Dynamic pricing of substitutable products with limited inventories under logit demand

This article considers the dynamic pricing of two substitutable products over a predetermined, finite selling season. The initial inventory levels of the products are fixed exogenously and there are no replenishment opportunities during the season. It is assumed that each arriving customer chooses from available products based on the multinomial logit choice model, which captures the effect of prices on consumer choice. Every time a product runs out of stock, the set of choices shrinks, capturing the effect of stockouts on consumer choice. It is shown that, under the optimal pricing policy, the marginal value of an item is increasing in the remaining time and decreasing in its own stock level and the other product's stock level. Despite such non-surprising behavior on the part of marginal values, the optimal price itself is not simply monotonic in the remaining time or the other product's stock level. For example, a product's optimal price may increase if the remaining time decreases or if the total inventory grows. It is shown that such optimal behavior can be understood through alternative gauges such as the optimal price difference between the two products and the optimal purchase probabilities.