Non-linear pricing by convex duality

Abstract

a b s t r a c tWe consider the pricing problem of a risk-neutral monopolist who produces (at a cost) and offers an infinitely divisible good to a single potential buyer that can be of a finite number of (single dimensional) types. The buyer has a non-linear utility function that is differentiable, strictly concave and strictly increasing. Using a simple reformulation and shortest path problem duality as in Vohra (2011) we transform the initial non-convex pricing problem of the monopolist into an equivalent optimization problem yielding a closed-form pricing formula under a regularity assumption on the probability distribution of buyer types. We examine the solution of the problem when the regularity condition is relaxed in different ways, or when the production function is non-linear and convex. For arbitrary type distributions, we offer a complete solution procedure.© 2015 Elsevier Ltd. All rights reserved. The settingNon-linear pricing is a basic problem of economic mechanism design under asymmetric information. Consider a monopolist who is producing an infinitely divisible good, e.g., sugar, and wishes to sell the good to a potential buyer with unknown valuation for his/her product. The seller's production function is assumed to be linear with a slope equal to c > 0. The seller is risk neutral, and therefore, seeks to maximize the expected revenue from the sale. The buyer can be one of a finite number of types t from the index set T = {1, . . . , m} with m > 2. The parameter t for the type of the buyer is assumed to represent the valuation of a potential buyer for the good. The buyer derives a utility equal to t · u(A t ) − p t from acquisition of a quantity A t (allocation to buyer of type t) of the good, where u is a differentiable, strictly concave, strictly increasing function (u ′′ (x) < 0, u ′ (x) > 0 for every x) with u(0) = 0 and a strictly decreasing (u ′ ) −1 , and p t is the price paid for acquisition of the quantity A t ≥ 0. The crux of the problem is that a potential buyer's type (or valuation of the good) t is private, i.e., unknown to the seller. However, the seller's beliefs about t are given by a probability mass function f on the discrete set T . The problem of the seller is to devise a mechanism that will maximize expected revenue while it elicits a truthful declaration of type by the seller and ensures his/her participation.

Two-echelon supply chain coordination under information asymmetry with multiple types

“…One variation is to consider a continuous range of retailer types such as in Corbett and De Groote 2000;Corbett, Zhou et al 2004. Pinar 2015 analyses the model with structurally different cost functions.…”