Semi-explicit Optimal Pricing for Consumer Choice Models with Network Effects

We obtain exact semi-explicit solutions to the optimal pricing problem for the multinomial logit (MNL) consumer choice model with network effects, through a novel conditioning argument and the use of the Lambert W function. Then we manage to characterize the exact optimal solution in the presence of negative, positive and weak or positive and strong network effects, and also in the setting of heterogeneous market parameters. Furthermore, we provide a semi-explicit expression of equilibrium sales quantities in terms of equilibrium prices, based on which we establish sufficient conditions for the uniqueness of market equilibria. In addition, we provide an exact semi-explicit expression for the steady-state choice probability, based on which we discuss the uniqueness of the steady-state choice probability