Variational Methods in Economics

PrefaceChapter 1 1.1. Introduction 1.2. Growth Models in EconomicsChapter 2: Calculus of Variations-Classical Theory 2.1. Introduction 2.2. Historical Foundations and Some Classical Problems 2.3. Functions 2.4. Phase Diagrams And Phase Spaces 2.5. The Simplest Problem in the Calculus of Variations 2.6. The Euler Equation and Corner Conditions 2.7. The Euler Equation in Special Cases 2.8. Examples 2.9. Ramsey's Problem 2.10. Ramsey's Problem-Finite Planning Horizons 2.11. Ramsey's Problem-Infinite Planning Horizons 2.12. Samuelson's Catenary Turnpike Theorem 2.13. Generalization to n Functions 2.14. Parametric Representation 2.15. The Weierstrass and Legendre Necessary Conditions 2.16. An n-Sector Generalization of Ramsey's Model 2.17. Sufficient Conditions Based on the Concavity of F 2.18. Classical Sufficiency Conditions 2.19. Sensitivity Analysis-End Point Changes 2.20. Weierstrass' Sufficient Conditions 2.21. Variable End Points-Free Boundaries 2.22. Variable End Points-Transversality 2.23. Transversality at Infinity 2.24. Transversality in the Model of Samuelson and Solow 2.25. The Connection with Dynamic Programming ProblemsChapter 3: Classical Theory-Constraints 3.1. Introduction 3.2. Isoperimetric Constraints 3.3. The Problem of Lagrange 3.4. A Simplified 'Proof of the Multiplier Rule for Fixed End Points 3.5. Transversality 3.6. Free Boundary Conditions 3.7. Examples 3.8. Preliminary Results 3.9. The Comparison Functions 3.10. The Convex Polyhedral Cone K 3.11. The Convex Cone K* 3.12. The Multiplier Rule and the Weierstrass and Legendre Necessary Conditions 3.13. Unbounded Intervals of Integration 3.14. Normality 3.15. Sufficiency Based on the Concavity of H 3.16. The Problem of Bolza ProblemsChapter 4: Optimal Control Theory 4.1. Introduction 4.2. The Basic Control Problem 4.3. Necessary Conditions-The Maximum Principle 4.4. The Conversion of a Nonautonomous System into an Autonomous System 4.5. An Heuristic Derivation of the Necessary Conditions 4.6. The Ramsey Model: Finite Planning Horizon 4.7. A Phase Diagram Using Conjugate Variables 4.8. The Ramsey Model: Infinite Planning Horizon 4.9. A Situation in Which it may be Optimal to Reach Bliss in a Finite Time 4.10. The Ramsey Model with Discounting 4.11. Optimal transfer from K0 to K1 with Z → 0 as K → K1 4.12. The Conversion of Classical Problems into Control Problems 4.13. The Synthesis Problem 4.14. The Model of Samuelson and Solow Again ProblemsChapter 5: Connections with the Classical Theory 5.1. Introduction 5.2. Valentine's Procedure 5.3. The Conversion of a Control Problem into a Problem of Lagrange 5.4. Necessary Conditions 5.5. The Time Variation of H 5.6. Examples 5.7. Sufficiency 5.8. Linear Control Problems 5.9. Constrained State Variables-Introduction 5.10. The Control Problem and the Equivalent Problem of Lagrange 5.11. Necessary Conditions for the Problem of Lagrange 5.12. An Interior Segment 5.13. A Boundary Segment 5.14. Transition Conditions ProblemsChapter 6: Two-Sector Models of Optimal Economic Growth 6.1. Introduction 6.2. The Capital Good Freely Transferable Between Industries 6.3. The Capital Good Immobile Between Industries 6.4. Final Remarks ProblemsAppendix I: Uniqueness of the Utility Functional and the Utility FunctionAppendix II: Implicit Function TheoremsAppendix III. Existence Theorems for Systems of Ordinary Differential EquationsReferences