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

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A consumer who wants to consume a good in a particular period may nevertheless attempt to buy it earlier if he is concerned that in delaying he would find the good already sold. This paper considers a model in which the good may be offered in two periods; the period in which all consumers most value the good (period 2), and an earlier period (period 1). Examining the profit-maximizing strategy of the firm under unbounded demand, we find that even with no cost of making the product available early, the firm does not profit, and usually loses, by making the product available early. Interestingly, the price that maximizes profits induces all arrivals to occur early, or all arrivals to occur late, depending on the parameters. The firm would not set a price which induces consumers to arrive in both periods. In particular, if the firm controls the penalty for arriving early, then it should set a high penalty so that no one arrives early. The Nash equilibrium behavior of consumers, when deciding if and when to arrive is more complicated than one may suppose, and can generate some unexpected behavior. For example, when there is unbounded demand, most potential consumers decide not to arrive at all. Additionally, the arrival rate may decline with the surplus a person gets from buying the good. Surprisingly, we find that an increase in the number of units for sale increases the number of consumers who arrive early. Moreover, we find that the profit-maximizing price increases with the number of units offered for sale. This too is unexpected as an increase in supply often results in price reduction. In our case, an increase in the number of units on sale also increases demand, and the seller may profit by increasing the price. In the single-unit case, we give closed solutions for the equilibrium customer behavior and profit-maximizing firm strategy and conduct sensitivity analysis.