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 2. an aggregate state of the world over which agents have an infinitesimal influence. We first review the connections between the eductive viewpoint that puts emphasis for example on strongly rational expectations equilibrium and the standard game-theoretical rationalizability concepts. Besides the Cobweb tâtonnement outcomes, which mimic an eductive reasoning subject to homogenous expectations, we define, characterize (and prove the convexity of) the sets of rationalizable states and point-rationalizable states, which respectively incorporate heterogenous point-expectations and heterogenous stochastic expectations. In the case where our model displays strategic complementarities, we find unsurprisingly that all the eductive criteria under scrutiny support rather similar conclusions, particularly when the equilibrium is unique. With strategic susbstitutabilities, 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, again, heterogenity of expectations does not matter. However, when cycles of order 2 do exist, our different criteria predict different set of outcomes, although all are tied with cycles of order 2. Under differentiability assumptions, the Poincaré-Hopf method leads to global results for strong rationality of equilibrium. At the local level, the different criteria under scrutiny can be adapted to the analysis of expectational coordination. They leads to the same stabilty conclusions, only when there are local strategic complementarities or strategic substitutabilities. However, so far as the analysis of local expectational coordination is concerned, it is argued and shown that the stochastic character of expectations can most often be forgotten.