Optimal Control Problems Related to the Robinson-Solow-Srinivasan Model
1st Edition
Published on 9. August 2022
Book
Paperback/Softback
XI, 348 pages
978-981-16-2254-0 (ISBN)
Description
This book is devoted to the study of classes of optimal control problems arising in economic growth theory, related to the Robinson-Solow-Srinivasan (RSS) model. The model was introduced in the 1960s by economists Joan Robinson, Robert Solow, and Thirukodikaval Nilakanta Srinivasan and was further studied by Robinson, Nobuo Okishio, and Joseph Stiglitz. Since then, the study of the RSS model has become an important element of economic dynamics. In this book, two large general classes of optimal control problems, both of them containing the RSS model as a particular case, are presented for study. For these two classes, a turnpike theory is developed and the existence of solutions to the corresponding infinite horizon optimal control problems is established.
The book contains 9 chapters. Chapter 1 discusses turnpike properties for some optimal control problems that are known in the literature, including problems corresponding to the RSS model. The first class of optimal control problems is studied in Chaps. 2-6. In Chap. 2, infinite horizon optimal control problems with nonautonomous optimality criteria are considered. The utility functions, which determine the optimality criterion, are nonconcave. This class of models contains the RSS model as a particular case. The stability of the turnpike phenomenon of the one-dimensional nonautonomous concave RSS model is analyzed in Chap. 3. The following chapter takes up the study of a class of autonomous nonconcave optimal control problems, a subclass of problems considered in Chap. 2. The equivalence of the turnpike property and the asymptotic turnpike property, as well as the stability of the turnpike phenomenon, is established. Turnpike conditions and the stability of the turnpike phenomenon for nonautonomous problems are examined in Chap. 5, with Chap. 6 devoted to the study of the turnpike properties for the one-dimensional nonautonomous nonconcave RSS model. The utility functions, which determine the optimality criterion, are nonconcave. The class of RSS models is identified with a complete metric space of utility functions. Using the Baire category approach, the turnpike phenomenon is shown to hold for most of the models. Chapter 7 begins the study of the second large class of autonomous optimal control problems, and turnpike conditions are established. The stability of the turnpike phenomenon for this class of problems is investigated further in Chaps. 8 and 9.
The book contains 9 chapters. Chapter 1 discusses turnpike properties for some optimal control problems that are known in the literature, including problems corresponding to the RSS model. The first class of optimal control problems is studied in Chaps. 2-6. In Chap. 2, infinite horizon optimal control problems with nonautonomous optimality criteria are considered. The utility functions, which determine the optimality criterion, are nonconcave. This class of models contains the RSS model as a particular case. The stability of the turnpike phenomenon of the one-dimensional nonautonomous concave RSS model is analyzed in Chap. 3. The following chapter takes up the study of a class of autonomous nonconcave optimal control problems, a subclass of problems considered in Chap. 2. The equivalence of the turnpike property and the asymptotic turnpike property, as well as the stability of the turnpike phenomenon, is established. Turnpike conditions and the stability of the turnpike phenomenon for nonautonomous problems are examined in Chap. 5, with Chap. 6 devoted to the study of the turnpike properties for the one-dimensional nonautonomous nonconcave RSS model. The utility functions, which determine the optimality criterion, are nonconcave. The class of RSS models is identified with a complete metric space of utility functions. Using the Baire category approach, the turnpike phenomenon is shown to hold for most of the models. Chapter 7 begins the study of the second large class of autonomous optimal control problems, and turnpike conditions are established. The stability of the turnpike phenomenon for this class of problems is investigated further in Chaps. 8 and 9.
