Optimal Public Provision of Private Goods

How should a policymaker allocate a good to consumers via a public option when they are also able to purchase the good from a competitive private market? I consider a designer who has preferences over the outcomes of both the public option and the private market, but can design only the public option. However, her design affects the distribution of consumers who purchase in the private market---and hence equilibrium outcomes. I find that the optimal design involves rationing the public option with a small number of tiers, where the probability of allocation is constant in each tier. I derive first-order conditions that characterize how each tier should be set in the optimal design. Finally, I show that tiered rationing remains optimal under a variety of different assumptions.