A Two-Product Newsvendor Problem with Partial Demand Substitution

Abstract

We show that a two-product newsvendor problem with partial demand substitution is equivalent to the classical newsvendor problem with the same economic parameters but an adjusted demand—the effective demand. By comparing the adjusted demand and the primary demand stochastically, we examine the impacts of demand substitution on the expected profit and optimal order quantities. We demonstrate that demand substitution reduces the newsvendor's demand variability in the sense of convex order. Furthermore, as the degree of substitution increases or the two products’ demands become less dependent in the sense of supermodular order, the newsvendor's effective demand becomes less uncertain, which implies a higher expected profit. Under rather general assumptions, we show that the distribution function for the newsvendor's effective demand satisfies the single-crossing property as the degree of substitution changes. This allows us to rank the optimal order quantities for two different degrees of substitution. To further develop insights, we analyze the case where the demand dependence structure is modeled using copula. We show that the optimal profit is decreasing and convex in demand dependence. Furthermore, by exploring the interaction effect of demand dependence and the degree of substitution, we show that a firm can achieve positive synergy by simultaneously increasing the substitution degree and decreasing demand dependence.