In McLean and Postlewaite (2002), we analyzed pure exchange economies with asymetrically informed agents. We defined a notion of informational size and showed that, when the aggregate information of all agents resolves nearly all the uncertainty regarding the state of nature, the conflict between incentive compatibility and (ex post) efficiency can be made small if agents have sufficiently small informational size. This paper investigates the relationship between informational size and efficiency for the case in which nontrivial aggregate uncertainty is present, i.e., when significant uncertainty about the world persists even when the information of all agents is known. We prove the existence of incentive compatible, individually rational and nearly ex post efficient allocations without assuming negligible aggregate uncertainty when agents have small informational size relative to informational variability. We further show that the conflict between incentive compatibility and efficiency asymptotically vanishes when an economy is replicated.