ISBN: 3540239065
TITLE: Basov
AUTHOR: Multi-dimensional Screening
TOC:

I Mathematieal Preliminaries 1
1 Vector Calculus 3
1.1 The Main Operations of Vector Calculus: div, grad, and Delta 3
1.2 Conservative Vector Fields 5
1.3 Curvilinear Integrals and the Geometric Meaning of the Existence of a Potential 7
1.4 Multiple and Repeated Integrals 8
1.5 The Flow oF a Vector Field and the Gauss-Ostrogradsky Theorem 12
1.6 The Circulation of a Vector Field and the Green Formula 15
1.7 Exercises 17
1.8 Bibliographic Notes 18
2 Partial Differential Equations 19
2.1 The First Order Partial Differential Equations 19
2.1.1. The Complete Integral and the General Integral 20
2.1.2 The Singular Integral 21
2.1.3 The Quasilinear Equations and the Method of Characteristics 22
2.1.4 Compatible Systems of the First Order PDEs 24
2.1.5 The Method of Characteristics for a Non-quasilinear First Order PDE 26
2.1.6 Examples 27
2.2 The Second Order Partial Differential Equations 30
2.2.1 Classification of the Linear Second Order Partial Differential Equations 30
2.2.2 Boundary Value Problems for Elliptic Equations 31
2.2.3 Examples 32
2.3 Group Theoretic Analysis of the Systems of Partial Differential Equations 38
2.3.1 One Parameter Lie Groups 38
2.3.2 Invariante of PDEs. Systems of PDEs, and Boundary Problems under Lie Groups 41
2.3.3 Calculating a Lie Group of a PDE 44
2.3.4 Calculating Invariants of the Lie Group 45
2.3 5 Examples 46
2.4 Exercises 47
2.5 Bibliographic Notes 49
3 Theory of Generalized Convexity 51
3.1 Definition and Properties of the Generalized Fenchel Conjugates 52
3.2 Generalized Convexity and Cyclic Monotonicity 55
3.3 Examples 58
3.4 Exercises 59
3.5 Bibliographic Notes 59
4 Calculus of Variations and the Optimal Control 61
4.1 Banach Spaces and Polish Spaces 61
4.2 Hilbert Spaces 65
4.3 Dual Space for a Normed Space and a Hilbert Space 66
4.4 Frechet Derivative of a Mapping between Normed Spaces 69
4.5 Functionals and Gateaux Derivatives 71
4.6 Euler Equation 73
4.7 Optimal Control 74
4.8 Examples 76
4.9 Exercises 78
4.10 Bibliographic Notes 79
5 Miscellaneous Techniques 81
5.1 Distributions and Generalized Solutions for the Partial Differential Equations 81
5.1.1 A Motivating Example 82
5.1.2 The Set of Test Functions and its Dual 83
5.1.3 Examples of Distributions 84
5.1.4 The Derivative of a Distribution 87
5.1 5 The Product of a Distribution and a Test Function and the Product of Distributions 88
5.1.6 The Resultant of a Distribution and a Dilation Operator 89
5.1.7 Adjoint Linear Differential Operators and Generalized Solutions of the Partial Differential Equations 91
5.2 Sobolev Spaces and Poincare Theorem 92
5.3 Sweeping Operators and Balayage of Measures 94
5.4 Coercive Functionals 96
5.5 Optimization by Vector Space Methods 96
5.6 Calculus of Variation Problem with Convexity Constraints 98
5.7 Supermodularity and Monotone Comparative Statics 99
5.8 Hausdorff Metric on Compact Sets of a Metric Space 103
5.9 Generalized Envelope Theorems 107
5.10 Exercises 109
5.11 Bibliographic Notes 110
II Economics of Multi-dimensional Screening 111
6 The Unidimensional Screening Model 115
6.1 Spence-Mirrlees Condition and Implementability 116
6.2 The Concept of the Information Rent 119
6.3 Three Approaches to the Unidimensional Relaxed Problem 119
6.3.1 The Direct Approach 119
6.3.2 The Dual Approach 120
6.3.3 The Hamiltonian Approach 121
6.4 Hamiltonian Approach to the Unidimensional Complete Problem 122
6.5 Type Dependent Participation Constraint 124
6.6 Random Participation Constraint 126
6.7 Examples 127
6.8 Exercises 133
6.9 Bibliographic Notes 134
7 The Multi-dimensional Screening Model 135
7.1 The Genericity of Exclusion 137
7.2 Generalized Convexity and Implementability 141
7.2.1 Is Bunching Robust in the Multi-dimensional Case? 143
7.3 Path Independence of Information Rents 144
7.4 Cost Based Tariffs 145
7.5 Direct Approach and Its Limitations 148
7.6 Dual Approach for m = n 151
7.6.1 The Relaxed Problem 152
7.6.2 An Alternative Approach to the Relaxed Problem 153
7.6.3 The Complete Problem 154
7.6.4 The Geometry of the Participation Region 155
7.6.5 A Sufficient Condition for Bunching 156
7.6.6 The Extension of the Dual Approach for n > rn 156
7.7 Hamiltonian Approach and the First Order Characterization of the Solution 158
7.7.1 The Economic Meaning of the Lagrange Multipliers 160
7.8 Symmetry Analysis of the First Order Conditions 161
7.9 Some Remarks on the Hamiltonian Approach to the Complete Problem 166
7.10 Examples and Economic Applications 167
7.11 Exercises 173
7.12 Bibliographic Notes 173
8 Beyond the Quasilinear Case 175
8.1 The Unidimensional Case 176
8.2 The Multi-dimensional Case 179
8.2.1 Implementability of a Surplus Function 180
8.2.2 Implementability of an Allocation 181
8.3 The First Order Characterization of the Solution of the Relaxed Problem 184
8.4 Exercises 186
8.5 Bibliographic Notes 187
9 Existence, Uniqueness, and Regularity Properties of the Solution 189
9.1 Existence and Uniqueness of the Solution of the Relaxed Problem 189
9.2 Existence of a Solution for the Complete Problem 191
9.3 Continuity of the Solution 192
9.4 Bibliographie Notes 194
10 Conclusions 195
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