Joint Inventory and Pricing Decisions for an Assortment

We seek optimal inventory levels and prices of multiple products in a given assortment in a newsvendor model (single period, stochastic demand) under price-based substitution, but not stockout-based substitution. We address a demand model involving multiplicative uncertainty, motivated by market share models often used in marketing. The pricing problem that arises is known not to be well behaved in the sense that, in its deterministic version, the objective function is not jointly quasi-concave in prices. However, we find that the objective function is still reasonably well behaved in the sense that there is a unique solution to the first-order conditions, and this solution is optimal for our problem.