Hadley / Kemp / Bliss Variational Methods in Economics

1. Auflage 2014
ISBN: 978-1-4832-7528-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 1, 388 Seiten, Web PDF

Reihe: Advanced Textbooks in Economics

Advanced Textbooks in Economics, Volume 1: Variational Methods in Economics focuses on the application of variational methods in economics, including autonomous system, dynamic programming, and phase spaces and diagrams. The manuscript first elaborates on growth models in economics and calculus of variations. Discussions focus on connection with dynamic programming, variable end points-free boundaries, transversality at infinity, sensitivity analysis-end point changes, Weierstrass and Legendre necessary conditions, and phase diagrams and phase spaces. The text then ponders on the constraints of classical theory, including unbounded intervals of integration, free boundary conditions, comparison functions, normality, and the problem of Bolza. The publication explains two-sector models of optimal economic growth, optimal control theory, and connections with the classical theory. Topics include capital good immobile between industries, constrained state variables, linear control problems, conversion of a control problem into a problem of Lagrange, and the conversion of a nonautonomous system into an autonomous system. The book is a valuable source of information for economists and researchers interested in the variational methods in economics.

Autoren/Hrsg.

Hadley, G.

Kemp, M. C.

Bliss, C. J.

Inhaltsverzeichnis

1;Front Cover;1
2;Variational Methods in Economics;4
3;Copyright Page;5
4;Table of Content;8
5;Preface;6
6;CHAPTER 1. GROWTH MODELS IN ECONOMICS;12
6.1;1.1. Introduction;12
6.2;1.2. Growth models in economics;12
7;CHAPTER 2. CALCULUS OF VARIATIONS-CLASSICAL THEORY;17
7.1;2.1. Introduction;17
7.2;2.2. Historical foundations and some classical problems;18
7.3;2.3. Functions;21
7.4;2.4. Phase diagrams and phase spaces;28
7.5;2.5. The simplest problem in the calculus of variations;33
7.6;2.6. The Euler equation and corner conditions;37
7.7;2.7. The Euler equation in special cases;51
7.8;2.8. Examples;52
7.9;2.9. Ramsey's problem;61
7.10;2.10. Ramsey's problem-finite planning horizons;65
7.11;2.11. Ramsey's problem-infinite planning horizons;73
7.12;2.12. Samuelson's catenary turnpike theorem;82
7.13;2.13. Generalization to n functions;89
7.14;2.14. Parametric representation;93
7.15;2.15. The Weierstrass and Legendre necessary conditions;97
7.16;2.16. An n-sector generalization of Ramsey's model;100
7.17;2.17. Sufficient conditions based on the concavity of F;107
7.18;2.18. Classical sufficiency conditions;116
7.19;2.19. Sensitivity analysis-end point changes;128
7.20;2.20. Weierstrass' sufficient conditions;131
7.21;2.21. Variable end points-free boundaries;138
7.22;2.22. Variable end points-transversality;142
7.23;2.23. Transversality at infinity;148
7.24;2.24. Transversality in the model of Samuelson and Solow;151
7.25;2.25. The connection with dynamic programming;158
7.26;Problems;163
8;CHAPTER 3. CLASSICAL THEORY . CONSTRAINTS;186
8.1;3.1. Introduction;186
8.2;3.2. Isoperimetric constraints;188
8.3;3.3. The problem of Lagrange;194
8.4;3.4. A simplified 'proof' of the multiplier rule for fixed end points;197
8.5;3.5. Transversality;203
8.6;3.6. Free boundary conditions;208
8.7;3.7. Examples;208
8.8;3.8. Preliminary results;212
8.9;3.9. The comparison functions;219
8.10;3.10. The convex polyhedral cone K;223
8.11;3.11. The convex cone K;226
8.12;3.12. The multiplier rule and the Weierstrass and Legendre necessary conditions;228
8.13;3.13. Unbounded intervals of integration;234
8.14;3.14. Normality;236
8.15;3.15. Sufficiency based on the concavity of H;239
8.16;3.16. The problem of Bolza;241
8.17;Problems;244
9;CHAPTER 4. OPTIMAL CONTROL THEORY;249
9.1;4.1. Introduction;249
9.2;4.2. The basic control problem;250
9.3;4.3. Necessary conditions-the maximum principle;252
9.4;4.4. The conversion of a nonautonomous system into an autonomous system;257
9.5;4.5. An heuristic derivation of the necessary conditions;260
9.6;4.6. The Ramsey model: finite planning horizon;266
9.7;4.7. A phase diagram using conjugate variables;268
9.8;4.8. The Ramsey model: infinite planning horizon;272
9.9;4.9. A situation in which it may be optimal to reach bliss in a finite time;275
9.10;4.10. The Ramsey model with discounting;278
9.11;4.11. Optimal transfer from Ko to K1 with K . 0 as K . K1;281
9.12;4.12. The conversion of classical problems into control problems;281
9.13;4.13. The synthesis problem;282
9.14;4.14. The model of Samuelson and Solow again;284
9.15;Problems;291
10;CHAPTER 5. CONNECTIONS WITH THE CLASSICAL THEORY;293
10.1;5.1. Introduction;293
10.2;5.2. Valentine's procedure;295
10.3;5.3. The conversion of a control problem into a problem of Lagrange;296
10.4;5.4. Necessary conditions;297
10.5;5.5. The time variation of H;303
10.6;5.6. Examples;305
10.7;5.7. Sufficiency;308
10.8;5.8. Linear control problems;310
10.9;5.9. Constrained state variables-introduction;315
10.10;5.10. The control problem and the equivalent problem of Lagrange;317
10.11;5.11. Necessary conditions for the problem of Lagrange;319
10.12;5.12. An interior segment;323
10.13;5.13. A boundary segment;326
10.14;5.14. Transition conditions;330
10.15;Problems;333
11;CHAPTER 6. TWO-SECTOR MODELS OF OPTIMAL ECONOMIC GROWTH;337
11.1;6.1. Introduction;337
11.2;6.2. The capital good freely transferable between industries;338
11.3;6.3. The capital good immobile between industries;358
11.4;6.4. Final remarks;372
11.5;Problems;374
12;APPENDIX I: Uniqueness of the utility functional and the utility function;375
13;APPENDIX II: Implicit function theorems;378
14;APPENDIX III: Existence theorems for systems of ordinary differential equations;381
15;REFERENCES;385
16;Index;388