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.
Author(s): Alexander J. Zaslavski
Series: Monographs in Mathematical Economics, 4
Publisher: Springer
Year: 2021
Language: English
Pages: 386
City: Singapore
Preface
Contents
1 Introduction
1.1 The Turnpike Phenomenon
1.2 Nonconcave (Nonconvex) Problems
1.3 Examples
1.4 Stability of the Turnpike Phenomenon
1.5 Nonautonomous Control Systems Without Constraints
1.6 Nonautonomous Constrained Control Systems
1.7 The Robinson–Solow–Srinivasan Model
1.8 Overtaking Optimal Programs for the RSS Model
1.9 Turnpike Properties of the RSS Model
1.10 Autonomous Optimal Control Problems
2 Turnpike Conditions for Optimal Control Systems
2.1 Preliminaries
2.2 Main Results
2.3 TP Implies ATP and Property (P)
2.4 Auxiliary Results
2.5 Proof of Theorem 2.2
2.6 Proof of Theorem 2.3
2.7 Proof of Theorem 2.5
2.8 Proof of Theorem 2.4
2.9 An Example
3 Nonautonomous Problems with Perturbed Objective Functions
3.1 Preliminaries
3.2 Main Results
3.3 Proof of Theorem 3.1
3.4 Proof of Theorem 3.2
3.5 Proof of Proposition 3.4
3.6 Proof of Theorem 3.3
3.7 Proof of Theorem 3.5
4 Nonautonomous Problems with Discounting
4.1 Preliminaries
4.2 Main Results
4.3 Proofs of Theorems 4.1 and 4.4
4.4 Proof of Theorem 4.3
5 Stability of the Turnpike Phenomenon for Nonautonomous Problems
5.1 Preliminaries and Stability Results
5.2 Auxiliary Results
5.3 Proofs of Theorems 5.1 and 5.2
5.4 Proofs of Theorems 5.3 and 5.4
5.5 Proof of Theorem 5.5
5.6 Proofs of Theorems 5.6 and 5.7
5.7 An Example
6 Stability of the Turnpike for Nonautonomous Problems with Discounting
6.1 Preliminaries and the Main Results
6.2 Auxiliary Results
6.3 Proofs of Theorems 6.2 and 6.3
6.4 Proof of Theorem 6.4
7 Turnpike Properties for Autonomous Problems
7.1 Preliminaries and the Main Results
7.2 Auxiliary Results
7.3 Proof of Theorem 7.2
7.4 Proof of Theorem 7.3
7.5 Proofs of Theorems 7.4 and 7.5
7.6 Proofs of Theorems 7.6, 7.7 and 7.8
7.7 The Robinson–Solow–Srinivasan Model
7.8 A Model of Economic Dynamics
7.9 Equivalence of Optimality Criteria
7.10 Proof of Theorem 7.22
7.11 Weak Turnpike Theorems
7.12 Proof of Theorem 7.23
8 Autonomous Problems with Perturbed Objective Functions
8.1 Preliminaries and the Main Results
8.2 Auxiliary Results
8.3 Proofs of Theorems 8.1 and 8.2
8.4 Proofs of Theorems 8.3 and 8.4
8.5 Optimal Control Systems with Discounting
8.6 An Auxiliary Result for Theorems 8.9 and 8.10
8.7 Proof of Theorem 8.9
8.8 Proof of Theorem 8.10
8.9 Existence of Overtaking Optimal Programs
9 Stability Results for Autonomous Problems
9.1 Preliminaries and the Main Results
9.2 Examples
9.3 Auxiliary Results
9.4 Proof of Theorem 9.3
9.5 Proof of Theorem 9.4
9.6 Optimal Control Problems with Discounting
9.7 An Auxiliary Result
9.8 Proof of Theorem 9.15
10 Models with Unbounded Endogenous Economic Growth
10.1 Introduction
10.2 Existence of Unbounded Growth and Balanced Growth Estimates
10.3 Auxiliary Results
10.4 Proof of Theorem 10.5
10.5 Proof of Theorem 10.6
References
Index
