Multiproduct Price Optimization and Competition Under the Nested Logit Model with Product-Differentiated Price Sensitivities

We study firms that sell multiple differentiated substitutable products and customers whose purchase behavior follows a Nested Logit model, of which the Multinomial Logit model is a special case. Customers make purchasing decision sequentially under the Nested Logit model: they first select a nest of products and subsequently purchase a product within the selected nest. We consider the general Nested Logit model with product-differentiated price sensitivities and general nest coefficients. The problem is to price the products to maximize the expected total profit. We show that the adjusted markup, defined as price minus cost minus the reciprocal of price sensitivity, is constant for all products within a nest at optimality. This reduces the problem’s dimension to a single variable per nest. We also show that each nest has an adjusted nest-level markup that is nest invariant, which further reduces the problem to a single variable optimization of a continuous function over a bounded interval. We provide conditions for this function to be uni-modal. We also use this result to simplify the oligopolistic multi-product price competition and characterize the Nash equilibrium. Furthermore, we extend to more general attraction functions including the linear utility and the multiplicative competitive interaction model, and show that the same technique can significantly simplify the multi-product pricing problems under the Nested Attraction model.