We consider an environment where potential buyers of an indivisible good have
liquidity constraints, in that they cannot pay more than their `budget'
regardless of their valuation. A buyer's valuation for the good as well as her
budget are her private information. We derive constrained-efficient and revenue
maximizing auctions for this setting. In general, the optimal auction
requires ‘pooling’ both at the top and in the middle despite the maintained
assumption of a monotone hazard rate. Further, the auctioneer will never find
it desirable to offer lump sum subsidies to bidders with low budgets.

On a technical note, our analysis is based on the `reduced form’ representation of auctions, which enables one to exploit a polymatroid representation of auctions. This polymatroid representation is useful in other applications, time permitting, will be outlined.