementarities, unsurprisingly, all our "eductive" criteria support similar conclusions, particularly when the equilibrium is unique. With strategic substitutabilities, the success of expectational coordination, in the case where a unique equilibrium does exists, relates with the absence of cycles of order 2 of the "Cobweb" mapping: in this case, full expectational coordination would be achieved. However, when cycles of order 2 do exist, our different criteria predict larger sets of outcomes, although all tied with cycles. Under differentiability assumptions, the Poincaré-Hopf method leads to other global stability results. At the local level, the different criteria under scrutiny can be adapted. They lead to the same expectational stability conclusions, only when there are local strategic complementarities or strategic substitutabilities. However, for the local stability analysis, it is demonstrated that the stochastic character of expectations can most often be forgotten.