How Advance Sales Can Reduce Profits: When to Buy, When to Sell, and What Price to Charge

Problem definition: Consider consumers who prefer to consume a good later rather than earlier. If the price is constant, then we would expect consumers to wait to buy the good. That does not hold if consumers are concerned that others will buy the good early, so that a shortage will later occur. When will consumers arrive when they fear a shortage? What is the profit-maximizing policy of a monopolist? Might the firm lose profits by offering advance sales? The timing of consumer arrivals is much studied. Little consideration, however, has addressed how anticipated shortages affect arrival times. The application is important: managers want to know when consumers will arrive, when they should make the product available, and what price to charge to maximize profits. Methodology/results: We use game theory. We analyze analytically outcomes when a single item is for sale: we give closed solutions for the equilibrium customer behavior and profit-maximizing firm strategy and conduct sensitivity analysis. For generalization concerning more than one unit, we give some analytical results and provide many numerical solutions. When the price is constant over time, then even with no operating cost of doing so, offering advance sales reduces profits. If, however, the firm must offer both advance sales and later sales, then the profit-maximizing price induces all arrivals at the same time (either early or late, depending on the parameters). An increase in the number of units offered for sale increases the profit-maximizing price and increases the firm’s expected profit. The equilibrium strategy of consumers can generate some unexpected behavior. The arrival rate may increase with the price of the good. For a given price, an increase in the number of units for sale increases the number of consumers who arrive early. Managerial implications: The firm should offer the good only at the time consumers most desire it, and not earlier. Additionally, the profit-maximizing price can be derived from our analysis. This price is not the price which maximizes the expected number of arrivals. Funding: This work was supported by the Israel Science Foundation [Grant 852/22]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1218 .