Multidimensional screening.
This book is devoted to the mathematical background of multidimensional screening problem, that is the problem when the private information of the consumer cannot be capture in one characteristic. The general formulation of the problem of multidimensional screening is following. Assume a monopolist who faces a continuum of consumers produces a good of with \(n\) quality dimensions. The cost of production is given by function \(c(\cdot)\). Each consumer is interested in consuming at most one unit of the good and has a utility \(u(\alpha,x)-t\), where \(\alpha\in R^{m}\) is her unobservable type distributed on an open, bounded, convex set \(\Omega\subset R^{m}\) according to density function \(f(\cdot)\); \(t\) is the amount of money transferred to the monopolist. Consumers have an outside option of value \(s_0(\alpha)\). The monopolist announces a non-linear tariff \(t(\cdot)\). The amount \(t(x)\) determines how much a consumer has to pay for a good quality characteristics \(x\). So we have the following model. The monopolist selects a continuous \(t:\;R^{n}_{+}\to R\) to solve: \(\max\limits_{t(\cdot)}\int_{\Omega}(t(x(\alpha))-c(x(\alpha)))f(\alpha)\,d\alpha\), where \(c(x)\) is cost of producing a good with quality \(x\) and \(x(\alpha)\) satisfies \(x(\alpha)\in \text{ arg\,max} (u(\alpha,x)-t(x))\), \(\max(u(\alpha,x)-t(x))\geq s_0(\alpha)\). The book consists of two part. The first part gives a mathematical preliminaries. The Chapter~1 contains a review of vector calculus and introduces the basic operations and notions such as: the divergence, the gradient, conservative vector fields, curvilinear integrals, multiple and repeated integrals, flow of vector field, circulation of a vector field. The Chapter~2 contains the basic results and techniques of the theory of partial differential equations of the first and second order. The notions and properties of complete and general integrals, singular integral, method of characteristics are presented for the first order partial differential equations. Group theoretic analysis of partial differential equations is discussed. The Chapter~3 is devoted to the theory of generalized convexity. The author studies the relation between generalized convexity of the function and cyclic monotonicity of its sub-differential. The Chapter~4 deals with calculus of variations and the optimal control theory. The notions and properties of Frechet and Gateaux derivatives and Euler equation are presented. The Chapter~5 presents techniques which may be useful in screening applications. It is considered following topics: distributions as solutions of differential equations; Sobolev spaces and Poincare theorem; sweeping operators and balayage of measures; optimization by vector space methods; supermodularity and monotone comparative statics; generalized envelope theorems. The second part of book considers the economics of multidimensional screening. The Chapter~6 presents the unidimensional screening model. The Spence-Mirrlees conditions and notion of implementability are presented. Three approaches (direct approach, dual approach and Hamilton approach) are discussed. Type dependent participation constraint and random participation constraint are considered. The Chapter~7 deals with multidimensional screening problem. The genericity of exclusion, generalized convexity and implementability are discussed. Then the direct approach, dual approach and Hamilton approach are investigated. In the Chapter~8 the author considers the case, when the utility of a consumer is not quasi-linear in money. In the unidimwnsional case it is provided a complete charaterization of the solution, and it is shown that the allocation and the payment of the top type lie on the Pareto frontier. In the multidimensional case the author describes a scheme that allows to check implementability of any allocation. In the Chapter~9 the problem of existence, uniqueness and continuity of the solution is studied. Some conclusions are presented in the Chapter~10.
scientific article; zbMATH DE number 2135751