In an auction with a buy price, the seller provides bidders with an option to end the auction early by accepting a transaction at a posted price. The "Buy-It-Now" option on eBay is a leading example of an auction with a buy price. This paper develops a model of an auction with a buy price in which bidders use the auction's reserve price and buy price to formulate a reference price. The model both explains why a revenue-maximizing seller would want to augment her auction with a buy price and demonstrates that the seller sets a higher reserve price when she can affect the bidders' reference price through the auction's reserve price and buy price than when she can affect the bidders' reference price through the auction's reserve price only. Introducing a small reference-price effect can shrink the range of buy prices bidders are willing to exercise. The comparative statics properties of bidding behavior are in sharp contrast to equilibrium behavior in other models where the existence and size of the auction's buy price have no effect on bidding behavior.
Shunda, Nicholas, "Auctions with a Buy Price: The Case of Reference-Dependent Preferences" (2007). Economics Working Papers. Paper 200742.