Optimal Capacity, Product Substitution, Linear Demand Models, and Uncertainty

Abstract

We study the capacity, pricing, and production decisions of a monopolist producing two substitutable products with flexible capacity. Although the capacity decision needs to be made ex ante, under demand uncertainty, pricing and production decisions can be postponed until after uncertainty is resolved. We show how key demand parameters (the nature of uncertainty, market size, and market risk) impact the optimal capacity decision under the linear demand function. In particular, we show that if the demand shock is multiplicative, then in terms of the “invest or not” decision, the firm will be immune to forecast errors in parameters of the underlying demand distribution. Furthermore, incorrectly modeling the demand shock as additive, when, in fact, it is multiplicative, may lead to overinvestment. On the other hand, although the concept of a growth in market size leads to similar conclusions under both additive and multiplicative demand shocks, how market risk affects the optimal capacity decision depends critically on the form of the demand shock as well as its correlation structure. Our analysis provides insights and principles on the optimal capacity investment decision under various demand settings.

ACKNOWLEDGMENTS

We are grateful to the Associate Editor and the two anonymous reviewers for valuable comments and suggestions that have led to significant improvements in the analysis and exposition of the article.

Notes

¹ Though e present our model in terms of resource flexibility, our modeling framework and results also apply to other flexible operations strategies, such as “delayed product differentiation” (the most well-known example of delayed product differentiation is Benetton, where the dyeing operation is postponed until after more accurate demand information is gathered; see Signorelli and Heskett 1989).

² Considering substitutable products is especially important for the study of flexible resources, because most often the products produced by flexible resources will be substitutable; see the following section for examples.

³ The trade-off between dedicated and more expensive flexible resources has been well studied in the operations management literature, and the firm's optimal “capacity portfolio” (i.e., capacities and mix of flexible and dedicated resources; Van Mieghem 2003) has been analytically characterized for a price-taker firm (Van Mieghem 1998) as well as for a price-setter monopolist (Bish and Wang 2004; Fine and Freund 1990) that produces products that are neither substitutable nor complementary. Lus and Muriel (2006) have performed a numerical study to analyze the capacity portfolio for substitutable or complementary products. Although in reality firms might find it preferable to invest in both dedicated and flexible resources, this complicates the analytics considerably (see Bish and Wang 2004 for the characterization of the optimal capacity portfolio under responsive pricing). Considering only one flexible resource allows us to move away from the flexible-dedicated resource trade-off and analyze our main questions of interest in isolation.

⁴ As mentioned in Introduction and Motivation, we use the term resource flexibility in the broad sense to represent flexible capacity as well as other operations strategies, such as delayed product differentiation. The latter problem can be studied using a similar framework: The total inventory to be acquired is determined when demands are uncertain, and the split of the total inventory between the substitutable products (e.g., sweaters of different colors in the Benetton example) is determined ex post uncertainty.

⁵ Note that for the special case of the perfect positive correlation between ξ₁, ξ₂ that we consider in Proposition 3, the joint pdf g(·, ·) reduces to that of a single variable.

⁶ It is possible to add market strength factors to the demand model in (2), with respective multipliers θ₁ and θ₂ that are known at the outset. To simplify the exposition, we omit these factors from the additive demand model. The analysis extends to this case.

⁷ Observe that the above assumptions relate the support region of (Θ₁, Θ₂) to parameters ν and β; i.e., to the level of substitution between the products. Hence, their pdf, f(·, ·), also depends on the product substitution parameters. Because the substitution parameters ν and β are constants in our model, we omit the dependence of the pdf on these parameters to simplify the exposition. In addition, for the special cases of the perfect positive and perfect negative correlation between Θ₁, Θ₂, that we study subsequently, the joint pdf f(·, ·) reduces to that of a single variable.

⁸ Singh and Vives (1984) consider a deterministic version of the problem; hence, market strength factors reduce to constants in their model.

⁹ It is important to reiterate the restriction that β < ν,′ that is, β can take on values in the ∊ -neighborhood of ν (see Bazaraa et al. 1993, p. 561, for definition of ∊-neighborhood), which does not include point ν. Otherwise, as observed in Choi (1991), when β = ν, the demand function reduces to d _i = θ _i [∊ _i − ν(p _i − p _{3 − i} )], i = 1,2; that is, as long as p ₁ = p ₂, demands become independent of prices, and infinite profits can be realized by charging infinite prices, leading to unbounded objective function values for Problem P ₂ (hence for Problem P ₁ ).

¹⁰ As a note, this result does not extend to complementary products (−ν < β < 0).

1¹ As stated above, θ _i = 1, i = 1, 2, in the additive case.

¹² In July 2008, a new Toyota Prius had a waiting time of 6 months in Blacksburg, Virginia.

¹³ Specifically, stochastic order increase of or in A(G), A(−), M(G), and M(−); and stochastic order increase of ξ (= ξ₁ = ξ₂) or Θ (= Θ ₁ = Θ ₂) in A(+) and M(+). Note that for A(−) and M(−), a stochastic increase of and corresponds to a larger value of the constant a, where ξ₁+ ξ₂ = a and Θ₁+ Θ₂ = a.

¹⁵ See Bish et al. (2008) for the results with∗.

¹⁴ Specifically, convex order increase of or in A(G), A(−), M(G), and M(−); and convex order increase of ξ or Θ in A(+) and M(+).

In the additive case θ ₁ = θ ₂ = 1.

In what follows, Π_{Ω j} ∗ ^A(G)(K) denotes Π∗ ^A(G)(K) as defined in (B₂), given Ω _j ^A(G), j = 1 … 7.

¹⁸ Π_{Ω j} ∗ ^M(G)(K) denotes the expression of Π∗ ^M(G)(K) in Ω _j ^M(G), j = 1 …7.