Multiproduct pricing under the multinomial logit model with local network effects

Motivated by direct interactions with practitioners and real-world data, we study a monopoly firm selling multiple substitute products to customers characterized by their different social network degrees. Under the multinomial logit model framework, we assume that the utility a customer with a larger network degree derives from the seller's products is subject to more impact from her neighbors and describe the customers' choice behavior by a Bayesian Nash game. We show that a unique equilibrium exists as long as these network effects are not too large. Furthermore, we study how the seller should optimally set the prices of the products in this setting. Under the homogeneous product-related parameter assumption, we show that if the seller optimally price-discriminates all customers based on their network degrees, the products' markups are the same for each customer type. Building on this, we characterize the sufficient and necessary condition for the concavity of the pricing problem, and show that when the problem is not concave, we can convert it to a single-dimensional search and solve it efficiently. We provide several further insights about the structure of optimal prices, both theoretically and numerically. Furthermore, we show that we can simultaneously relax the multinomial logit model and homogeneous product-related parameter assumptions and allow customer in- and out-degrees to be arbitrarily distributed while maintaining most of our conclusions robust.

Automated summary

Motivated by interactions with practitioners and real-world data, this study examines a monopoly firm selling multiple substitute products to customers with different social network degrees. Using the multinomial logit model and assuming that customers with larger network degrees are more influenced by their neighbors, the study describes customers' choice behavior using a Bayesian Nash game. The study finds that a unique equilibrium exists if network effects are not too large. Furthermore, it investigates the optimal pricing strategy for the firm in this setting, providing insights and solutions for concave pricing problems and relaxing some assumptions.