General Equilibrium Theory of Value

Comments 73 CHAPTER 8: The Fine Picture 74 8.1 Introduction 74 8.2 Aggregate Demand at a No-trade Equilibrium 74 8.3 Regularity of the No-trade Equilibria 75 8.4 The Set of Equilibrium Allocations 75 8.5 Economies with a Unique Equilibrium 78 8.6 Degree of the Natural Projection 79 8.7 The Set of Regular Equilibria 79 8.8 Conclusion 81 8.9 Notes and Comments 81 CHAPTER 9: Production with Decreasing Returns 82 9.1 Introduction 82 9.2 Production Sets: Definitions 82 9.3 Production Sets: Main Properties 84 9.4 The Firm's Objective Function 89 9.5 The Strict Decreasing Returns to Scale Firm 90 9.6 The Net Supply Function as a Primitive Concept 92 9.7 Conclusion 94 9.8 Notes and Comments 95 CHAPTER 10: Equilibrium with Decreasing Returns 96 10.1 Introduction 96 10.2 The General Equilibrium Model with Private Ownership of Decreasing Returns to Scale Firms 96 10.3 Production Adjusted Demand Functions 98 10.4 The Equivalent Exchange Model 101 10.5 Properness of the Natural Projection 103 10.6 Conclusion 107 10.7 Notes and Comments 107 CHAPTER 11: Production with Constant Returns 108 11.1 Introduction 108 11.2 Production Sets 109 11.3 The Net Supply Correspondence 112 11.4 Three Examples 115 11.5 Net Supply Correspondence of a Smooth Constant Returns to Scale Firm 118 11.6 The Graph of the Net Supply Correspondence 120 11.7 Conclusion 123 11.8 Notes and Comments 123 CHAPTER 12: Equilibrium with Constant Returns 124 12.1 Introduction 124 12.2 Decreasing and Constant Returns: General Case 124 12.3 Constant Returns: Reduced Form 125 12.4 Equilibria of the Model N 126 12.5 The Equilibrium Manifold Approach 126 12.6 The Equilibrium Manifold for the Model N 127 12.7 The Natural Projection 133 12.8 Regular and Critical Equilibria 134 12.9 Degrees of the Natural Projection 137 12.10 Regular and Singular Economies 138 12.11 Uniqueness of Equilibrium over ?(T) 139 12.12 The Natural Projection as a Finite Covering of the Set of Regular Economies 141 12.13 Values of the Natural Projection Degrees 143 12.14 Conclusion 144 12.15 Notes and Comments 144 Postscript 145 APPENDIX A: Notation 149 A.1 Points, Vectors, Inner Product 149 A.2 Gradient 149 A.3 Second-Order Derivatives and the Hessian Matrix of a Smooth Function 150 APPENDIX B: Point-set Topology 151 B.1 Proper Maps 151 APPENDIX C: Smooth Manifolds 152 C.1 The Implicit Function Theorem 152 C.2 Smooth Manifolds and Submanifolds 152 C.3 Smooth Mappings, Immersions, and Submersions 153 APPENDIX D: Singularities of Smooth Maps 155 D.1 Critical and Regular Points 155 D.2 Singular and Regular Values 155 D.3 Sard's Theorem 156 D.4 The Regular Value Theorem 156 D.5 The Case where dimX = dimY 156 D.6 Coverings 157 D.7 Surjectivity of Maps with Non-Zero Modulo 2 Degree 157 APPENDIX E: Convexity 159 E.1 Convex and Strictly Convex Sets 159 E.2 Quasi-concave Functions 159 E.3 Smooth Quasi-concavity and Second-Order Derivatives 162 E.4 Bordered Hessian of a Smoothly Quasi-concave Function 164 E.5 Recession Cone of a Convex Set 165 APPENDIX F: Miscellany 166 F.1 Dimension of Semi-algebraic Sets 166 References 167 Index 171