Decision SupportThe effect of demand uncertainty in a price-setting newsvendor model
Introduction
The practice of suppliers of exerting various promotional efforts is common in many industries in such forms as national and/or local media advertisements, sponsorships, product placements, customer rebates and financing, or their combinations. All their efforts are aimed to increase the consumers’ awareness about their brands and hence the sales of their products, and to entice their retailers or franchisees to order more. Similarly, retailers may also engage in various demand improving activities, such as point-of-sale displays, loyal reward programs, and sales incentive plans. However, an interesting question arises: will these efforts increase the retailer’s order quantities if the market demand becomes stochastically larger (see Shaked and Shanthikumar, 2006; or Section 2) as a result of certain demand improving activities? If the retailer is a price setter, then the answer to the above question is unclear: the retailer may order more, equal, or less. The intuition behind this result can be easily explained. As the market becomes larger, the retailer may set a higher price to earn a higher margin per unit sold while ordering less to reduce the left-over inventory risk.
This result stands in sharp contrast to an existing observation that the retailer will order more if a sales effort improves the demand in a newsvendor setting, where the retail price is exogenous. Such difference between the newsvendor settings with and without endogenous pricing decision is also one of the motivations of the current paper. That is, our first intention here is to investigate the impacts of a stochastically larger demand on the retailer’s optimal pricing and order quantity decisions as well as her maximum expected profit. Such efforts will be useful in evaluating the impacts of various sales endeavors.
While various sales efforts may enhance the market demand, there are some efforts exerted by companies that reduce demand uncertainty. For example, in a Harvard Business School case, Leitax (a disguised digital camera manufacturer) reported a 50% increase in forecast accuracy as a result of the implementation of the so-called “consensus forecasting program”. Another interesting question then arises: Should a firm order more or less and price higher or lower if the demand uncertainty is reduced? Accordingly, this is the second motivation of this paper.
Specifically, we consider a single period inventory-pricing problem in which a newsvendor needs to decide both the order quantity of a single product and the selling price before the stochastic, price-sensitive demand is realized. As usual, we model the demand uncertainty (or randomness) in either the additive or multiplicative form. Sales efforts may influence the demand through this randomness, or forecast accuracy improvements may affect the magnitude of the randomness.
Joint pricing and ordering problems have received much attention in the research area of operations management in the last decade. For comprehensive surveys, see Yano and Gilbert, 2003, Elmaghraby and Keskinocak, 2003. Some recent papers that consider price-setting newsvendor problem include Yao et al., 2006, Chen and Bell, 2009, Granot and Yin, 2007, Granot and Yin, 2008, where the demand function adopted takes either an additive form or a multiplicative form. However, as noted by many previous studies, the optimal behaviors under the additive model setting are quite different from those under the multiplicative model setting (see, e.g., Petruzzi and Dada, 1999, Yao et al., 2006). To study the effects of demand randomness on the newsvendor’s optimal decisions and her expected profit systematically, we consider both additive and multiplicative demand models in this paper.
A common tool in studying the effects of uncertainties on the optimal decisions and objective functions is stochastic comparison. Song (1994) indicates that the buffer cost may not increase when leadtime becomes longer, whereas a more variable leadtime demand always leads to a higher system cost. Ridder et al. (1998) show that the newsvendor’s cost may be lower when the exogenous demand becomes more variable. As a special case of second order stochastic dominance, the mean-preserving transformation is also commonly used for stochastic comparisons. For example, Gerchak and Mossman (1992) show that a more variable demand may lead to a higher or lower optimal order quantity, and Gerchak and He (2003) study the benefits of risk-pooling, which always increase with the variabilities of individual random demands. In the current paper, we study the effect of demand uncertainty for a price-setting newsvendor under both first and second order stochastic dominances (see Definition 1, Definition 2 in Section 2).
Through numerically examples, Lau and Lau (2002) study the effects of demand uncertainty when the retailer is a price-setting newsvendor in a manufacturer-retailer channel. However, to the best of our knowledge, Li and Atkins (2005) is the only paper that analytically studies the effect of demand uncertainty in a price-setting newsvendor model. More specifically, they investigate the joint effects of coordination and information when demand becomes more variable. Their demand variability is defined under a second order stochastic dominance. Indeed, our results with respect to the second order stochastic dominance are consistent with those by Li and Atkins (2005). However, to analyze the joint effects of coordination and information, they assume a linear demand function and zero shortage cost. Our results are established under a general price-setting newsvendor framework, that is, our demand functions are more general, and we include a shortage cost. This is possible partially because we do not deal with the effect of coordination. As well understood in the inventory-pricing literature, the inclusion of a shortage cost may significantly complicate the analysis. Moreover, to our knowledge, some of our results are new in the literature, such as the following: (i) a stochastically larger demand (in the first order dominance sense) may even lead to a lower order quantity and profit when price is endogenous, which is in sharp contrast to the well-known result for the problem with exogenous price; (ii) a stochastically larger demand (in the second order dominance sense) leads to a higher selling price in general for the additive demand case but to a lower selling price under some mild conditions for the multiplicative demand case. Our results under the second stochastic dominance thus represents significant extensions of Li and Atkins (2005). In particular, under mild conditions, we verify that a less (higher) variable demand will lead to a higher (lower) and lower (higher) selling price for the additive and multiplicative demand case, respectively. An attempt to explain the rationale for such pricing behaviors is also made.
The remainder of this paper is organized as follows. We describe the basic model and review the related definitions of stochastic comparison in Section 2. We then study the effects of demand uncertainty in a price-setting newsvendor facing additive and multiplicative demand forms in Sections 3 Additive demand model, 4 Multiplicative demand model, respectively. Section 5 concludes our paper.
Section snippets
The model
Consider a newsvendor who faces a price-dependent random demand. At the beginning of a selling season, the newsvendor needs to make a joint decision on selling price and order quantity to maximize the expected profit. Let c be the unit ordering cost (or purchase cost). During the selling season, the demand is met through the newsvendor’s stock; any unmet demand incurs a shortage cost b ⩾ 0 in addition to the loss of profit; and the left-over inventory at the end of the selling season is salvaged
Additive demand model
Let z = q − d(p) be the stocking factor. From (3), the newsvendor’s expected profit function is then
For any given p(>c), let r = (p + b − c)/(p + b − s).π(z, p) is concave in z. Hence, the optimal stocking factor is z∗(p) = F−1(r), and the corresponding optimal order quantity is q∗(p) = d(p) + F−1(r). Substituting z∗(p) = F−1(r) into (7) and rearranging the items, we haveTaking the first derivative of π(z∗(p), p) with
Multiplicative demand model
Let z = q/d(p) be the stocking factor. Thus, the newsvendor’s expected profit function isFor any given p(>c), the optimal stocking factor is z∗(p) = F−1(r) and the corresponding optimal order quantity is q∗(p) = d(p)F−1(r). Similar to the additive model, we have
Taking the first derivative of π(z∗(p), p) with respect to p, we have
Conclusion
We have investigated the effects of demand uncertainty on the optimal pricing-quantity decision and on the optimal expected profit in a price-setting newsvendor model. Table 1 summarizes the major results of the two demand models. Under the first order stochastic dominance, we find that a (stochastically) larger demand: (i) leads to a higher price in general for the additive demand model, but may lead to a lower price for the multiplicative demand model; (ii) may not necessarily lead to a
Acknowledgements
This research was partially supported by Hong Kong RGC Grant No. 410906, NSFC Grant Nos. 70801035 and 70901059, and the Fundamental Research Funds for the Central Universities (Grant No. 105-275171).
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