Multidimensional Screening

Contents......Page 11
I Mathematical Preliminaries......Page 15
1.1 The Main Operations of Vector Calculus: div, grad, and Δ......Page 17
1.2 Conservative Vector Fields......Page 19
1.3 Curvilinear Integrals and the Potential......Page 21
1.4 Multiple and Repeated Integrals......Page 22
1.5 The Flow of a Vector Field and the Gauss-Ostrogradsky Theorem......Page 26
1.6 The Circulation of a Vector Field and the Green Formula......Page 29
1.7 Exercises......Page 30
1.8 Bibliographic Notes......Page 32
2.1 The First Order Partial Differential Equations......Page 33
2.1.1 The Complete Integral and the General Integral......Page 34
2.1.2 The Singular Integral......Page 35
2.1.3 The Method of Characteristics......Page 36
2.1.4 Compatible Systems of the First Order PDEs......Page 38
2.1.5 The Method of Characteristics for a Non-quasilinear First Order PDF......Page 40
2.1.6 Examples......Page 41
2.2.1 Linear Second Order Partial Differential Equations......Page 44
2.2.2 Boundary Value Problems for Elliptic Equations......Page 45
2.2.3 Examples......Page 46
2.3.1 One Parameter Lie Groups......Page 52
2.3.2 Invariance of PDEs, Systems of PDEs, and Boundary Problems under Lie Groups......Page 55
2.3.3 Calculating a Lie Group of a PDE......Page 58
2.3.4 Calculating Invariants of the Lie Group......Page 59
2.3.5 Examples......Page 60
2.4 Exercises......Page 61
2.5 Bibliographic Notes......Page 63
3 Theory of Generalized Convexity......Page 64
3.1 The Generalized Fenchel Conjugates......Page 65
3.2 Generalized Convexity and Cyclic Monotonicity......Page 68
3.3 Examples......Page 71
3.4 Exercises......Page 72
3.5 Bibliographic Notes......Page 73
4.1 Banach Spaces and Polish Spaces......Page 74
4.2 Hilbert Spaces......Page 78
4.3 Dual Space for a Normed Space and a Hilbert Space......Page 79
4.4 Prechet Derivative of a Mapping between Normed Spaces......Page 82
4.5 Functionals and Gateaux Derivatives......Page 84
4.6 Euler Equation......Page 86
4.7 Optimal Control......Page 87
4.8 Examples......Page 89
4.9 Exercises......Page 91
4.10 Bibliographic Notes......Page 92
5.1 Distributions as Solutions of Differential Equations......Page 94
5.1.1 A Motivating Example......Page 95
5.1.2 The Set of Test Functions and Its Dual......Page 96
5.1.3 Examples of Distributions......Page 97
5.1.4 The Derivative of a Distribution......Page 100
5.1.5 The Product of a Distribution and a Test Function and the Product of Distributions......Page 101
5.1.6 The Resultant of a Distribution and a Dilation......Page 102
5.1.7 Adjoint Linear Differential Operators and Generalized Solutions of the Partial Differential Equations......Page 104
5.2 Sobolev Spaces and Poincare Theorem......Page 105
5.3 Sweeping Operators and Balayage of Measures......Page 107
5.5 Optimization by Vector Space Methods......Page 109
5.6 Calculus of Variation Problem with Convexity Constraints......Page 111
5.7 Supermodularity and Monotone Comparative Statics......Page 112
5.8 Hausdorff Metric on Compact Sets of a Metric Space......Page 116
5.9 Generalized Envelope Theorems......Page 120
5.10 Exercises......Page 122
5.11 Bibliographic Notes......Page 123
II Economics of Multidimensional Screening......Page 124
6 The Unidimensional Screening Model......Page 127
6.1 Spence-Mirrlees Condition and Implementability......Page 128
6.3.1 The Direct Approach......Page 131
6.3.2 The Dual Approach......Page 132
6.3.3 The Hamiltonian Approach......Page 133
6.4 Hamiltonian Approach to the Complete Problem......Page 134
6.5 Type Dependent Participation Constraint......Page 136
6.6 Random Participation Constraint......Page 138
6.7 Examples......Page 139
6.8 Exercises......Page 145
6.9 Bibliographic Notes......Page 146
7 The Multidimensional Screening Model......Page 147
7.1 The Genericity of Exclusion......Page 149
7.2 Generalized Convexity and Implementability......Page 153
7.2.1 Is Bunching Robust in the Multidimensional Case?......Page 155
7.3 Path Independence of Information Rents......Page 157
7.4 Cost Based Tariffs......Page 158
7.5 Direct Approach and Its Limitations......Page 160
7.6 Dual Approach for m = n......Page 163
7.6.1 The Relaxed Problem......Page 164
7.6.2 An Alternative Approach to the Relaxed Problem......Page 165
7.6.3 The Complete Problem......Page 166
7.6.4 The Geometry of the Participation Region......Page 167
7.6.5 A Sufficient Condition for Bunching......Page 168
7.6.6 The Extension of the Dual Approach for n > m......Page 169
7.7 Hamiltonian Approach and the First Order Conditions......Page 170
7.8 Symmetry Analysis of the First Order Conditions......Page 173
7.9 The Hamiltonian Approach to the Complete Problem......Page 178
7.10 Examples and Economic Applications......Page 180
7.12 BibHographic Notes......Page 186
8 Beyond the Quasilinear Case......Page 188
8.1 The Unidimensional Case......Page 189
8.2 The Multidimensional Case......Page 192
8.2.1 Implementabihty of a Surplus Function......Page 193
8.2.2 Implementabihty of an Allocation......Page 194
8.3 The First Order Conditions for the Relaxed Problem......Page 197
8.5 Bibhographic Notes......Page 200
9.1 Existence and Uniqueness for the Relaxed Problem......Page 201
9.2 Existence of a Solution for the Complete Problem......Page 203
9.3 Continuity of the Solution......Page 204
9.4 Bibliographic Notes......Page 206
10 Conclusions......Page 207
11 References......Page 209