We study a seller that starts with an initial inventory of goods, has a target horizon over which to sell the goods, and is subject to a set of financial milestone constraints on the revenues and sales that need to be achieved at different time points along the sales horizon. We characterize the revenue maximizing dynamic pricing policy for the seller and highlight the effect of revenue and sales milestones on its structure. We show that the optimal policy can be written in feedback form, wherein the price at each point in time is selected so as to track the most stringent among all downstream milestones. Building on that observation, we propose a discrete-review policy that aims to dynamically track the appropriate milestone constraint even in settings with no advance demand model information, and show that this simple and practical policy is near-optimal in settings with large initial capacity and long sales horizons. One motivating application comes from the sales of new multi-unit, residential real-estate developments, where intermediate milestone constraints play an important role in their financing and construction. A second motivation is the popular practice, in the absence of demand model information, to make pricing decisions so as to track upcoming sales and revenue milestones, which our analysis shows to be near-optimal as long as the tracking target is selected appropriately.
Besbes, Omar, and Costis Maglaras. "Dynamic Pricing with Financial Milestones: Feedback-Form Policies." Management Science 58, no. 9 (September 2012): 1715-1731.
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