Pai, Mallesh and Rakesh Vohra (2014), "Optimal Auctions with Financially Constrained Buyers." Journal of Economic Theory, 150, 383-425.

We study the sale of an indivisible good to liquidity constrained buyers: they cannot pay more than their “budget” regardless of their valuation. Both valuation and budget are private information. We derive the symmetric revenue maximizing and constrained efficient auctions in this setting. We show an implementation via a modified all-pay auction. The highest bidder need not win the good outright, or, stated differently, the auction has “pooling,” despite the usual regularity conditions. Subsidizing low budget buyers cannot increase revenue. From a technical standpoint, we contribute to auction design with multidimensional private information by working directly with reduced-form allocation rules.