On Concavity of the Monopolist's Problem Facing Consumers with Nonlinear Price Preferences

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. © 2019 Wiley Periodicals, Inc.

The Implementation Duality

“…Adding such conditions to our Assumption 1 opens the possibility to investigate questions that go beyond those addressed in this paper. For instance, McCann and Zhang (2017) used the implementation duality to show how the conditions from Figalli, Kim, and McCann (2011), under which the principal's problem can be reduced to a convex maximization program, can be extended to the non-quasilinear case.…”

A Primer on Generated Jacobian Equations: Geometry, Optics, Economics

“…Nöldeke and Samuelson also noted in [15] that the duality structure given by a generating function yields what is known in the computer science and economics literature as a Galois connection. Another relevant work concerning nonquasilinear utility functions is that of McCann and Zhang [14]. In [14] the authors study a monopolist problem with an underlying generating function that is not necessarily quasilinear.…”

“…Another relevant work concerning nonquasilinear utility functions is that of McCann and Zhang [14]. In [14] the authors study a monopolist problem with an underlying generating function that is not necessarily quasilinear. They identify necessary and sufficient conditions for the concavity of this problem.…”

“…They identify necessary and sufficient conditions for the concavity of this problem. The work [14] shows that the monopolist problem (or principalagent problem) is equivalent to a concave maximization problem on a convexity constraint, the necessary and sufficient conditions on being a Ma-Trudinger-Wang-type condition. Previous related work by Figalli, Kim, and Mc-Cann proved a similar result for quasilinear utilities.…”

Existence in multidimensional screening with general nonlinear preferences

“…Recently, Nöldeke-Samuelson [22] provided a general existence result assuming that the agent and product space are compact, by implementing a duality argument based on Galois connections. McCann-Zhang [17] not only showed a general existence result assuming the single-crossing type condition and boundedness of the agent-type and product-type spaces, but generalized uniqueness and convexity results of Figalli-Kim-McCann [11] to the non-quasilinear case, by using G-convexity arguments. In this paper, we also explore existence using G-convex analysis, which will be introduced in Subsection 2.1, but with less restriction on boundedness of the product domain and without assuming the generalized single-crossing condition [16].…”