We construct an elementary mechanism [dutta, b., sen, a., vohra, r., 1995. Nash implementation through elementary mechanisms in economic environments. Review of economic design 1, 173–203] that nash implements the constrained walrasian correspondence. We extend it to incomplete and non-exclusive information economies by enlarging the message space of agents. In addition, measurability restrictions on allocations with respect to prices proper to constrained rational expectations equilibria are imposed in the outcome function. We show that by imposing such restrictions, the mechanism bayesian implements the constrained rational expectations equilibrium correspondence. This result shows game-theoretic connections between these two market equilibrium concepts. However, these connections are obtained at the price of strong restrictions on the behavior of agents.