It is shown that any informationally decentralized mechanism that realizes fair allocations over the class of classical pure exchange environments has a message space of dimension no smaller than the number of agents times the number of commodities. Since the equal income Walrasian mechanism, in which all agents take prices parametrically and maximize utility subject to the average income constraint, realizes fair outcomes over the class of classical pure exchange environments and has a message space of that dimension it is informationally efficient. Further, it is shown that it is the unique informationally efficient mechanism realizing fair allocations.