The rise of information technology has made it relatively simple to change prices over the life of a product.  Such “dynamic pricing” is important for many industries: In retailing, sellers can carefully manage end-of season sales; in the airline industry, firms can adjust the price of a particular flight to the number of seats sold, and in advertising, platforms can sell ad slots in advance and tailor to price to the level of demand.

This raises the question: How should a retailer choose when to start a sale, or when to raise the price of a flight? In the traditional dynamic pricing (or “revenue management”) model, the seller faces a sequence of customers who arrive over time and buy if the price is less than their willingness to pay (and otherwise walk away). A price reduction then raises the probability of a sale today but lowers the revenue from that sale and also lowers future revenue, since there are fewer items left tomorrow. This problem was first studied over fifty years ago and was solved by looking at the problem backwards. Suppose there is an ad firm trying to price a single front-page ad slot for a particular date, and that it can choose one price each day. First, we ask: What is the highest expected profit the seller can gain if it only has one day left to sell the slot (i.e. the day before the broadcast date)? Next, we move backwards and ask: What is the highest expected profit the seller can gain if it only has two days left to sell the ad, given that we already know the expected profit the seller makes if it fails to sell the good, and must therefore try to sell it the final day. Continuing like this, if one is given the distribution of values for incoming customers, one can solve for the optimal sequence of prices.

In their recent paper entitled “Revenue Management with Forward-Looking Buyers“, Professors Simon Board (UCLA) and Andrzej Skrzypacz (Stanford University) ask what happens if customers can time their purchases to take advantage of variations in prices. For example, if a customer values a coat at $200 and the price is $195, she only gains $5 of utility if she purchases immediately; she may therefore risk the chance of a stock-out and wait, in the hope that the price will fall. Such strategic behavior is important for firms. For example, JC Penney’s customers became so accustomed to endless sales promotions, that revenue dropped by 25 percent when it experimented with a flatter pricing policy. Such behavior is also likely to become more prevalent with the rise on online price comparison tools that help customers time their purchases (e.g. Kayak with airline fares).

In their paper, Board and Skrzypacz suppose that customers prefer to buy sooner rather than later, but will wait if prices are expected to fall sufficiently quickly. They then allow the seller to choose any feasible selling mechanism, allowing it to run a sequence of auctions, issue coupons to buyer who arrive early, or let the price paid by one buyer depend on the reports of other buyers who are waiting to buy. Despite all these options, they show that the seller optimally chooses a sequence of posted prices that are characterized by an intuitive differential equation. This result is surprising since one can no longer use simple backwards induction: when buyers are strategic, their behavior earlier in the game depends on the prices they expect later on. Nevertheless, the paper shows how take account of these effects using the tools of mechanism design.

The paper can thus help design sales policies in a wide variety of markets in which dynamic pricing is prevalent, such as online advertising, package holidays, and concert tickets. It also sheds light on common business practices. For example, profits are higher if buyers are forward-looking, which explains why firms such as Nordstrom benefit from having a predictable sales cycle and suggests that retailers should embrace price alerts. Additionally, it illuminates the “puzzle” of why most goods are sold via posted prices rather than auctions, and helps us understand when auctions may perform better.