This paper examines the validity of one very special version of Coase's Theorem. The version we examine is that in any economy in which the property rights are fully allocated, competition will lead to efficient allocations. One repercussion of this result is that one way to "solve" the public goods problem would be to allocate property rights fully, transforming the economy to a private goods one and let markets do their work. This is particularly appealing due to its decentralized nature, but one must question the claim that the market will lead to efficient outcomes in this case. That is, the privatized economy created above is of a very special type which, as it turns out is highly susceptible to strategic behavior. We show that the "mechanism" suggested above is not likely to work well in economies with either pure public goods or "global" externalities. Basically, the free-rider problem manifests itself as one of monopoly power in this private goods setting. On the other hand, if the public goods or externalities are "local" in nature, there is reason to hope that this (and perhaps other) mechanism(s) will work well. The work is related to the recent literature on the foundations of Walrasian Equilibrium in that it points up a relationship between the appropriateness of Walrasian equilibrium as a solution concept, the incentives for strategic play, the aggregate level of complementarities in the economy and the problem of coordinating economic activity.