Efficient and optimal mechanisms with private information acquisition costs

In auctions with private information acquisition costs, we completely characterize (socially) efficient and (revenue) optimal two-stage mechanisms, with the first stage being an entry right allocation mechanism and the second stage being a traditional private good provision mechanism. Both efficiency and revenue optimality require that the second-stage selling mechanism be ex post efficient and the number of entry slots be endogenously determined. We show that both efficient and optimal entry can be truthfully implemented in dominant strategies, and can also be implemented via all-pay, though not uniform-price or discriminatory-price, auctions.