More details
Product info
Paperback
Series
Edition
1st ed. 2021
Language
English
Place of publication
Singapore
Singapore
Target group
Professional and scholarly
Illustrations
1
1 s/w Abbildung
XI, 348 p. 1 illus.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 20 mm
Weight
546 gr
ISBN-13
978-981-16-2254-0 (9789811622540)
DOI
10.1007/978-981-16-2252-6
Schweitzer Classification
Other editions
Additional editions
Alexander J. Zaslavski
Optimal Control Problems Related to the Robinson-Solow-Srinivasan Model
Book
08/2021
1st Edition
Springer
€139.09
Shipment within 3-4 weeks
Content
Chapter 1. Introduction1.1 The turnpike phenomenon1.2. Nonconcave (nonconvex) problems1.3. Examples1.4. Stability of the turnpike phenomenon1.5 The Robinson-Solow-Srinivasan model1.6. Overtaking optimal programs for the Robinson-Solow-Srinivasan model1.7. Turnpike properties of the Robinson-Solow-Srinivasan modelChapter 2. Good programs for infinite horizon optimal control problems2.1. Preliminaries and the main results2.2. Upper semicontinuity of cost functions2.3 The nonstationary Robinson-Solow-Srinivasan model2.4. Auxiliary results for Theorems 2.4, 2.5 and 2.72.5. Properties of the function $U$2.6. Proof of Theorem 2.42.7. Proof of Theorem 2.52.8. Proof of Theorem 2.72.9. Problems with discounting2.10. The Robinson-Solow-Srinivasan model with discounting2.11. Auxiliary results for Theorem 2.232.12. Proof of Theorem 2.23Chapter 3. One-dimensional concave Robinson-Solow-Srinivasan model3.1. Preliminaries and main results3.2. Auxiliary results3.3. Proof of Theorem 3.143.4. Stability results3.5. Proof of Theorem 3.26Chapter 4. Autonomous nonconcave optimal control problems4.1. Preliminaries4.2. A controlability lemma4.3. The turnpike property implies the asymptotic turnpike property4.4. Auxiliary results4.5. The asymptotic turnpike property implies the turnpike property4.6. A weak turnpike property4.7. A turnpike result4.8. Auxiliary results for Theorem 4.114.9. Proof of Theorem 4.114.10. Stability results4.11. A subclass of modelsChapter 5. Turnpike phenomenon for nonautonomous problems5.1 Preliminaries5.2. Turnpike properties5.3. Examples5.4. The turnpike property implies (P1) and (P2)5.5. Auxiliary results5.6. Completion of the proof of Theorem 5.25.7. A turnpike result5.8. An auxiliary result for Theorem 5.85.9. Proof of Theorem 5.8 5.10. Stability results5.11. Proof of Theorem 5.11Chapter 6. Generic results6.1. One-dimensional nonconcave Robinson-Solow-Srinivasan model6.2. The main results6.3. Auxiliary results6.4. Auxiliary results for Theorem 6.6.6.5. Proof of Theorem 6.66.6. Proof of Theorem 6.10Chapter 7. Turnpike phenomenon for a class of optimal control problems7.1. Preliminaries and main results7.2. Auxiliary results7.3. Proof of Theorem 7.27.4. Proof of Theorem 7.37.5. Proofs of Theorems 7.4 and 7.57.7. The Robinson-Solow-Srinivasan model7.8. A general model of economic dynamics7.9. Equivalence of optimality criterions7.10. Proof of Theorem 7.227.11 Weak turnpike results7.12. Proof of Theorem 7.23Chapter 8. Problems with perturbed objective functions8.1 Preliminaries and main results8.2. Auxiliary results8.3 Proofs of Theorems 8.1 and 8.28.4. Proofs of Theorems 8.3 and 8.48.5. Problems with discounting8.6. Auxiliary results for Theorems 8.10 and 8.118.7. Proof of Theorem 8.108.8. Proof of Theorem 8.118.9. Existence of overtaking optimal programsChapter 9. Stability of the turnpike phenomenon9.1. Preliminaries and main results9.2. Examples9.3. Auxiliary results9.4. Proof of Theorem 9.39.5. Proof of Theorem 9.49.6. Problems with discounting9.7. Auxiliary results for Theorem 9.159.8. Proof of Theorem 9.15