Multidimensional Screening

By: Suren Basov

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Hardcover | 11 January 2005

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218 Pages

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In many industries the tariffs are not strictly proportional to the quantity purchased, i. e, they are nonlinear. Examples of nonlinear tariffs include railroad and electricity schedules and rental rates for durable goods and space. The major justification for the nonlinear pricing is the existence of private information on the side of consumers. In the early papers on the subject, private information was captured either by assuming a finite number of types (e. g. Adams and Yellen, 1976) or by a unidimensional continuum of types (Mussa and Rosen, 1978). Economics of the unidimen sional problems is by now well understood. The unidimensional models, however, do not cover all the situations of practical interest. Indeed, often the nonlinear tariffs specify the payment as a function of a variety of characteristics. For example, railroad tariffs spec ify charges based on weight, volume, and distance of each shipment. Dif ferent customers may value each of these characteristics differently, hence the customer's type will not in general be captured by a unidimensional characteristic and a problem of multidimensional screening arises. In such models the consumer's private information (her type) is captured by an m-dimensional vector, while the good produced by the monopolist has n quality dimensions.

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Non-Fiction Engineering & Technology Technology in General Economics Economic Theory & Philosophy Mathematics Optimisation Game Theory Econometrics Reference, Information & Interdisciplinary Subjects Research & Information Information theory

Cybernetics & Systems Theory Science Physics Mathematical Physics

Mathematical Preliminaries	p. 1
Vector Calculus	p. 3
The Main Operations of Vector Calculus: div, grad, and ¿	p. 3
Conservative Vector Fields	p. 5
Curvilinear Integrals and the Geometric Meaning of the Existence of a Potential	p. 7
Multiple and Repeated Integrals	p. 8
The Flow of a Vector Field and the Gauss-Ostrogradsky Theorem	p. 12
The Circulation of a Vector Field and the Green Formula	p. 15
Exercises	p. 17
Bibliographic Notes	p. 18
Partial Differential Equations	p. 19
The First Order Partial Differential Equations	p. 19
The Complete Integral and the General Integral	p. 20
The Singular Integral	p. 21
The Quasilinear Equations and the Method of Characteristics	p. 22
Compatible Systems of the First Order PDEs	p. 24
The Method of Characteristics for a Non-quasilinear First Order PDE	p. 26
Examples	p. 27
The Second Order Partial Differential Equations	p. 30
Classification of the Linear Second Order Partial Differential Equations	p. 30
Boundary Value Problems for Elliptic Equations	p. 31
Examples	p. 32
Group Theoretic Analysis of the Systems of Partial Differential Equations	p. 38
One Parameter Lie Groups	p. 38
Invariance of PDEs. Systems of PDEs, and Boundary Problems under Lie Groups	p. 41
Calculating a Lie Group of a PDE	p. 44
Calculating Invariants of the Lie Group	p. 45
Examples	p. 46
Exercises	p. 47
Bibliographic Notes	p. 49
Theory of Generalized Convexity	p. 51
Definition and Properties of the Generalized Fenchel Conjugates	p. 52
Generalized Convexity and Cyclic Monotonicity	p. 55
Examples	p. 58
Exercises	p. 59
Bibliographic Notes	p. 59
Calculus of Variations and the Optimal Control	p. 61
Banach Spaces and Polish Spaces	p. 61
Hilbert Spaces	p. 65
Dual Space for a Normed Space and a Hilbert Space	p. 66
Frechet Derivative of a Mapping between Normed Spaces	p. 69
Functionals and Gateaux Derivatives	p. 71
Euler Equation	p. 73
Optimal Control	p. 74
Examples	p. 76
Exercises	p. 78
Bibliographic Notes	p. 79
Miscellaneous Techniques	p. 81
Distributions and Generalized Solutions for the Partial Differential Equations	p. 81
A Motivating Example	p. 82
The Set of Test Functions and its Dual	p. 83
Examples of Distributions	p. 84
The Derivative of a Distribution	p. 87
The Product of a Distribution and a Test Function and the Product of Distributions	p. 88
The Resultant of a Distribution and a Dilation Operator	p. 89
Adjoint Linear Differential Operators and Generalized Solutions of the Partial Differential Equations	p. 91
Sobolev Spaces and Poincare Theorem	p. 92
Sweeping Operators and Balayage of Measures	p. 94
Coercive Functionals	p. 96
Optimization by Vector Space Methods	p. 96
Calculus of Variation Problem with Convexity Constraints	p. 98
Supermodularity and Monotone Comparative Statics	p. 99
Hausdorff Metric on Compact Sets of a Metric Space	p. 103
Generalized Envelope Theorems	p. 107
Exercises	p. 109
Bibliographic Notes	p. 110
Economics of Multi-dimensional Screening	p. 111
The Unidimensional Screening Model	p. 115
Spence-Mirrlees Condition and Implementability	p. 116
The Concept of the Information Rent	p. 119
Three Approaches to the Unidimensional Relaxed Problem	p. 119
The Direct Approach	p. 119
The Dual Approach	p. 120
The Hamiltonian Approach	p. 121
Hamiltonian Approach to the Unidimensional Complete Problem	p. 122
Type Dependent Participation Constraint	p. 124
Random Participation Constraint	p. 126
Examples	p. 127
Exercises	p. 133
Bibliographic Notes	p. 134
The Multi-dimensional Screening Model	p. 135
The Genericity of Exclusion	p. 137
Generalized Convexity and Implementability	p. 141
Is Bunching Robust in the Multi-dimensional Case?	p. 143
Path Independence of Information Rents	p. 144
Cost Based Tariffs	p. 145
Direct Approach and Its Limitations	p. 148
Dual Approach for m = n	p. 151
The Relaxed Problem	p. 152
An Alternative Approach to the Relaxed Problem	p. 153
The Complete Problem	p. 154
The Geometry of the Participation Region	p. 155
A Sufficient Condition for Bunching	p. 156
The Extension of the Dual Approach for n > m	p. 156
Hamiltonian Approach and the First Order Characterization of the Solution	p. 158
The Economic Meaning of the Lagrange Multipliers	p. 160
Symmetry Analysis of the First Order Conditions	p. 161
Some Remarks on the Hamiltonian Approach to the Complete Problem	p. 166
Examples and Economic Applications	p. 167
Exercises	p. 173
Bibliographic Notes	p. 173
Beyond the Quasilinear Case	p. 175
The Unidimensional Case	p. 176
The Multi-dimensional Case	p. 179
Implementability of a Surplus Function	p. 180
Implementability of an Allocation	p. 181
The First Order Characterization of the Solution of the Relaxed Problem	p. 184
Exercises	p. 186
Bibliographic Notes	p. 187
Existence, Uniqueness, and Regularity Properties of the Solution	p. 189
Existence and Uniqueness of the Solution of the Relaxed Problem	p. 189
Existence of a Solution for the Complete Problem	p. 191
Continuity of the Solution	p. 192
Bibliographic Notes	p. 194
Conclusions	p. 195
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Multidimensional Screening

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