Inventory and Pricing in Global Operations: Learning From Observed Demands

We develop a two-period model applicable to global sourcing by considering a firm that operates in two markets: one is located in the U.S. and the second is in a country having a selling season that does not overlap with the U.S.' selling season. Demand for each market depends linearly on the selling 'price and includes an unknown scale parameter. We assume that the firm learns from sales in the first market to assist decision making in the second. We also assume a single procurement opportunity, but allow the firm to ship leftovers from the first market to the second if doing so is profitable. Our results include the characterization of the optimal recourse policy, which represents the firm's decisions made at the beginning of the second selling season after it observes both sales in the first market and a realized value of the foreign exchange rate. Additionally, we provide a sufficient condition for reducing the optimization problem to a maximization over a single variable that we interpret as the safety stock for the first market. Further,we provide evidence that the sufficient condition is a rather mild one, likely to be satisfied in practical applications. We also establish a lower bound on the optimal value of the first-market safety stock, thereby truncating the search region of the last decision variable. This lower bound represents the optimal safety stock for the first market if that decision were made myopically, without regard to its effect on the profit associated with the second market. This is a working paper. Please do not cite or quote without permission. Comments on the paper are welcome. 1.0 Introduction Consider a single. monopolistic finn that operates in two separate markets and desires to establish the optimal quantity to procure and selling price to set for its product in each market. The firm offers the same product for sale in both markets. but the selling seasons are non-overlapping. Consequently. the finn is provided the opportunity to transfer some or all of the leftovers remaining from the first market to the second market for possible sale in the second selling season. In addition. the respective market demand functions -each of which are price dependent and include a scale parameter that is unknown at the start of the first selling season -are correlated perfectly due to homogenous customer preferences. As a result. the firm can revise its characterization of the unknown demand parameter applicable to the second market by observing sales in the first market. In particular. if there are leftovers in the first market. then the finn's sales and demand are equivalent; thus. the finn deduces the value of the unknown parameter. thereby eliminating the associated uncertainty in the demand function for the second market. If. however, there are no leftovers in the first market, the firm cannot deduce the value of the unknown parameter. Consequently, that parameter continues to contribute uncertainty to the demand function for the second market. although the firm can update its characterization of that uncertainty based on information obtained from the first market. We assume that the firm commits to its procurement quantities for both markets concurrently. at the beginning of the first selling season. We do not require that the firm receive both its procurement quantities at the beginning of the first selling season, only that it establish at that time a contractual arrangement governing the specified amount to be delivered at the start of each selling seasOI1. One motivation for this restriction is the desire to establish a modeling framework for the firm that negotiates a cost diScount schedule by bringing larger procurement quantities to the bargaining table. Perhaps a more appropriate motivation, though. is that this particular procedure is common in the fashion goods and related industries for which often there exists only one_procurement opportunity. A key implication of having only one procurement opportunity is that the decision of a procurement quantity for the second market is made with limited information. Since the firm learns from its operations in the first market, it fS likely that it would choose a procurement quantity differently if it were to make its decision after rather than before the tirst selling season. As recourse, though, the firm has two options. First, it can affect demand in the second market by changing its selling price. And second, if at the beginning of the second selling season the firm reaches the conclusion that the amount of stock procured for the second market is less than it desires, it can supplement its supply by transferring a portion of the leftovers remaining from the first market. However, this option might provide only limited recourse since the tirm is constrained by the number of leftovers remaining from the first market. If no leftovers remain from the first market, then t~e firm's resulting management situation represents an application of a yield-management problem. This is because, in such a case, although the firm's capacity is tixed, it can affect the trade-off between expected leftovers and expected shortages by altering market demand through its selling price. (Weatherford and Bodily (1992) provide a general classification scheme for yield-management problems; the one described here is similar to an airline or hotel booking problem in which there is only one price class and demand.is uncertain.) In order to keep general the formulation of this model, we assume that the two markets are separated by international boundaries, thereby establishing global sourcing as the primary application. Non-international applications -e.g., a scenario in which a firm operating in a single country uses an initial test market before offering a new product for sale in a second market -are obtained by fixing the foreign exchange rate at one. The management situation described above is an extension to a problem formulated by Kouvelis and Gutierrez (1992), who developed an optimal (profit-maximizing) strategy for an international firm that sells a fashion good in two, non-overlapping markets. In their model, Kouvelis and Gutierrez incorporate foreign exchange risk and provide the alternative of transferring to the second market some portion of the leftovers remaining from the first market; but, they assume that the selling price is given (for each market) and that the firm is