Existence, Uniqueness, Concavity and Geometry of the Monopolist’s Problem Facing Consumers with Nonlinear Price Preferences

A monopolist wishes to maximize her profits by finding an optimal price menu. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his 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 thesis, two existence results will be provided, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e., quasilinear) function of the price paid. This has been an open problem for several decades before the first multi-dimensional result obtained here and independently by N\"oldeke and Samuelson in 2015. Additionally, a necessary and sufficient condition for the convexity or concavity of this principal's (bilevel) optimization problem is investigated. 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. Moreover, the analytic and geometric interpretations of a specific condition relevant to the concavity of the problem have been explored. Finally, various examples are given to explain the interaction between preferences of agents' utility and monopolist's profit which ensure statements equivalent to the concavity of the principal-agent problem. In particular, an example with quasilinear preferences on $n$-dimensional hyperbolic spaces is given with explicit solutions to show uniqueness without concavity. Similar results on spherical and Euclidean spaces are also provided. Additionally, the solutions of hyperbolic and spherical cases converge to those of Euclidean spaces as curvature goes to 0.Ph.D