Suppose you are selling your house and 10 potential buyers are lined up.  For whatever reason you cannot hold an auction (in fact sellers rarely do) but what you can do is make take-it-or-leave-it price demands.  To be clear:  this means that you can approach buyers in sequence proposing to each a price.  If a buyer accepts you are committed to sell and if he rejects you are committed to refuse sale to this buyer.  All buyers are ex ante identical, meaning that you while you don’t know their maximum willingness to pay, you have the same beliefs about each of them.  How do you determine the profit-maximizing price?

It is somewhat surprising that despite the symmetry, in order to maximize profits you will discriminate and charge them different prices.  What you will do is randomly order them and offer a descending sequence of prices.  The buyer who was randomly put first in the order (unlucky?) will be charged the highest price and this is an essential part of your optimal pricing policy.

Although it sounds surprising at first the intuition is pretty simple, it’s an application of the idea of option value.  When you have only one buyer left you will charge him some price p.  This price balances a tradeoff between high prices conditional on sale and the risk of having the offer rejected.  Since this is the last buyer the cost of that downside is that you will not make a sale.

Now the same tradeoff determines your price to the second-to-last buyer.  Except now the cost of having your offer rejected is lower because you will have another chance to sell.  So you are willing to take a larger chance of a rejected offer and therefore set a higher price.  Now continuing up the list, at every step the option value associated with a rejected offer increases and therefore so does the price.

OK that was easy.  Now consider a model where the seller posts prices and the buyers choose when to arrive.  This should break the symmetry if higher value buyers arrive earlier or later than lower value buyers.  And they will for two reasons.  First, nobody with a willingness to pay that is below the opening price will want to be first.  Second, even among buyers with a high willingness to pay, the higher it is the more you value the increased chance to buy relative to lower prices later.  (There is a “single-crossing property.”) The seller adjusts to this by further steepening the price path, etc.

Thanks to Toomas Hinnosaar for conversations on the topic.  Here is a paper by Liad Blumrosen and Thomas Holenstein on optimal posted prices.

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