Turnpike Theory for the Robinson–Solow–Srinivasan Model 3030603067, 9783030603069

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Springer Optimization and Its Applications 166

Alexander J. Zaslavski

Turnpike Theory for the Robinson–Solow– Srinivasan Model

Springer Optimization and Its Applications Volume 166

Series Editors Panos M. Pardalos , University of Florida My T. Thai , University of Florida Honorary Editor Ding-Zhu Du, University of Texas at Dallas Advisory Editors Roman V. Belavkin, Middlesex University John R. Birge, University of Chicago Sergiy Butenko, Texas A&M University Vipin Kumar, University of Minnesota Anna Nagurney, University of Massachusetts Amherst Jun Pei, Hefei University of Technology Oleg Prokopyev, University of Pittsburgh Steffen Rebennack, Karlsruhe Institute of Technology Mauricio Resende, Amazon Tamás Terlaky, Lehigh University Michael N. Vrahatis, University of Patras Van Vu, Yale University Guoliang Xue, Arizona State University Yinyu Ye, Stanford University

Aims and Scope Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences. The series Springer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS. More information about this series at http://www.springer.com/series/7393

Alexander J. Zaslavski

Turnpike Theory for the Robinson–Solow–Srinivasan Model

Alexander J. Zaslavski Department of Mathematics Technion – Israel Institute of Technology Haifa, Israel

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-030-60306-9 ISBN 978-3-030-60307-6 (eBook) https://doi.org/10.1007/978-3-030-60307-6 Mathematics Subject Classification: 49J24, 49J99, 49K15, 49K40, 90C31 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The growing importance of the turnpike theory and infinite horizon optimal control has been recognized in recent years. This is not only due to impressive theoretical developments but also because of numerous applications to engineering, economics, life sciences, etc. This book is devoted to the study of a class of optimal control problems arising in mathematical economics, related to the Robinson–Solow–Srinivasan model. In the 1960s this model was introduced by Robinson, Solow, and Srinivasan and was studied by Robinson, Okishio, and Stiglitz. In 2005 this model was revisited by M. Ali Khan and T. Mitra and till now is an important and interesting topic in mathematical economics. As usual, for the Robinson–Solow–Srinivasan model, the existence of optimal solutions over infinite horizon and the structure of solutions on finite intervals are under consideration. In our books [117, 121], we study a class of discrete-time optimal control problems which describe many models of economic dynamics except of the Robinson–Solow–Srinivasan model. This happens because some assumptions posed in [117, 121], which are true for many models of economic dynamics, do not hold for the Robinson–Solow–Srinivasan model. Namely, for many models of economic dynamics, the turnpike is a singleton, and a local controllability property holds in a neighborhood of the turnpike. For the Robinson–Solow–Srinivasan model, the turnpike is a singleton too, but the local controllability property does not hold. This makes the situation more difficult and less understood. Nevertheless, we show in this book that the turnpike theory presented in [117] is extended for the Robinson–Solow–Srinivasan model. In Chapter 1 we discuss turnpike properties for some classes of discretetime optimal control problems. In Chapter 2 we present the description of the Robinson–Solow–Srinivasan model and discuss its basic properties. In particular, we show the existence of weakly optimal programs and good programs and prove an average turnpike property. Infinite horizon optimal control problems, related to the Robinson–Solow–Srinivasan model, are studied in Chapter 3, where we establish a convergence of good programs to the golden-rule stock, show the existence of overtaking optimal programs and analyze their convergence to the golden-rule stock, and consider some properties of good programs.

v

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Turnpike properties for the Robinson–Solow–Srinivasan model are analyzed in Chapter 4. To have these properties means that the approximate solutions of the problems are essentially independent of the choice of an interval and endpoint conditions. It is shown that these turnpike properties hold and that they are stable under perturbations of an objective function. In Chapter 5 we study infinite horizon optimal control problems related to the Robinson–Solow–Srinivasan model with a nonconcave utility function. In particular, we establish the existence of good programs and optimal programs using different optimality criterions. In Chapter 6 we study infinite horizon optimal control problems with nonautonomous optimality criterions. The utility functions, which determine the optimality criterion, are nonconcave. The class of models contains, as a particular case, the Robinson–Solow–Srinivasan model. We establish the existence of good programs and optimal programs. Chapter 7 contains turnpike results for a class of discrete-time optimal control problems. These control problems arise in economic dynamics and describe the one-dimensional nonstationary Robinson–Solow–Srinivasan model. In Chapter 8 we continue to study the Robinson–Solow–Srinivasan model and compare different optimality criterions. In particular, we are interested in good programs and agreeable and weakly maximal programs. Chapter 9 is devoted to the study of the turnpike properties for the Robinson– Solow–Srinivasan model. The utility functions, which determine the optimality criterion, are nonconcave. We show that the turnpike properties hold and that they are stable under perturbations of an objective function. Moreover, we consider a class of RSS models which is identified with a complete metric space of utility functions. Using the Baire category approach, we show that the turnpike phenomenon holds for most of the models. In Chapter 10 we consider the one-dimensional autonomous Robinson–Solow– Srinivasan model, a multiplicity of optimal programs under certain conditions, and properties of the optimal policy correspondence. The continuous-time Robinson– Solow–Srinivasan model is studied in Chapter 11. The author believes that this book will be useful for researches interested in the turnpike theory and infinite horizon optimal control and their applications. Haifa, Israel December 28, 2019

Alexander J. Zaslavski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Turnpike Phenomenon for Convex Discrete-Time Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Turnpike Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Turnpike Results for Problems in Metric Spaces . . . . . . . . . . . . . . . . . . 1.4 The Robinson–Solow–Srinivasan Model . . . . . . . . . . . . . . . . . . . . . . . . . .

1 16 18 20

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 28 30 37

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 42 45 48 50 54 55 56 60 66 67

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 80 85 96

2

1

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4.5 4.6 4.7 4.8 4.9

Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensions of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 4.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 4.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100 103 105 108 116

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 130 138 141 148 149 152 154 159 160 161 163 164 165 169 170 175 178

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 184 186 193 201 204 208 210 212 214 219 223

7

One-Dimensional Robinson–Solow–Srinivasan Model . . . . . . . . . . . . . . . . . 7.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Proof of Theorem 7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 231 254

5

Contents

ix

7.4 7.5 7.6

Proof of Theorem 7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Proof of Theorem 7.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Proofs of Theorems 7.12 and 7.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 268 269 270 271 272 274 277 279

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 285 291 304 307 312 313 317 321 327 329 335

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 341 343 346 347 348 349

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369 369 372 374 383 389 391 407 410 413

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11.10 11.11 11.12 11.13

Discount Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 11.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 11.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal Programs over Infinite Horizon. . . . . . . . . . . . . . . . . . . . . . . . . . .

417 418 423 429

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Chapter 1

Introduction

In this chapter we discuss turnpike properties and optimality criterions over infinite horizon for three classes of discrete-time dynamic optimization problems. The first class contains convex unconstrained dynamic optimization problems, the second one is a class of constrained problems without convexity (concavity) assumptions, and the third class of problems is related to the the Robinson–Solow–Srinivasan model.

1.1 The Turnpike Phenomenon for Convex Discrete-Time Problems The study of the existence and the structure of solutions of optimal control problems and dynamic games defined on infinite intervals and on sufficiently large intervals has been a rapidly growing area of research [2, 4, 11, 16, 19, 23–26, 36, 62, 68, 69, 71, 93, 100, 106, 117, 120, 121, 124] which has various applications in engineering [1, 58, 125, 126], in models of economic growth [3, 6, 10, 17, 28, 32–34, 37–50, 61, 67, 72, 76, 80, 82, 89, 103, 108, 110, 116], in infinite discrete models of solidstate physics related to dislocations in one-dimensional crystals [7, 90], and in the theory of thermodynamical equilibrium for materials [21, 59, 64–66]. Discrete-time optimal control problems were considered in [5, 12, 15, 22, 27, 35, 51–54, 92, 94, 95, 98, 99, 104, 105, 111, 114, 115, 122], finite-dimensional continuous-time problems were analyzed in [13, 14, 18, 20, 55, 57, 60, 63, 74, 96, 97, 101, 102, 107, 112, 113, 119], infinite-dimensional optimal control was studied in [8, 9, 30, 75, 81, 88, 123], while solutions of dynamic games were discussed in [29, 31, 56, 91, 109, 118]. In this book we are interested in optimal control problems related to the Robinson– Solow–Srinivasan model, which was introduced in the 1960s by Robinson [77], Solow [83], and Srinivasan [84] and was studied by Robinson, Okishio, and Stiglitz [73, 78, 85–87]. Recently, the Robinson–Solow–Srinivasan model was studied by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_1

1

2

1 Introduction

M. Ali Khan and T. Mitra [37–45], M. Ali Khan and Piazza [46–49], M. Ali Khan and A. J. Zaslavski [51–54], and A. J. Zaslavski [92, 94, 98–101, 103–108, 110, 111, 114–116]. In this section we discuss turnpike properties and optimality criterions over infinite horizon for a class of convex dynamic unconstrained optimization problems. n Euclidean space with the inner product xy = nLet R be the n-dimensional n which induces the norm x y , x, y ∈ R i i i=1 |x| =

 n 

1/2 xi2

, x = (x1 , . . . , xn ) ∈ R n .

i=1

Let v : R n × R n → R 1 be bounded from below function. We consider the minimization problem T −1

v(xi , xi+1 ) → min,

(P0 )

i=0

such that {xi }Ti=0 ⊂ R n and x0 = z, xT = y, where T is a natural number and the points y, z ∈ R n . The interest in discrete-time optimal problems of type (P0 ) stems from the study of various optimization problems which can be reduced to it, e.g., continuous-time control systems which are represented by ordinary differential equations whose cost integrand contains a discounting factor [57], tracking problems in engineering [1, 58, 125], the study of Frenkel–Kontorova model [7, 90] and the analysis of a long slender bar of a polymeric material under tension in [21, 59, 64–66]. In this section we suppose that the function v : R n × R n → R 1 is strictly convex and differentiable and satisfies the growth condition v(y, z)/(|y| + |z|) → ∞ as |y| + |z| → ∞.

(1.1)

We intend to study the behavior of solutions of the problem (P0 ) when the points y, z and the real number T vary and T is sufficiently large. Namely, we are interested to study a turnpike property of solutions of (P0 ) which is independent of the length of the interval T , for all sufficiently large intervals. To have this property means, roughly speaking, that solutions of the optimal control problems are determined mainly by the objective function v and are essentially independent of T , y, and z. Turnpike properties are well-known in mathematical economics. The term was first coined by Samuelson in 1948 (see [82]) where he showed that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path (also called a von Neumann path). This property was further investigated for optimal trajectories of models of economic dynamics (see, e.g., [61, 67, 80] and the references mentioned there). Many turnpike results are collected in [93, 117].

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems

3

In order to meet our goal, we consider the auxiliary optimization problem v(x, x) → min, x ∈ R n .

(P1 )

It follows from the strict convexity of v and (1.1) that the problem (P1 ) has a unique solution x. ¯ Let ∇v(x, ¯ x) ¯ = (l1 , l2 ),

(1.2)

where l1 , l2 ∈ R n . Since x¯ is a solution of (P1 ), it follows from (1.2) that for each h ∈ Rn, l1 h + l2 h = (l1 , l2 )(h, h) = lim t −1 [v(x¯ + th, x¯ + th) − v(x, ¯ x)] ¯ ≥ 0. t→0+

Thus (l1 + l2 )h ≥ 0 for all h ∈ R n , l2 = −l1 and ∇v(x, ¯ x) ¯ = (l1 , −l1 ),

(1.3)

For each (y, z) ∈ R n × R n , set

L(y, z) = v(y, z) − v(x, ¯ x) ¯ − ∇v(x, ¯ x)(y ¯ − x, ¯ z − x) ¯ = v(y, z) − v(x, ¯ x) ¯ − l1 (y − z).

(1.4)

It is not difficult to verify that the function L : R n × R n → R 1 is differentiable and strictly convex. It follows from (1.1) and (1.4) that L(y, z)/(|y| + |z|) → ∞ as |y| + |z| → ∞.

(1.5)

Since the functions v and L are both strictly convex, it follows from (1.4) that L(y, z) ≥ 0 for all (y, z) ∈ R n × R n

(1.6)

L(y, z) = 0 if and only if y = x, ¯ z = x¯

(1.7)

and

4

1 Introduction

[70, 79]. We claim that the function L : R n × R n → R 1 has the following property: n n (C) If a sequence {(yi , zi )}∞ i=1 ⊂ R × R satisfies the equality lim L(yi , zi ) = 0,

i→∞

then lim (yi , zi ) = (x, ¯ x). ¯

i→∞

n n Assume that a sequence {(yi , zi )}∞ i=1 ⊂ R × R satisfies limi→∞ L(yi , zi ) = 0. ∞ In view of (1.5), the sequence {(yi , zi )}i=1 is bounded. Let (y, z) be its limit point. Then it is easy to see that the equality

L(y, z) = lim L(yi , zi ) = 0 i→∞

holds and by (1.7) (y, z) = (x, ¯ x). ¯ This implies that (x, ¯ x) ¯ = limi→∞ (yi , zi ). Thus the property (C) holds, as claimed. Consider an auxiliary minimization problem T −1

L(xi , xi+1 ) → min,

(P2 )

i=0

such that {xi }Ti=0 ⊂ R n and x0 = z, xT = y, where T is a natural number and the points y, z ∈ R n . It follows from (1.4) that for any integer T ≥ 1 and any sequence {xi }Ti=0 ⊂ R n , we have T −1

L(xi , xi+1 ) =

i=0

T −1 i=0

=

T −1

v(xi , xi+1 ) − T v(x, ¯ x) ¯ −

T −1

l1 (xi − xi+1 )

i=0

v(xi , xi+1 ) − T v(x, ¯ x) ¯ − l1 (x0 − xT ).

(1.8)

i=0

Relation (1.8) implies that the problems (P0 ) and (P2 ) are equivalent. Namely, {xi }Ti=0 ⊂ R n is a solution of the problem (P0 ) if and only if it is a solution of the problem (P2 ). Let T be a natural number and Δ ≥ 0. A sequence {xi }Ti=0 ⊂ R n is called (Δ)-optimal if for any sequence {xi }Ti=0 ⊂ R n satisfying xi = xi , i = 0, T , the inequality

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems T −1

v(xi , xi+1 ) ≤

T −1

i=0

5

   +Δ v xi , xi+1

i=0

holds. Clearly, if a sequence {xi }Ti=0 ⊂ R n is (0)-optimal, then it is a solution of the problems (P0 ) and (P2 ) with z = x0 and y = xT . We prove the following existence result. Proposition 1.1 Let T > 1 be an integer and y, z ∈ R n . Then the problem (P0 ) has a solution. Proof It is sufficient to show that the problem (P2 ) has a solution. Consider a sequence {xi }Ti=0 ⊂ R n such that x0 = z, xT = y. Set M1 =

T −1

   L xi , xi+1

i=0

and M2 = inf

T −1 

L(xi , xi+1 ) : {xi }Ti=0 ⊂ R n , x0 = z, xT = y .

(1.9)

i=0

Clearly, 0 ≤ M2 ≤ M1 . We may assume without loss of generality that M2 < M1 .

(1.10)

(k)

There exists a sequence {xi }Ti=0 ⊂ R n , k = 1, 2, . . . such that for any natural number k, (k)

(k)

x0 = z, xT = y

(1.11)

and lim

k→∞

T −1



(k) (k) L xi , xi+1 = M2 .

i=0

In view of (1.10)–(1.12), we may assume that

(1.12)

6

1 Introduction T −1



(k) < M1 for all integers k ≥ 1. L xi(k) , xi+1

(1.13)

i=0

By (1.13) and (1.5), there is M3 > 0 such that (k)

|xi | ≤ M3 for all i = 0, . . . , T and all integers k ≥ 1.

(1.14)

In view of (1.14), extracting subsequences, using diagonalization process and reindexing, if necessary, we may assume without loss of generality that for each i ∈ {0, . . . , T }, there exists (k)

xi = lim xi .

(1.15)

x0 = z, xT = y.

(1.16)

k→∞

By (1.11) and (1.15),

It follows from (1.15) and (1.12) that T −1

L( xi , xi+1 ) = M2 .

i=0

Together with (1.16) and (1.9), this implies that { xi }Ti=0 is a solution of the problem P2 . This completes the proof of Proposition 1.1. Denote by Card(A) the cardinality of a set A. The following result establishes a turnpike property for approximate solutions of the problem (P0 ). Proposition 1.2 Let M1 , M2 ,  be positive numbers. Then there exists a natural number k0 such that for each integer T > 1 and each (M1 )-optimal sequence {xi }Ti=0 ⊂ R n satisfying |x0 | ≤ M2 , |xT | ≤ M2

(1.17)

the following inequality holds: ¯ + |xi+1 − x| ¯ > }) ≤ k0 . Card({i ∈ {0, . . . , T − 1} : |xi − x| Proof By condition (C) there is δ > 0 such that for each (y, z) ∈ R n ×R n satisfying L(y, z) ≤ δ we have

(1.18)

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems

|y − x| ¯ + |z − x| ¯ ≤ .

7

(1.19)

Set ¯ + M2 }. M3 = sup{L(y, z) : y, z ∈ R n and |y| + |z| ≤ |x|

(1.20)

Choose a natural number k0 > δ −1 (M1 + 2M3 ).

(1.21)

Assume that an integer T > 1 and that an (M1 )-optimal sequence {xi }Ti=0 ⊂ R n satisfies (1.17). Set ¯ i = 1, . . . , T − 1. y0 = x0 , yT = xT , yi = x,

(1.22)

Since the sequence {xi }Ti=0 is (M1 )-optimal, it follows from (1.22) that T −1

v(xi , xi+1 ) ≤

i=0

T −1

v(yi , yi+1 ) + M1 .

i=0

Together with (1.7), (1.8), and (1.22), this implies that T −1

L(xi , xi+1 ) ≤

i=0

T −1

L(yi , yi+1 ) + M1 = L(x0 , x) ¯ + L(x, ¯ xT ) + M1 .

i=0

Combined with (1.17) and (1.20), this implies that T −1

L(xi , xi+1 ) ≤ M1 + 2M3 .

i=0

It follows from the choice of δ (see (1.18) and (1.19)), (1.21), and the inequality above that   ¯ + |xi+1 − x| ¯ > } Card {i ∈ {0, . . . , T − 1} : |xi − x|   ≤ Card {i ∈ {0, . . . , T − 1} : L(xi , xi+1 ) > δ} ≤ δ −1

T −1 i=0

Proposition 1.2 is proved.

L(xi , xi+1 ) ≤ δ −1 (M1 + 2M3 ) ≤ k0 .

8

1 Introduction

Proposition 1.2 implies the following turnpike result for exact solutions of the problem (P0 ). Proposition 1.3 Let M,  be positive numbers. Then there exists a natural number k0 such that for each integer T > 1, each y, z ∈ R n satisfying |y|, |z| ≤ M, and each optimal sequence {xi }Ti=0 ⊂ R n of the problem (P0 ), the following inequality holds:   ¯ + |xi+1 − x| ¯ > } ≤ k0 . Card {i ∈ {0, . . . , T − 1} : |xi − x| It is easy now to see that the optimal solution {xi }Ti=0 of the problem (P0 ) spends most of the time in an -neighborhood of x. ¯ By Proposition 1.3 the number of all integers i ∈ {0, . . . , T − 1} such that xi does not belong to this -neighborhood, does not exceed the constant k0 which depends only on M, , and does not depend on T . Following the tradition, the point x¯ is called the turnpike. Moreover we can show that the set {i ∈ {0 . . . , T } : |xi − x| ¯ > } is contained in the union of two intervals [0, k1 ]∪[T −k1 , T ], where k1 is a constant depending only on M, . We also study the infinite horizon problem associated with the problem (P0 ). By (1.1) there is M∗ > 0 such that v(y, z) > |v(x, ¯ x)| ¯ +1

(1.23)

for any (y, z) ∈ R n × R n satisfying |y| + |z| ≥ M∗ . We suppose that the sum over empty set is zero. Proposition 1.4 Let M0 > 0. Then there exists M1 > 0 such that for each integer T ≥ 1 and each sequence {xi }Ti=0 ⊂ R n satisfying |x0 | ≤ M0 , T −1

v(xi , xi+1 ) ≥ T v(x, ¯ x) ¯ − M1 .

(1.24)

i=0

Proof Put M1 = |l1 |(M0 + M∗ ). Assume that an integer T ≥ 1 and a sequence {xi }Ti=0 ⊂ R n satisfies |x0 | ≤ M0 .

(1.25)

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems

9

If |xi | > M∗ , i = 1, . . . , T , then by (1.23) T −1

v(xi , xi+1 ) ≥ T v(x, ¯ x) ¯

i=0

and (1.24) holds. Therefore we may assume that there exists a natural number q such that q ≤ T , |xq | ≤ M∗ .

(1.26)

We may assume without loss of generality that |xi | > M∗ for all integers i satisfying q < i ≤ T .

(1.27)

By (1.23) and (1.27), T −1

(v(xi , xi+1 ) − v(x, ¯ x)) ¯ =

i=0

q−1  (v(xi , xi+1 ) − v(x, ¯ x)) ¯ i=0

 + {v(xi , xi+1 ) − v(x, ¯ x) ¯ : an integer i satisfies q ≤ i < T }

≥

q−1  (v(xi , xi+1 ) − v(x, ¯ x)). ¯ i=0

It follows from the equation above, (1.6), (1.8), (1.25), (1.26), and the choice of M1 that T −1

(v(xi , xi+1 ) − v(x, ¯ x)) ¯ ≥

i=0

q−1 

(v(xi , xi+1 ) − v(x, ¯ x)) ¯

i=0

=

q−1 

L(xi , xi+1 ) + l1 (x0 − xq ) ≥ −|l1 |(|x0 | + |xq |)

i=0

≥ −|l1 |(M0 + M∗ ) = −M1 . Proposition 1.4 is proved. Fix a number M˜ > 0 such that ˜ Proposition 1.4 holds with M0 = M∗ and M1 = M. n Proposition 1.5 Let {xi }∞ i=0 ⊂ R . Then either the sequence

(1.28)

10

1 Introduction

T −1 

∞ 

v(xi , xi+1 ) − v(x, ¯ x) ¯

T =1

i=0

is bounded or lim

T −1

T →∞



 ¯ x) ¯ = ∞. v(xi , xi+1 ) − v(x,

(1.29)

i=0

Proof It follows from (1.23) that if for all sufficiently large natural numbers i, |xi | ≥ M∗ , then (1.29) holds. Therefore we may assume without loss of generality that there exists a strictly increasing sequence of natural numbers {tk }∞ k=1 such that |xtk | < M∗ for all integers k ≥ 1.

(1.30)

T −1 By Proposition 1.4 the sequence { i=0 (v(xi , xi+1 )−v(x, ¯ x))} ¯ ∞ T =1 is bounded from below. Assume that this sequence is not bounded from above. In order to complete the proof, it is sufficient to show that (1.29) holds. Let Q be any positive number. Then there exists a natural number T0 such that T 0 −1

  ˜ v(xi , xi+1 ) − v(x, ¯ x) ¯ > Q + M.

(1.31)

i=0

Choose a natural number k such that tk > T0 + 4.

(1.32)

T > tk .

(1.33)

Let an integer

By (1.30), (1.32), and (1.33), there exists an integer S such that T > S ≥ T0 ,

(1.34)

|xS | ≤ M∗ ,

(1.35)

|xt | > M∗ for all integers t satisfying

(1.36)

S > t ≥ T0 . It follows from (1.23), (1.28), (1.31), (1.34)–(1.36), and Proposition 1.4 that

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems T −1

(v(xi , xi+1 ) − v(x, ¯ x)) ¯ =

i=0

+ +



T 0 −1

11

(v(xi , xi+1 ) − v(x, ¯ x)) ¯

i=0

{v(xi , xi+1 ) − v(x, ¯ x) ¯ : i is an integer and T0 ≤ i < S}

T −1

(v(xi , xi+1 ) − v(x, ¯ x)) ¯

i=S

> Q + M˜ +

T −1

(v(xi , xi+1 ) − v(x, ¯ x)) ¯ > Q.

i=S

Thus for any integer T > tk , T −1

(v(xi , xi+1 ) − v(x, ¯ x)) ¯ > Q.

i=0

Since Q is any positive number, (1.29) holds. Proposition 1.5 is proved. n is called good [28, 93, 117] if the sequence A sequence {xi }∞ i=0 ⊂ R ∞ T −1  ¯ x) ¯ }T =1 is bounded. { i=0 v(xi , xi+1 ) − v(x,

Proposition 1.6 n 1. A sequence {xi }∞ i=0 ⊂ R is good if and only if ∞ 

L(xi , xi+1 ) < ∞.

i=0 n 2. If a sequence {xi }∞ ¯ i=0 ⊂ R is good, then it converges to x. n Proof Assume that a sequence {xi }∞ i=0 ⊂ R is good. Then there exists M0 > 0 such that T −1

  v(xi , xi+1 ) − v(x, ¯ x) ¯ < M0 for all integers T ≥ 1.

(1.37)

i=0

By (1.23) and (1.37), there exists a strictly increasing sequence of natural numbers {tk }∞ k=1 such that |xtk | < M for all natural numbers k. Let k be a natural number. By (1.8), (1.37), and (1.38),

(1.38)

12

1 Introduction

M0 >

t k −1

k −1   t v(xi , xi+1 ) − v(x, ¯ x) ¯ = L(xi , xi+1 ) + l1 (x0 − xtk )

i=0

i=0

≥

t k −1

L(xi , xi+1 ) − |l1 |(|x0 | + |xtk |)

i=0

≥

t k −1

L(xi , xi+1 ) − |l1 |(|x0 | + M∗ )

i=0

and t k −1

L(xi , xi+1 ) ≤ M0 + |l1 |(|x0 | + M∗ ).

i=0

Since the inequality above holds for all natural numbers k, we conclude that ∞ 

L(xi , xi+1 ) ≤ M0 + |l1 |(|x0 | + M∗ ).

i=0

In view of (C), the sequence {xi }∞ ¯ and assertion 2 is proved. i=0 converges to x, Assume that M1 :=

∞ 

L(xi , xi+1 ) < ∞.

(1.39)

i=0

By (1.5) there is M2 > 0 such that |xi | < M2 for all integers i ≥ 0.

(1.40)

In view of (1.8), (1.39), and (1.40), for all natural numbers T , T −1

−1   T v(xi , xi+1 ) − v(x, ¯ x) ¯ = L(xi , xi+1 ) + l1 (x0 − xT )

i=0

i=0

≤ M1 + 2|l1 |M2 . Together with Proposition 1.5, this implies that the sequence {xi }∞ i=0 is good. Proposition 1.6 is proved. n Proposition 1.7 Let x ∈ R n . Then there exists a sequence {xi }∞ i=0 ⊂ R such that ∞ n x0 = x and for each sequence {yi }i=0 ⊂ R satisfying y0 = x, the inequality

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems ∞ 

L(xi , xi+1 ) ≤

i=0

∞ 

13

L(yi , yi+1 )

i=0

holds. Proof Set  M0 = inf

∞ 

L(yi , yi+1 ) :

{yi }∞ i=0

⊂ R and y0 = x . n

(1.41)

i=0 n Clearly, M0 is well-defined and M0 ≥ 0. There exists a sequence {xi }∞ i=0 ⊂ R , k = 1, 2, . . . such that (k)

(k)

x0 = x, k = 1, 2, . . . , lim

k→∞

(1.42)

∞ 

 (k) = M0 . L xi(k) , xi+1

(1.43)

i=0

By (1.5) and (1.43), there exists M1 > 0 such that (k)

|xi | < M1 for all integers i ≥ 0 for all integers k ≥ 1.

(1.44)

In view of (1.44) using diagonalization process, extracting subsequences, and reindexing, we may assume without loss of generality that for any integer i ≥ 0, there is (k)

xi = lim xi .

(1.45)

x0 = x.

(1.46)

k→∞

By (1.42) and (1.45),

It follows from (1.6), (1.43), and (1.45) that for any natural number T , T −1 i=0

L(xi , xi+1 ) = lim

k→∞

T −1

∞  



 (k) (k) ≤ lim = M0 . L xi(k) , xi+1 L xi(k) , xi+1

i=0

k→∞

i=0

Since T is an arbitrary natural number, we conclude that ∞  i=0

L(xi , xi+1 ) ≤ M0 .

14

1 Introduction

Together with (1.41) and (1.46), this implies that ∞ 

L(xi , xi+1 ) = M0 .

i=0

This completes the proof of Proposition 1.7. In our study we use the following optimality criterion introduced in the economic literature [6, 28, 89] and used in the optimal control [19, 93, 117]. n A sequence {xi }∞ i=0 ⊂ R is called overtaking optimal if lim sup T →∞

T −1 

v(xi , xi+1 ) −

i=0

T −1

 v(yi , yi+1 ) ≤ 0

i=0

n for any sequence {yi }∞ i=0 ⊂ R satisfying y0 = x0 . n Proposition 1.8 Let {xi }∞ i=0 ⊂ R . Then the following assertions are equivalent:

1. the sequence {xi }∞ i=0 is overtaking optimal; 2. ∞ 

L(xi , xi+1 ) ≤

i=0

∞ 

L(yi , yi+1 )

i=0

n for any sequence {yi }∞ i=0 ⊂ R satisfying y0 = x0 .

Proof Assume that the sequence {xi }∞ i=0 is overtaking optimal. Clearly, it is good. By Proposition 1.6, ∞ 

L(xi , xi+1 ) < ∞.

i=0 n Let a sequence {yi }∞ i=0 ⊂ R satisfies

y0 = x0 .

(1.47)

We show that ∞  i=0

We may assume that

L(xi , xi+1 ) ≤

∞  i=0

L(yi , yi+1 ).

1.1 The Turnpike Phenomenon for Convex Discrete-TimeProblems ∞ 

15

L(yi , yi+1 ) < ∞.

i=0

Then in view of (C), ¯ lim xi = x. ¯ lim yi = x,

i→∞

(1.48)

i→∞

Since the sequence {xi }∞ i=0 is overtaking optimal, it follows from (1.47), (1.8), and (1.48) that 0 ≥ lim sup T →∞

= lim sup T →∞

= lim sup T →∞

=

∞ 

T −1 

v(xi , xi+1 ) −

T −1

i=0

T −1 

 v(yi , yi+1 )

i=0

L(xi , xi+1 ) + l1 (x0 − xT ) −

i=0

T −1 

T −1

 L(yi , yi+1 ) − l1 (y0 − yT )

i=0

L(xi , xi+1 ) −

i=0

L(xi , xi+1 ) −

i=0

T −1



L(yi , yi+1 ) + l1 (yT − xT )

i=0 ∞ 

L(yi , yi+1 ).

i=0

Thus assertion 2 holds. Assume that assertion 2 holds. Let us show that the sequence {xi }∞ i=0 is overtaking optimal. Clearly, ∞ 

L(xi , xi+1 ) < ∞.

i=0

By Proposition 1.6 the sequence {xi }∞ i=0 is good and ¯ lim xi = x.

(1.49)

i→∞

n Assume that a sequence {yi }∞ i=0 ⊂ R satisfies

y0 = x0 .

(1.50)

We show that lim sup T →∞

T −1  i=0

v(xi , xi+1 ) −

T −1 i=0

 v(yi , yi+1 ) ≤ 0.

16

1 Introduction

We may assume without loss of generality that the sequence {yi }∞ i=0 is good. Then by Proposition 1.6, lim yi = x, ¯

i→∞

∞ 

L(yi , yi+1 ) < ∞.

(1.51)

i=0

It follows from (1.8), (1.49)–(1.51), and assertion 2 that lim sup

T −1 

T →∞

i=0

= lim sup T →∞

=

∞ 

v(xi , xi+1 ) −

T −1 

∞  i=0

 v(yi , yi+1 )

i=0

L(xi , xi+1 ) + l1 (x0 − xT ) −

i=0

L(xi , xi+1 ) −

i=0

=

T −1

∞ 

 L(yi , yi+1 ) + l1

i=0

L(xi , xi+1 ) −

∞ 

T −1

 L(yi , yi+1 ) − l1 (y0 − yT )

i=0



lim yT − lim xT

T →∞

T →∞

L(yi , yi+1 ) ≤ 0.

i=0

Thus assertion 1 holds, and Proposition 1.8 is proved. Proposition 1.7 and 1.8 imply the following existence result. Proposition 1.9 For any x ∈ R n , there exists an overtaking optimal sequence n {xi }∞ i=0 ⊂ R such that x0 = x.

1.2 The Turnpike Phenomenon In the previous section, we proved the turnpike result and the existence of overtaking optimal solutions for rather simple class of discrete-time problems. The problems of this class are unconstrained, and their objective functions are convex and differentiable. In [117] the turnpike property and the existence of solutions over infinite horizon were established for several classes of constrained optimal control problems without convexity (concavity) assumptions. In particular, in Chapter 2 of [117], we study the structure of approximate solutions of an autonomous discretetime control system with a compact metric space of states X. This control system is described by a bounded upper semicontinuous function v : X × X → R 1 which determines an optimality criterion and by a nonempty closed set Ω ⊂ X × X which determines a class of admissible trajectories (programs). We study the problem

1.2 The Turnpike Phenomenon T −1

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=0 ⊂ Ω, x0 = z, xT = y,

17

(P )

i=0

where T ≥ 1 is an integer and the points y, z ∈ X. In the classical turnpike theory, the objective function v possesses the turnpike property (TP) if there exists a point x¯ ∈ X (a turnpike) such that the following condition holds: For each positive number , there exists an integer L ≥ 1 such that for each integer T ≥ 2L and each solution {xi }Ti=0 ⊂ X of the problem (P), the inequality ρ(xi , x) ¯ ≤  is true for all i = L, . . . , T − L. It should be mentioned that the constant L depends neither on T nor on y, z. The turnpike phenomenon has the following interpretation. If one wishes to reach a point A from a point B by a car in an optimal way, then one should turn to a turnpike, spend most of time on it, and then leave the turnpike to reach the required point. In the classical turnpike theory [28, 67, 80, 89], the space X is a compact convex subset of a finite-dimensional Euclidean space, the set Ω is convex, and the function v is strictly concave. Under these assumptions the turnpike property can be established, and the turnpike x¯ is a unique solution of the maximization problem v(x, x) → max, (x, x) ∈ Ω. this situation it is shown that for each program In T −1 {xt }∞ , either the sequence { ¯ x)} ¯ ∞ t=0 v(xt , xt+1 ) − T v(x, t=0 T =1 is bounded (in this ∞ case the program {xt }t=0 is called (v)-good), or it diverges to −∞. Moreover, it is also established that any (v)-good program converges to the turnpike x. ¯ In the sequel this property is called as the asymptotic turnpike property. T −1 In [117] we study the problems (P) with the constraint {(xi , xi+1 )}i=0 ⊂ Ω where Ω is an arbitrary nonempty closed subset of the metric space X × X. Clearly, these constrained problems are more difficult and less understood than their unconstrained prototypes in the previous section. They are also more realistic from the point of view of mathematical economics. In general a turnpike is not necessarily a singleton. Nevertheless problems of the type (P) for which the turnpike is a singleton are of great importance because of the following reasons: there are many models of economic growth for which a turnpike is a singleton; if a turnpike is a singleton, then approximate solutions of (P) have very simple structure, and this is very important for applications; if a turnpike is a singleton, then it can be easily calculated as a solution of the problem v(x, x) → max, (x, x) ∈ Ω. The turnpike property is very important for applications. Suppose that our objective function v has the turnpike property and we know a finite number of “approximate” solutions of the problem (P). Then we know the turnpike x, ¯ or at least its approximation, and the constant L (see the definition of TP) which is an estimate for the time period required to reach the turnpike. This information can be useful if we need to find an “approximate” solution of the problem (P) with a new time interval [m1 , m2 ] and the new values z, y ∈ X at the endpoints m1 and m2 . Namely, instead of solving this new problem on the “large” interval [m1 , m2 ], we can find an “approximate” solution of the problem (P) on the “small” interval

18

1 Introduction

[m1 , m1 + L] with the values z, x¯ at the endpoints and an “approximate” solution of the problem (P) on the “small” interval [m2 − L, m2 ] with the values x, ¯ y at the endpoints. Then the concatenation of the first solution, the constant sequence xi = x, ¯ i = m1 + L, . . . , m2 − L, and the second solution is an “approximate” solution of the problem (P) on the interval [m1 , m2 ] with the values z, y at the endpoints. Sometimes as an “approximate” solution of the problem (P), we can 2 choose any admissible sequence {xi }m i=m1 satisfying xm1 = z, xm2 = y and xi = x¯ for all i = m1 + L, . . . , m2 − L.

1.3 Turnpike Results for Problems in Metric Spaces In [117] we study the turnpike phenomenon for a class of discrete-time optimal control problems which is considered below. Let (X, ρ) be a compact metric space, Ω be a nonempty closed subset of X × X, and v : X × X → R 1 be a bounded upper semicontinuous function. A sequence {xt }∞ t=0 ⊂ X is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt+1 ) ∈ Ω for all nonnegative integers t. A sequence {xt }Tt=0 where T ≥ 1 is an integer is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt+1 ) ∈ Ω for all integers t ∈ [0, T − 1]. We consider the problems T −1

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=0 ⊂ Ω, x0 = y,

i=0

and T −1

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=0 ⊂ Ω, x0 = y, xT = z,

i=0

where T ≥ 1 is an integer and the points y, z ∈ X. We suppose that there exist a point x¯ ∈ X and a positive number c¯ such that the following assumptions hold: (i)  (x, ¯ x) ¯ is an interior point of Ω; T −1 (ii) ¯ x) ¯ + c¯ for any natural number T and any program t=0 v(xt , xt+1 ) ≤ T v(x, {xt }Tt=0 . The property (ii) implies that for each program {xt }∞ t=0 , either the sequence T −1  t=0

∞ v(xt , xt+1 ) − T v(x, ¯ x) ¯ T =1

1.3 Turnpike Results for Problems in Metric Spaces

19

 −1 is bounded, or limT →∞ [ Tt=0 v(xt , xt+1 ) − T v(x, ¯ x)] ¯ = −∞. A program {xt }∞ is called (v)-good if the sequence t=0 T −1 

∞ v(xt , xt+1 ) − T v(x, ¯ x) ¯ T =1

t=0

is bounded. We also suppose that the following assumption holds. (iii) (the asymptotic turnpike property) For any (v)-good program {xt }∞ t=0 , limt→∞ ρ(xt , x) ¯ = 0. Note that the properties (i)–(iii) hold for models of economic dynamics considered in the classical turnpike theory. For each positive number M, denote by XM the set of all points x ∈ X for which there exists a program {xt }∞ t=0 such that x0 = x and that for all natural numbers T the following inequality holds: T −1

v(xt , xt+1 ) − T v(x, ¯ x) ¯ ≥ −M.

t=0

It is not difficult to see that ∪{XM : M ∈ (0, ∞)} is the set of all points x ∈ X for which there exists a (v)-good program {xt }∞ t=0 satisfying x0 = x. Let T ≥ 1 be an integer and Δ ≥ 0. A program {xi }Ti=0 ⊂ R n is called (Δ)optimal if for any program {xi }Ti=0 satisfying x0 = x0 , the inequality T −1 i=0

v(xi , xi+1 ) ≥

T −1

   −Δ v xi , xi+1

i=0

holds. In Chapter 2 of [117], we prove the following turnpike result for approximate solutions of our first optimization problem stated above. Theorem 1.10 Let , M be positive numbers. Then there exist a natural number L and a positive number δ such that for each integer T > 2L and each (δ)-optimal program {xt }Tt=0 which satisfies x0 ∈ XM there exist nonnegative integers τ1 , τ2 ≤ ¯ ≤  for all t = τ1 , . . . , T −τ2 and if ρ(x0 , x) ¯ ≤ δ, then τ1 = 0. L such that ρ(xt , x) An analogous turnpike result for approximate solutions of our second optimization problem is also proved in Chapter 2 of [117]. A program {xt }∞ optimal if for each program {yt }∞ t=0 is called (v)-overtaking t=0 T −1 satisfying y0 = x0 , the inequality lim supT →∞ t=0 [v(yt , yt+1 )−v(xt , xt+1 )] ≤ 0 holds.

20

1 Introduction

In Chapter 2 of [117], we prove the following result which establishes the existence of an overtaking optimal program. Theorem 1.11 Assume that x ∈ X and that there exists a (v)-good program {xt }∞ t=0 such that x0 = x. Then there exists a (v)-overtaking optimal program {xt∗ }∞ t=0 such that x0∗ = x. In [117] we also study the stability of the turnpike phenomenon and show that the turnpike property is stable under perturbations of the objective function v. Note that the stability of the turnpike property is crucial in practice. One reason is that in practice we deal with a problem which consists of a perturbation of the problem we wish to consider. Another reason is that the computations introduce numerical errors.

1.4 The Robinson–Solow–Srinivasan Model 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}. R+

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi ,

i=1

and let x y, x > y, x ≥ y have their usual meaning. Namely, for a given pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , we say that x ≥ y, if xi ≥ yi for all i = 1, . . . , n, x > y if x ≥ y and x = y, and x y if xi > yi for all i = 1, . . . , n. n Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n . Let a = (a1 , . . . , an ) 0, b = (b1 , . . . , bn ) 0, b1 ≥ b2 · · · ≥ bn , d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. We assume the following: There exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci .

1.4 The Robinson–Solow–Srinivasan Model

21

A sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0 n n × R+ , x(t + 1) ≥ (1 − d)x(t), (x(t), y(t)) ∈ R+

0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(1.52)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences

 T2 −1 2 {x(t)}Tt=T , {y(t)} t=T1 1 n and for each integer t satisfying T ≤ t < T , is called a program if x(T2 ) ∈ R+ 1 2 relations (1.52) are valid. Assume that w : [0, ∞) → R 1 is a continuous strictly increasing concave and differentiable function which represents the preferences of the planner. Define n n Ω = {(x, x  ) ∈ R+ × R+ : x  − (1 − d)x ≥ 0

and a(x  − (1 − d)x) ≤ 1} n given by and a correspondence Λ : Ω → R+

  n : 0 ≤ y ≤ x and ey ≤ 1 − a(x  − (1 − d)x) . Λ(x, x  ) = y ∈ R+ For every (x, x  ) ∈ Ω, set   u(x, x  ) = max w(by) : y ∈ Λ(x, x  ) . n such that ( x, x ) is a solution to the problem: The golden-rule stock is x ∈ R+  maximize u(x, x ) subject to (i) x  ≥ x; (ii) (x, x  ) ∈ Ω. By Theorem 2.3, which will be proved in Chapter 2, there exists a unique goldenrule stock

x = (1/(1 + daσ ))e(σ ). It is not difficult to see that x is a solution to the problem w(by) → max, y ∈ Λ( x, x ). A program {x(t), y(t)}∞ t=0 is good if there is a real number M such that T  (w(by(t)) − w(b y )) ≥ M for every nonnegative integer T . t=0

22

1 Introduction

A program {x(t), y(t)}∞ t=0 is bad if T  (w(by(t)) − w(b y )) = −∞. lim

T →∞

t=0

We will show in Chapter 2 that every program that is not good is bad and that n , there exists a good program {x(t), y(t)}∞ satisfying for every point x0 ∈ R+ t=0 x(0) = x0 . In Chapter 3 we prove the following results obtained in [92]. Theorem 1.12 Assume that the function w is strictly concave. Then for every good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

Set ξσ = 1 − d − (1/aσ ). Theorem 1.13 Assume that ξσ = −1. Then x, x) lim (x(t), y(t)) = (

t→∞

for every good program {x(t), y(t)}∞ t=0 . A program {x ∗ (t), y ∗ (t)}∞ t=0 is overtaking optimal if lim sup

T 

T →∞ t=0

[w(by(t)) − w(by ∗ (t))] ≤ 0

∗ for every program {x(t), y(t)}∞ t=0 which satisfies x(0) = x (0). In Chapter 3 we prove the following result obtained in [92].

Theorem 1.14 Assume that for every good program {x(t), y(t)}∞ t=0 , lim (x(t), y(t)) = ( x, x ).

t→∞

n , there is an overtaking optimal program Then for every point x0 ∈ R+ ∞ {x(t), y(t)}t=0 such that x(0) = x0 .

Corollary 1.15 Assume that the function w is strictly concave. Then for every point n , there exists an overtaking optimal program {x(t), y(t)}∞ satisfying x0 ∈ R+ t=0 x(0) = x0 .

1.4 The Robinson–Solow–Srinivasan Model

23

n , there is an Corollary 1.16 Assume that ξσ = −1. Then for every point x0 ∈ R+ ∞ overtaking optimal program {x(t), y(t)}t=0 such that x(0) = x0 .

Theorem 1.13 shows that if ξσ = −1, then the asymptotic turnpike property (iii) of Section 1.3 holds with x being the turnpike. It will also be shown that property (ii) of Section 1.3 holds. But property (i) from Section 1.3 does not hold: ( x, x ) is not an interior point of the set Ω. This makes the situation more difficult and less understood. The next auxiliary result (see Proposition 3.13) together with certain monotonicity properties of the model will help us to overcome these difficulties. n Proposition 1.17 Let  > 0. Then there exists δ > 0 such that for each x, x  ∈ R+ satisfying

x  ≤ δ, x − x , x  − n such that there exist x¯ ≥ x  , y ∈ R+

(x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯ y − x  ≤ , x¯ − x  ≤ . It should be mentioned that in the book, sequences are denoted as {x(t)}∞ t=0 because this notation is used in the literature on the Robinson–Solow–Srinivasan model.

Chapter 2

The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties

In this chapter we present the description of the Robinson–Solow–Srinivasan model and discuss its basic properties. In particular, we show the existence of weakly optimal programs and good programs and prove an average turnpike property.

2.1 The Robinson–Solow–Srinivasan Model In this book we use the following notation. 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}. R+

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi ,

i=1

and let x y, x > y, x ≥ y have their usual meaning. Namely, for a given pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , we say that x ≥ y, if xi ≥ yi for all i = 1, . . . , n, x > y if x ≥ y and x = y, and x y if xi > yi for all i = 1, . . . , n. n Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_2

25

26

2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties

Let a = (a1 , . . . , an ) 0, b = (b1 , . . . , bn ) 0, d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. These parameters define an economy capable of producing a finite number n of alternative types of machines. For every i = 1, . . . , n, one unit of machine of type i requires ai > 0 units of labor to construct it, and together with one unit of labor, each unit of it can produce bi > 0 units of a single consumption good. Thus, the production possibilities of the economy are represented by an (labor) inputcoefficients vector, a = (a1 , . . . , an ) 0, and an output-coefficients vector, b = (b1 , . . . , bn ) 0. Without loss of generality, we assume that the types of machines are numbered such that b1 ≥ b2 · · · ≥ bn . We assume that all machines depreciate at a rate d ∈ (0, 1). Thus the effective labor cost of producing a unit of output on a machine of type i is given by (1 + dai )/bi : the direct labor cost of producing unit output and the indirect cost of replacing the depreciation of the machine in this production. We consider the reciprocal of the effective labor cost, the effective output that takes the depreciation into account, and denote it by ci for the machine of type i. In this chapter we assume that there is a unique machine type σ at which effective labor cost (1 + dai )/bi is minimal, or at which the effective output per man bi /(1 + dai ) is maximal. Thus assume the following: there exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci .

(2.1)

For each nonnegative integer t, let x(t) = (x1 (t), . . . , xn (t)) ≥ 0 denote the amounts of the n types of machines that are available in time-period t, and let z(t + 1) = (z1 (t + 1), . . . , zn (t + 1)) ≥ 0 be the gross investments in the n types of machines during period t + 1. Hence, z(t + 1) = (x(t + 1) − x(t)) + dx(t), the sum of net investment and of depreciation. Let y(t) = (y1 (t), . . . , yn (t)) be the amounts of the n types of machines used for production of the consumption good, by(t), during period t + 1. Let the total labor force of the economy be stationary and positive. We normalize it to be unity. It is clear that gross investment, z(t + 1), representing the production of new machines of the various types, requires az(t + 1) units of labor in period t. Also y(t) representing the use of available machines for manufacture of the consumption good requires ey(t) units of labor in period t. Thus, the availability of labor constrains employment in the consumption and investment sectors by az(t+1)+ey(t) ≤ 1. Note that the flow of consumption and of investment (new machines) is in gestation during the period and available at the end of it. We now give a formal description of this technological structure. A sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0 n n × R+ , (x(t), y(t)) ∈ R+

x(t + 1) ≥ (1 − d)x(t), 0 ≤ y(t) ≤ x(t)

2.1 The Robinson–Solow–Srinivasan Model

27

and a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1. We associate with every program {x(t), y(t)}∞ t=0 its gross investment sequence {z(t + 1)}∞ such that t=0 z(t + 1) = x(t + 1) − (1 − d)x(t), t = 0, 1, . . . and a consumption sequence {by(t)}∞ t=0 . The following result was obtained in [37]. Proposition 2.1 For every program {x(t), y(t)}∞ t=0 , there exists a constant m(x(0)) > 0, depending only on x(0), such that x(t) ≤ m(x(0))e for all nonnegative integers t. Proof For every integer t > 0, we have ax(t) ≤ 1 + (1 − d)ax(t − 1) ≤

t−1 

(1 − d)τ + (1 − d)t ax(0).

τ =0

The inclusion d ∈ (0, 1) implies that ax(t) ≤ d −1 + ax(0). There exists j ∈ {1, . . . , n} such that aj ≤ ai , i = 1, . . . , n. Using the inequality ai > 0, i = 1, . . . , n, we deduce that for all i = 1, . . . , n,   xi (t) ≤ aj−1 d −1 + ax(0) := m(x(0)). Proposition 2.1 is proved. Let w : [0, ∞) → R 1 be a continuous strictly increasing concave and differentiable function which represents the preferences of the planner. In this chapter we use the following optimality criterion. A program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly optimal if lim inf T →∞

T  

 w(by(t)) − w(by ∗ (t)) ≤ 0

t=0

∗ for every program {x(t), y(t)}∞ t=0 satisfying x(0) = x (0).

28

2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties

Set

 n n × R+ : x  − (1 − d)x ≥ 0 Ω = (x, x  ) ∈ R+  and a(x  − (1 − d)x) ≤ 1 .

n given by We have a correspondence Λ : Ω → R+

   n Λ(x, x  ) = y ∈ R+ : 0 ≤ y ≤ x and ey ≤ 1 − a x  − (1 − d)x . For any (x, x  ) ∈ Ω, define    u(x, x  ) = max w(by) : y ∈ Λ x, x  .

2.2 A Golden-Rule Stock n such that ( A golden-rule stock is x ∈ R+ x, x ) is a solution to the problem:  maximize u(x, x ) subject to (i) x  ≥ x; (ii) (x, x  ) ∈ Ω. For i = 1, . . . , n, set   qi . i = w  bσ (1 + daσ )−1 qi = ai bi /(1 + dai ), p

Set y = (1 + daσ )−1 e(σ ). The following useful lemma was obtained in [37]. Lemma 2.2 w(b y ) ≥ w(by) + p x  − p x for any (x, x  ) ∈ Ω and for any y ∈  Λ(x, x ). Proof Let (x, x  ) ∈ Ω, y ∈ Λ(x, x  ). Set z = x  − (1 − d)x. It is not difficult to see that q (x  − x) b y − by − q (x  − x) = cσ − by − = cσ − by − q (x  − (1 − d)x) + d qx

2.2 A Golden-Rule Stock

29

= cσ (1 − ey − az) + cσ ey + cσ az − by − q z + d qx = cσ (1 − ey − az) +

n 

n  (cσ − bi )yi + (cσ − ci )ai zi + d qx

i=1

= cσ (1 − ey − az) +

i=1

n n   (cσ − ci )yi + (cσ − ci )ai zi + d q (x − y). i=1

(2.2)

i=1

Applying (2.1) we obtain that by − b y ≤ qx − q x. The inequality above implies that y )(by − b y) w(by) − w(b y ) ≤ w  (b ≤ w  (b y )( qx − q x) = p x − p x  . This completes the proof of Lemma 2.2. The next result was established in [37]. Theorem 2.3 There exists a unique golden-rule stock x = (1 + daσ )−1 e(σ ). Proof Set x = y = (1 + daσ )−1 e(σ ). It is not difficult to see that ( x, x ) ∈ Ω and y ∈ Λ( x, x ). Applying Lemma 2.2 we obtain that x is a golden-rule stock. Let us show that x is a unique golden-rule stock. Suppose to the contrary that ˜ x˜  ) and (x, ˜ x˜  ) is another solution with a corresponding y˜ ∈ Λ(x, ˜ z˜ = x˜  − (1 − d)x. Since the function w(·) is strictly increasing, we have by˜ = b y = cσ . On substituting x, ˜ y, ˜ and z˜ for x, y, and z in (2.2), we obtain that the right-hand side of (2.2) equals zero. This implies that each of its four terms is zero,

30

2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties

y˜i = z˜ i = 0 for all i ∈ {1, . . . , n} \ {σ }, x˜i = y˜i for all i ∈ {1, . . . , n} and that y˜σ + aσ z˜ σ = 1. ˜ we obtain that Coupling the first assertion with the relation cσ = by, y˜σ = (1 + daσ )−1 , and therefore from the third assertion that x˜ = (1 + daσ )−1 e(σ ). It follows from the last assertion that z˜ σ = d(1 + daσ )−1 and that x˜  = z˜ + (1 − d)x˜   = d(1 + daσ )−1 + (1 − d)(1 + daσ )−1 e(σ ) = (1 + daσ )−1 e(σ ). This completes the proof of Theorem 2.3.

2.3 Good Programs The results of this section were obtained in [37]. We use the following notion of good programs introduced by Gale [28] and used in optimal control [19, 65, 93, 117]. 1 A program {x(t), y(t)}∞ t=0 is called good if there exists M ∈ R such that T  (w(by(t)) − w(b y )) ≥ M for all integers T ≥ 0. t=0

A program is called bad if

2.3 Good Programs

31

lim

T →∞

T  (w(by(t)) − w(b y )) = −∞. t=0

n . Then there exists a good program Proposition 2.4 Let x0 ∈ R+

{x(t), y(t)}∞ t=0 which satisfies x(0) = x0 . Proof For every nonnegative integer t, set z(t + 1) = d x. Set y(0) = 0, and for all nonnegative integer t ≥ 0, define y(t + 1) = (1 − d)y(t) + d x. It is easy to see that that the sequence {y(t)}∞ x. t=0 is increasing and converges to Set x(0) = x0 , and for every nonnegative integer t ≥ 0, define x(t + 1) = (1 − d)x(t) + z(t + 1). Clearly, {x(t), y(t)}∞ t=0 is a program. It is easy to see that for all integers t > 1, x) by(t) − b x = (1 − d)t (by(1) − b and that for all natural numbers t, by(t) ≥ db x. This implies that for all integers t ≥ 0, x − by(t)) ≤ w  (db x )(b x − by(1))(1 − d)t−1 . w(b x ) − w(by(t)) ≤ w (by(t))(b Therefore the sequence {w(b x ) − w(by(t))}∞ t=0 is summable, and the program ∞ {x(t), y(t)}t=0 is good. Proposition 2.5 Let {x(t), y(t)}∞ t=0 be a program. Then there exists a constant M(x(0)) ≥ 0 such that for every pair of nonnegative integers t1 ≤ t2 ,

32

2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties t2 

(w(by(t)) − w(b y )) ≤ M(x(0)).

t=t1

Proof Let m(x(0)) be as guaranteed by Proposition 2.1. Lemma 2.2 implies that for every pair of nonnegative integers t1 ≤ t2 , t2 

(w(by(t)) − w(b y ))

t=t1

≤p (x(t1 ) − x(t2 + 1)) ≤p x(t1 ) ≤ m(x(0))

n 

p j .

j =1

In order to complete the proof, it is sufficient to define n   M(x(0)) = m(x(0))w bσ (1 + daσ )−1 ai bi (1 + dai )−1 . i=1

Proposition 2.5 easily implies the following result. Proposition 2.6 Every program which is not good is bad. For any (x, x  ) ∈ Ω and any y ∈ Λ(x, x  ), set (x − x  ) − (w(by) − w(b y )). δ(x, y, x  ) = p

(2.3)

We say that a program {x(t), y(t)}∞ t=0 has the average turnpike property if lim T −1

T →∞

T −1

(x(t), y(t)) = ( x, y ).

t=0

Proposition 2.7 Assume that a program {x(t), y(t)}∞ t=0 is good. Then it has the average turnpike property. Proof For every natural number T , set x(T ¯ ) = T −1

T −1 t=0

x(t), y(T ¯ ) = T −1

T −1

y(t).

t=0

Proposition 2.1 implies that the sequence {x(T ¯ ), y(T ¯ )}∞ T =1 has a convergent ∞ ∞ subsequence. Denote its limit by (x , y ). It is not difficult to see that

2.3 Good Programs

33

(x ∞ , x ∞ ) ∈ Ω, y ∞ ∈ Λ(x ∞ , x ∞ ). Since the program {x(t), y(t)}∞ t=0 is good and the function w is concave, there exists a number G such that for all natural numbers T , T −1 G ≤ T −1

T −1

(w(by(t)) − w(b y )) ≤ w(by(T ¯ )) − w(b y ).

t=0

This implies that y) w(by ∞ ) ≥ w(b and that x ∞ is a golden-rule stock. Since the golden-rule stock is unique and the function w is strictly increasing, we conclude that x, y∞ = y. x∞ = This completes the proof of Proposition 2.7. The next result easily follows from Lemmas 2.2 and (2.3). Proposition 2.8 Assume that {x(t), y(t)}∞ t=0 is a program. Then for every integer t ≥ 0, δ(x(t), y(t), x(t + 1)) ≥ 0 and for every natural number T , T 

(w(by(t)) − w(b y ))

t=0

=p (x(0) − x(T + 1)) −

T 

δ(x(t), y(t), x(t + 1)).

t=0

Proposition 2.8 implies the following result. Proposition 2.9 A program {x(t), y(t)}∞ t=0 is good if and only if ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

t=0 n , define For every x0 ∈ R+

34

2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties

Δ(x0 ) = inf

∞ 

δ(x(t), y(t), x(t + 1)) : {x(t), y(t)}∞ t=0

t=0

 is a program such that x(0) = x0 . n . Then Proposition 2.10 Let x0 ∈ R+

0 ≤ Δ(x0 ) < ∞ and there exists a program {x  (t), y  (t)}∞ t=0 such that x  (0) = x0 , Δ(x0 ) =

∞ 

δ(x  (t), y  (t), x  (t + 1)).

t=0

Proof In view of Proposition 2.1, there exists a good program {x(t), y(t)}∞ t=0 such that x(0) = x0 . Proposition 2.8 implies that Δ(x0 ) < ∞. Evidently, Δ(x0 ) ≥ 0. For every natural number i, there exists a program {x (i) (t), y (i) (t)}∞ t=0 such that x (i) (0) = x0 and ∞    δ x (i) (t), y (i) (t), x (i) (t + 1) ≤ Δ(x0 ) + i −1 .

(2.4)

t=0

Proposition 2.1 implies that there exists a number m0 > 0 such that x (i) (t) ≤ m0 e for all integers i ≥ 1 and all integers t ≥ 0. Extracting subsequences and using the diagonalization process, we obtain the existence of a strictly increasing sequence of natural numbers {ik }∞ k=1 such that for every nonnegative integer t ≥ 0, there exist x  (t) = lim x (ik ) (t), y  (t) = lim y (ik ) (t). k→∞

k→∞

2.3 Good Programs

35

Clearly, for every integer t ≥ 0,     δ x  (t), y  (t), x  (t + 1) = lim δ x (ik ) (t), y (ik ) (t), x (ik ) (t + 1) . k→∞

It is easy to see that {x  (t), y  (t)}∞ t=0 is a program such that x(0) = x0 and ∞    δ x  (t), y  (t), x  (t + 1) ≥ Δ(x0 ). t=0

It is not difficult to see that Δ(x0 ) = lim

k→∞

∞    δ x (i) (t), y (i) (t), x (i) (t + 1) t=0



≥ lim

T →∞

≥ lim

T →∞

T 

lim

k→∞

 δ(x (t), y (t), x (t + 1)) (i)

(i)

(i)

t=0

∞ T       δ x  (t), y  (t), x  (t + 1) = δ x  (t), y  (t), x  (t + 1) . t=0

t=0

This completes the proof of Proposition 2.10. Proposition 2.11 Assume that a program {x(t), y(t)}∞ t=0 satisfies Δ(x(0)) =

∞ 

δ(x(t), y(t), x(t + 1)).

t=0

Then it is weakly optimal. Proof In view of Proposition 2.9, the program {x(t), y(t)}∞ t=0 is good. Assume that it is not weakly optimal. Then there exist a program {x  (t), y  (t)}∞ t=0 satisfying x  (0) = x(0), a number  > 0, and a natural number t such that for all integers T ≥ t , T    w(by  (t)) − w(by(t)) > . t=0

36

2 The Description of the Robinson–Solow–Srinivasan Model and Its Basic Properties

Clearly, the program {x  (t), y  (t)}∞ t=0 is good. Proposition 2.7 implies that the  (t), y  (t)}∞ have the average turnpike property. and {x programs {x(t), y(t)}∞ t=0 t=0 Proposition 2.8 implies that for all integers T ≥ t , 
y, x > y, x ≥ y have their usual meaning. n Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n . Let a = (a1 , . . . , an ) >> 0, b = (b1 , . . . , bn ) >> 0, b1 ≥ b2 · · · ≥ bn , d ∈ (0, 1),

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_3

39

40

3 Infinite Horizon Optimization

ci = bi /(1 + dai ), i = 1, . . . , n.

(3.1)

We assume the following: There exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci .

(3.2)

Recall that a sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0 n n × R+ , x(t + 1) ≥ (1 − d)x(t), (x(t), y(t)) ∈ R+

0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(3.3)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences

 T2 −1 2 {x(t)}Tt=T , {y(t)} t=T1 1 n and for each integer t satisfying T ≤ t < T , is called a program if x(T2 ) ∈ R+ 1 2 relation (3.3) is valid. Assume that w : [0, ∞) → R 1 is a continuous strictly increasing concave and differentiable function which represents the preferences of the planner. Define n × R n : x  − (1 − d)x ≥ 0 Ω = {(x, x  ) ∈ R+ +

and a(x  − (1 − d)x) ≤ 1}

(3.4)

n given by and a correspondence Λ : Ω → R+ n : 0 ≤ y ≤ x and ey ≤ 1 − a(x  − (1 − d)x)}. Λ(x, x  ) = {y ∈ R+

(3.5)

For every (x, x  ) ∈ Ω, set u(x, x  ) = max{w(by) : y ∈ Λ(x, x  )}.

(3.6)

n such that ( x, x ) is a solution to the Recall that the golden-rule stock is x ∈ R+ problem: maximize u(x, x  ) subject to (i) x  ≥ x; (ii) (x, x  ) ∈ Ω. By Theorem 2.3, there exists a unique golden-rule stock

x = (1/(1 + daσ ))e(σ ). It is not difficult to see that x is a solution to the problem

(3.7)

3.1 Overtaking Optimal Programs

41

w(by) → max, y ∈ Λ( x, x ). Set y = x.

(3.8)

i = w  (b x ) qi . qi = ai bi /(1 + dai ), p

(3.9)

For i = 1, . . . , n, set

In view of Lemma 2.2, x w(b x ) ≥ w(by) + p x  − p

(3.10)

for every (x, x  ) ∈ Ω and for every y ∈ Λ(x, x  ). A program {x(t), y(t)}∞ t=0 is good if there is a real number M such that T  (w(by(t)) − w(b y )) ≥ M for every nonnegative integer T . t=0

A program {x(t), y(t)}∞ t=0 is bad if lim

T →∞

T  (w(by(t)) − w(b y )) = −∞. t=0

By Proposition 2.6, every program that is not good is bad. Proposition 2.4 implies n , there exists a good program {x(t), y(t)}∞ satisfying that for every point x0 ∈ R+ t=0 x(0) = x0 . In this chapter we prove the following results obtained in [92]. Theorem 3.1 Assume that the function w is strictly concave. Then for every good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

Set ξσ = 1 − d − (1/aσ ). Theorem 3.2 Assume that ξσ = −1. Then x, x) lim (x(t), y(t)) = (

t→∞

(3.11)

42

3 Infinite Horizon Optimization

for every good program {x(t), y(t)}∞ t=0 . In this book we use a notion of an overtaking optimal program introduced by Atsumi [6], Gale [28], and von Weizsacker [89] and a notion of a weakly optimal program introduced by Brock [17]. These optimality criterions are used in optimal control [19, 93, 117]. ∞ is weakly optimal if for every program {x(t), y(t)}∞ A program {x(t), ¯ y(t)} ¯ t=0 t=0 which satisfies x(0) = x(0), ¯ the inequality lim inf T →∞

T  [w(by(t)) − w(by(t))] ¯ ≤0 t=0

is true. A program {x ∗ (t), y ∗ (t)}∞ t=0 is overtaking optimal if lim sup

T 

T →∞ t=0

[w(by(t)) − w(by ∗ (t))] ≤ 0

∗ for every program {x(t), y(t)}∞ t=0 which satisfies x(0) = x (0). In this chapter we prove the following result obtained in [92].

Theorem 3.3 Assume that for every good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

n , there is an overtaking optimal program Then for every point x0 ∈ R+ ∞ {x(t), y(t)}t=0 such that x(0) = x0 .

Corollary 3.4 Assume that the function w is strictly concave. Then for every point n , there exists an overtaking optimal program {x(t), y(t)}∞ satisfying x0 ∈ R+ t=0 x(0) = x0 . n , there is an Corollary 3.5 Assume that ξσ = −1. Then for every point x0 ∈ R+ ∞ overtaking optimal program {x(t), y(t)}t=0 such that x(0) = x0 .

3.2 Auxiliary Results for Theorems 3.1–3.3 Proposition 3.6 Let m0 be a positive number. Then there exists a positive number −1 ) m1 such that for every integer T > 0 and every program ({x(t)}Tt=0 , {y(t)}Tt=0 satisfying x(0) ≤ m0 e, the inequality x(t) ≤ m1 e is valid for all integers t ∈ {0, . . . , T }. For the proof of this result, see Proposition 2.1.

3.2 Auxiliary Results for Theorems 3.1–3.3

43

For every (x, x  ) ∈ Ω and every y ∈ Λ(x, x  ), define (see (2.3)) (x − x  ) − (w(by) − w(b y )). δ(x, y, x  ) = p

(3.12)

∞ Let a program {x(t), y(t)}∞ t=0 be given. We denote by ω({x(t), y(t)}t=0 ) the set 3n of all points (u, v, η) ∈ R for which there exists a strictly increasing sequence of natural numbers {tk }∞ k=1 for which

lim (x(tk ), y(tk ), x(tk + 1)) = (u, v, η).

k→∞

Proposition 3.6 implies that the set ω({x(t), y(t)}∞ t=0 ) is well-defined and it is a 3n compact subset of R . Lemma 3.7 Assume that a program {x(t), y(t)}∞ t=0 is good and that (u0 , v0 , η0 ) ∈ ω({x(t), y(t)}∞ t=0 ).

(3.13)

(u0 , η0 ) ∈ Ω, v0 ∈ Λ(u0 , η0 ),

(3.14)

δ(u0 , v0 , η0 ) = 0

(3.15)

Then

and there exists a sequence {u(t), v(t)}∞ t=−∞ such that (u(0), v(0), u(1)) = (u0 , v0 , η0 )

(3.16)

and (u(s), v(s), u(s + 1)) ∈ ω({x(t), y(t)}∞ t=0 ) for all integers s.

(3.17)

Proof Since the program {x(t), y(t)}∞ t=0 is good, there exists a real number M0 ∈ R 1 for which T 

(w(by(t)) − w(b y )) ≥ M0 for all nonnegative integers T .

(3.18)

t=0

Proposition 3.6 implies that there exists a positive number M1 such that x(t), y(t) ≤ M1 for all nonnegative integers T . By (3.18), for every nonnegative integer T ,

(3.19)

44

3 Infinite Horizon Optimization

M0 ≤

T  (w(by(t)) − w(b y )) t=0

T  = [w(by(t)) − w(b y) + p x(t + 1) − p x(t)] i=0

+ p x(0) − p x(T + 1). Combined with (3.12) and (3.19), the relation above implies that for every nonnegative integer T , T 

δ(x(t), y(t), x(t + 1))

t=0

=

T 

[w(b y ) − w(by(t)) + p x(t) − p x(t + 1)]

t=0

x(0) − p x(T + 1) ≤ −M0 + 2|| p||M1 . ≤ −M0 + p

Therefore ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

(3.20)

t=0

By (3.13), there exists a strictly increasing sequence of natural numbers {tk }∞ k=1 such that lim (x(tk ), y(tk ), x(tk + 1)) = (u0 , v0 , η0 ).

k→∞

(3.21)

Since the set {(x, y, x  ) : (x, x  ) ∈ Ω, y ∈ Λ(x, x  )} is closed and contains the sequence {x(t), y(t), x(t + 1)}∞ t=0 , it follows from (3.21) that (u0 , η0 ) ∈ Ω and v0 ∈ Λ(u0 , η0 ).

3.3 Proofs of Theorems 3.1 and 3.2

45

Therefore relation (3.14) is true. By (3.20), (3.21), and the continuity of the function δ(·, ·, ·), equality (3.15) holds. For every integer k ≥ 1, define a sequence  ∞ x (k) (s), y (k) (s)

s=−tk

⊂ Rn × Rn

by   (k) x (s), y (k) (s) = (x(s + tk ), y(s + tk )) for all integers s ≥ −tk .

(3.22)

Extracting subsequences, re-indexing, and using diagonalization process, we obtain that there exists a strictly increasing sequence of natural numbers {kj }∞ j =1 such that for every integer s, there exists the limit (u(s), v(s)) = lim (x (kj ) (s), y (kj ) (s)). j →∞

(3.23)

In view of (3.22) and (3.23), (u(s), v(s), u(s + 1)) ∈ ω({x(t), y(t)}∞ t=0 ) for all integers s.

(3.24)

It follows from (3.21), (3.22), and (3.23) that (u(0), v(0), u(1)) = lim (x (kj ) (0), y (kj ) (0), x (kj ) (1)) j →∞

= lim (x(tkj ), y(tkj ), x(tkj + 1)) = (u0 , v0 , η0 ). j →∞

This completes the proof of Lemma 3.7.

3.3 Proofs of Theorems 3.1 and 3.2 Proof of Theorem 3.1 Suppose that the function w is strictly concave. Let {x(t), ∞ y(t)}∞ t=0 be a given good program. Assume that (u0 , v0 , η0 ) ∈ ω({x(t), y(t)}t=0 ). Lemma 3.7 implies that (u0 , η0 ) ∈ Ω, v0 ∈ Λ(u0 , η0 ), δ(u0 , v0 , η0 ) = 0. Together with Lemma 2.13, this implies that u0 = η0 = x and δ( x , v0 , x ) = 0. Since v0 ≤ x , relation (3.12) and the strict monotonicity of the function w imply x . Hence that v0 = ω({x(t), y(t)}∞ x, x, x )}. t=0 ) = {(

46

3 Infinite Horizon Optimization

Theorem 3.1 is proved. The following lemma is used in the proof of Theorem 3.2. 2n Lemma 3.8 Assume that ξσ = −1 and that a sequence {x(t), y(t)}∞ t=−∞ ⊂ R+ satisfies the following conditions: There exists a positive number S0 such that

x(t), y(t) ≤ S0 for all integers t,

(3.25)

x(t + 1) ≥ (1 − d)x(t) and 0 ≤ y(t) ≤ x(t) for all integers t,

(3.26)

a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1 for all integers t,

(3.27)

δ(x(t), y(t), x(t + 1)) = 0 for all integers t.

(3.28)

Then x(t) = y(t) = x for all integers t. Proof Lemma 2.13 and (3.26)–(3.28) imply that for each integer t, yi (t) = xi (t) = 0 for all i ∈ {1, . . . , n} \ {σ }

(3.29)

xσ (t + 1) = 1/aσ + ξσ xσ (t), yσ (t) ≤ xσ (t).

(3.30)

and that

It follows from (3.12) and (3.28)–(3.30) imply that for every integer t, a(x(t + 1) − (1 − d)x(t)) + ex(t) = aσ (xσ (t + 1) − (1 − d)xσ (t)) + xσ (t) = aσ (1/aσ + ξσ xσ (t) − (1 − d)xσ (t)) + xσ (t) = aσ (1/aσ − (1/aσ )xσ (t)) + xσ (t) = 1. It follows from (3.12), (3.28)–(3.30), strict monotonicity of w, and the relation above that for every integer t, we have yσ (t) = xσ (t).

(3.31)

It follows from our assumptions that there are two cases: |ξσ | < 1

(3.32)

3.3 Proofs of Theorems 3.1 and 3.2

47

and |ξσ | > 1.

(3.33)

Assume that (3.32) is valid. Fix an integer s. By induction, using (3.30) one can show that for every natural number t, we have xσ (s) = (ξσ )t xσ (s − t) + (1/aσ )

t−1 

ξσi .

(3.34)

i=0

It is clear that for t = 1 Equation (3.34) follows from relation (3.30). Assume that τ ≥ 1 is an integer and that Equation (3.34) is true for t = τ . In view of (3.34) with t = τ and (3.30), we have xσ (s) = (ξσ )τ xσ (s − τ ) + (1/aσ )

τ −1 

ξσi

i=0

= ξστ [1/aσ + ξσ xσ (s − τ − 1)] + (1/aσ )

τ −1 

ξσi

i=0

= ξστ +1 xσ (s − τ − 1) + (1/aσ )

τ 

ξσi .

i=0

Hence (3.34) is valid for t = τ + 1. Therefore we have shown that Equation (3.34) is true for every natural number t. It follows from (3.11), (3.25), (3.32), and (3.34) that

 t−1 ∞   t i xσ (s) = lim (ξσ ) xσ (s − t) + (1/aσ ) ξσ = (1/aσ ) ξσi t→∞

i=0

= (1/aσ )(1 − ξσ )

−1

= (1/aσ )(d + 1/aσ )

t=0 −1

= (daσ + 1)−1 = xσ .

Together with (3.29) and (3.31), this implies that y(s) = x(s) = x for all integers s.

(3.35)

Assume that (3.33) holds. It follows from (3.30) that for every integer t, we have xσ (t) = ξσ−1 [xσ (t + 1) − 1/aσ ].

(3.36)

Fix an integer s. By induction we show that for every natural number t, we have

48

3 Infinite Horizon Optimization

−t

xσ (s) = (ξσ ) xσ (s + t) − (1/aσ )

t 

ξσ−i .

(3.37)

i=1

Evidently, for t = 1 equality (3.37) follows from relation (3.36). Assume that τ ≥ 1 is an integer and that equality (3.37) is valid for t = τ . In view of (3.37) with t = τ and (3.36), we have xσ (s) = (ξσ )−τ xσ (s + τ ) − (1/aσ )

τ 

ξσ−i

i=1

= ξσ−τ [ξσ−1 (xσ (s + τ + 1) − 1/aσ )] − (1/aσ )

τ 

ξσ−i

i=1

= ξσ−τ −1 xσ (s + τ + 1) − (1/aσ )

τ +1 

ξσ−i .

i=i

Hence (3.37) is true for t = τ + 1. Thus we have shown that (3.37) holds for every natural number t. By (3.11), (3.25), (3.33), and (3.37), we have

xσ (s) = lim

t→∞

= =

−t

(ξσ ) xσ (s + t) − (1/aσ )

t 

 ξσ−i

= −(1/aσ )

i=1

∞ 

ξσ−i

t=1

−(1/aσ )ξσ−1 (1 − ξσ−1 )−1 = (−1/aσ )(ξσ − 1)−1 (1/aσ )(d + 1/aσ )−1 = (daσ + 1)−1 = xσ .

Together with (3.29) and (3.31), the relation above implies that (3.35) is valid for all integers s. Lemma 3.8 is proved. Now Theorem 3.2 follows from Lemmas 3.7 and 3.8.

3.4 Proof of Theorem 3.3 n . Proposition 2.4 implies that there is a good program from x . Assume that x0 ∈ R+ 0 By Proposition 2.9, a program {x(t), y(t)}∞ is good if and only if t=0 ∞  t=0

Put

δ(x(t), y(t), x(t + 1)) < ∞.

3.4 Proof of Theorem 3.3

49

 Δ(x0 ) = inf

∞ 

δ(x(t), y(t), x(t + 1)) : {x(t), y(t)}∞ t=0

t=0



is a program such that x(0) = x0 .

(3.38)

Evidently, Δ(x0 ) < ∞. Proposition 2.10 implies that there exists a program {x ∗ (t), y ∗ (t)}∞ t=0 such that x ∗ (0) = x0 and

∞ 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1)) = Δ(x0 ).

(3.39)

t=0

We claim that the program {x ∗ (t), y ∗ (t)}∞ t=0 is overtaking optimal. Let {x(t), y(t)}∞ be a program satisfying t=0 x(0) = x0 .

(3.40)

Assume that {x(t), y(t)}∞ t=0 is not good. In view of Proposition 2.4, it is bad and lim

T →∞

T 

(w(by(t)) − w(b y )) = −∞.

(3.41)

t=0

Since {x ∗ (t), y ∗ (t)}∞ t=0 is good, there exists a positive number M0 for which T 

(w(by ∗ (t)) − w(b y )) ≥ M0 for all nonnegative integers T .

(3.42)

t=0

By (3.41) and (3.42), for every nonnegative integer T , we have T 

(w(by(t)) − w(by ∗ (t)) =

t=0

≤

T T   (w(by(t)) − w(b y )) − (w(by ∗ (t)) − w(b y )) t=0

t=0

T  (w(by(t)) − w(b y )) − M0 → −∞ as T → ∞.

(3.43)

t=0

Assume that {x(t), y(t)}∞ t=0 is a good program. Then x, x ), lim (x(t), y(t)) = (

t→∞

lim (x ∗ (t), y ∗ (t)) = ( x, x ).

t→∞

(3.44)

By (3.12), (3.38), (3.39), (3.40), and (3.44), for every nonnegative integer T , we have

50

3 Infinite Horizon Optimization

T 

[w(by(t)) − w(by ∗ (t))] =

t=0

=

T 

[w(by(t)) − w(b y )] −

t=0 T 

T  [w(by ∗ (t)) − w(b y )] t=0

[w(by(t)) − w(b y) + p x(t + 1) − p x(t)] + p x(0) − p x(T + 1)

t=0

−

T 

[w(by ∗ (t)) − w(b y) + p x ∗ (t + 1) − p x ∗ (t)] − p x ∗ (0) + p x ∗ (T + 1)

t=0

=−

T 

δ(x(t), y(t), x(t + 1)) +

t=0

T 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1))

t=0 ∗

− p [x(T + 1) − x (T + 1)] → −

∞ 

δ(x(t), y(t), x(t + 1)) +

t=0

∞ 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1))

t=0

= Δ(x0 ) −

∞ 

δ(x(t), y(t), x(t + 1)) ≤ 0

t=0

as T → ∞. This completes the proof of Theorem 3.3.

3.5 Examples Assume that ξσ = −1

(3.45)

and that the function w is linear. In this section we present an example of a good program which does not converge to the golden-rule stock. This example was constructed in [92]. Lemma 2.13 implies that {(x, x  ) ∈ Ω : there exists y ∈ Λ(x, x  ) such that δ(x, y, x  ) = 0} = {(x, x  ) ∈ Ω : xi = xi = 0 for all i ∈ {1, . . . , n} \ {σ }, xσ = (1/aσ ) + ξσ xσ }.

(3.46)

By (3.11) and (3.46), we have d + 1/aσ = 2, daσ + 1 = 2aσ

(3.47)

3.5 Examples

51

and x = y = 1/(1 + daσ )e(σ ) = 1/(2aσ )e(σ ).

(3.48)

Let κ ≥ 0, and define (x, y, x  ) ∈ R 3n by x = y = κe(σ ), x  = (1/aσ + ξσ κ)e(σ ) = (1/aσ − κ)e(σ ).

(3.49)

By (3.47) and (3.49), we have x  − (1 − d)x = (1/aσ − κ − (1 − d)κ)e(σ ) = (1/aσ − κ/aσ )e(σ ).

(3.50)

In view of (3.50), a(x  − (1 − d)x) = aσ (1 − κ)aσ−1 = 1 − κ.

(3.51)

By (3.47) and (3.49)–(3.51), (x, x  ) ∈ Ω if and only if κ ≤ 1.

(3.52)

It follows from (3.49) and (3.52) 1 − a(x  − (1 − d)x) − ex = 1 − (1 − κ) − κ = 0. Combined with (3.52), the relation above implies that (x, x  ) ∈ Ω and x ∈ Λ(x, x  ) if and only if κ ≤ 1.

(3.53)

Since (3.53) is true for every nonnegative number κ, it is easy to see that (x  , x) ∈ Ω and x  ∈ Λ(x  , x) if and only if 1/aσ − κ ≤ 1 and κ ≤ 1/aσ . (3.54) Let κ ∈ [1/aσ − 1, 1]

(3.55)

be given. In view of (3.47) such that κ exists and it is nonnegative. For every nonnegative integer t set y(2t) = x(2t) = κe(σ ), y(2t + 1) = x(2t + 1) = (1/aσ − κ)e(σ ). Relations (3.53)–(3.56) imply that (x(t), x(t + 1)) ∈ Ω, y(t) ∈ Λ(x(t), x(t + 1))

(3.56)

52

3 Infinite Horizon Optimization

for every integer t. Hence {x(t), y(t)}∞ t=0 is a program. By (3.46), (3.55), and (3.56), we have δ(x(t), x(t), x(t + 1)) = 0 for every nonnegative integer t.

(3.57)

Thus {x(t), y(t)}∞ t=0 is a good program. It is clear that ∞ ω({x(t)}∞ t=0 , {y(t)}t=0 ) = {(κe(σ ), κe(σ ), (1/aσ − κ)e(σ ))

((1/aσ − κ)e(σ ), (1/aσ − κ)e(σ ), κe(σ ))} and it is not a singleton if κ = (2aσ )−1 . Therefore Theorem 3.3 does not hold if ξσ = −1 and w is linear. Let us show the existence of a weakly optimal program which is not overtaking optimal. Set n = 1, w(z) = z for all z ≥ 0, b1 = 1, d ∈ (0, 1), a1 = (2 − d)−1 .

(3.58)

By (3.11) and (3.58), we have 1/2 < a1 < 1, ξ1 = 1 − d − 1/a1 = −1. Assume that κ = 1, and define a program {x(t), y(t)}∞ t=0 by (3.56). Proposition 2.11 and (3.57) imply that the program {x(t), y(t)}∞ is weakly optimal. There t=0 are two cases: • {x(t), y(t)}∞ t=0 is overtaking optimal; • {x(t), y(t)}∞ t=0 is not overtaking optimal. It is sufficient to consider only the first case. Set x(0) ¯ = 1, y(0) ¯ = x , x(t) ¯ = y(t) ¯ = x for all natural numbers t.

(3.59)

It is easy to see that x(1)−(1 ¯ − d)x(0) ¯ = (1 + da1 )−1 − (1 − d)=(d − da1 + d 2 a1 )(1 + da1 )−1 > 0, a1 (x(1) ¯ − (1 − d)x(0)) ¯ + y(0) ¯ = a1 (d − da1 + d 2 a1 )(1 + da1 )−1 + (1 + da1 )−1 = 1 + a1 (−da1 + d 2 a1 )(1 + da1 )−1 < 1. ∞ is a program. The relations above imply that the sequence {x(t), ¯ y(t)} ¯ t=0 ∞ We claim that {x(t), ¯ y(t)} ¯ t=0 is a weakly optimal program.

3.5 Examples

53

For every integer T ≥ 1, it follows from (3.48), (3.56), (3.58), and (3.59) that 2T −1 

y(t) = T (κ + (1/a1 − κ))

t=0

= (T /a1 ) =

2T −1 

y(t). ¯

(3.60)

t=0 ∞ is not a weakly optimal program. Then there is a Assume that {x(t), ¯ y(t)} ¯ t=0 ∞   program {x (t), y (t)}t=0 for which

¯ = x(0), x  (0) = x(0) lim inf

T 

T →∞

[y  (t) − y(t)] ¯ > 0.

t=0

Thus there exist a positive number  and an integer T0 ≥ 1 such that for every natural number T ≥ T0 , we have T −1

[y  (t) − y(t)] ¯ ≥ .

(3.61)

t=0

It follows from (3.60) and (3.61) that for every integer T ≥ T0 , we have 2T −1 

[−y(t) + y  (t)] =

t=0

2T −1 

[y  (t) − y(t)] ¯ ≥

t=0

and T  lim sup [y  (t) − y(t)] ≥ . T →∞ t=0

This contradicts our assumption that {x(t), y(t)}∞ t=0 is an overtaking optimal ∞ is a weakly program. The contradiction we have reached proves that {x(t), ¯ y(t)} ¯ t=0 optimal program. By (3.46), (3.47), (3.56), (3.59), and (3.60), for every integer T ≥ 1, 2T  t=0

y(t) −

2T  t=0

y(t) ¯ = y(2T ) − y(2T ¯ ) = 1 − x = 1 − (2aσ−1 ) > 0.

54

3 Infinite Horizon Optimization

Therefore

lim sup T →∞

T  t=0

y(t) −

T 

 y(t) ¯ >0

t=0

∞ is not an overtaking optimal program. and {x(t), ¯ y(t)} ¯ t=0

3.6 Convergence Results In this section we prove the results obtained in [51]. Theorem 3.9 Assume that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Then for each program {x(t), y(t)}∞ t=0 , the following conditions are equivalent: ∞ (i) t=0 δ(x(t), y(t), x(t + 1)) = Δ(x(0)) (see (3.38)). (ii) {x(t), y(t)}∞ t=0 is overtaking optimal. (iii) {x(t), y(t)}∞ t=0 is weakly optimal. Theorem 3.10 Assume that at least one of the following conditions holds: (a) w is strictly concave. (b) ξσ =

−1. Let M0 ,  > 0. Then there exists a natural number T0 such that for each overtaking optimal program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e and each integer t ≥ T0 , x(t) − x , y(t) − x  ≤ . Theorem 3.11 Assume that at least one of the following conditions holds: (a) w is strictly concave. (b) ξσ =

−1. Let  > 0. Then there exists δ > 0 such that for each overtaking optimal program {x(t), y(t)}∞ x  ≤ δ, the following inequality holds: t=0 satisfying x(0) − x(t) − x , y(t) − x ≤  for all integers t ≥ 0.

3.7 Proof of Theorem 3.9

55

3.7 Proof of Theorem 3.9 Let {x(t), y(t)}∞ t=0 be a program. In the proof of Theorem 3.3, it was proved that (i) implies (ii). Evidently (ii) implies (iii). Let us show that (iii) implies (i). Assume that {x(t), y(t)}∞ t=0 is weakly optimal. By Proposition 2.14, the program {x(t), y(t)}∞ is good. Proposition 2.9 implies that t=0 ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

(3.62)

t=0

In view of Proposition 2.10, there exists a program {x ∗ (t), y ∗ (t)}∞ t=0 such that x ∗ (0) = x(0),

∞ 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1)) = Δ(x(0)).

(3.63)

t=0 ∞ ∗ ∗ Since {x(t), y(t)}∞ t=0 and {x (t), y (t)}t=0 are good programs, we have

x, x ). lim (x(t), y(t)) = lim (x ∗ (t), y ∗ (t)) = (

t→∞

t→∞

(3.64)

By Proposition 2.8 and (3.63), for all integers T ≥ 0, T ∞   (w(by(t)) − w(b y )) = p (x(0) − x(T + 1)) − δ(x(t), y(t), x(t + 1)), t=0 T 

t=0

(w(by ∗ (t)) − w(b y )) = p (x ∗ (0) − x ∗ (T + 1)) −

t=0

T 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1))

t=0

and T  (w(by ∗ (t)) − w(by(t))) = p (x(T + 1) − x ∗ (T + 1)) t=0

+

T  t=0

δ(x(t), y(t), x(t + 1)) −

∞ 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1)).

t=0

Since the program {x(t), y(t)}∞ t=0 is weakly optimal, it follows from (3.62)–(3.64) that

56

3 Infinite Horizon Optimization

0 ≥ lim inf T →∞

=

∞ 

T  [w(by ∗ (t)) − w(by(t))] t=0

δ(x(t), y(t), x(t + 1)) −

t=0

∞ 

δ(x ∗ (t), y ∗ (t), x ∗ (t + 1))

t=0 ∗

+ p lim (x(T + 1) − x (T + 1)) = T →∞

∞ 

δ(x(t), y(t), x(t + 1)) − Δ(x(0)).

t=0

Thus ∞ 

δ(x(t), y(t), x(t + 1)) = Δ(x(0))

t=0

(see (3.38)). This completes the proof of Theorem 3.9.

3.8 Auxiliary Results for Theorems 3.10 and 3.11 Proposition 3.12 Let M0 > 0. Then n and x ≤ M0 e} < ∞. sup{Δ(x) : x ∈ R+

Proof By Proposition 3.6 there is M1 > 0 such that for each program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e, we have x(t) ≤ M1 e for each integer t ≥ 0. Set z(0) = 0, z(t + 1) = (1 − d)z(t) + (1/aσ )e(σ ), t = 0, 1, . . . .

(3.65)

It is easy to see that

lim z(T ) =

T →∞

aσ−1

T −1



(1 − d)

t

e(σ ) = (daσ )−1 e(σ ).

(3.66)

t=0

Clearly there exists a natural number T0 such that x. z(T0 ) ≥

(3.67)

n satisfies x ≤ M e. Set Assume now that x ∈ R+ 0

x(0) = x, x(t + 1) = (1 − d)x(t) + (1/aσ )e(σ ), t = 0, . . . , T0 − 1,

(3.68)

3.8 Auxiliary Results for Theorems 3.10 and 3.11

57

y(t) = 0, t = 0, . . . , T0 − 1. T0 −1 0 , {y(t)}t=0 ) is a program and that It is not difficult to see that ({x(t)}Tt=0

x(t) ≥ z(t), t = 0, . . . , T0 .

(3.69)

x. x(T0 ) ≥ z(T0 ) ≥

(3.70)

By (3.67) and (3.69),

For all natural numbers t ≥ T0 , set y(t) = x , x(t + 1) = (1 − d)x(t) + d x.

(3.71)

It is easy to see that {x(t), y(t)}∞ t=0 is a program. In view of Proposition 2.8 and (3.7), for all natural numbers T > T0 , T 

δ(x(t), y(t), x(t + 1)) = p (x(0) − x(T + 1)) −

t=0

T  (w(by(t)) − w(b y )) t=0

≤ M0 p e −

T 0 −1

[w(by(t)) − w(b y )].

(3.72)

t=0

It follows from (3.72) that for all natural numbers T > T0 , T 

δ(x(t), y(t), x(t + 1)) ≤ M0 p e + T0 w(b y ) − T0 w(0).

t=0

This inequality implies that Δ(x) ≤

∞ 

δ(x(t), y(t), x(t + 1)) ≤ M0 p e + T0 w(b y ) − T0 w(0).

t=0

This completes the proof of Proposition 3.12. n Proposition 3.13 Let  > 0. Then there exists δ > 0 such that for each x, x  ∈ R+ satisfying

x ≤ δ x − x , x  − n such that there exist x¯ ≥ x  , y ∈ R+

(x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯

58

3 Infinite Horizon Optimization

y − x  ≤ , x¯ − x  ≤ . Proof Choose a positive number δ such that 16δn + 2δan < min{, 8−1 d(1 + daσ )−1 }.

(3.73)

n satisfy Let x, x  ∈ R+

x  ≤ δ. x − x , x  −

(3.74)

zi = max{xi − (1 − d)xi , 0}, i = 1, . . . .n,

(3.75)

y = min{xσ , (1 − az)}e(σ ), x¯ = (1 − d)x + z.

(3.76)

Define z ∈ R n as follows:

By (3.75) and (3.76), x¯ ≥ (1 − d)x, x¯ ≥ x  .

(3.77)

a(x¯ − (1 − d)x) = az.

(3.78)

In view of (3.76),

It follows from (3.74) and (3.75) that for each i ∈ {1, . . . , n} \ {σ } |xi |, |xi | ≤ δ, −δ ≤ xi − (1 − d)xi ≤ δ, 0 ≤ zi ≤ δ.

(3.79)

By (3.74), |[xσ − (1 − d)xσ ] − d(daσ + 1)−1 | ≤ |xσ − (1 + daσ )−1 | + (1 − d)|xσ − (daσ + 1)−1 | ≤ 2δ. Together with (3.73)–(3.75), this inequality implies that xσ − (1 − d)xσ > 0, zσ = xσ − (1 − d)xσ , |zσ − d(1 + daσ )−1 | ≤ 2δ.

(3.80)

In view of (3.73), (3.74), (3.76), (3.79), and (3.80), x¯ − x  ≤ (1 − d)x − (1 − d) x  + z − d x ≤ δ + z − d x  ≤ δ + 2δn < .

(3.81)

3.8 Auxiliary Results for Theorems 3.10 and 3.11

59

It follows from (3.73), (3.79), and (3.80) that |az − ad x | ≤ az − d x  ≤ a2δn < 8−1 (1 + daσ )−1

(3.82)

az ≤ ad x + 8−1 (1 + daσ )−1 = aσ d(1 + daσ )−1 + (8(1 + daσ ))−1 < 1.

(3.83)

and

Combined with (3.77) and (3.78), this relation implies that a(x¯ − (1 − d)x) < 1 and (x, x) ¯ ∈ Ω.

(3.84)

By (3.76), (3.78), (3.83), and (3.84), y ∈ Λ(x, x). ¯ In view of (3.73), (3.74), (3.76), and (3.82), xσ  ≤ |1 − az − (1 + daσ )−1 | + |xσ − xσ | y − x  = yσ − ≤ δ + |daσ (1 + daσ )−1 − az| = δ + |az − ad x | ≤ δ + ||a||2δn < . This completes the proof of Proposition 3.13. Proposition 3.14 Assume that for each good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

n satisfying x − x  ≤ δ, Let  > 0. Then there is δ > 0 such that for each x ∈ R+ the inequality Δ(x) ≤  holds.

Proof Choose a positive number δ1 such that δ1  p < /8 and δ1 < /8, n satisfying y − y  ≤ δ1 . |w(by) − w(b y )| ≤ /8 for each y ∈ R+

(3.85) (3.86)

n satisfying By Proposition 3.13 there is δ ∈ (0, δ1 /4) such that for each x, x  ∈ R+ n   ||x − x ||, ||x − x || ≤ δ, there exist x¯ ≥ x , y ∈ R+ such that

(x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯ y − x , x¯ − x  ≤ δ1 /4.

(3.87)

n satisfy Let x ∈ R+

x − x  ≤ δ.

(3.88)

n , y ∈ R n such that In view of this inequality and the choice of δ, there are x¯ ∈ R+ +

(x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯

60

3 Infinite Horizon Optimization

x¯ ≥ x , x¯ − x , y − y  ≤ δ1 /4.

(3.89)

Define x(0) = x, y(0) = y, x(1) = x, ¯ x(t + 1) = (1 − d)x(t) + d x , y(t) = y

(3.90)

for all integers t ≥ 1. Relations (3.89) and (3.90) imply that {x(t), y(t)}∞ t=0 is a good program and x. lim x(t) =

(3.91)

t→∞

By Proposition 2.8, (3.85), (3.86), and (3.88)–(3.91), Δ(x) ≤

∞ 

δ(x(t), y(t), x(t + 1))

t=0

= lim

T →∞

p (x(0) − x(T + 1)) −

T 

 (w(by(t)) − w(b y ))

t=0

=p (x − x ) − (w(by(0)) − w(b y )) ≤  pδ + |w(by(0)) − w(b y )| ≤  pδ + |w(by) − w(b y )| < /2. Proposition 3.14 is proved.

3.9 Proofs of Theorems 3.10 and 3.11 Lemma 3.15 Let S0 ,  > 0. Then there exists δ > 0 such that for each sequence 2n {x(t), y(t)}∞ t=−∞ ⊂ R+ satisfying x(t), y(t) ≤ S0 for all integers t, (x(t), x(t + 1)) ∈ Ω, y(t) ∈ Λ(x(t), x(t + 1)) for all integers t, δ(x(t), y(t), x(t + 1)) ≤ δ for all integers t,

(3.92) (3.93) (3.94)

the following inequality holds: x(t) − x , y(t) − x  ≤  for all integers t.

(3.95)

3.9 Proofs of Theorems 3.10 and 3.11

61

Proof Let us assume the converse. Then for each natural number k, there exist a 2n sequence {x (k) (t), y (k) (t)}∞ t=−∞ ⊂ R+ which satisfies x (k) (t), y (k) (t) ≤ S0 for all integers t,

(3.96)

(x (k) (t), x (k) (t + 1)) ∈ Ω, y (k) (t) ∈ Λ(x (k) (t), x (k+1) (t + 1)) for all integers t, (3.97) δ(x (k (t), y (k) (t), x (k) (t + 1)) ≤ 1/k for all integers t (3.98) and an integer τk such that x , y (k) (τk ) − x } ≥ . max{x (k) (τk ) −

(3.99)

We may assume without loss of generality that τk = 0 for all natural numbers k.

(3.100)

Extracting subsequences, re-indexing, and using diagonalization process, we obtain that there exists a strictly increasing sequence of natural numbers {kj }∞ j =1 such that for each integer s, there is (u(s), v(s)) = lim (x (kj ) (s), y (kj ) (s)). j →∞

(3.101)

Since the set 3n {(x, y, x  ) ∈ R+ : (x, x  ) ∈ Ω, y ∈ Λ(x, x  )}

is closed, it follows from (3.100) and (3.101) that (u(s), u(s + 1)) ∈ Ω, v(s) ∈ Λ(u(s), u(s + 1)) for all integers s.

(3.102)

In view of (3.96) and (3.101), for all integers s, u(s), v(s) ≤ S0 .

(3.103)

Since the function δ(·, ·, ·) is nonnegative and continuous, it follows from (3.98) and (3.101) that δ(u(s), v(s), u(s + 1)) = 0 for all integers s.

(3.104)

Relations (3.99)–(3.101) imply that max{u(0) − x , v(0) − x } ≥ .

(3.105)

62

3 Infinite Horizon Optimization

If w is strictly concave, then it follows from (3.102), (3.103), and Lemma 2.13 that u(s) = v(s) = x for all integers s.

(3.106)

If ξσ = −1, then it follows from (3.102)–(3.104) and Lemma 3.8 that for all integers s, u(s) = v(s) = x . Thus (3.106) holds in both cases. This contradicts (3.105). The contradiction we have reached proves Lemma 3.15. Choose M0 > 1 + (1 + daσ )−1 .

(3.107)

Proof of Theorem 3.11 By Proposition 3.6, there is S0 > 0 such that for each program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e, x(t) ≤ S0 e for all integers t ≥ 0.

(3.108)

In view of Lemma 3.15, there is 1 ∈ (0, ) such that for each sequence 2n {x(t), y(t)}∞ t=−∞ ⊂ R+ satisfying x(t), y(t) ≤ S0 n for all integers t,

(3.109)

(x(t), x(t + 1)) ∈ Ω, y(t) ∈ Λ(x(t), x(t + 1)), δ(x(t), y(t), x(t + 1)) ≤ 1 (3.110) for all integers t, the following inequality holds: x(t) − x , y(t) − y  ≤ /8 for all integers t.

(3.111)

By Proposition 3.14 and Theorems 3.1 and 3.2, there is δ1 > 0 such that for each n satisfying x − x ∈ R+ x  ≤ δ1 , Δ(x) ≤ 1 /4.

(3.112)

It follows from Proposition 3.13 and the continuity of the function δ(·, ·, ·) that there n satisfying ||x − x || ≤ δ2 , there exist x¯ ≥ x and is δ2 > 0 such that for each x ∈ R+ n y ∈ R+ which satisfy ( x , x) ¯ ∈ Ω, y ∈ Λ( x , x), ¯

(3.113)

p  + 1)−1 . δ( x , y, x) ¯ ≤ 1 /4, x¯ − x ≤ (1 /4)( Set δ = min{δ1 , δ2 , 1}.

(3.114)

3.9 Proofs of Theorems 3.10 and 3.11

63

Assume that {x(t), y(t)}∞ t=0 is an overtaking optimal program such that x(0) − x  ≤ δ.

(3.115)

By Theorems 3.1, 3.2, and 3.9, the choice of δ1 (see (3.112)), (3.114), and (3.115), ∞ 

δ(x(t), y(t), x(t + 1)) = Δ(x(0)) ≤ 1 /4.

(3.116)

t=0

In view of (3.114), (3.115), and the choice of δ2 (see (3.113)), there exist x¯ ≥ x(0) n such that and y¯ ∈ R+ ( x , x) ¯ ∈ Ω, y¯ ∈ Λ( x , x), ¯ δ( x , y, ¯ x) ¯ ≤ 1 /4,

(3.117)

p  + 1)−1 . x¯ − x(0) ≤ (1 /4)( Set z = x¯ − x(0).

(3.118)

∞ Define a sequence {x(t), ¯ y(t)} ¯ t=−∞ by

x(t) ¯ = x for all integers t < 0, y(t) ¯ = x for all integers t < −1, y(−1) ¯ = y, ¯ x(0) ¯ = x, ¯ x(t) ¯ = x(t) + (1 − d)t z for all integers t ≥ 1, y(t) ¯ = y(t) for all integers t ≥ 0.

(3.119)

It follows from (3.117), (3.118), and (3.119) that (x(t), ¯ x(t ¯ + 1)) ∈ Ω, y(t) ¯ ∈ Λ(x(t), ¯ x(t ¯ + 1)) for all integers t.

(3.120)

By (3.107), (3.119), (3.120), and the choice of S0 (see (3.108)), x(t), ¯ y(t) ¯ ≤ S0 n for all integers t.

(3.121)

In view of (3.116), (3.119), and Theorems 3.1 and 3.2, ¯ = lim x(t) = x. lim x(t)

i→∞

t→∞

(3.122)

64

3 Infinite Horizon Optimization

Proposition 2.8 and (3.119) imply that for any natural number T , T 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1)) = p (x(0) ¯ − x(T ¯ + 1)) −

t=0

T  (w(by(t)) ¯ − w(b y )) t=0

=p (x(0) − x(T + 1)) −

T 

(w(by(t)) − w(b y ))

t=0

+ p(x(0) ¯ − x(0)) + p (x(T + 1) − x(T ¯ + 1)) ≤

T 

δ(x(t), y(t), x(t + 1)) +  px(0) ¯ − x(0)

t=0

+ p x(T + 1) − x(T ¯ + 1). Combined with (3.116), (3.117), and (3.122), this relation implies that ∞ 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1))

t=0

≤

∞ 

δ(x(t), y(t), x(t + 1)) +  px(0) ¯ − x(0) ≤ 1 /4 + 1 /4.(3.123)

t=0

It follows from (3.117), (3.119), and (3.123) that for all integers t, δ(x(t), ¯ y(t), ¯ x(t ¯ + 1)) ≤ 1 .

(3.124)

In view of (3.120), (3.121), (3.124), and the choice of 1 (see (3.109)–(3.111)), x(t) ¯ − x , y(t) ¯ − x  ≤ /8

(3.125)

for all integers t. Relations (3.117)–(3.119) imply that for all integers t ≥ 0, x(t) ¯ − x(t) ≤ 1 /4 < /4. By (3.119) and (3.125), for all integers t ≥ 0, y(t) − x  ≤ /8. In view of (3.125) and (3.126), for all integers t ≥ 0, x(t) − x  ≤ x(t) − x(t) ¯ + x(t) ¯ − x  ≤ /4 + /8 < . This completes the proof of Theorem 3.11.

(3.126)

3.9 Proofs of Theorems 3.10 and 3.11

65

Proof of Theorem 3.10 By Proposition 3.6 there is M1 > 0 such that for each program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e, the following inequality holds: x(t) ≤ M1 e for all integers t ≥ 0.

(3.127)

By Theorem 3.11 there exists δ > 0 such that for each overtaking optimal program x  ≤ δ, the following inequality holds: {x(t), y(t)}∞ t=0 satisfying x(0) − x(t) − x , y(t) − x  ≤  for all integers t ≥ 0.

(3.128)

We show that there exists a natural number τ0 such that the following property holds: (P) For each overtaking optimal program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e, there exists an integer t such that x  < δ. 0 ≤ t ≤ τ0 and x(t) − Let us assume the contrary. Then for each natural number k, there exists an overtaking optimal program {x (k) (t), y (k) (t)}∞ t=0 such that x (k) (0) ≤ M0 e, x (k) (t) − x  ≥ δ, t = 0, . . . , k.

(3.129)

Proposition 3.12 implies that there is D0 > 0 such that n satisfying z ≤ M0 e. Δ(z) ≤ D0 for each z ∈ R+

(3.130)

In view of (3.129), (3.130), and Theorems 3.1, 3.2, and 3.9 for each natural number k, ∞ 

δ(x (k) (t), y (k) (t), x (k) (t + 1)) = Δ(x (k) (0)) ≤ D0 .

(3.131)

t=0

By the choice of M1 and (3.129), x (k) (t), y (k) (t) ≤ M1 e for all integers t ≥ 0 and all natural numbers k. Extracting subsequences, re-indexing, and using diagonalization process, we obtain that there exists a strictly increasing sequence of natural numbers {kj }∞ j =1 such that for each integer s ≥ 0, there exists (x(s), ˜ y(s)) ˜ = lim (x (kj ) (s), y (kj (s)). j →∞

(3.132)

∞ is a program. In view of (3.132), (3.140), It is not difficult to see that {x(t), ˜ y(t)} ˜ t=0 and the continuity of the function δ(·, ·, ·),

66

3 Infinite Horizon Optimization ∞ 

δ(x(t), ˜ y(t), ˜ x(t ˜ + 1)) ≤ D0 .

t=0 ∞ is a good program. Theorems 3.1 and 3.2 imply This implies that {x(t), ˜ y(t)} ˜ t=0 that

˜ = x. lim x(t)

t→∞

On the other hand, it follows from (3.129) and (3.132) that x(t) ˜ − x  ≥ δ for all integers t ≥ 0. The contradiction we have reached proves that there is a natural number τ0 such that property (P) holds. Now assume that {x(t), y(t)}∞ t=0 is an overtaking optimal program satisfying x(0) ≤ M0 e. By property (P) there is an integer t0 ∈ [0, τ0 ] such that x  < δ. x(t0 ) −

(3.133)

Clearly the program {x(t + t0 ), y(t + t0 )}∞ t=0 is also overtaking optimal. In view of (3.133) and the choice of δ (see (3.128)) x , y(t + t0 ) − x  ≤  for all integers t ≥ 0. x(t + t0 ) − This completes the proof of Theorem 3.10.

3.10 The Structure of Good Programs in the RSS Model We prove the following three theorems which were obtained in [98]. Theorem 3.16 Let a program {x(t), y(t)}∞ t=0 be good. Then for each i {1, . . . , n} \ {σ }, ∞ 

xi (t) < ∞,

t=0 ∞  (xσ (t) − yσ (t)) < ∞ t=0

and the sequence {

T −1 t=0

xσ (t) − T (1 + daσ )−1 }∞ T =1 is bounded.

∈

3.11 Proofs of Theorems 3.16–3.18

67

Theorem 3.17 Let the function w be linear. Then a program {x(t), y(t)}∞ t=0 is good if and only if for each i ∈ {1, . . . , n} \ {σ }, ∞ 

xi (t) < ∞,

t=0 ∞  (xσ (t) − yσ (t)) < ∞ t=0

and the sequence {

T −1 t=0

xσ (t) − T (1 + daσ )−1 }∞ T =1 is bounded.

Theorem 3.18 Let w ∈ C 2 , w  (b x ) = 0 and let for every good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Then a program {x(t), y(t)}∞ t=0 is good if and only if for each i ∈ {1, . . . , n} \ {σ }, ∞ 

xi (t) < ∞,

t=0 ∞  (xσ (t) − yσ (t)) < ∞, t=0 ∞ 

(yσ (t) − xσ )2 < ∞,

t=0

and the sequence {

T −1 t=0

xσ (t) − T (1 + daσ )−1 }∞ T =1 is bounded.

3.11 Proofs of Theorems 3.16–3.18 Proof of Theorem 3.16 Since the program {x(t), y(t)}∞ t=0 is good, it follows from Proposition 2.9 that ∞  t=0

For t = 0, 1, . . . , set

δ(x(t), y(t), x(t + 1)) < ∞.

(3.134)

68

3 Infinite Horizon Optimization

z(t) = x(t + 1) − (1 − d)x(t).

(3.135)

Since w is concave [70, 79] for each z ∈ [0, ∞), x )(z − b x ). w(z) − w(b x ) ≤ w (b

(3.136)

Since w  (b x ) = 0 and w is strictly increasing, it is easy to see that x ) > 0. w  (b

(3.137)

By (2.2) (see the proof of Lemma 2.2), (3.8), (3.9), (3.12), (3.135), and (3.136), for each integer t ≥ 0, δ(x(t), y(t), x(t + 1)) = p (x(t) − x(t + 1)) − (w(by(t)) − w(b y )) ≥p (x(t) − x(t + 1)) + w  (b x )(b x − by(t)) x )[b x − by(t) + q (x(t) − x(t + 1))] = w  (b

n  x ) cσ (1−ey(t)−az(t))+ (cσ −ci )yi (t) = w  (b i=1

 n  (cσ −ci )ai zi (t)+d q (x(t)−y(t)) . (3.138) + i=1

Combined with (3.2), (3.3), (3.134), and (3.137), relation (3.138) implies that ∞ >

∞ 



δ(x(t), y(t), x(t + 1)) ≥ w (b x)

t=0

∞ 

cσ (1 − ey(t) − az(t))

t=0

 n n   + (cσ − ci )yi (t) + (cσ − ci )ai zi (t) + d q (x(t) − y(t)) . (3.139) i=1

i=1

Relations (3.2), (3.3), (3.135), (3.137) and (3.139) imply that for all i ∈ {1, . . . , n} \ {σ } ∞ 

yi (t) < ∞,

(3.140)

(xi (t) − yi (t)) < ∞, i = 1, . . . , n.

(3.141)

t=0 ∞  t=0

In view of (3.140) and (3.141),

3.11 Proofs of Theorems 3.16–3.18 ∞ 

69

xi (t) < ∞ for all i ∈ {1, . . . , n} \ {σ }.

(3.142)

t=0

Relations (3.2), (3.3), (3.137), and (3.139) imply that ∞ 

(1 − ey(t) − az(t)) < ∞.

t=0

This inequality, (3.3), and (3.135) imply that ∞  (1 − yσ (t) − aσ (xσ (t + 1) − (1 − d)xσ (t))) < ∞. t=0

Therefore there is M1 > 0 such that for each natural number T , M1 >

T −1

(1 − yσ (t) − aσ (xσ (t + 1) − xσ (t) + dxσ (t)))

t=0

=T −

T −1

yσ (t) − aσ (xσ (T ) − xσ (0)) − daσ

t=0

T −1

xσ (t) ≥ 0.

t=0

It follows from this inequality, (3.141), (3.142), and Proposition 3.6 that there is M2 > 0 such that for each natural number T ,   T −1       xσ (t) (1 + daσ ) < M2 . T −   t=0

Theorem 3.16 is proved. Proof of Theorem 3.17 Assume that a program {x(t), y(t)}∞ t=0 is good. Then by Theorem 3.16, ∞ 

xi (t) < ∞ for all i ∈ {1, . . . , n} \ {σ },

(3.143)

t=0 ∞  (xσ (t) − yσ (t)) < ∞, t=0

and the sequence {T (1 + daσ )−1 −

T −1 t=0

xσ (t)}∞ T =1 is bounded.

(3.144)

70

3 Infinite Horizon Optimization

Now assume that {x(t), y(t)}∞ is a program, that (3.143) and (3.144) hold, and t=0  T −1 xσ (t)}∞ that the sequence {T (1 + daσ )−1 − t=0 T =1 is bounded. We prove that the ∞ program {x(t), y(t)}t=0 is good. For t = 0, 1, . . . , set z(t) = x(t + 1) − (1 − d)x(t).

(3.145)

T −1 xσ (t)}∞ Since the sequence {T (1 + daσ )−1 − t=0 T =1 is bounded, it follows from Proposition 3.6 and (3.144) that the sequence  T −

T −1

yσ (t) − daσ

t=0

=

T −1 

T −1

∞ xσ (t) − a(xσ (T ) − xσ (0)) T =1

t=0

∞ (1 − yσ (t) − aσ (xσ (t + 1) − (1 − d)xσ (t))) T =1

t=0

is also bounded. Together with (3.3) and (3.145), this implies that ∞  (1 − yσ (t) − aσ zσ (t)) < ∞. t=0

Combined with (3.3) and (3.143), this inequality implies that ∞  (1 − ey(t) − az(t)) < ∞.

(3.146)

t=0

It follows from (3.9), the linearlity of w, (3.8), (3.145), and (2.2) (see Lemma 2.2) that for each integer t ≥ 0, δ(x(t), y(t), x(t + 1)) = p (x(t) − x(t + 1)) − (w(by(t)) − w(b y )) =p (x(t) − x(t + 1)) − w (b x )(by(t) − b x) = w  (b x )[−by(t) + b x + q (x(t) − x(t + 1))]

n  = w  (b x ) cσ (1 − ey(t) − az(t)) + (cσ − ci )yi (t) i=1

 n  + (cσ − ci )ai zi (t) + d q (x(t) − y(t)) . i=1

Clearly w  (b x ) > 0. By (3.2), (3.3), (3.143)–(3.147),

(3.147)

3.11 Proofs of Theorems 3.16–3.18 ∞ 

71

δ(x(t), y(t), x(t + 1)) < ∞.

t=0

Together with Proposition 2.9, this inequality implies that {x(t), y(t)}∞ t=1 is a good program. Theorem 3.17 is proved. Proof of Theorem 3.18 Assume that {x(t), y(t)}∞ t=0 is a program. Let t ≥ 0 be an integer. Set z(t) = x(t + 1) − (1 − d)x(t).

(3.148)

In view of (3.12), δ(x(t), y(t), x(t + 1)) = p (x(t) − x(t + 1)) − (w(by(t)) − w(b y )).

(3.149)

Since w ∈ C 2 , it follows from Taylor’s theorem that there exists y }, max{by(t), b y }] γt ∈ [min{by(t), b

(3.150)

such that y )(by(t) − b y ) + 2−1 w  (γt )(by(t) − b y )2 . w(by(t)) − w(b y ) = w  (b

(3.151)

Relations (3.8), (3.9), (3.148), (3.149), (3.151), and (2.2) from Lemma 2.2 imply that x )(b y − by(t)) − 2−1 w  (γt )(by(t) − b y )2 δ(x(t), y(t), x(t + 1)) = w (b +w  (b y ) q (x(t) − x(t + 1)) = −2−1 w  (γt )(by(t) − b y )2 + w  (b y )[b y − by(t) + q (x(t) − x(t + 1))]

= −2−1 w  (γt )(by(t) − b y )2 + w  (b y ) cσ (1 − ey(t) − az(t)) +

n 

(cσ − ci )yi (t)

i=1

+

n 

 (cσ − ci )ai zi (t) + d q (x(t) − y(t)) .

(3.152)

i=1

Since w is concave and increasing and w (b x ) = 0, we have x ) = w  (b y ) > 0, w  (γt ) ≤ 0. w  (b Assume that {x(t), y(t)}∞ t=0 is good. Then

(3.153)

72

3 Infinite Horizon Optimization

lim (x(t), y(t)) = ( x, x ).

(3.154)

δ(x(t), y(t), x(t + 1)) < ∞.

(3.155)

t→∞

By Proposition 2.9, ∞  t=0

Since w (b y ) = 0 and w is concave, we conclude that y ) < 0. w  (b

(3.156)

Since w ∈ C 2 , it follows from (3.154), (3.156), and (3.150) that there exists a natural number t0 such that for each integer t ≥ t0 , y ). w  (γt ) ≤ 2−1 w  (b

(3.157)

It follows from (3.152), the choice of t0 , and (3.157) that for each integer t ≥ t0 , y )(by(t) − b y )2 δ(x(t), y(t), x(t + 1)) ≥ −(4)−1 w  (b

+w  (b y ) cσ (1 − ey(t) − az(t)) +

n  (cσ − ci )yi (t) i=1

+

n 



(cσ − ci )ai zi (t) + d q (x(t) − y(t)) .

(3.158)

i=1

It follows from (3.2), (3.3), (3.148), (3.153), (3.155), (3.156), and (3.158) that ∞ 

(xi (t) − yi (t)) < ∞, i = 1, . . . , n,

(3.159)

t=0 ∞ 

yi (t) < ∞, i ∈ {1, . . . , n} \ {σ },

(3.160)

t=0 ∞ 

(by(t) − b y )2 < ∞.

(3.161)

t=0

Relations (3.159) and (3.160) imply that ∞  t=0

xi (t) < ∞, i ∈ {1, . . . , n} \ {σ }.

(3.162)

3.11 Proofs of Theorems 3.16–3.18

73

By Theorem 3.16, the sequence {T (1 + daσ )−1 − (3.160) and (3.161), ∞ 

T −1 t=0

xσ (t)}∞ T =1 is bounded. By

(yσ (t) − yσ )2 < ∞.

(3.163)

t=0

Assume that (3.159), (3.162), and (3.163) hold and that the the sequence  T (1 + daσ )

−1

−

T −1

∞ xσ (t) T =1

t=0

is bounded. We show that {x(t), y(t)}∞ t=0 is a good program. By (3.159), (3.162), and (3.163), x, x ). lim (x(t), y(t)) = (

(3.164)

t→∞

Since w  (b y ) = 0 and w is concave, we have y ) < 0. w  (b

(3.165)

By (3.150), (3.164), and (3.165), there exists a natural number t0 such that for each integer t ≥ t0 w  (γt ) ≥ 2w  (b y ).

(3.166)

It follows from (3.152), (3.166), and the choice of t0 that for each integer t ≥ t0 , y )(by(t) − b y )2 δ(x(t), y(t), x(t + 1)) ≤ −w (b

+w  (b y ) cσ (1 − ey(t) − az(t)) +

n  (cσ − ci )yi (t) i=1

+

n 



(cσ − ci )ai zi (t) + d q (x(t) − y(t)) .

(3.167)

i=1

By (3.2), (3.3), (3.148), (3.153), (3.159), and (3.162), w (b y)

n n ∞  ∞    (cσ − ci )yi (t) < ∞, w  (b y) (cσ − ci )ai zi (t) < ∞, t=0 i=1

t=0 i=1

y) w  (b

∞  t=0

d q (x(t) − y(t)) < ∞.

(3.168)

74

3 Infinite Horizon Optimization

By (3.162) and (3.163), ∞ 

(by(t) − b y )2 < ∞.

(3.169)

cσ (1 − ey(t) − az(t)) < ∞.

(3.170)

t=0

We show that ∞  t=0

Clearly, it is sufficient to show that the sequence T −1 

∞ (1 − yσ (t) − aσ (xσ (t + 1) − (1 − d)xσ (t))) T =1

t=0

T −1 is bounded. Since the sequence {T (1 + daσ )−1 − t=0 xσ (t)}∞ T =1 is bounded, it follows from Proposition 3.6 and (3.159) that the sequence T −1 

∞ (1 − yσ (t) − aσ (xσ (t + 1) − xσ (t)) + aσ dxσ (t)) T =1

t=0

=

T −1 

∞ (1 − yσ (t) − aσ dxσ (t)) − aσ (x(T ) − xσ (0))

T =1

t=0

is bounded. Thus (3.170) holds. Relations (3.15), (3.159), (3.162), (3.165), (3.167), (3.169), and (3.170) imply that ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

t=0

Together with Proposition 2.9, this inequality implies that {x(t), y(t)}∞ t=0 is a good program. Theorem 3.18 is proved.

Chapter 4

Turnpike Results for the Robinson–Solow–Srinivasan Model

In this chapter we study the turnpike properties for the Robinson–Solow–Srinivasan model. To have these properties means that the approximate solutions of the problems are essentially independent of the choice of an interval and endpoint conditions. We show that these turnpike properties hold and that they are stable under perturbations of an objective function.

4.1 The Main Results 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}.

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi ,

i=1

and let x >> y, x > y, x ≥ y have their usual meaning. n Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n . Let a = (a1 , . . . , an ) >> 0, b = (b1 , . . . , bn ) >> 0, d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_4

(4.1) 75

76

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

We assume the following: There exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci .

(4.2)

Recall that a sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0, n n × R+ , x(t + 1) ≥ (1 − d)x(t), (x(t), y(t)) ∈ R+

0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(4.3)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences

 T2 −1 2 {x(t)}Tt=T , {y(t)} t=T 1 1 n and for each integer t satisfying T ≤ t < T is called a program if x(T2 ) ∈ R+ 1 2 relations (4.3) are valid. Assume that w : [0, ∞) → R 1 is a continuous strictly increasing concave and differentiable function which represents the preferences of the planner. Define n n Ω = {(x, x  ) ∈ R+ × R+ : x  − (1 − d)x ≥ 0

and a(x  − (1 − d)x) ≤ 1} n given by and a correspondence Λ : Ω → R+ n : 0 ≤ y ≤ x and ey ≤ 1 − a(x  − (1 − d)x)}, (x, x  ) ∈ Ω. Λ(x, x  ) = {y ∈ R+

For every (x, x  ) ∈ Ω, set u(x, x  ) = max{w(by) : y ∈ Λ(x, x  )}. n such that ( x, x ) is a solution to the Recall that the golden-rule stock is x ∈ R+ problem: maximize u(x, x  ) subject to (i) x  ≥ x; (ii) (x, x  ) ∈ Ω. By Theorem 2.3, there exists a unique golden-rule stock

x = (1/(1 + daσ ))e(σ ). It is not difficult to see that x is a solution to the problem

(4.4)

4.1 The Main Results

77

w(by) → max, y ∈ Λ( x, x ). Set y = x.

(4.5)

i = w  (b x ) qi . qi = ai bi /(1 + dai ), p

(4.6)

For i = 1, . . . , n, set

In view of Lemma 2.2, x w(b x ) ≥ w(by) + p x  − p

(4.7)

for every (x, x  ) ∈ Ω and for every y ∈ Λ(x, x  ). For every (x, x  ) ∈ Ω and every y ∈ Λ(x, x  ), define (x − x  ) − (w(by) − w(b y )). δ(x, y, x  ) = p

(4.8)

By (4.7) and (4.8), δ(x, y, x  ) ≥ 0 for every (x, x  ) ∈ Ω and every y ∈ Λ(x, x  ).

(4.9)

A program {x(t), y(t)}∞ t=0 is good if there is a real number M such that T  (w(by(t)) − w(b y )) ≥ M for every nonnegative integer T . t=0

A program {x(t), y(t)}∞ t=0 is bad if lim

T →∞

T  (w(by(t)) − w(b y )) = −∞. t=0

In the next proposition, which follows from Propositions 2.4, 2.6, and 2.9, we collect the properties of good programs. n Proposition 4.1 (i) Any program that is not good is bad. (ii) For any x0 ∈ R+ ∞ such that x(0) = x . (iii) A program there exists a good program {x(t), y(t)} 0 ∞t=0 {x(t), y(t)}∞ t=0 δ(x(t), y(t), x(t + 1)) < ∞. t=0 is good if and only if n . Set Let x0 ∈ R+

78

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Δ(x0 ) = inf

 ∞

δ(x(t), y(t), x(t + 1)) : {x(t), y(t)}∞ t=0 is a

t=0

program such that x(0) = x0 }

(4.10)

Since there exists a good program from x0 , it follows from Proposition 4.1 that Δ(x0 ) < ∞.

(4.11)

Recall that a program {x ∗ (t), y ∗ (t)}∞ t=0 is called overtaking optimal if lim sup

T 

T →∞ t=0

[w(by(t)) − w(by ∗ (t))] ≤ 0

∗ for every program {x(t), y(t)}∞ t=0 satisfying x(0) = x (0). ∞ ∗ ∗ A program {x (t), y (t)}t=0 is called weakly optimal if for each program ∗ {x(t), y(t)}∞ t=0 satisfying x(0) = x (0), the following inequality holds:

lim inf T →∞

T  [w(by(t)) − w(by ∗ (t))] ≤ 0. t=0

n and T ≥ 1 be a natural number. Set Let z ∈ R+

U (z, T ) = sup

T −1 

T −1 w(by(t)) : ({x(t)}Tt=0 , {y(t)}t=0 )

t=0

is a program such that x(0) = z}

(4.12)

n , T , T be integers, 0 ≤ T < T . Clearly U (z, T ) is a finite number. Let x0 , x1 ∈ R+ 1 2 1 2 Define ⎧ 2 −1 ⎨T 

T2 −1 2 w(by(t)) : {x(t)}Tt=T , {y(t)} U (x0 , x1 , T1 , T2 ) = sup t=T 1 1 ⎩ t=T1

is a program such that x(T1 ) = x0 , x(T2 ) ≥ x1 } .

(4.13)

(Here we suppose that a supremum over empty set is −∞.) Clearly U (x0 , x1 , T1 , T2 ) < ∞. n and any integer T ≥ 1, U (z, T ) = U (z, 0, 0, T ). It is also clear that for any z ∈ R+

4.1 The Main Results

79

In this chapter we assume that the following asymptotic turnpike property holds: (ATP) Each good program {x(t), y(t)}∞ t=0 converges to the golden-rule stock ( x, x) : x, x ). lim (x(t), y(t)) = (

t→∞

With Card(A) we denote in the sequel the cardinality of a finite set A. In this chapter we prove the following two turnpike results obtained in [53]. Theorem 4.2 Let M,  be positive numbers and Γ ∈ (0, 1). Then there exists a n satisfying natural number L such that for each integer T > L, each z0 , z1 ∈ R+ T −1 z0 ≤ Me and az1 ≤ Γ d −1 , and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − M,

t=0

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ L. Theorem 4.3 Let M,  be positive numbers and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ such that for each integer T > 2L, n satisfying z ≤ Me and az ≤ Γ d −1 , and each program each z0 , z1 ∈ R+ 0 1 T −1 ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − γ ,

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Corollary 4.4 Let M,  be positive numbers and Γ ∈ (0, 1). Then there exists a n satisfying natural number L such that for each integer T > L, each z0 , z1 ∈ R+ −1 z0 ≤ Me and az1 ≤ Γ d , and each pair of programs T −1 T −1 ) and ({xb (t)}Tt=0 , {yb (t)}t=0 ) ({xa (t)}Tt=0 , {ya (t)}t=0

which satisfy

80

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

xa (0) = xb (0) = z0 , xa (T ) ≥ z1 , xb (T ) ≥ z1 , T −1

w(bya (t)) ≥ U (z0 , z1 , 0, T ) − M,

t=0 T −1

w(byb (t)) ≥ U (z0 , z1 , 0, T ) − M,

t=0

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{xa (t) − xb (t), ya (t) − yb (t)} > } ≤ L.

4.2 Auxiliary Results for Theorems 4.2 and 4.3 We suppose that the sum over empty set is zero. Proposition 4.5 Let Γ ∈ (0, 1). Then there exists a natural number k(Γ ) such that n and each z ∈ R n satisfying az ≤ Γ d −1 , there is a program for each z0 ∈ R+ 1 1 + k(Γ ) k(Γ )−1 ({x(t)}t=0 , {y(t)}t=0 ) such that x(0) = z0 , x(k(Γ )) ≥ z1 .  i Proof Since ∞ i=0 (1 − d) = 1/d, there is a natural number k(Γ ) such that k(Γ )−1 

(1 − d)i > Γ d −1 .

i=0 n and z ∈ R n satisfies az ≤ Γ d −1 . Put Assume that z0 ∈ R+ 1 1 +

z2 = Γ −1 dz1 , x(0) = z0 , x(t + 1) = (1 − d)x(t) + z2 and y(t) = 0 for all integers t ≥ 0. It is easy to see that {x(t), y(t)}∞ t=0 is a program and that x(k(Γ )) ≥

k(Γ )−1  i=0

Proposition 4.5 is proved.

(1 − d)i z2 ≥ Γ d −1 z2 = z1 .

4.2 Auxiliary Results for Theorems 4.2 and 4.3

81

In the sequel with each Γ ∈ (0, 1), we associate a natural number k(Γ ) for which the assertion of Proposition 4.5 holds. n and each natural Proposition 4.6 There is m > 0 such that for each z ∈ R+ number T ,

U (z, T ) ≥ T w(b x ) − m. Proof Put y(0) = 0, y(t + 1) = (1 − d)y(t) + d x for all integers t ≥ 0.

(4.14)

It is not difficult to see that the sequence {y(t)}∞ t=0 is non-decreasing and converges to x as t → ∞. Put x − by(1))|w  (db x )| + |w(by(0)) − w(b x )| + |w(by(1)) − w(b x )|. m = d −1 (b (4.15) Let z ∈

n. R+

Set x(0) = z,

(4.16)

x(t + 1) = (1 − d)x(t) + d x for all integers t ≥ 0. It is not difficult to see that {x(t), y(t)}∞ t=0 is a program. It follows from the definition of the sequence {y(t)}∞ that for all natural numbers t, t=0 by(t) ≥ db x,

(4.17)

x ). by(t) − b x = (1 − d)t−1 (by(1) − b

(4.18)

and that for all integers t ≥ 2,

Since the function w is concave and strictly increasing, it follows from (4.17) and (4.18) that for all integers t ≥ 2, x − by(t)) ≤ w  (db x )(b x − by(t)) w(b x ) − w(by(t)) ≤ w (by(t))(b ≤ w  (db x )(b x − by(1))(1 − d)t−1 . By the relation above, (4.15), and (4.16), for each natural number T , U (z, T ) ≥

T −1 t=0

+

w(by(t)) =



{w(by(t)) : t ∈ {0, 1} and t ≤ T − 1}

 {w(by(t)) : t is an integer and 2 ≤ t < T }

82

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

≥ T w(b x ) − |w(by(0)) − w(b x )| − |w(by(1)) − w(b x )|  − {|w(b x ) − w(by(t))| : t is an integer and 2 ≤ t < T } ≥ T w(b x ) − |w(by(0)) − w(b x )| − |w(by(1)) − w(b x )| − w  (db x )(b x − by(1))

T +1

(1 − d)t−1

t=2

≥ T w(b x ) − |w(by(0)) − w(b x )| − |w(by(1)) − w(b x )| − w  (db x )(b x − by(1))d −1 = T w(b x ) − m. Proposition 4.6 is proved. Proposition 4.7 Let Γ ∈ (0, 1). Then there exists m > 0 such that for each z0 ∈ n , each z ∈ R n satisfying az ≤ Γ d −1 , and each natural number T > k(Γ ), R+ 1 1 + x ) − m. U (z0 , z1 , 0, T ) ≥ T w(b Proof By Proposition 4.6 there is m0 > 0 such that n and each natural number T . U (z, T ) ≥ T w(b x ) − m0 for each z ∈ R+

(4.19)

Put x )| + |w(0)|). m = m0 + k(Γ )(|w(b n , z ∈ R n satisfies az ≤ Γ d −1 and a natural number Assume that z0 ∈ R+ 1 1 + T −k(Γ ) −k(Γ )−1 , {y(t)}Tt=0 ) such that T > k(Γ ). By (4.19) there is a program ({x(t)}t=0

x(0) = z0 ,

T −k(Γ )−1

w(by(t)) ≥ (T − k(Γ ))w(b x ) − m0 .

(4.20)

t=0

By the choice of k(Γ ), Proposition 4.5, and the inequality az1 ≤ Γ d −1 , there is a −1 program ({x(t)}Tt=T −k(Γ ) , {y(t)}Tt=T −k(Γ ) ) such that x(T ) ≥ z1 . By (4.19), (4.20), and the choice of m, U (z0 , z1 , 0, T ) ≥

T −1 t=0

w(by(t)) ≥

T −k(Γ )−1

w(by(t)) − k(Γ ))|w(0)|

t=0

≥ (T − k(Γ ))w(b x ) − m0 − k(Γ )|w(0)|

4.2 Auxiliary Results for Theorems 4.2 and 4.3

83

≥ T w(b x ) − k(Γ )|w(b x )| − m0 − k(Γ )|w(0)| ≥ T w(b x ) − m. Proposition 4.7 is proved. Proposition 4.8 Let m0 > 0. Then there exists m2 > 0 such that for each natural T −1 ) which satisfies x(0) ≤ m0 e, the number T and each program ({x(t)}Tt=0 , {y(t)}t=0 following inequality holds: T −1

[w(by(t)) − w(b x )] ≤ m2 .

t=0

Proof By Proposition 3.6 there exists m1 > 0 such that for each natural number T T −1 and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) ≤ m0 e, we have x(t) ≤ m1 e for all t = 0, . . . , T .

(4.21)

p m1 n. m2 ≥ 2

(4.22)

Choose a number

−1 ) satisfies Assume that T is a natural number and a program ({x(t)}Tt=0 , {y(t)}Tt=0 x(0) ≤ m0 e. Then (4.21) holds. By (4.5) and (4.7), for each integer t ∈ [0, T − 1],

w(b y ) ≥ w(by(t)) + p (x(t + 1)) − p x(t). Together with (4.21) and (4.22), this implies that T −1

[w(by(t)) − w(b x )] ≤

t=0

T −1

[ px(t) − p x(t + 1)]

t=0

=p x(0) − p x(T ) ≤ 2 p nm1 ≤ m2 . Proposition 4.8 is proved. It is easy to see that the following auxiliary result holds. Proposition 4.9 Assume that T1 , T2 are nonnegative integers, T1 < T2 , T2 −1 2 ({x(t)}Tt=T , {y(t)}t=T ) 1 1 n . Then ({x(t) + (1 − d)t−T1 u}T2 , {y(t)}T2 −1 ) is also a is a program, and u ∈ R+ t=T1 t=T1 program.

84

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

n , and let Proposition 4.10 Let τ be a natural number, M > 0, x0 , x1 ∈ R+

 −1 {x(t)}τt=0 , {y(t)}τt=0 be a program such that x(0) = x0 , x(τ ) ≥ x1 ,

τ −1 

w(by(t)) ≥ U (x0 , x1 , 0, τ ) − M.

(4.23)

t=0

Then for each pair of integers S1 , S2 satisfying 0 ≤ S1 < S2 ≤ τ,

(4.24)

the following inequality holds: S 2 −1

w(by(t)) ≥ U (x(S1 ), x(S2 ), S1 , S2 ) − M.

t=S1

Proof Let us assume the contrary. Then there exists a pair of integers S1 , S2 satisfying (4.24) such that S 2 −1

w(by(t)) < U (x(S1 ), x(S2 ), S1 , S2 ) − M.

(4.25)

t=S1 2 2 −1 By (4.13) and (4.25), there exists a program ({x  (t)}St=S , {y  (t)}St=S ) such that 1 1

x  (S1 ) = x(S1 ), x  (S2 ) ≥ x(S2 ), S 2 −1 t=S1

w(by(t))
M.

t=S1

This contradicts (4.23). The contradiction we have reached proves Proposition 4.10.

4.3 Four Lemmas Lemma 4.11 Let m0 , m1 > 0,  > 0. Then there is a natural number τ such that −1 for each program ({x(t)}τt=0 , {y(t)}τt=0 ) satisfying x(0) ≤ m0 e,

τ −1 

w(by(t)) ≥ τ w(b x ) − m1 ,

t=0

there is an integer t ∈ [0, τ − 1] such that y(t) − x , x(t) − x  ≤ . Proof Let us assume the contrary. Then for each natural number k, there exists a (k) (k) program ({xt }kt=0 , {yt }k−1 t=0 ) such that (0) ≤ m0 e,

k−1 

w(by (k) (t)) ≥ kw(b x ) − m1 ,

(4.29)

  (k) (k) max yt − x , xt − x  > , t = 0, . . . , k.

(4.30)

x

(k)

t=0

In view of (4.29) and Proposition 3.6, there is m2 > m0 such that for each natural number k, x (k) (t) ≤ m2 e, t = 0, . . . , k.

(4.31)

86

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

By Proposition 4.8, there is m3 > 0 such that for each natural number T and each T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying x(0) ≤ m2 e, the following inequality holds: T −1

(w(by(t)) − w(b y )) ≤ m3 .

(4.32)

t=0

Let k be a natural number, and let an integer s satisfy 0 < s < k. It follows from (4.31) and the choice of m3 (see (4.32)) that k−1  [w(by (k) (t)) − w(b y )] ≤ m3 . t=s

Combined with (4.29) this implies that s−1 

[w(by (k) (t)) − w(b y )] =

t=0

k−1  [w(by (k) (t)) − w(b y )] t=0

−

k−1  [w(by (k) (t)) − w(b y )] ≥ −m1 − m3 . t=s

Thus for each pair of natural numbers s, k satisfying s < k, s−1 

[w(by (k) (t)) − w(b y )] ≥ −m1 − m3 .

(4.33)

t=0

By extracting subsequence and using (4.31) and diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {kj }∞ j =1 and ∗ (t)}∞ ⊂ R n such that , {y sequences {x ∗ (t)}∞ t=0 t=0 x (kj ) (t) → x ∗ (t), y (kj ) (t) → y ∗ (t) as j → ∞ for all integers j ≥ 0.

(4.34)

It is not difficult to see that {x ∗ (t), y ∗ (t)}∞ t=0 is a program. In view of (4.31) and (4.34), x ∗ (t) ≤ m2 e for all integers t ≥ 0. By (4.33) and (4.34), for all natural numbers s, s−1  t=0

(w(by ∗ (t)) − w(b x )) ≥ −m3 − m1 .

(4.35)

4.3 Four Lemmas

87

This implies that {x ∗ (t), y ∗ (t)}∞ t=0 is a good program and by (ATP), x , x ∗ (t) → x as t → ∞. y ∗ (t) →

(4.36)

On the other hand, it follows from (4.30) and (4.34) that for all integers t ≥ 0, x , x ∗ (t) − x } ≥ . max{y ∗ (t) − This contradicts (4.36). The contradiction we have reached proves Lemma 4.11. 2n Lemma 4.12 Let S0 > 0, and let a sequence {x(t), y(t)}∞ t=−∞ ⊂ R+ satisfy the following conditions:

x(t), y(t) ≤ S0 for all integers t, x(t + 1) ≥ (1 − d)x(t), 0 ≤ y(t) ≤ x(t) for all integers t, a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1 for all integers t, δ(x(t), y(t), x(t + 1)) = 0 for all integers t. Then x(t) = y(t) = x for all integers t. Proof Assume that the lemma does not hold. Then we may assume without loss of generality that (x(0), y(0)) = ( x, x ). Set κ = min{1, max{x(0) − x , y(0) − x }}.

(4.37)

By Proposition 3.13, there exists a strictly increasing sequence {δk }∞ k=1 ⊂ (0, κ) such that for each natural number k, the following property holds: n , x − (P1) If x, x  ∈ R+ x , x  − x  ≤ 4δk , then there exist x¯ ≥ x  , y ∈ R n such that x  ≤ 8−k κ, δ(x, y, x) ¯ ≤ 8−k κ. (x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯ y − x  ≤ 8−k κ, x¯ − By Lemma 4.11, for each natural number k, there exist natural numbers Tk , Sk > 4 such that (1 − d)min{Tk ,Sk } < min{δk , 8−k κ},

(4.38)

x , y(Tk ) − x , x(−Sk ) − x , y(−Sk ) − x  ≤ min{δk , 8−k κ}. x(Tk ) − (4.39)

88

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Set τ0 = 0, τ1 = T1 + S1 , τk+1 = τk + Tk+1 + Sk+1 + 1 for any integer k ≥ 1. (4.40) By induction, we construct a program {u(t), v(t)}∞ . Set t=0 u(t) = x(t − S1 ), t = 0, . . . , τ1 , v(t) = y(t − S1 ), t = 0 . . . , τ1 − 1.

(4.41)

Now assume that k is a natural number, we defined u(t), t = 0, . . . , τk , v(t), t = τk −1 k 0, . . . , τk − 1 such that ({u(t)}τt=0 , {v(t)}t=0 ) is a program, x  ≤ 2 min{δk , 8−k κ}, u(τk ) − τ k −1

δ(u(t), v(t), u(t + 1)) ≤ (1 + 4 p )

k−1 

(4.42) 8−j ,

(4.43)

j =0

t=0

and if k ≥ 2, then for j = 1, . . . , k − 1, max{v(τj + Sj +1 + 1) − x , u(τj + Sj +1 + 1) − x } ≥ 2−1 κ.

(4.44)

(It is not difficult to see that for k = 1, all these assumptions hold.) Define u(t), t = τk + 1, . . . , τk+1 , v(t), t = τk , . . . , τk+1 − 1. First define u(τk + 1), v(τk ). By (4.39), (4.42), and (P1), there exist u(τk + 1) ≥ x(−Sk+1 ), v(τk ) ∈ R n

(4.45)

such that (u(τk ), u(τk + 1)) ∈ Ω, v(τk ) ∈ Λ(u(τk ), u(τk + 1)),

(4.46)

x  ≤ 8−k κ, v(τk ) − x  ≤ 8−k κ, u(τk + 1) −

(4.47)

δ(u(τk ), v(τk ), u(τk + 1)) ≤ 8−k κ.

(4.48)

For t = τk + 1, . . . , τk+1 − 1, set v(t) = y(t − τk − Sk+1 − 1),

(4.49)

and for t = τk + 2, . . . , τk+1 , set u(t) = x(t − τk − Sk+1 − 1) + (1 − d)t−τk −1 [u(τk + 1) − x(−Sk+1 )]. τ

τ

k+1 k+1 By Proposition 4.9, (4.45), (4.46), (4.49), and (4.50), ({u(t)}t=0 , {v(t)}t=0 program.

(4.50) −1

) is a

4.3 Four Lemmas

89

By (4.37)–(4.40), (4.47), and (4.50), x  = x(Tk+1 ) + (1 − d)Tk+1 +Sk+1 [u(τk + 1) − x(−Sk+1 )] − x u(τk+1 ) − x  + (1 − d)Tk+1 +Sk+1 (u(τk + 1) − x  +  x − x(−Sk+1 )) ≤ x(Tk+1 ) − ≤ min{δk+1 , 8−k−1 κ} + (1 − d)Tk+1 +Sk+1 (8−k κ + 8−k κ) ≤ 2 min{δk+1 , 8−k−1 κ} and x  ≤ 2 min{δk+1 , 8−k−1 κ}. u(τk+1 ) −

(4.51)

By (4.38), (4.39), (4.47), (4.49) and (4.50), v(τk + 1 + Sk+1 ) = y(0), u(τk + 1 + Sk+1 ) − x(0) = (1 − d)Sk+1 u(τk + 1) − x(−Sk+1 ) x  +  x − x(−Sk+1 )) ≤ (1 − d)Sk+1 (u(τk + 1) − ≤ (1 − d)Sk+1 (3 · 8−k κ) ≤ 8−1 κ.

(4.52)

By (4.52), x  = y(0) − x , v(τk + 1 + Sk+1 ) − x  ≥  x − x(0) − x(0) − u(τk + 1 + Sk+1 ) u(τk + 1 + Sk+1 ) − ≥  x − x(0) − 8−1 κ. Together with (4.37) this implies that x , u(τk + 1 + Sk+1 ) − x } max{v(τk + 1 + Sk+1 ) − ≥ max{y(0) − x  − 8−1 κ,  x − x(0) − 8−1 κ} ≥ κ/2. By (4.43) and (4.48), τk+1 −1



δ(u(t), v(t), u(t + 1))

t=0

=

τ k −1 t=0

δ(u(t), v(t), u(t + 1)) + δ(u(τk ), v(τk ), u(τk + 1))

(4.53)

90

4 Turnpike Results for the Robinson–Solow–Srinivasan Model τk+1 −1



+

δ(u(t), v(t), u(t + 1))

t=τk +1

≤ (1 + 4 p )

k−1 

8−j + 8−k κ +

j =0

τk+1 −1



δ(u(t), v(t), u(t + 1)).

(4.54)

t=τk +1

By (4.8), (4.39), (4.40), (4.45), (4.47), (4.49)–(4.51), and the equality δ(x(t), y(t), x(t + 1)) = 0 which holds for all integers t, τk+1 −1



τk+1 −1

δ(u(t), v(t), u(t + 1)) =

t=τk +1



[w(b x ) − w(bv(t)) + p (u(t) − u(t + 1))]

t=τk +1

τk+1 −1

=



[w(b x ) − w(by(t − τk − Sk+1 − 1))] + p (u(τk + 1) − u(τk+1 ))

t=τk +1 Tk+1 −1

=



[w(b x ) − w(by(t)) + p (x(t) − x(t + 1))]

t=−Sk+1

(u(τk + 1) − u(τk+1 )) −p (x(−Sk+1 ) − x(Tk+1 )) + p = − p (x(−Sk+1 ) − x(Tk+1 )) + p (u(τk + 1) − u(τk+1 )) ≤  p 2 · 8−k−1 κ +  p(2 · 8−k−1 κ + 8−k−1 κ) ≤  p 8−k κ. Together with (4.54) this implies that τk+1 −1

 t=0

δ(u(t), v(t), u(t + 1)) ≤ (1 + 4 p )

k 

8−j .

(4.55)

j =0

It follows from (4.51), (4.53), and (4.55) that in such a manner, we constructed by induction a program {(u(t), v(t)}∞ t=0 such that for each integer k ≥ 1, (4.42) and (4.43) hold and for each integer j ≥ 1 (4.44) holds. By Proposition 4.1(iii) and (4.43), {u(t), v(t)}∞ t=0 is a good program. In view of (ATP), u(t) → x , v(t) → x as t → ∞. This contradicts (4.44) which holds for any integer j ≥ 1. The contradiction we have reached proves Lemma 4.12.

4.3 Four Lemmas

91

Lemma 4.13 Let  > 0. Then there exists γ > 0 such that for each natural number T −1 T and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) − x , x(T ) − x ≤ γ , δ(x(t), y(t), x(t + 1)) ≤ γ , t = 0, 1, . . . , T − 1, the following inequality holds: x(t) − x , y(t) − x  ≤ , t = 0, . . . , T − 1.

(4.56)

Proof Let {γk }∞ k=1 be a strictly decreasing sequence of positive numbers such that γ1 < 8−1 , lim γk = 0. k→∞

(4.57)

Assume that the lemma does not hold. Then for each number k, there exist a natural Tk −1 k number Tk and a program ({x (k) (t)}Tt=0 , {y (k) (t)}t=0 ) such that x (k) (0) − x  ≤ γk , x (k) (Tk ) − x  ≤ γk , δ(x (k) (t), y (k) (t), x (k) (t + 1)) ≤ γk , t = 0, . . . , Tk − 1,

(4.58) (4.59)

and there is a natural number Sk ∈ [0, Tk − 1]

(4.60)

x , y (k) (Sk ) − x } > . max{x (k) (Sk ) −

(4.61)

such that

By (4.58) and Proposition 3.6, there exists M > 0 such that for all natural numbers k, x (k) (t) ≤ Me for all t = 0, . . . , Tk .

(4.62)

Extracting a subsequence and re-indexing, we may assume that one of the following cases holds: (1) (2) (3) (4)

sup{Tk : k = 1, 2, . . . } < ∞; sup{Sk : k = 1, 2, . . . } < ∞, Tk − Sk → ∞ as k → ∞; Sk → ∞ as k → ∞, sup{Tk − Sk : k = 1, 2, . . . } < ∞. Sk → ∞, Tk − Sk → ∞ as k → ∞.

92

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Assume that the case (1) holds. Extracting a subsequence and re-indexing, we may assume without loss of generality that Tk = T1 , Sk = S1 for all natural numbers k; that for each integer t ∈ [0, T1 ], there exists x(t) = limk→∞ x (k) (t); and that for each t ∈ [0, T1 − 1], there is y(t) = limk→∞ y (k) (t). It is not difficult to see that T1 −1 1 ({x(t)}Tt=0 , {y(t)}t=0 ) is a program. In view of (4.57)–(4.59) and (4.61), δ(x(t), y(t), x(t + 1)) = 0, t = 0, 1, . . . , T1 − 1,

(4.63)

x, x(T1 ) = x(0) = x , y(S1 ) − x } ≥ . max{x(S1 ) − Set x(t) = x for all integers t ∈ (−∞, 0) ∪ (T1 , ∞), y(t) = x for all integers t ∈ (−∞, 0) ∪ (T1 , ∞). Clearly {x(t), y(t)}∞ t=−∞ satisfies the assumptions of Lemma 4.12 which in its turn implies that x(t) = y(t) = x for all integers t. This contradicts (4.63). The contradiction we have reached proves that the case (1) does not hold. Assume that the case (2) holds. Extracting a subsequence and re-indexing, we may assume without loss of generality that Sk = S1 for all natural numbers k

(4.64)

and that for each integer t ≥ 0, there exist x(t) = lim x (k) (t), y(t) = lim y (k) (t). k→∞

k→∞

(4.65)

It is clear that {x(t), y(t)}∞ t=0 is a program. By (4.57)–(4.59), (4.61), (4.62), (4.64), and (4.65), x(t) ≤ Me for all integers t ≥ 0, δ(x(t), y(t), x(t + 1)) = 0 for all integers t ≥ 0, x(0) = x, x , y(S1 ) − x } ≥ . max{x(S1 ) −

(4.66)

4.3 Four Lemmas

93

Set x(t) = x for all integers t < 0, y(t) = x for all integers t < 0. Clearly {x(t), y(t)}∞ t=−∞ satisfies the assumptions of Lemma 4.12 which in its turn implies that x(t) = y(t) = x for all integers t. This contradicts (4.66). The contradiction we have reached proves that the case (2) does not hold. Assume that the case (3) holds. Extracting a subsequence and re-indexing, we may assume that Tk − Sk = T1 − S1 for all natural numbers k.

(4.67)

For each natural number k, set x˜ (k) (t) = x (k) (t + Tk ), t = −Tk . . . . , 0,

(4.68)

y˜ (k) (t) = y (k) (t + Tk ), t = −Tk , . . . , −1. Extracting a subsequence and re-indexing, we may assume without loss of generality that there exist x(t) = lim x˜ (k) (t) for each integer t ≤ 0 and k→∞

y(t) = lim y˜ (k) (t) for each integer t < 0. k→∞

(4.69)

By (4.57)–(4.59), (4.61), (4.62), and (4.67)–(4.69), x(t) ≤ Me for al integers t ≤ 0,

(4.70)

δ(x(t), y(t), x(t + 1)) = 0 for all integers t < 0, x(0) = x, x , y(S1 − T1 ) − x } ≥ . max{x(S1 − T1 ) − Set x(t) = x for all integers t > 0, y(t) = x for all integers t ≥ 0. Clearly {x(t), y(t)}∞ t=−∞ satisfies the assumptions of Lemma 4.12 which in its turn implies that x(t) = y(t) = x for all integers t. This contradicts (4.70). The contradiction we have reached proves that the case (3) does not hold.

94

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Assume that the case (4) holds. For each natural number k, define sequences Tk −Sk −1 k −Sk , {y˜ (k) (t)}t=−S by {x˜ (k) (t)}Tt=−S k k x˜ (k) (t) = x (k) (t + Sk ), t = −Sk , . . . . , Tk − Sk ,

(4.71)

y˜ (k) (t) = y (k) (t + Sk ), t = −Sk , . . . , Tk − Sk − 1. Extracting a subsequence and re-indexing, we may assume without loss of generality that for each integer t, there exists x∗ (t) = lim x˜ (k) (t), y∗ (t) = lim y˜ (k) (t). k→∞

k→∞

(4.72)

By (4.57), (4.59), (4.61), (4.62), (4.71), and (4.72) for each integer t, x∗ (t) ≤ Me,

(4.73)

δ(x∗ (t), y∗ (t), x∗ (t + 1)) = 0, x , y∗ (0) − x } ≥ . max{x∗ (0) − Clearly {x∗ (t), y∗ (t)}∞ t=−∞ satisfies the assumptions of Lemma 4.12 which in its turn implies that x∗ (t) = y∗ (t) = x for all integers t. This contradicts (4.73). The contradiction we have reached proves that the case (4) does not hold. Therefore in the all four cases, we have reached a contradiction which proves that Lemma 4.13 holds. Lemma 4.14 Let  > 0. Then there exists γ > 0 such that for each natural number T −1 T and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) − x  ≤ γ , x(T ) − x ≤ γ , T −1

w(by(t)) ≥ U (x(0), x(T ), 0, T ) − γ ,

(4.74) (4.75)

t=0

the following inequality holds:

T −1 t=0

δ(x(t), y(t), x(t + 1)) ≤ .

Proof Choose a positive number δ0 such that 2( p  + 1)δ0 < /8,

(4.76)

n |w(b x ) − w(bz)| < /8 for any z ∈ R+ satisfying z − x  ≤ δ0 .

By Proposition 3.13, there is γ ∈ (0, 1) such that γ < δ0 and the following property holds:

4.3 Four Lemmas

95

n satisfy x − (P2) If x, x  ∈ R+ x , x  − x  ≤ γ , then there exist x¯ ≥ x  , n y ∈ R such that

x  ≤ δ0 . (x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯ y − x  ≤ δ0 , x¯ − T −1 ) satisfy (4.74) Let T be a natural number, and let a program ({x(t)}Tt=0 , {y(t)}t=0 T −1 T , { y(t)} ˜ ). Set and (4.75). We construct a program ({x(t)} ˜ t=0 t=0

x(0) ˜ = x(0).

(4.77)

There are two cases: T = 1 and T ≥ 2. Consider the case with T = 1. By (4.74) and (P2), there exist x(1) ˜ and y(0) ˜ such that x(1) ˜ ≥ x(1), y(0) ˜ ∈ Rn,

(4.78)

˜ − x  ≤ δ0 . (x(0), ˜ x(1)) ˜ ∈ Ω, y(0) ˜ ∈ Λ(x(0), ˜ x(1)), ˜ y(0) ˜ − x  ≤ δ0 , x(1) By (4.8) and (4.74)–(4.78), δ(x(0), y(0), x(1)) = w(b x ) − w(by(0)) + p (x(0) − x(1)) ≤ w(b x ) − w(by(0)) ˜ +p (x(0) − x(1)) + γ ≤ w(b x ) − w(by(0)) ˜ +  p2γ + γ < δ0 (2 p  + 1) + /8 < . Thus in the case T = 1, the lemma holds. Assume that T > 1. By (4.74), (4.75), (4.77), and (P2), there exist x(1), ˜ y(0) ˜ such that x(1) ˜ ≥ x , y(0) ˜ ∈ Rn,

(4.79)

(x(0), ˜ x(1)) ˜ ∈ Ω, y(0) ˜ ∈ Λ(x(0), ˜ x(1)), ˜ y(0) ˜ − x  ≤ δ0 , x(1) ˜ − x  ≤ δ0 . If an integer t satisfies 1 < t ≤ T − 1, then we put ˜ − x ). y(t ˜ − 1) = x , x(t) ˜ = x + (1 − d)t−1 (x(1) T −1 T −2 ˜ It is clear that ({x(t)} ˜ t=0 , {y(t)} t=0 ) is a program. n such that By (4.74) and (P2), there exist z ≥ x(T ), y(T ˜ − 1) ∈ R+

( x , z) ∈ Ω, y(T ˜ − 1) ∈ Λ( x , z),

(4.80)

96

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

y(T ˜ − 1) − x  ≤ δ0 , z − x  ≤ δ0 .

(4.81)

˜ − x ]. x(T ˜ ) = z + (1 − d)T −1 [x(1)

(4.82)

Set

T −1 T , {y(t)} ˜ By (4.79)–(4.82), ({x(t)} ˜ t=0 ) is a program. By (4.77), (4.79), (4.81), t=0 and (4.82),

x(0) ˜ = x(0), x(T ˜ ) ≥ x(T ).

(4.83)

In view of (4.75) and (4.83), T −1

w(by(t)) ≥

t=0

T −1

w(by(t)) ˜ − γ.

(4.84)

t=0

By (4.8), (4.74), (4.76), (4.79)–(4.81), and (4.84), T −1

δ(x(t), y(t), x(t + 1)) =

t=0

≤

T −1

[−w(by(t)) + w(b x )] + p (x(0) − x(T ))

t=0 T −1

[w(b x ) − w(by(t))] ˜ + γ +  p (x(0) − x  + x(T ) − x )

t=0

≤ |w(b x ) − w(by(0))| ˜ + |w(b x ) − w(by(T ˜ − 1))| + 2 p γ ≤ /4 + 2 p δ0 < . Lemma 4.14 is proved.

4.4 Proof of Theorem 4.2 By Proposition 3.6, there is M1 > 0 such that for each natural number T and each T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying x(0) ≤ Me, the following inequality holds: x(t) ≤ M1 e for all integers t ∈ [0, T ].

(4.85)

n , each z ∈ R n By Proposition 4.7, there exists M2 > 0 such that for each z0 ∈ R+ 1 + satisfying az1 ≤ Γ d −1 , and each natural number T > k(Γ ),

x ) − M2 . U (z0 , z1 , 0, T ) ≥ T w(b

(4.86)

4.4 Proof of Theorem 4.2

97

By Lemma 4.13, there is 1 ∈ (0, ) such that for each natural number T and each T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x  ≤ 1 , x(0) − x  ≤ 1 , x(T ) −

(4.87)

δ(x(t), y(t), x(t + 1)) ≤ 1 , t = 0, . . . , T − 1,

(4.88)

the following inequality holds: x(t) − x , y(t) − x  ≤ , t − 0, . . . , T − 1.

(4.89)

By Lemma 4.11, there exists a natural number L0 such that for each program

 L0 −1 0 {x(t)}L t=0 , {y(t)}t=0 satisfying x(0) ≤ M1 e, L 0 −1

w(by(t)) ≥ L0 w(b y ) − M2 − M − 4 p nM1 ,

(4.90) (4.91)

t=0

there is an integer t ∈ [0, L0 − 1] such that y(t) − x , x(t) − x  ≤ 1 .

(4.92)

Choose a natural number p)]. L > 8L0 + 4k(Γ ) + (2L0 + 1)[2 + 1−1 (M + M2 + 2M1 n

(4.93)

Assume that an integer T > L, n , z0 ≤ Me, az1 ≤ Γ d −1 z 0 , z 1 ∈ R+

(4.94)

T −1 and that a program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfies

x(0) = z0 , x(T ) ≥ z1 ,

T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − M.

(4.95)

t=0

In view of (4.94) and (4.95), the relation (4.85) holds. By (4.86), (4.94), (4.95), and the inequality T > L > k(Γ ),

98

4 Turnpike Results for the Robinson–Solow–Srinivasan Model T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − M ≥ T w(b x ) − M2 − M.

(4.96)

t=0

It follows from (4.8), (4.85) and (4.96) that T −1

δ(x(t), y(t), x(t + 1)) =

t=0

T −1

[w(b x ) − w(by(t))] + p (x(0) − x(T ))

t=0

p. ≤ M + M2 + 2M1 n

(4.97)

It follows from (4.8), (4.85), and (4.97) that for each pair of integers S1 , S2 satisfying 0 ≤ S1 < S2 ≤ T , |

S 2 −1

[w(by(t)) − w(b x )]| ≤

t=S1

S 2 −1

δ(x(t), y(t), x(t + 1)) + 2nM1  p

t=S1

p. ≤ M + M2 + 4nM1 

(4.98)

It follows from the choice of L0 (see (4.85), (4.90)–(4.92)), (4.94), (4.95), and (4.98) that the following property holds: (P3) For each integer S satisfying 0 ≤ S ≤ T − L0 , there is an integer t ∈ [S, S + L0 − 1] such that y(t) − x , x(t) − x  ≤ 1 . Set t0 = 0. By (P3), (4.93), and the relation T > L, there is an integer t1 ∈ [L0 , 2L0 − 1] such that x , x(t1 ) − x  ≤ 1 . y(t1 ) − Assume that an integer j ≥ 1 and we defined integers ti , i = 0, . . . , j such that tj < T , ti+1 ∈ [ti + L0 , ti + 2L0 − 1] for i = 0, . . . , j − 1 and that x , x(ti ) − x  ≤ 1 y(ti ) −

(4.99)

for i = 1, . . . , j . If tj + 2L0 > T , then put tj +1 = T , and complete the construction of the sequence. If tj + 2L0 ≤ T , then by (P3) there is an integer tj +1 ∈ [tj + L0 , tj + 2L0 − 1] such that

4.4 Proof of Theorem 4.2

99

x(tj +1 ) − x , y(tj +1 ) − x  ≤ 1 . In such a way, we construct by induction a finite sequence of nonnegative integers {ti }ki=0 such that t0 = 0, tk = T , 1 ≤ ti+1 − ti ≤ 2L0 for i = 0, . . . , k − 1 and that for i = 1, . . . , k − 1, the inequalities (4.99) hold. Assume that an integer i satisfies 1 ≤ i ≤ k − 2 and that ti+1 −1



δ(x(t), y(t), x(t + 1)) ≤ 1 .

j =ti

Together with (4.99) and the choice of 1 (see (4.87)–(4.89)), this implies that x(t) − x , y(t) − x  ≤ , t = ti , . . . , ti+1 − 1.

(4.100)

By (4.100), {t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ⊂ {0, . . . , t1 } ∪ {tk−1 , . . . , T } ∪ {{ti , . . . , ti+1 } : ti+1 −1

i ∈ {1, . . . , k − 2} and



δ(x(t), y(t), x(t + 1)) > 1 }.

j =ti

This implies that Card{t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ 2(2L0 + 1) + (2L0 + 1)Card{i ∈ {1, . . . , k − 2} : ti+1 −1



δ(x(t), y(t), x(t + 1)) > 1 }.

j =ti

By (4.97), ti+1 −1

Card{i ∈ {1, . . . , k − 2} :



δ(x(t), y(t), x(t + 1)) > 1 }

t=ti

≤ 1−1

T −1

δ(x(t), y(t), x(t + 1)) ≤ 1−1 (M + M2 + 2M1 n p ).

t=0

Together with (4.93) and (4.101), this implies that

(4.101)

100

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Card{t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ (2L0 + 1)[2 + 1−1 (M + M2 + 2M1 n p)] < L. This completes the proof of Theorem 4.2.

4.5 Proof of Theorem 4.3 By Proposition 3.6, there is M1 > 0 such that for each natural number T and each T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying x(0) ≤ Me, the following inequality holds: x(t) ≤ M1 e for all integers t ∈ [0, T ].

(4.102)

n , each z ∈ R n By Proposition 4.7, there exists M2 > 0 such that for each z0 ∈ R+ 1 + −1 satisfying az1 ≤ Γ d , and each natural number T > k(Γ ),

x ) − M2 . U (z0 , z1 , 0, T ) ≥ T w(b

(4.103)

By Proposition 4.8, there exists M3 > 0 such that for each natural number T T −1 ) satisfying x(0) ≤ M1 e, the following and each program ({x(t)}Tt=0 , {y(t)}t=0 inequality holds: T −1

[w(by(t)) − w(b y )] ≤ M3 .

(4.104)

t=0

By Lemma 4.13, there is 1 > 0 such that for each natural number T and each T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x  ≤ 1 , x(0) − x  ≤ 1 , x(T ) −

(4.105)

δ(x(t), y(t), x(t + 1)) ≤ 1 , t = 0, . . . , T − 1,

(4.106)

the following inequality holds: x(t) − x , y(t) − x  ≤ , t = 0, . . . , T − 1.

(4.107)

By Lemma 4.11, there exists γ ∈ (0, min{1, , 1 })

(4.108)

4.5 Proof of Theorem 4.3

101

T −1 such that for each natural number T and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies

x(0) − x  ≤ γ , x(T ) − x ≤ γ , T −1

(4.109)

w(by(t)) ≥ U (x(0), x(T ), 0, T ) − γ ,

t=0

the following inequality holds: T −1

δ(x(t), y(t), x(t + 1)) ≤ 1 .

(4.110)

t=0

By Lemma 4.11, there exists a natural number L0 such that for each program

 L0 −1 0 {x(t)}L t=0 , {y(t)}t=0 satisfying x(0) ≤ M1 e,

L 0 −1

w(by(t)) ≥ L0 w(b y ) − M2 − M3 − 1,

(4.111)

t=0

there is an integer t ∈ [0, L0 − 1] such that y(t) − x , x(t) − x ≤ γ .

(4.112)

L = L0 + k(Γ ).

(4.113)

n , z0 ≤ Me, az1 ≤ Γ d −1 z 0 , z 1 ∈ R+

(4.114)

Put

Assume that an integer T > 2L,

T −1 and that a program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfies

x(0) = z0 , x(T ) ≥ z1 , T −1 t=0

w(by(t)) ≥ U (z0 , z1 , 0, T ) − γ .

(4.115) (4.116)

102

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

In view of (4.114) and (4.115), the relation (4.102) holds. By (4.108), (4.113), (4.114), and (4.116), T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − γ ≥ T w(b x ) − M2 − 1.

(4.117)

t=0

It follows from the choice of M3 (see (4.104)) and (4.102) that T −1

[w(by(t)) − w(b x )] ≤ M3 ,

t=L0

T −L 0 −1 

[w(by(t)) − w(b x )] ≤ M3 .

(4.118)

t=0

By (4.117) and (4.118), L 0 −1

[w(by(t)) − w(b x )] =

t=0

T −1

[w(by(t)) − w(b x )] −

T −1

[w(by(t)) − w(b x )]

t=L0

t=0

≥ −M2 − 1 − M3 , T −1

[w(by(t)) − w(b x )] =

t=T −L0

T −1

[w(by(t)) − w(b x )]

(4.119)

(4.120)

t=0

−

T −L 0 −1 

[w(by(t)) − w(b x )] ≥ −M2 − 1 − M3 .

t=0

It follows from (4.102), (4.119), (4.120), and the choice of L0 that there exist integers τ1 ∈ [0, L0 − 1], τ2 ∈ [T − L0 , T − 1] such that x , y(τi ) − x  ≤ γ , i = 1, 2. x(τi ) −

(4.121)

x  ≤ γ , then we put (If x(0) − x  ≤ γ , then we put τ1 = 0, and if x(T ) − τ2 = T ). By (4.116) and Proposition 4.10, τ 2 −1

w(by(t)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − γ .

(4.122)

t=τ1

In view of (4.121), (4.122), and the choice of γ (see (4.109), (4.110)) τ 2 −1 t=τ1

δ(x(t), y(t), x(t + 1)) ≤ 1 .

(4.123)

4.6 Extensions of Theorem 4.3

103

It follows from (4.108), (4.121), (4.123), and the choice of 1 (see (4.105)–(4.107)) that x(t) − x , y(t) − x  ≤ , t = τ1 , . . . , τ2 − 1. Theorem 4.3 is proved.

4.6 Extensions of Theorem 4.3 In the sequel we use the following helpful result. Lemma 4.15 Let a number M0 > max{(ai d)−1 : i = 1, . . . , n}, (x, x  ) ∈ Ω and x ≤ M0 e. Then x  ≤ M0 e. Proof Clearly, a(x  − (1 − d)x) ≤ 1, and for each i = 1, . . . , n, xi ≤ ai−1 + (1 − d)xi ≤ d(ai d)−1 + (1 − d)M0 ≤ dM0 + (1 − d)M0 = M0 . Lemma 4.15 is proved. In this chapter we prove the following three turnpike results obtained in [108]. Theorem 4.16 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M,  be positive numbers and Γ ∈ (0, 1). Then there exist a natural number n L and a positive number γ such that for each integer T > 2L, each z0 , z1 ∈ R+ −1 satisfying z0 ≤ Me and az1 ≤ Γ d −1 , and each program ({x(t)}Tt=0 , {y(t)}Tt=0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 , τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), 0, L) − γ for all τ ∈ {0, . . . , T − L}

t=τ

and T −1

w(by(t)) ≥ U (x(T − L), z1 , 0, L) − γ ,

t=T −L

there are integers τ1 , τ2 such that

104

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 4.17 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M,  be positive numbers. Then there exist a natural number L and a positive n satisfying z ≤ Me, number γ such that for each integer T > 2L, each z0 ∈ R+ 0 T −1 T and each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z0 , τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), 0, L) − γ for all τ ∈ {0, . . . , T − L}

t=τ

(4.124)

and T −1

w(by(t)) ≥ U (x(T − L), L) − γ ,

(4.125)

t=T −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 4.17 implies the following result. Theorem 4.18 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M,  be positive numbers. Then there exist a natural number L and a positive n satisfying z ≤ Me, number γ such that for each integer T > 2L, each z0 ∈ R+ 0 T −1 T and each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z0 ,

(4.126)

4.7 Proof of Theorem 4.16

105 T −1

w(by(t)) ≥ U (z0 , T ) − γ ,

(4.127)

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1.

(4.128) (4.129)

x  ≤ γ then τ1 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0 and if x(T ) −

4.7 Proof of Theorem 4.16 We may assume without loss of generality that M > max{(ai d)−1 : i = 1, . . . , n}.

(4.130)

Since a x = aσ (1 + daσ )−1 , we may assume without loss of generality that a x < Γ d −1 , x  ≤ } < Γ d −1 . sup{ay : y ∈ R n and y −

(4.131) (4.132)

Theorem 4.3 implies that there are an integer L0 ≥ 1 and a number γ > 0 such that the following property holds: n which (P4) for every natural number T > 2L0 , every pair of points z0 , z1 ∈ R+ T −1 T −1 satisfy z0 ≤ Me, az1 ≤ Γ d , and every program ({x(t)}t=0 , {y(t)}t=0 ) satisfying x(0) = z0 , x(T ) ≥ z1 , T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − γ ,

(4.133) (4.134)

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L0 ], τ2 ∈ [T − L0 , T ],

(4.135)

x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1,

(4.136)

x  ≤ γ , then τ2 = T . if x(0) − x  ≤ γ , then τ1 = 0 and if x(T ) −

(4.137)

106

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Fix L = 3L0 + 1.

(4.138)

n satisfy Assume that a pair of points z0 , z1 ∈ R+

z0 ≤ Me, az1 ≤ Γ d −1 ,

(4.139)

and let T > 2L be a natural number. Assume that a program

−1 {x(t)}Tt=0 , {y(t)}Tt=0



satisfies (4.133); that for every integer τ ∈ {0, . . . , T − L}, we have τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), 0, L) − γ ;

(4.140)

w(by(t)) ≥ U (x(T − L), z1 , 0, L) − γ .

(4.141)

t=τ

and that T −1 t=T −L

We show that there exist integers τ1 ∈ [0, L] and τ2 ∈ [T − l, T ] for which (4.136) and (4.137) are valid. Lemma 4.15, (4.130), and (4.139) imply that x(t) ≤ Me, t = 0, . . . , T .

(4.142)

T −1 Consider the program ({x(t)}Tt=T −L , {y(t)}t=T −L ). Property (P4), applied to this program, (4.133), (4.138), (4.139), (4.141), and (4.142) imply that

x(t) − x , y(t) − x  ≤ , t = T − L + L0 , . . . , T − L0 − 1,

(4.143)

if x(T ) − x  ≤ γ , then x(t) − x , y(t) − x  ≤ , t = T − L + L0 , . . . , T − 1.

(4.144)

If x(t) − x , y(t) − x  ≤ , t = 0, . . . , T − L + L0 , then the assertion of the theorem holds. Hence we may assume that there exists an integer t0 ∈ {0, . . . , T − L + L0 − 1} for which

4.7 Proof of Theorem 4.16

107

max{x(t0 ) − x , y(t0 ) − x } > .

(4.145)

We may assume without loss of generality that x(t) − x , y(t) − x ≤  for all integers t satisfying t0 < t < T − L + L0 .

(4.146)

We show that t0 ≤ 3L0 .

(4.147)

Assume the contrary and consider the program

 t0 +L−2L0 −1 0 +L−2L0 {x(t)}tt=t . , {y(t)} −2L −2L t=t 0 0 0 0 It follows from (4.138), (4.143), and (4.146) that x(t0 + L − 2L0 ) − x  ≤ .

(4.148)

By (4.132) and (4.148), we have ax(t0 + L − 2L0 ) ≤ Γ d −1 .

(4.149)

By (4.138), (4.140), (4.142), (4.144), (4.149), and property (P4) applied to the program

 t0 +L−2L0 −1 0 +L−2L0 {x(t)}tt=t , , {y(t)} −2L −2L t=t 0 0 0 0 we have x(t) − x , y(t) − x  ≤  for all t = t0 − 2L0 + L0 , . . . , t0 + L − 3L0 − 1. In view of the relation above, (4.138), (4.143), (4.144), and (4.146), we have x(t) − x , y(t) − x  ≤  for all integers t satisfying t0 − L0 ≤ t ≤ T − L0 − 1. This contradicts inequality (4.145). The contradiction we have reached proves that t0 ≤ 3L0 . Combined with relations (4.143), (4.144), and (4.146), this implies that x(t) − x , y(t) − x  ≤ , t = 3L0 + 1, . . . , T − L0 − 1,

(4.150)

108

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

if x(T ) − x  ≤ γ , then x(t) − x , y(t) − x  ≤ , t = 3L0 + 1, . . . , T − 1.

(4.151)

Assume that x(0) − x ≤ γ .

(4.152)

Relations (4.132), (4.138), 4.149, and (4.151) imply that x  ≤ , ax(L + L0 ) ≤ Γ d −1 , x(L + L0 ) − x(L) − x  ≤ , ax(L) ≤ Γ d −1 .

(4.153)

It follows from (4.138), (4.140), (4.142), (4.152), (4.153), and property (P4) applied L+L0 L0 +L−1 L−1 ) that to the programs ({x(t)}L t=0 , {y(t)}t=0 ) and ({x(t)}t=L0 , {y(t)}t=L0 x(t) − x , y(t) − x  ≤ , t = 0, . . . , L − L0 − 1, x(t) − x , y(t) − x  ≤ , t = 2L0 , . . . , L − 1. Together with relations (4.150) and (4.151), this completes the proof of Theorem 4.16.

4.8 Stability Results n → R 1 , define For every positive number M and every function φ : R+

φM = sup{|φ(z)| : z ∈ R n and 0 ≤ z ≤ Me}. n → R 1 , i = T , . . . , T − 1 be Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and wi : R+ 1 2 n n, bounded on bounded subsets of R+ functions. For every pair of points z0 , z1 ∈ R+ define

 T 2 −1 

T2 −1 wt (y(t)) : U {wt }t=T1 , z0 , z1 = sup t=T1



 T2 T2 −1 {x(t)}t=T1 , {y(t)}t=T1 is a program such that x(T1 ) = z0 , x(T2 ) ≥ z1 ,

4.8 Stability Results

109

 T 2 −1 

T2 −1 U {wt }t=T1 , z0 = sup wt (y(t)) : t=T1



T2 −1 2 {x(t)}Tt=T , {y(t)}t=T 1 1



 is a program such that x(T1 ) = z0 .

(Here we assume that supremum over empty set is −∞.) It is not difficult to see that the following result holds. n → R1, Lemma 4.19 Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and wi : R+ n i = T1 , . . . , T2 − 1 be bounded on bounded subsets of R+ upper semicontinuous functions. Then the following assertions hold. n , there exists a program ({x(t)}T2 , {y(t)}T2 −1 ) such 1. For every point z0 ∈ R+ t=T1 t=T1 that

x(T1 ) = z0 ,

T 2 −1

T2 −1 wt (y(t)) = U ({wt }t=T , z0 ). 1

t=T1 n such that U ({w }T2 −1 , z , z ) is finite, there 2. For every pair of points z0 , z1 ∈ R+ t t=T1 0 1 T2 −1 2 exists a program ({x(t)}Tt=T , {y(t)}t=T ) such that x(T1 ) = z0 , x(T2 ) ≥ z1 and 1 1 T 2 −1

T2 −1 wt (y(t)) = U ({wt }t=T , z0 , z1 ). 1

t=T1

The following stability results were obtained in [108]. Theorem 4.20 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, each n satisfying z ≤ Me and az ≤ Γ d −1 , each finite sequence of functions z 0 , z 1 ∈ R+ 0 1 n n wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·))M ≤ γ˜ −1 ) such for every integer i ∈ {0, . . . , T − 1} and every program ({x(t)}Tt=0 , {y(t)}Tt=0 that

x(0) = z0 , x(T ) ≥ z1 ,

110

4 Turnpike Results for the Robinson–Solow–Srinivasan Model τ +L−1 

τ +L−1 wt (y(t)) ≥ U ({wt }t=τ , x(τ ), x(τ + L)) − γ˜

t=τ

for every τ ∈ {0, . . . , T − L} and T −1 t=T −L

T −1 wt (y(t)) ≥ U ({wt }t=T −L , x(T − L), z1 ) − γ˜ ,

there exist integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ˜ , then τ2 = T . Moreover if |x(0) − x  ≤ γ˜ , then τ1 = 0, and if x(T ) − Proof Theorem 4.20 follows easily from Theorem 4.16. Namely, let a natural number L and γ > 0 be as guaranteed by Theorem 4.16. Set γ˜ = γ (4−1 (L + 1))−1 . Now it easy to see that the assertion of Theorem 4.20 holds. Theorem 4.21 Suppose that for each good program {u(t), v(t)}∞ t=0 , lim (u(t), v(t)) = ( x, x ).

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, each z0 ∈ n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , i = R+ 0 i + n and such that 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ˜ −1 ) which satisfies for each i ∈ {0, . . . , T −1} and each program ({x(t)}Tt=0 , {y(t)}Tt=0

x(0) = z0 , τ +L−1 

τ +L−1 wt (y(t)) ≥ U ({wt }t=τ , x(τ ), x(τ + L)) − γ˜ ,

t=τ

for each integer τ ∈ {0, . . . , T − L} and

4.8 Stability Results

111

T −1 t=T −L

T −1 wt (y(t)) ≥ U ({wt }t=T −L , x(T − L)) − γ˜

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Proof Let a natural number L and γ > 0 be as guaranteed by Theorem 4.17. Set γ˜ = 4−1 γ (L + 1)−1 . It is easy now to see that Theorem 4.21 holds. Theorem 4.20 implies the following result. Theorem 4.22 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and Γ ∈ (0, 1). Then there exist a natural number L, a positive number γ and λ > 1 such that for each integer T > n satisfying z ≤ Me and az ≤ Γ d −1 , each finite sequence of 2L, each z0 , z1 ∈ R+ 0 1 n functions wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets n and such that w − w(b(·)) of R+ i M ≤ γ for each i ∈ {0, . . . , T − 1}, each T −1 sequence {αi }i=0 ⊂ (0, 1] such that for each i, j ∈ {0, . . . , T − 1} satisfying |j − −1 ) i| ≤ L the inequality αi αj−1 ≤ λ holds and each program ({x(t)}Tt=0 , {y(t)}Tt=0 such that x(0) = z0 , x(T ) ≥ z1 , τ +L−1 

τ +L−1 αt wt (y(t)) ≥ U ({αt wt }t=τ , x(τ ), x(τ + L)) − γ ατ

t=τ

for each integer τ ∈ {0, . . . , T − L} and T −1 t=T −L

T −1 αt wt (y(t)) ≥ U ({αt wt }t=T −L , z1 ) − γ αT −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1.

112

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(0) − x  ≤ γ , then τ2 = T . Theorem 4.22, applied with z1 = 0, implies the following result. Theorem 4.23 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L, a positive number γ and λ > 1 such that for each integer T > 2L, each n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , z0 ∈ R+ 0 i + n and such that i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ −1 ⊂ (0, 1] such that for each for each i ∈ {0, . . . , T − 1}, each sequence {αi }Ti=0 i, j ∈ {0, . . . , T − 1} satisfying |j − i| ≤ L the inequality αi αj−1 ≤ λ holds and

T −1 ) such that each program ({x(t)}Tt=0 , {y(t)}t=0

x(0) = z0 , τ +L−1 

τ +L−1 αt wt (y(t)) ≥ U ({αt wt }t=τ , x(τ ), x(τ + L)) − γ ατ

t=τ

for each integer τ ∈ {0, . . . , T − L} and T −1 t=T −L

T −1 αt wt (y(t)) ≥ U ({αt wt }t=T −L , x(T − L)) − γ αT −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 4.23 implies the following result. Theorem 4.24 Suppose that for each good program {u(t), v(t)}∞ t=0 , lim (u(t), v(t)) = ( x, x ).

t→∞

4.8 Stability Results

113

Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L, a positive number γ and λ > 1 such that for each integer T > 2L, n satisfying z ≤ Me, each finite sequence of upper semicontinuous each z0 ∈ R+ 0 n → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets functions wi : R+ n and such that of R+ wi − w(b(·))M ≤ γ −1 ⊂ (0, 1] such that for each for each i ∈ {0, . . . , T − 1}, each sequence {αi }Ti=0 i, j ∈ {0, . . . , T − 1} satisfying |j − i| ≤ L the inequality αi αj−1 ≤ λ holds and

T −1 ) such that each program ({x(t)}Tt=0 , {y(t)}t=0

x(0) = z0 and T −1

T −1 αt wt (y(t)) = U ({αt wt }t=0 , x(0))

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − x  ≤ γ , then τ2 = T . Theorem 4.20 implies the following result. Theorem 4.25 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, each n satisfying z ≤ Me and az ≤ Γ d −1 , each finite sequence of functions z 0 , z 1 ∈ R+ 0 1 n n wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·))M ≤ γ˜ T −1 for each i ∈ {0, . . . , T − 1} and each program ({x(t)}Tt=0 , {y(t)}t=0 ) such that

x(0) = z0 , x(T ) ≥ z1 ,

114

4 Turnpike Results for the Robinson–Solow–Srinivasan Model T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 , z1 ) − γ˜

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ˜ , then τ2 = T . Moreover if x(0) − x  ≤ γ˜ , then τ1 = 0, and if x(T ) − Theorem 4.25, applied with z1 = 0, implies the following result. Theorem 4.26 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, each z0 ∈ n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , i = R+ 0 i + n and such that 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ˜ −1 ) which satisfies for each i ∈ {0, . . . , T −1} and each program ({x(t)}Tt=0 , {y(t)}Tt=0

x(0) = z0 , T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 ) − γ˜

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ˜ , then τ2 = T . Moreover if x(0) − x  ≤ γ˜ , then τ1 = 0, and if x(T ) − The following results were obtained in [114]. Theorem 4.27 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

4.8 Stability Results

115

Let M > max{(ai d)−1 : i = 1, . . . , n}, M0 > 0,  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ˜ such that for each n satisfying z ≤ Me and az ≤ Γ d −1 , each finite integer T > L, each z0 , z1 ∈ R+ 0 1 n sequence of functions wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on n and such that bounded subsets of R+ wi − w(b(·))M ≤ γ˜ T −1 ) such that for each i ∈ {0, . . . , T − 1} and each program ({x(t)}Tt=0 , {y(t)}t=0

x(0) = z0 , x(T ) ≥ z1 , T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 , z1 ) − M0

t=0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L. Theorem 4.27, applied with z1 = 0, implies the following result. Theorem 4.28 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n}, M0 > 0 and  > 0. Then there exist a natural number L and a positive number γ˜ such that for each integer T > L, each n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , z0 ∈ R+ 0 i + n and such that i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ˜ −1 ) which satisfies for each i ∈ {0, . . . , T −1} and each program ({x(t)}Tt=0 , {y(t)}Tt=0

x(0) = z0 , T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 ) − M0

t=0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L.

116

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

4.9 Proof of Theorem 4.27 Clearly, a x = aσ (1 + daσ )−1 < d −1 .

(4.154)

In view of (4.154), we may assume without any loss of generality that a x + a < Γ d −1 .

(4.155)

By Theorem 4.25, there exist a natural number L0 and a positive number γ˜0 < 1 such that the following property holds: n → R1, (Pi) for each integer T > 2L0 , each finite sequence of functions wi : R+ n i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·))M ≤ γ˜0 n satisfying z ≤ Me and az ≤ Γ d −1 for each i ∈ {0, . . . , T − 1}, each z0 , z1 ∈ R+ 0 1 T −1 and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies

x(0) = z0 , x(T ) ≥ z1 , T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 , z1 ) − γ˜0 ,

(4.156)

t=0

we have x(t) − x , y(t) − x  ≤  for all t = L0 , . . . , T − L0 − 1. By Theorem 4.2, there exists a natural number L1 ≥ 2 such that the following property holds: n satisfying z ≤ Me and az ≤ (Pii) for each integer T > L1 , each z0 , z1 ∈ R+ 0 1 T −1 Γ d −1 , and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − M0 − 4,

(4.157)

t=0

we have Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L1 . Choose a positive number

4.9 Proof of Theorem 4.27

117

γ˜ < γ˜0 (8L1 )−1

(4.158)

and a natural number L > 8L0 + 11L1 + (22L1 + 8L0 + 5)(2 + M0 γ˜0−1 ).

(4.159)

Assume that an integer T > L,

(4.160)

n z0 ≤ Me, az1 ≤ Γ d −1 , z 0 , z 1 ∈ R+

(4.161)

n → R 1 , i = 0, . . . , T − 1 are bounded on bounded subsets of R n functions wi : R+ + and satisfy

wi − w(b(·))M ≤ γ˜ for all i ∈ {0, . . . , T − 1},

(4.162)

T −1 ) satisfies and a program ({x(t)}Tt=0 , {y(t)}t=0

x(0) = z0 , x(T ) ≥ z1 ,

T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 , z1 ) − M0 .

(4.163) (4.164)

t=0

In order to complete the proof, it is sufficient to show that Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L.

(4.165)

By (4.161), (4.163), Lemma 4.15, and the inequality M > max{(ai d)−1 : i = 1, . . . , n}, we have x(t) ≤ Me, t = 0, . . . , T .

(4.166)

In view of (4.159) and (4.160), T − 4L1 > 0. By (4.164), T −1 t=T −4L1

T −1 wt (y(t)) ≥ U ({wt }t=T −4L1 , x(T − 4L1 ), z1 ) − M0 ,

(4.167)

−1 By Lemma 4.15 and (4.162), for each program ({u(t)}Tt=T −4L1 , {v(t)}Tt=T −4L1 ) satisfying u(T − 4L1 ) ≤ Me,

118

4 Turnpike Results for the Robinson–Solow–Srinivasan Model T −1

|

T −1

wt (v(t)) −

t=T −4L1

w(bv(t))| ≤ 4L1 γ˜ .

(4.168)

w(by(t))| ≤ 4L1 γ˜ ,

(4.169)

t=T −4L1

In view of (4.166) and (4.168), T −1

|

T −1

wt (y(t)) −

t=T −4L1

t=T −4L1

T −1 |U ({wt }t=T −4L1 , x(T − 4L1 ), z1 ) − U (x(T − 4L1 ), z1 , T − 4L1 , T )| ≤ 4L1 γ˜ .

(4.170) By (4.167)–(4.170), we have T −1

T −1

w(by(t)) ≥

t=T −4L1

wt (y(t)) − 4L1 γ˜

t=T −4L1

T −1 ≥ U ({wt }t=T −4L1 , x(T − 4L1 ), z1 ) − M0 − 4L1 γ˜

≥ U (x(T − 4L1 ), z1 , T − 4L1 , T ) − 4L1 γ˜ − M0 − 4L1 γ˜ .

(4.171)

By (4.158) and (4.171), we have T −1

w(by(t)) ≥ U (x(T − 4L1 ), z1 , T − 4L1 , T ) − M0 − 1.

(4.172)

t=T −4L1

By (4.161), (4.164), (4.166), (4.172), and property (Pii) (applied to the program 4L1 −1 1 ({x(t + T − 4L1 )}4L t=0 , {y(t + T − 4L1 )}t=0 )), we have x , |y(t) − x } > }) ≤ L1 . Card({t ∈ {T − 4L1 , . . . , T − 1} : max{x(t) − By the inequality above, there is an integer S such that T − 3L1 ≤ S ≤ T − L1 ,

(4.173)

max{x(S) − x , y(S) − x } ≤ . Relation (4.155) implies that n satisfying z − x  ≤ . az < Γ d −1 for each z ∈ R+

It follows from (4.164), (4.173), and Proposition 4.9 that

(4.174)

4.9 Proof of Theorem 4.27 S−1 

119

wt (y(t)) ≥ U ({wt }S−1 t=0 , z0 , x(S)) − M0 .

(4.175)

t=0

We show that the following property holds: (Piii) for each integer τ0 satisfying x  ≤ , 4L1 ≤ τ0 ≤ S, x(τ0 ) −

(4.176)

there is an integer τ1 such that τ0 − 3L1 ≤ τ1 ≤ τ0 − L1 , x , y(τ1 ) − x } ≤ . max{x(τ1 ) − Assume that an integer τ0 satisfies (4.176). By Lemma 4.15, (4.162) and (4.176), τ0 −1 0 , {v(t)}t=τ ) satisfying for each program ({u(t)}τt=τ 0 −4L1 0 −4L1 u(τ0 − 4L1 ) ≤ Me, we have |

τ 0 −1

wt (v(t)) −

t=τ0 −4L1

τ 0 −1

w(bv(t))| ≤ 4L1 γ˜ .

(4.177)

w(by(t))| ≤ 4L1 γ˜ ,

(4.178)

t=τ0 −4L1

In view of (4.166) and (4.177), |

τ 0 −1 t=τ0 −4L1

wt (y(t)) −

τ 0 −1 t=τ0 −4L1

τ0 −1 |U ({wt }t=τ , x(τ0 − 4L1 ), x(τ0 )) 0 −4L1

− U (x(τ0 − 4L1 ), x(τ0 ), τ0 − 4L1 , τ0 )| ≤ 4L1 γ˜ .

(4.179)

By (4.175), (4.176), and Proposition 4.9, τ 0 −1 t=τ0 −4L1

τ0 −1 wt (y(t)) ≥ U ({wt }t=τ , x(τ0 − 4L1 ), x(τ0 )) − M0 . 0 −4L1

It follows from (4.158) and (4.178)–(4.180) that

(4.180)

120

4 Turnpike Results for the Robinson–Solow–Srinivasan Model τ 0 −1

w(by(t)) ≥

t=τ0 −4L1

τ 0 −1

wt (y(t)) − 4L1 γ˜

t=τ0 −4L1

τ0 −1 ≥ U ({wt }t=τ , x(τ0 − 4L1 ), x(τ0 )) − M0 − 4L1 γ˜ 0 −4L1

≥ U (x(τ0 − 4L1 ), x(τ0 ), τ0 − 4L1 , τ0 ) − M0 − 8L1 γ˜ ≥ U (x(τ0 − 4L1 ), x(τ0 ), τ0 − 4L1 , τ0 ) − M0 − 1.

(4.181)

By (4.166), (4.174), (4.176), (4.181), and the property (Pii) (applied to the 4L1 −1 1 program ({x(t + τ0 − 4L1 )}4L t=0 , {y(t + τ0 − 4L1 )}t=0 )), we have x , y(t) − x } > }) ≤ L1 . Card({t ∈ {τ0 − 4L1 , . . . , τ0 − 1} : max{x(t) − By the inequality above, there is an integer τ1 such that x , y(τ1 ) − x } ≤ . τ0 − 3L1 ≤ τ1 ≤ τ0 − L1 , max{x(τ1 ) − Thus the property (Piii) holds. The property (Piii) and (4.173) imply that there are a natural number k and a strictly increasing sequence of nonnegative integers {Si }ki=1 such that x  ≤ , i = 1, . . . , k, Sk = S, S1 < 4L1 , x(Si ) −

(4.182)

L1 ≤ Si+1 − Si ≤ 3L1 for all integers i satisfying 1 ≤ i < k.

(4.183)

It follows from (4.176), (4.182), and (4.183) that the following property holds: (Piv) for each integer t ∈ [4L1 , S], there exist integers t˜1 , t˜2 ∈ [S1 , S] such that L1 ≤ t˜2 − t˜1 ≤ 3L1 , t ∈ [t˜1 , t˜2 ], x  ≤ , i = 1, 2. x(t˜i ) − Set t0 = 0. By induction we construct a finite strictly increasing sequence of integers q {ti }i=0 such that tq = S;

(4.184)

(Pv) for each integer i satisfying 0 ≤ i < q − 1, ti+1 −1



t

−1

i+1 wt (y(t)) < U ({wt }t=t i

, x(ti ), x(ti+1 )) − γ˜0 ;

t=ti

(Pvi) if an integer i satisfies 0 ≤ i < q, ti+1 − ti ≥ 2 and (4.185), then

(4.185)

4.9 Proof of Theorem 4.27

121

ti+1 −2



t

−2

i+1 wt (y(t)) ≥ U ({wt }t=t i

, x(ti ), x(ti+1 − 1)) − γ˜0 .

(4.186)

t=ti

Assume that an integer p ≥ 0 and we have already defined a strictly increasing p sequence of integers {ti }i=0 such that tp < S and that for each integer i satisfying 0 ≤ i < p, (4.185) and (4.186) hold. (Clearly, for p = 0 our assumption holds.) We define tp+1 . There are two cases: S−1 

wt (y(t)) ≥ U ({wt }S−1 t=tp , x(tp ), x(S)) − γ˜0 ;

(4.187)

wt (y(t)) < U ({wt }S−1 t=tp , x(tp ), x(S)) − γ˜0 .

(4.188)

t=tp S−1  t=tp

If (4.187) holds, then we set q = p + 1, tq = S, and in this case, the construction is completed, and (Pv) and (Pvi) hold. Assume that (4.188) holds. Set tp+1 = min{τ ∈ {tp + 1, . . . , S} : τ −1 

−1 wt (y(t)) < U ({wt }τt=t , x(tp ), x(τ )) − γ˜0 }. p

t=tp

Clearly, tp+1 is well-defined and tp+1 > tp . If tp+1 = S, then we set q = p + 1, the construction is completed, and it is not difficult to see that (Pv) and (Pvi) hold. If tp+1 < S, then it is easy to see that the assumption made for p is also true for p + 1. Clearly our construction is completed after a final number of steps, and let tq be its last element. It follows from the construction that tq = S and that the properties (Pv) and (Pvi) hold. By (4.148), Proposition 4.9, and property (Pv), M0 ≥ U ({wt }S−1 t=0 , z0 , x(S)) −

S−1 

wt (y(t))

t=0

≥



t

−1

i+1 {U ({wt }t=t i

ti+1 −1

, x(ti ), x(ti+1 )) −



wt (y(t)) :

t=ti

i is an integer, 0 ≤ i < q − 1} > (q − 1)γ˜0 ,

122

4 Turnpike Results for the Robinson–Solow–Srinivasan Model

q < M0 γ˜0−1 + 1.

(4.189)

Set A = {i ∈ {0, . . . , q − 1} : ti+1 − ti ≥ 12L1 + 6L0 + 1}.

(4.190)

Let i ∈ A.

(4.191)

(ti+1 − 1) − ti ≥ 12L1 + 6L0 .

(4.192)

By (4.190) and (4.191),

By the properties (Pv) and (Pvi) and (4.192), relation (4.186) holds. By (4.192) and (Piv), there exist integers t˜i , t˜i+1 such that 4L1 + ti ≤ t˜i ≤ ti + 7L1 , ti+1 − 1 ≥ t˜i+1 ≥ ti+1 − 3L1 − 1, x  ≤ , x(t˜i+1 ) − x  ≤ . x(t˜i ) −

(4.193) (4.194)

In view of (4.192) and (4.193), t˜i+1 − t˜i ≥ ti+1 − ti − 10L1 − 1 ≥ 2L1 + 6L0 .

(4.195)

It follows from (4.186), (4.193) and Proposition 4.9 that t˜i+1 −1

 t=t˜i

t˜

i+1 wt (y(t)) ≥ U ({wt }t= t˜

−1

i

, x(t˜i ), x(t˜i+1 )) − γ˜0 .

(4.196)

By (4.158), (4.162), (4.166), (4.174), (4.194)–(4.196), and (Pi) (applied to the t˜i+1 −t˜i t˜i+1 −t˜i −1 program ({x(t + t˜i )}t=0 , {y(t + t˜i )}t=0 )), x(t) − x , y(t) − x  ≤ , t = t˜i + L0 , . . . , t˜i+1 − L0 − 1. Together with (4.193) this implies that x(t)− x , y(t)− x  ≤ , t = ti +7L1 +L0 , . . . , ti+1 −3L1 −L0 −2 for all i ∈ A. By (4.184), (4.190), and (4.197), {t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }

(4.197)

4.9 Proof of Theorem 4.27

123

⊂ {S, . . . , T − 1} ∪ (∪{{ti , . . . , ti+1 } : i ∈ {0, . . . , q − 1} \ A}) ∪ (∪{{ti , . . . , ti + 7L1 + L0 } ∪ {ti+1 − 3L1 − L0 − 1, ti+1 } : i ∈ A}).

(4.198)

By (4.159), (4.173), (4.189), (4.190), and (4.198), Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ 3L1 + q(12L1 + 6L0 + 2) + q(7L1 + L0 + 1) + q(3L1 + L0 + 2) ≤ 3L1 + q(22L1 + 8L0 + 5) ≤ 3L1 + (22L1 + 8L0 + 5)(1 + M0 γ˜0−1 + 1) < L. This completes the proof of Theorem 4.27.

Chapter 5

The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

We study infinite horizon optimal control problems related to the Robinson–Solow– Srinivasan model with a nonconcave utility function. In particular, we establish the existence of good programs and optimal programs using different optimality criterions.

5.1 Good Programs 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}. R+

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi ,

i=1

and let x >> y, x > y, x ≥ y have their usual meaning. n Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n . Let a = (a1 , . . . , an ) >> 0, b = (b1 , . . . , bn ) >> 0, and d ∈ (0, 1]. A sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0, n n × R+ , x(t + 1) ≥ (1 − d)x(t), (x(t), y(t)) ∈ R+

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_5

125

126

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(5.1)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences

 T2 −1 2 {x(t)}Tt=T , {y(t)} t=T 1 1 n and for each integer t satisfying T ≤ t < T , is called a program if x(T2 ) ∈ R+ 1 2 relations (5.1) hold. Let w : [0, ∞) → [0, ∞) be a continuous strictly increasing function which represents the preferences of the planner. n and every natural number T , set For every point x0 ∈ R+

U (x0 , T ) = sup

T −1 

w(by(t)) :



−1 {x(t)}Tt=0 , {y(t)}Tt=0

t=0

is a program such that x(0) = x0 }.

(5.2)

In the sequel we assume that supremum of empty set is −∞. n , and let T be a natural number. Set Let x0 , x˜0 ∈ R+ U (x0 , x˜0 , T ) = sup

T −1 

w(by(t)) :



−1 {x(t)}Tt=0 , {y(t)}Tt=0

t=0

is a program such that x(0) = x0 , x(T ) ≥ x˜0 }.

(5.3)

The next proposition follows immediately from the continuity of w. n and every integer T > 0, there exists a Proposition 5.1 For every point x0 ∈ R+ T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) such that x(0) = x0 and T −1

w(by(t)) = U (x0 , T ).

t=0

Set n n × R+ : x  ≥ (1 − d)x and a(x  − (1 − d)x) ≤ 1}. Ω = {(x, x  ) ∈ R+ n given by We have a correspondence Λ : Ω → R+

  n Λ x, x  = {y ∈ R+ : 0 ≤ y ≤ x and

(5.4)

5.1 Good Programs

127

ey ≤ 1 − a(x  − (1 − d)x)}, (x, x  ) ∈ Ω.

(5.5)

Let M0 be a positive number, and let T ≥ 1 be an integer. Set (M0 , T ) = sup{ U

T −1

w(by(t)) :

t=0



−1 is a program such that x(0) ≤ M0 e}. {x(t)}Tt=0 , {y(t)}Tt=0

(5.6)

(M0 , T ) is finite. The next proposition follows immediately from It is clear that U the continuity of w. Proposition 5.2 For every positive number M0 and every integer T ≥ 1, T −1 there exists a program ({x(t)}Tt=0 , {y(t)}t=0 ) such that x(0) ≤ M0 e and T −1 t=0 w(by(t)) = U (M0 , T ). In the sequel we use the following simple auxiliary result. Lemma 5.3 Let a number M0 > max{(ai d)−1 : i = 1, . . . , n}, (x, x  ) ∈ Ω, and let x ≤ M0 e. Then x  ≤ M0 e. For the proof see Lemma 4.15. In this chapter we show the existence of a positive constant μ such that the following properties hold: T −1 (a) For each program {x(t), y(t)}∞ t=0 either the sequence { t=0 [w(by(t)) − μ]}∞ T =1 is bounded or lim

T −1

T →∞

[w(by(t)) − μ] = −∞;

t=0

n there exists a program {x(t), y(t)}∞ such that x(0) = x (b) for each x0 ∈ R+ 0 t=0 T −1 and that the sequence { t=0 [w(by(t)) − μ]}∞ is bounded. T =1

For any (x, x  ) ∈ Ω, define     u x, x  = max w(by) : y ∈ Λ(x, x  ) . In this section we state several results obtained in [94]. Our first result allows us to define the constant μ. Theorem 5.4 Let M1 , M2 > max{(dai )−1 : i = 1, . . . , n}. Then there exist finite limits (Mi , p)/p, i = 1, 2 lim U

p→∞

128

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

and (M1 , p)/p = lim U (M2 , p)/p. lim U

p→∞

p→∞

Theorem 5.4 will be proved in Section 5.4. Define (M, p)/p μ = lim U p→∞

(5.7)

where M > max{(dai )−1 : i = 1, . . . , n}. By Theorem 5.4, μ is well-defined and does not depend on M. The next theorem will be also proved in Section 5.4. Theorem 5.5 Let M0 > max{(dai )−1 : i = 1, . . . , n}. Then there exists M > 0 such that (M0 , p) − pμ| ≤ M for all integers p ≥ 1. |U Corollary 5.6 Let M0 > max{(dai )−1 : i = 1, . . . , n}. Then there exists a positive number M such that for each program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e and each integer T ≥ 1, T −1

[w(by(t)) − μ] ≤ M.

t=0

Note that Corollary 5.6 easily follows from Theorem 5.5. The next result will be proved in Section 5.5. Proposition 5.7 Let {x(t), y(t)}∞ t=0 be a program. Then either the sequence T −1 ∞ { t=0 [w(by(t)) − μ]}T =1 is bounded or lim

T −1

T →∞

[w(by(t)) − μ] = −∞.

t=0

1 A program {x(t), y(t)}∞ t=0 is called good if there exists M ∈ R such that T 

(w(y(t)) − μ) ≥ M for all T ≥ 0.

t=0

A program is called bad if lim

T →∞

T  (w(y(t)) − μ) = −∞. t=0

5.1 Good Programs

129

By Proposition 5.7 any program that is not good is bad. Set x(t) = (2nd max{ai : i = 1, . . . , n})−1 e, y(t) = min{(2n)−1 , (2nd max{ai : i = 1, . . . , n})−1 }e for all integers t ≥ 0. It is easy to see that {x(t), y(t)}∞ t=0 is a program. By Corollary 5.6, μ ≥ lim T

−1

T →∞

T −1

w(by(t)) > w(0).

t=0

Thus we have shown that μ > w(0).

(5.8)

The following theorem will be proved in Section 5.4. Theorem 5.8 Let M0 > max{(dai )−1 : i = 1, . . . , n}. Then there exists a positive n which satisfies x ≤ M e, there exists a number M such that for every x0 ∈ R+ 0 0 ∞ program {x(t), y(t)}t=0 such that x(0) = x0 ; that for every integer T1 ≥ 0 and every natural number T2 > T1 ,    T2 −1     w(by(t)) − μ(T − T ) 2 1  ≤ M;    t=T1 and that for every natural number T , T −1

w(by(t)) = U (x(0), x(T ), T ).

(5.9)

t=0

Theorem 5.8 establishes that for every initial state x0 ≥ 0, there exists a good program {x(t), y(t)}∞ t=0 such that x(0) = x0 . In addition this program satisfies (5.9) for every natural number T . This leads us to the following definition. A program {x(t), y(t)}∞ t=0 is called weakly maximal if equality (5.9) holds for every natural number T . Our final result which will be proved in Section 5.6 establishes a relation between good programs and weakly maximal programs. Theorem 5.9 Let {x(t), y(t)}∞ t=0 be a weakly maximal program such that lim supt→∞ by(t) > 0. Then the program {x(t), y(t)}∞ t=0 is good.

130

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

5.2 Auxiliary Results Lemma 5.10 Let δ > 0, and let M0 > max{(ai d)−1 : i = 1, . . . , n}.

(5.10)

Then there exists an integer T0 ≥ 4 such that for every natural number τ ≥ T0 , −1 every program ({x(t)}τt=0 , {y(t)}τt=0 ) which satisfies x(0) ≤ M0 e, by(τ − 1) ≥ δ,

(5.11)

n which satisfies and every x˜0 ∈ R+

x˜0 ≤ M0 e,

(5.12)

τ −1 τ , {y(t)} there exists a program ({x(t)} ˜ t=0 ˜ t=0 ) such that

˜ ) ≥ x(τ ). x(0) ˜ = x˜0 , x(τ

(5.13)

Proof Choose a natural number T0 ≥ 4 such that 2(1 − d)T0 M0 ≤ δn−1 (max{bi : i = 1, . . . , n})−1 (max{ai : i = 1, . . . , n})−1 . (5.14) −1 ({x(t)}τt=0 , {y(t)}τt=0 )

Assume that an integer τ ≥ T0 , that a program n satisfies (5.12). Define fies (5.11), and that a point x˜0 ∈ R+ x(0) ˜ = x˜0

satis-

(5.15)

and for t = 0, . . . , τ − 1, set y(t) ˜ = 0, x(t ˜ + 1) = (1 − d)x(t) ˜ + [x(t + 1) − (1 − d)x(t)] 

+ n−1 [1 − a(x(t + 1) − (1 − d)x(t))] a1−1 , . . . , an−1 .

