This book is devoted to the study of a class of optimal control problems arising in mathematical economics, related to the Robinson–Solow–Srinivasan (RSS) model. It will be useful for researches interested in the turnpike theory, infinite horizon optimal control and their applications, and mathematical economists. The RSS is a well-known model of economic dynamics that was introduced in the 1960s and as many other models of economic dynamics, the RSS model is determined by an objective function (a utility function) and a set-valued mapping (a technology map). The set-valued map generates a dynamical system whose trajectories are under consideration and the objective function determines an optimality criterion. The goal is to find optimal trajectories of the dynamical system, using the optimality criterion.
Chapter 1 discusses turnpike properties for some classes of discrete time optimal control problems. Chapter 2 present the description of the RSS model and discuss its basic properties. Infinite horizon optimal control problems, related to the RSS model are studied in Chapter 3. Turnpike properties for the RSS model are analyzed in Chapter 4. Chapter 5 studies infinite horizon optimal control problems related to the RSS model with a nonconcave utility function. Chapter 6 focuses on infinite horizon optimal control problems with nonautonomous optimality criterions. Chapter 7 contains turnpike results for a class of discrete-time optimal control problems. Chapter 8 discusses the RSS model and compares different optimality criterions. Chapter 9 is devoted to the study of the turnpike properties for the RSS model. In Chapter 10 the one-dimensional autonomous RSS model is considered and the continuous time RSS model is studied in Chapter 11.
Author(s): Alexander J. Zaslavski
Series: Springer Optimization and Its Applications, 166
Publisher: Springer
Year: 2021
Language: English
Pages: 447
City: Cham
Preface
Contents
1 Introduction
1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems
1.2 The Turnpike Phenomenon
1.3 Turnpike Results for Problems in Metric Spaces
1.4 The Robinson–Solow–Srinivasan Model
2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties
2.1 The Robinson–Solow–Srinivasan Model
2.2 A Golden-Rule Stock
2.3 Good Programs
2.4 The von Neumann Facet
3 Infinite Horizon Optimization
3.1 Overtaking Optimal Programs
3.2 Auxiliary Results for Theorems 3.1–3.3
3.3 Proofs of Theorems 3.1 and 3.2
3.4 Proof of Theorem 3.3
3.5 Examples
3.6 Convergence Results
3.7 Proof of Theorem 3.9
3.8 Auxiliary Results for Theorems 3.10 and 3.11
3.9 Proofs of Theorems 3.10 and 3.11
3.10 The Structure of Good Programs in the RSS Model
3.11 Proofs of Theorems 3.16–3.18
4 Turnpike Results for the Robinson–Solow–Srinivasan Model
4.1 The Main Results
4.2 Auxiliary Results for Theorems 4.2 and 4.3
4.3 Four Lemmas
4.4 Proof of Theorem 4.2
4.5 Proof of Theorem 4.3
4.6 Extensions of Theorem 4.3
4.7 Proof of Theorem 4.16
4.8 Stability Results
4.9 Proof of Theorem 4.27
5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function
5.1 Good Programs
5.2 Auxiliary Results
5.3 Properties of the Function U
5.4 Proofs of Theorems 5.4, 5.5 and 5.8
5.5 Proof of Proposition 5.7
5.6 Proof of Theorem 5.9
5.7 The RSS Model with Discounting
5.8 An Auxiliary Result for Theorem 5.18
5.9 Proof of Theorem 5.18
5.10 Weakly Agreeable Programs
5.11 Proof of Theorem 5.22
5.12 Auxiliary Results
5.13 Proof of Theorem 5.23
5.14 Proof of Theorem 5.24
5.15 Weakly Maximal Programs
5.16 Proof of Theorem 5.27
5.17 Proof of Theorem 5.28
5.18 Proof of Theorem 5.29
6 Infinite Horizon Nonautonomous Optimization Problems
6.1 The Model Description and Main Results
6.2 Upper Semicontinuity of Cost Functions
6.3 The Nonstationary Robinson–Solow–Srinivasan Model
6.4 Auxiliary Results for Theorems 6.4, 6.5, and 6.7
6.5 Properties of the Function U
6.6 Proof of Theorem 6.4
6.7 Proof of Theorem 6.5
6.8 Proof of Theorem 6.7
6.9 Overtaking Optimal Programs
6.10 A Subclass of Infinite Horizon Problems
6.11 Auxiliary Results for Theorems 6.23
6.12 Proof of Theorem 6.23
7 One-Dimensional Robinson–Solow–Srinivasan Model
7.1 Preliminaries and Main Results
7.2 Auxiliary Results
7.3 Proof of Theorem 7.8
7.4 Proof of Theorem 7.9
7.5 Proof of Theorem 7.7
7.6 Proofs of Theorems 7.12 and 7.13
8 Optimal Programs
8.1 Preliminaries
8.2 Optimality Criteria
8.3 Four Theorems
8.4 Proof of Theorem 8.6
8.5 Proof of Theorem 8.7
8.6 Proof of Theorem 8.8
8.7 Proof of Theorem 8.9
8.8 Maximal Programs
8.9 One-Dimensional Model
9 Turnpike for the RSS Model with Nonconcave Utility Functions
9.1 Preliminaries and Main Results
9.2 Auxiliary Results
9.3 Proof of Theorem 9.3
9.4 Proofs of Theorems 9.5 and 9.7
9.5 Generalizations of the Turnpike Results
9.6 Proof of Theorem 9.14
9.7 Stability Results
9.8 Proof of Theorem 9.19
9.9 An Auxiliary Result
9.10 Perturbations
9.11 Generic Results
10 An Autonomous One-Dimensional Model
10.1 The Model
10.2 A Golden Rule
10.3 Optimality and Value-Loss Minimization
10.4 Optimality Does Not Imply Value-Loss Minimization
10.5 Optimal Policy Function
10.6 Proofs
11 The Continuous-Time Robinson–Solow–Srinivasan Model
11.1 Infinite Horizon Problems
11.2 Proofs of Propositions 11.1–11.3
11.3 Auxiliary Results
11.4 Proofs of Theorems 11.4 and 11.5
11.5 Turnpike Results
11.6 Auxiliary Results
11.7 Proof of Theorem 11.4
11.8 Proof of Theorem 11.5
11.9 Stability of the Turnpike Phenomenon
11.10 Discount Case
11.11 Proof of Theorem 11.26
11.12 Proof of Theorem 11.29
11.13 Optimal Programs over Infinite Horizon
References
Index
