Optimal capacity, product substitution, linear demand models, and uncertainty.

INTRODUCTION AND MOTIVATION

In today's highly uncertain marketplace, firms are increasingly resorting to flexibility, both on supply and demand sides, to effectively match their supply with demand. An example of supply-side flexibility is resource flexibility (also known as product-mix flexibility), which refers to a resource with the capability to produce multiple products. (1) An example of demand-side flexibility arises from the use of postponed (responsive) pricing, under which the firm postpones its pricing decision until after market uncertainty is resolved (see Chod and Rudi 2005; Chod et al. 2006; and Van Mieghem and Dada 1999 for related discussion and examples). As a result, the operations management literature has shown great interest in the multi-product firm's capacity investment decision with flexible resources, and different variations of this problem have been studied. Two assumptions commonly made in these studies are to assume that the demand-price relationship is linear and that the demand uncertainty in the investment stage is of additive form (see Bish and Wang 2004; Chod and Rudi 2005; Fine and Freund 1990; Goyal and Netessine 2005, 2007; Lus and Muriel 2006; Van Mieghem 1998; see also Van Mieghem 2003 for an excellent review and discussion on the capacity investment decision problem in general). An exception is Chod and Rudi (2006), which considers an isoelastic demand function under multiplicative uncertainty but for two "independent" products (i.e., with no cross-price effect on each other), as opposed to the "substitutable" products that we consider here. (2)

In this article, our objective is to study how the various market conditions and assumptions on demand affect a monopolist firm's capacity investment decision under responsive pricing. In particular, our focus is on how key demand parameters, such as the nature of demand uncertainty (i.e., "additive" versus "multiplicative" shock), market size, and market risk, impact the optimal flexible capacity decision and expected profit for a firm producing substitutable (differentiated) products that satisfy the same consumer need under the linear demand function.

A particular challenge in modeling demand is to incorporate the forecast error at the capacity investment stage into the demand function. Though most of the related operations management literature utilizes the "additive" form of demand uncertainty (i.e., parallel shift) in the capacity investment stage, other functional forms of uncertainty, such as the "multiplicative" form, may also arise frequently in markets. (The terms additive and multiplicative uncertainty were first coined by Karlin and Carr 1962 and have been used commonly; see Agrawal and Seshadri 2000; Cowan 2004; and Petruzzi and Dada 1999 for detailed discussion on each type of uncertainty.) Each model of uncertainty is appropriate in different market settings, and our objective is to understand the effect of the type of uncertainty on the firm's capacity investment decision.

The capacity investment literature mostly considers the linear form of demand because this represents the relationship between price and demand reasonably well (at least for some range of prices) while preserving analytical tractability. Linear aggregate demand models can be derived by assuming that there exists a fictional representative consumer for the whole economy who determines the demands for the different products so as to maximize her utility surplus. In particular, the linear demand model results when the representative consumer's utility function is quadratic (or approximated by the second-order Taylor's approximation), strictly concave, and additively separable in consumption quantities of the other products in the market.

Our study generates the following insights and principles.

* We characterize structural properties of an optimal capacity decision under additive and multiplicative shock settings. These properties allow us to derive the necessary and sufficient optimality conditions under each setting.