On concavity of the monopolist's problem facing consumers with nonlinear price preferences

TL;DR

This paper characterizes when a monopolist's profit maximization problem with nonlinear consumer preferences is convex or concave, providing necessary and sufficient conditions that enhance analysis and computation of optimal pricing strategies.

Contribution

It establishes necessary and sufficient conditions for the convexity or concavity of the monopolist's optimization problem with nonlinear preferences, extending previous work with fewer assumptions.

Findings

Identifies conditions for convexity and concavity of the problem

Provides criteria that are both necessary and sufficient

Enhances understanding of stability and uniqueness in pricing strategies

Abstract

A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent $x$ will choose to buy that product $y(x)$ which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this paper, we provide a necessary and sufficient condition for the convexity or concavity of the principal's (bilevel) optimization problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e. quasilinear) function of the price paid. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences.