Pricing optimization and competition under the linear nested stochastic choice model

In this article, we investigate the pricing optimization of firms selling multiple alternatives to the market where consumer purchase behavior follows the linear nested stochastic choice (LNSC) model. As a special case of the nested stochastic choice (NSC) model, LNSC similarly features a two-step Luce procedure. Considering differentiated price sensitivities in a non-exact preference function form, the present research specifically shows that, for any product in each nest, the adjusted markup is constant under certain conditions; and the adjusted nest-level markup is constant among nests under another sufficient condition. The "loss-leader" effect is observed, which indicates that it may be optimal to price a product with a negative adjusted markup or even a negative margin to attract more attention to the corresponding nest. Based on these results, the pricing optimization can be simplified to a single-variable problem where the objective function is unimodal. Then, a special case with an exponential preference function is discussed along with its concavity of the total expected profit. The above results are also used to construct the oligopoly multiproduct price competition and characterize the Nash equilibrium. Finally, a series of sensitivity analyses are conducted to reveal the impacts of key parameters on the optimal solutions.