(5.16)

τ −1 τ , {y(t)} It is easy to see that ({x(t)} ˜ t=0 ˜ t=0 ) is a program. It follows from (5.16) that for all integers t = 0, . . . , τ − 1, we have

x(t ˜ + 1) − x(t + 1) ≥ (1 − d)(x(t) ˜ − x(t)). Together with (5.11), (5.12), and (5.15), the inequality above implies that ˜ − x(0)) ≥ (1 − d)τ −1 (−M0 e). x(τ ˜ − 1) − x(τ − 1) ≥ (1 − d)τ −1 (x(0)

5.2 Auxiliary Results

131

By this relation, (5.1), (5.10), (5.11), (5.14), and (5.16), we have x(τ ˜ ) − x(τ ) = (1 − d)(x(τ ˜ − 1) − x(τ − 1))



+ n−1 [1 − a(x(τ ) − (1 − d)x(τ − 1))] a1−1 , . . . , an−1 

≥ (1 − d)τ (−M0 e) + n−1 ey(τ − 1) a1−1 , . . . , an−1

≥ −(1 − d)T0 M0 e + n−1 (by(τ − 1))(max{bi : i = 1, . . . , n})−1 × (a1−1 , . . . , an−1 ) ≥ −(1 − d)T0 M0 e + δn−1 (max{bi : i = 1, . . . , n})−1 × (a1−1 , . . . , an−1 ) ≥ 0. This completes the proof of Lemma 5.10. Choose a positive number γ such that n   γ < 2−1 , 4γ < (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . , an−1 bi .

(5.17)

i=1

Lemma 5.11 Let M1 > 0, and let a number M0 satisfy (5.10). Then there exists a pair of integers L1 , L2 ≥ 4 such that for every natural number T ≥ L1 + L2 , every T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying x(0) ≤ M0 e,

T −1

w(by(t)) ≥ U (x(0), T ) − M1 ,

(5.18)

t=0

and every integer τ ∈ [L1 , T − L2 ], the following inequality holds: max{by(t) : t = τ, . . . , τ + L2 − 1} ≥ γ .

(5.19)

Proof Lemma 5.10 implies that there exists an integer L1 ≥ 4 such that the following property holds: (P1) If an integer S ≥ L1 , a program ({u(t)}St=0 , {v(t)}S−1 t=0 ) satisfies u(0) ≤ M0 e, bv(S − 1) ≥ γ ,

(5.20)

n satisfies u ˜ 0 ≤ M0 e, then there exists a program and if a point u˜ 0 ∈ R+ S−1 S , {v(t)} ˜ ) such that ({u(t)} ˜ t=0 t=0

u(0) ˜ = u˜ 0 , u(S) ˜ ≥ u(S).

(5.21)

132

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

Fix an integer L2 ≥ 1 for which L2 ≥ 4(M1 + w(M0 eb) + 2L1 w(1) + 1)    

× w (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . , an−1 eb  −w(γ ))−1 + 1 + 8(L1 + 1).

(5.22)

−1 Assume that an integer T ≥ L1 + L2 , a program ({x(t)}Tt=0 , {y(t)}Tt=0 ) satisfies (5.18), and an integer τ satisfies

L 1 ≤ τ ≤ T − L2 .

(5.23)

We show that (5.19) is valid. Let us assume the contrary. Then by(t) < γ , t = τ, . . . , τ + L2 − 1.

(5.24)

Clearly, one of the following cases holds: by(t) < γ , t = τ, . . . , T − 1;

(5.25)

max{by(t) : t = τ, . . . , T − 1} ≥ γ .

(5.26)

Now we define a natural number τ0 as follows. If (5.25) holds, then we put τ0 = T . If (5.26) is valid, then in view of (5.23), (5.24), and (5.26), there exists an integer τ0 ≥ 1 for which τ + L2 ≤ τ0 ≤ T − 1,

(5.27)

by(τ0 ) ≥ γ ,

(5.28)

by(t) < γ , t = τ, . . . , τ0 − 1.

(5.29)

It is easy to see that (5.29) is valid in both cases and that in both cases τ0 − τ ≥ L2 .

(5.30)

Assume that (5.25) holds. Recall that in this case τ0 = T . Set x(t) ˜ = x(t), t = 0, . . . , τ, y(t) ˜ = y(t), t = 0, . . . , τ − 1, y(τ ˜ ) = 0,

(5.31)

5.2 Auxiliary Results

133

  y(t) ˜ = (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . , an−1 e, t = τ +1, . . . , T −1,

(5.32)

 x(t ˜ + 1) = (1 − d)x(t) ˜ + (2n)−1 a1−1 , . . . , ai−1 , . . . , an−1 , t = τ, . . . , T − 1. t−1 T , {y(t)} It is easy to see that ({x(t)} ˜ t=0 ˜ t=0 ) is a program. By (5.17), (5.18), (5.23), (5.25), (5.31), (5.32), and the inequality L2 ≥ 4,

M1 ≥ U (x(0), T ) −

T −1

w(by(t)) ≥

t=0

=

T −1

w(by(t)) ˜ −

t=τ

T −1



T −1 t=0

w(by(t)) ˜ −

T −1

w(by(t))

t=0

w(by(t))

t=τ −1

≥ (T − 1 − τ ) w((2n)

 min

1, a1−1 , . . . , ai−1 , . . . an−1

n 

 bi )

i=1

− (T − τ )w(γ )





≥ (T − 1 − τ ) w (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . an−1

n 

 bi

 − w(γ )

i=1

− w(γ )

 ≥ (L2 /2) w (2n)

−1

 min

1, a1−1 , . . . , ai−1 , . . . an−1

n 

 bi

 − w(γ )

i=1

− w(1). The relation above implies that

 L2 ≤ 2(M1+w(1)) w (2n)

−1

 −1 n   −1 −1 −1 min 1, a1 , . . . , ai , . . . an bi −w(γ ) . i=1

This inequality contradicts (5.22). The contradiction we have reached proves that (5.25) does not hold. Therefore (5.26) holds, and the integer τ0 satisfies (5.27)–(5.29). Set x(t) ˜ = x(t), t = 0, . . . , τ, y(t) ˜ = y(t), t = 0, . . . , τ − 1, y(τ ˜ ) = 0,

(5.33)

  y(t) ˜ = (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . , an−1 e, t = τ + 1, . . . , τ0 − L1 − 1, (5.34) x(t ˜ + 1) = (1 − d)x(t) ˜ + (2n)−1 (a1−1 , . . . , ai−1 , . . . , an−1 ), t = τ, . . . , τ0 − L1 − 1.

134

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

τ0 −L1 τ0 −L1 −1 It is not difficult to see that ({x(t)} ˜ ˜ ) is a program. Lemma 5.3, t=0 , {y(t)} t=0 (5.10), (5.18), and (5.33) imply that

˜ 0 − L1 ) ≤ M0 e, x(τ0 − L1 ), x(τ

(5.35)

0 ≤ y(τ0 ) ≤ x(τ0 ) ≤ M0 e. τ0 +1 0 Let us consider the program ({(x(t)}t=τ , {y(t)}τt=τ ). Property (P1), (5.28), 0 −L1 0 −L1 and (5.35) imply that there exists a program

 τ0 +1 τ0 {x(t)} ˜ ˜ t=τ0 −L1 , {y(t)} t=τ0 −L1 which satisfies x(τ ˜ 0 + 1) ≥ x(τ0 + 1).

(5.36)

τ0 +1 τ0 ˜ It is easy to see that ({x(t)} ˜ t=0 ) is a program. If T > τ0 + 1, then we t=0 , {y(t)} define

y(t) ˜ = y(t), t = τ0 + 1, . . . , T − 1, x(t ˜ + 1) = (1 − d)x(t) ˜ + x(t + 1) − (1 − d)x(t), t = τ0 + 1, . . . , T − 1.

(5.37)

T −1 T , {y(t)} By (5.37) and (5.38), ({x(t)} ˜ t=0 ) is a program. Lemma 5.4, (5.10), t=0 ˜ (5.17), (5.18), (5.22), (5.27), (5.29), (5.33)–(5.35), (5.37), and (5.39) imply that

M1 ≥ U (x(0), T ) −

T −1

w(by(t)) ≥

t=0

=

T −1

w(by(t)) ˜ −

t=τ

≥

τ0 −L 1 −1

T −1

T −1

w(by(t)) ˜ −

t=0

w(by(t)) =

[w(by(t)) ˜ − w(by(t))] −

t=τ

w(by(t))

t=0

τ0 

t=τ

T −1

w(by(t)) ˜ −

t=τ τ0 

τ0 

w(by(t))

t=τ

w(by(t))

t=τ0 −L1

   

 ≥ (τ0 −L1 − τ − 1) w (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . an−1 eb − w(γ ) − w(γ ) − L1 w(γ ) − w(by(τ0 ))    

 ≥ (L2 − L1 − 1) w (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . an−1 be − w(γ ) − (L1 + 1)w(γ ) − w(M0 eb)

5.2 Auxiliary Results

135

   

 ≥ (L2 /2) w (2n)−1 min 1, a1−1 , . . . , ai−1 , . . . an−1 eb − w(γ ) − 2L1 w(1) − w(M0 eb). By the relation above, we have L2 ≤ 2(M1 + 2L1 w(1) +w(M0 eb))[w((2n)−1 min{1, a1−1 , . . . , ai−1 , . . . an−1 }eb) − w(γ )]−1 . This inequality contradicts (5.22). The contradiction we have reached proves (5.19). This completes the proof of Lemma 5.11. Lemma 5.12 Let M1 be a positive number, and let a number M0 satisfy (5.10). Then there exist a pair of integers L¯ 1 , L¯ 2 ≥ 1 and a positive number M2 such that T −1 ) satisfying for every integer T ≥ L¯ 1 + L¯ 2 and every program {x(t)}Tt=0 , {y(t)}t=0 x(0) ≤ M0 e,

T −1

w(y(t)) ≥ U (x(0), T ) − M1 ,

(5.38)

t=0

the following assertion holds: If integers T1 , T2 ∈ [0, T − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1

w(by(t)) ≥ U (x(T1 ), T2 − T1 ) − M2 .

(5.39)

t=T1

Proof Let a pair of integers L1 , L2 ≥ 4 be as guaranteed by Lemma 5.11. Lemma 5.10 implies that there is an integers L3 ≥ 4 for which that the following property holds: (P2) If a natural number S ≥ L3 , if a program ({u(t)}St=0 , {v(t)}S−1 t=0 ) satisfies u(0) ≤ M0 e, bv(S − 1) ≥ γ , n satisfies u ˜ 0 ≤ M0 e, then there is a program and if a point u˜ 0 ∈ R+ S−1 S ({u(t)} ˜ , { v(t)} ˜ ) satisfying t=0 t=0

˜ ≥ u(S). u(0) ˜ = u˜ 0 , u(S) Fix a pair of integers L¯ 1 , L¯ 2 ≥ 1 and a number M2 > 0 which satisfy L¯ 1 ≥ L1 , L¯ 2 > 2(L2 + L3 + 1),

(5.40)

136

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

 M2 > M1 + (L2 + L3 )w M0

n 

 bi .

(5.41)

i=1 T −1 ) Assume that a natural number T ≥ L¯ 1 + L¯ 2 , that a program ({x(t)}Tt=0 , {y(t)}t=0 satisfies (5.38), and that a pair of integers T1 , T2 satisfies

T1 , T2 ∈ [0, T − L¯ 2 ], L¯ 1 ≤ T2 − T1 .

(5.42)

We claim that inequality (5.39) holds. Lemma 5.3, (5.10), and (5.38) imply that x(t) ≤ M0 e for all integers t = 0, . . . , T .

(5.43)

Proposition 5.1 implies that there exists a program  x

(1)

(t)

T2 t=T1

 , y

(1)

(t)

T2 −1  t=T1

which satisfies x (1) (T1 ) = x(T1 ),

T 2 −1

 w by (1) (t) = U (x(T1 ), T2 − T1 ).

(5.44)

t=T1

Lemma 5.3, (5.10), (5.43), and (5.44) imply that x (1) (t) ≤ M0 e, t = T1 , . . . , T2 .

(5.45)

It follows from (5.40) and (5.42) that T1 + L1 ≤ T1 + L3 + L¯ 1 ≤ L3 + T2 ≤ T − L¯ 2 + L3 ≤ T − 2L2 − L3 .

(5.46)

In view of the choice of L1 , L2 , Lemma 5.11, (5.38), (5.40), and (5.46), max{by(t) : t = T2 + L3 , . . . , T2 + L3 + L2 − 1} ≥ γ . Hence there is an integer τ ∈ [T2 + L3 , . . . , T2 + L3 + L2 − 1]

(5.47)

by(τ ) ≥ γ .

(5.48)

for which

Define

5.2 Auxiliary Results

137

x(t) ˜ = x(t), t = 0, . . . , T1 , y(t) ˜ = y(t) for each integer t such that 0 ≤ t ≤ T1 − 1, x(t) ˜ = x (1) (t), t = T1 + 1, . . . , T2 , y(t) ˜ = y (1) (t), t = T1 , . . . , T2 − 1. T2 T2 −1 ˜ It is not difficult to see that ({x(t)} ˜ t=0 , {y(t)} t=0 ) is a program. Property (P2), (5.43), and (5.45)–(5.48) imply that there is a program

 τ +1  τ x (2) (t) , y (2) (t) t=T2



t=T2

such that x (2) (T2 ) = x (1) (T2 ), x (2) (τ + 1) ≥ x(τ + 1).

(5.49)

Set ˜ = y (2) (t), t = T2 , . . . , τ. x(t) ˜ = x (2) (t), t = T2 + 1, . . . , τ + 1, y(t)

(5.50)

τ +1 τ ) is a program. It follows from (5.49) and (5.50) that ˜ Clearly, ({x(t)} ˜ t=0 , {y(t)} t=0

x(τ ˜ + 1) ≥ x(τ + 1).

(5.51)

Set y(t) ˜ = y(t), t = τ + 1, . . . , T − 1, x(t ˜ + 1) = (1 − d)x(t) ˜ + x(t + 1) − (1 − d)x(t), t = τ + 1, . . . , T − 1.

(5.52)

T −1 T , {y(t)} By (5.51) and (5.52), ({x(t)} ˜ t=0 ) is a program. Relations (5.38), (5.43), t=0 ˜ (5.44), (5.47), and (5.52) imply that

M1 ≥ U (x(0), T ) −

T −1

w(by(t)) ≥

t=0

=

T −1 t=T1

w(by(t)) ˜ −

T −1 t=T1

T −1

w(by(t)) ˜ −

t=0

w(by(t)) ≥

T 2 −1 t=T1

T −1

w(by(t))

t=0

w(by(t)) ˜ −

T 2 −1 t=T1

w(by(t))

138

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

−

τ 

w(by(t))

t=T2

≥ U (x(T1 ), T2 − T1 ) −

T 2 −1

 w(by(t)) − (τ − T2 + 1)w M0

t=T1

n 

 bi .

i=1

Together with (5.41) and (5.47), the inequality above implies that T 2 −1

 w(by(t)) ≥ U (x(T1 ), T2 − T1 ) − M1 − (L3 + L2 )w M0

t=T1

n 

 bi

i=1

≥ U (x(T1 ), T2 − T1 ) − M2 . Lemma 5.12 is proved.

5.3 Properties of the Function U It is not difficult to see that the next result is valid. Proposition 5.13 Let T ≥ 1 be an integer, Δ ≥ 0, T1 , T2 be integers satisfying −1 0 ≤ T1 < T2 ≤ T and let ({x(t)}Tt=0 , {y(t)}Tt=0 ) be a program satisfying T −1

w(by(t)) ≥ U (x(0), x(T ), T ) − Δ.

t=0

Then T 2 −1

w(by(t)) ≥ U (x(T1 ), x(T2 ), T2 − T1 ) − Δ.

t=T1

We use the constant γ > 0 introduced in Section 5.2 which satisfies relation (5.17). Lemma 5.14 Let M0 > max{(dai )−1 : i = 1, . . . , n}.

(5.53)

Then there are an integer L ≥ 1 and a positive number M1 such that for every pair n which satisfies of points x0 , x˜0 ∈ R+

5.3 Properties of the Function U

139

x0 , x˜0 ≤ M0 e

(5.54)

and every natural number T ≥ L, the following inequality is valid: |U (x0 , T ) − U (x˜0 , T )| ≤ M1 .

(5.55)

Proof Let integers L1 , L2 ≥ 4 be as guaranteed by Lemma 5.11 with M1 = 1. Lemma 5.10 implies that there is a natural number L3 ≥ 4 such that the following property holds: (P3) If an integer S ≥ L3 , a program ({u(t)}St=0 , {v(t)}S−1 t=0 ) satisfies u(0) ≤ M0 e, bv(S − 1) ≥ γ , n satisfies u ˜ 0 ≤ M0 e, then there exists a program and if u˜ 0 ∈ R+

 S−1 S {u(t)} ˜ ˜ t=0 , {v(t)} t=0 which satisfies u(0) ˜ = u˜ 0 , u(S) ˜ ≥ u(S). Fix an integer L > 2(L1 + L2 + L3 + 1),

(5.56)

and set  M1 = (L1 + L2 + L3 )w M0

n 

 bi .

(5.57)

i=1 n satisfy (5.54) and that a natural number T ≥ L. Assume that x0 , x˜0 ∈ R+ −1 Proposition 5.1 implies that there exists a program ({x(t)}Tt=0 , {y(t)}Tt=0 ) satisfying

x(0) = x0 ,

T −1

w(by(t)) = U (x0 , T ).

(5.58)

t=0

Lemma 5.5, (5.54) and (5.58) imply that x(t) ≤ M0 e for all t = 0, . . . , T .

(5.59)

L 1 + L 3 < L − L 2 ≤ T − L2 .

(5.60)

By (5.56), we have

140

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

Lemma 5.10, (5.54), (5.56), (5.58), and (5.60) and the choice of L1 , L2 imply that max{by(t) : t = L3 + L1 , . . . , L3 + L1 + L2 − 1} ≥ γ . Thus there exists an integer τ ∈ {L3 + L1 , . . . , L3 + L1 + L2 − 1}

(5.61)

by(τ ) ≥ γ .

(5.62)

for which

+1 , {y(t)}τt=0 ). Property (P3), (5.54), (5.58), Let us consider the program ({x(t)}τt=0 (5.59), (5.61), and (5.62) imply that there is a program

 τ +1 τ {x(t)} ˜ ˜ t=0 t=0 , {y(t)} which satisfies x(0) ˜ = x˜0 , x(τ ˜ + 1) ≥ x(τ + 1).

(5.63)

Set y(t) ˜ = y(t), t = τ + 1, . . . , T − 1, x(t ˜ + 1) = (1 − d)x(t) ˜ + x(t + 1) − (1 − d)x(t), t = τ + 1, . . . , T − 1.

(5.64)

T −1 T , {y(t)} By (5.63) and (5.64), ({x(t)} ˜ t=0 ˜ t=0 ) is a program. Relations (5.57)–(5.59), (5.61), (5.63), and (5.64) imply that

U (x˜0 , T ) ≥

T −1

w(by(t)) ˜ =

t=0

= U (x0 , T ) −

T −1 t=0

τ  t=0

≥ U (x0 , T ) −

w(by(t)) − [

τ  t=0

w(by(t)) −

T −1

w(by(t)) −

t=0 τ 



T −1

w(by(t))] ˜

t=0

w(by(t)) ˜

t=0



w(by(t)) ≥ U (x0 , T ) − (τ + 1)w M0 

≥ U (x0 , T ) − (L3 + L1 + L2 )w M0

n  i=1

 bi

n 

 bi

i=1

= U (x0 , T ) − M1 .

5.4 Proofs of Theorems 5.4, 5.5 and 5.8

141

Hence we have shown that for every integer T ≥ L and every pair of points n which satisfy (5.54), we have x0 , x˜0 ∈ R+ U (x˜0 , T ) ≥ U (x0 , T ) − M1 . Lemma 5.14 is proved. Corollary 5.15 Let a number M0 satisfy (5.53). Then there exist a positive number M1 and an integer L ≥ 1 such that for every natural number T ≥ L and every n which satisfies x ≤ M e, x0 ∈ R+ 0 0 (M0 , T )| ≤ M1 . |U (x0 , T ) − U The next result follows from Lemmas 5.3 and 5.12, Corollary 5.15, and (5.53). Lemma 5.16 Let a number M0 satisfy (5.53), and let M1 be a positive number. Then there exist integers L¯ 1 , L¯ 2 ≥ 1 and a number M2 > 0 such that for every T −1 natural number T ≥ L¯ 1 + L¯ 2 and every program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying x(0) ≤ M0 e,

T −1

w(by(t)) ≥ U (x(0), T ) − M1 ,

t=0

the following assertion holds: If integers T1 , T2 ∈ [0, T − L¯ 2 ] satisfy T2 − T1 ≥ L¯ 1 , then T 2 −1

(M0 , T2 − T1 ) − M2 . w(by(t)) ≥ U

t=T1

5.4 Proofs of Theorems 5.4, 5.5 and 5.8 In the sequel we assume that the sum over empty set is zero. Choose M0 > max{(dai )−1 : i = 1, . . . , n}, M1 = 1.

(5.65)

Let natural numbers L¯ 1 , L¯ 2 and a positive number M2 be as guaranteed by Lemma 5.16.

142

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

n satisfy Let x0 ∈ R+

x0 ≤ M0 e.

(5.66)

Proposition 5.1 implies that for every integer k ≥ 1, there exists a program ({x (k) (t)}kt=0 , {y (k) (t)}k−1 t=0 ) which satisfies x

(k)

(0) = x0 ,

k−1 

 w by (k) (t) = U (x0 , k).

(5.67)

t=0

By Lemma 5.3 and (5.65)–(5.67), for every integer k ≥ 1, x (k) (t) ≤ M0 e, t = 0, . . . , k.

(5.68)

Lemma 5.16, the choice of L¯ 1 , L¯ 2 and M2 , (5.65), and (5.67) imply that the following property holds: (i) For every natural number k ≥ L¯ 1 + L¯ 2 and every pair of integers T1 , T2 ∈ [0, k − L¯ 2 ] satisfying T2 − T1 ≥ L¯ 1 , T 2 −1

 (M0 , T2 − T1 ) − M2 . w by (k) (t) ≥ U

(5.69)

t=T1

By (5.68), there exists a strictly increasing sequence of natural numbers {kj }∞ j =1 such that for every nonnegative integer t, there exist y (t) = lim y (kj ) (t). x (t) = lim x (kj ) (t), j →∞

j →∞

(5.70)

Evidently, { x (t), y (t)}∞ t=0 is a program. In view of (5.68) and (5.70), x (t) ≤ M0 e for all integers t ≥ 0.

(5.71)

In view of (5.67) and (5.70), x (0) = x0 .

(5.72)

Property (i), (5.70), and (5.71) imply that for every pair of integers T1 , T2 ∈ [0, ∞) which satisfies T2 − T1 ≥ L¯ 1 , we have (M0 , T2 − T1 ) ≥ U

T 2 −1 t=T1

(M0 , T2 − T1 ) − M2 . w(b y (t)) ≥ U

(5.73)

5.4 Proofs of Theorems 5.4, 5.5 and 5.8

143

Let p be an integer for which p ≥ L¯ 1 . We claim that for all sufficiently large natural numbers T , we have   T −1    −1  w(b y (t)) ≤ 2p−1 M2 . p U (M0 , p) − T −1  

(5.74)

t=0

Assume that T ≥ p is a natural number. Then there exist integers q, s for which q ≥ 1, 0 ≤ s < p, T = pq + s.

(5.75)

By (5.75), T −1

T −1

(M0 , p) w(b y (t)) − p−1 U

t=0

=T

⎛

pq−1  −1 ⎝

w(b y (t)) +



{w(b y (t)) :

t=0

⎞

(M0 , p) t is an integer such that pq ≤ t ≤ T − 1}⎠ − p−1 U = T −1

 {w(b y (t)) : t is an integer such that pq ≤ t ≤ T − 1}

q−1 (i+1)p−1

   (M0 , p) + T −1 pq (pq)−1 w(b y (t)) − p−1 U i=0

t=ip

 = T −1 {w(b y (t)) : t is an integer such that pq ≤ t ≤ T − 1} ⎡ ⎛ q−1 (i+1)p−1

   ⎝ + T −1 pq (pq)−1 ⎣ w(b y (t)) ⎞

i=0

t=ip

⎤

(M0 , p)⎦ − p−1 U (M0 , p)⎠ + q U (M0 , p). −U ¯ the equation above Together with (5.71), (5.73), (5.75), and the inequality p ≥ L, implies that   T −1    −1   −1 w(b y (t)) − p U (M0 , p) ≤ T −1 pw(M0 be) + (pq)−1 qM2 T   t=0

144

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

(M0 , p)|q/T − 1/p| ≤ T −1 pw(M0 be) +U (M0 , p)s(pT )−1 → M2 /p as T → ∞. + M2 /p + U Thus (5.74) is true for all sufficiently large natural numbers T . Since p is an arbitrary integer satisfying p ≥ L¯ 1 , we conclude that  T

−1

T −1

∞ w(b y (t)) T =1

t=0

is a Cauchy sequence. Evidently, there exists lim T −1

T →∞

T −1

w(b y (t)).

t=0

It follows from (5.74) that for every integer p ≥ L¯ 1 , we have (M0 , p) − lim T −1 |p−1 U T →∞

T −1

w(b y (t))| ≤ p−1 (2M2 ).

(5.76)

t=0

Since (5.76) is true for every integer p ≥ L¯ 1 , we conclude that lim T −1

T →∞

T −1

(M0 , p)/p. w(b y (t)) = lim U p→∞

t=0

(5.77)

Now it is easy to see that Theorem 5.4 is true. Let μ be defined by (5.7). We have (M0 , p)/p. μ = lim U p→∞

(5.78)

It follows from (5.76)–(5.78) that for every natural number p ≥ L¯ 1 , we have     −1 p U (M0 , p) − μ ≤ p−1 (2M2 ).

(5.79)

Inequality (5.79) implies the validity of Theorem 5.5. Now we are ready to complete the proof of Theorem 5.8. In view of (5.73) and (5.79), for every pair of integers T1 , T2 ∈ [0, ∞) which satisfies T2 − T1 ≥ L¯ 1 , we have

5.4 Proofs of Theorems 5.4, 5.5 and 5.8

145

     T  T   2 −1   2 −1     ≤ w(b y (t)) − μ(T − T ) w(b y (t)) − U (M , T − T ) 2 1  0 2 1      t=T1   t=T1   +  U (M0 , T2 − T1 ) − μ(T2 − T1 ) ≤ 3M2 . (5.80) By (5.71), for every nonnegative integer T1 and every integer T2 ∈ [T1 +1, T1 + L¯ 1 ], we have    T T 2 −1   2 −1   w(b y (t)) − μ(T2 − T1 ) ≤ w(b y (t)) + μL¯ 1 ≤ L¯ 1 w(M0 be) + μL¯ 1 .   t=T1  t=T1 Combined with (5.80) the inequality above implies that for every pair of integers T1 , T2 ≥ 0 which satisfies T2 > T1 , we have    T2 −1     ¯ 1 (w(M0 be) + μ). w(b y (t)) − μ(T − T ) (5.81) 2 1  ≤ 3M2 + L    t=T1 We claim that for every natural number T , T −1

w(b y (t)) = U (x(0), x(T ), T ).

(5.82)

t=0

Since μ > w(0) (see (5.8)), it follows from (5.81) that there exists a strictly increasing sequence {Ti }∞ i=1 such that T1 ≥ 4 and that w(b y (Ti − 1)) > 2−1 (μ + w(0)) for all natural numbers i.

(5.83)

Since the function w is continuous, there exists a positive number r0 such that b y (Ti − 1) > r0 for all natural numbers i. Extracting a subsequence and re-indexing if necessary, we may assume that there exist a positive number r1 and a natural number q ∈ {1, . . . , n} such that yq (Ti − 1) > r1 for all natural numbers i.

(5.84)

It is clear that it is sufficient to show that (5.82) holds for all T = Ti − 1, i = 1, 2, . . . . Let j be a natural number. We claim that (5.82) is true with T = Tj − 1. Assume T −1

T −2

j j the contrary. Then there exist a program ({x(t)} ¯ ¯ t=0 , {y(t)} t=0 ) and a number Δ > 0 which satisfy

146

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

x(0) ¯ = x (0), x(T ¯ j − 1) ≥ x (Tj − 1), Tj −2

 t=0

(5.85)

Tj −2

w(by(t)) ¯ >



w(b y (t)) + 2Δ.

(5.86)

t=0

Since the function w is continuous, there is a number δ0 ∈ (0, r1 /16)

(5.87)

n which satisfies z , z ≤ M e and such that for every pair of points z1 , z2 ∈ R+ 1 2 0 z1 − z2  ≤ δ0 n, we have

|w(bz1 ) − w(bz2 )| ≤ Δ/(2Tj + 2)−1 .

(5.88)

Fix a positive number   δ < min 1, Δ/16, δ0 , δ0 (max{ai : i = 1, . . . , n})−1 (4n)−1 .

(5.89)

In view of the construction of the program { x (t), y (t)}∞ t=0 (see (5.70)), there exists an integer k > Tj + 1 such that x (t) ≤ δ, t = 0, . . . , Tj + 1, x (k) (t) −

(5.90)

y (t) ≤ δ, t = 0, . . . , Tj + 1. y (k) (t) − Define x(t) ˜ = x(t), ¯ t = 0, . . . , Tj − 1,

(5.91)

y(t) ˜ = y(t), ¯ t = 0, . . . , Tj − 2, and set y (Tj − 1) − δ0 e(q), y(T ˜ j − 1) =

(5.92)

x (Tj ) − (1 − d) x (Tj − 1) + (1 − d)x(T ¯ j − 1) x(T ˜ j) = + δ0 (max{ai : i = 1, . . . , n})−1 (4n)−1 e.

(5.93)

It follows from (5.84), (5.87), and (5.92) that y(T ˜ j − 1) ≥ 0. By (5.85), (5.91), and (5.92),

5.4 Proofs of Theorems 5.4, 5.5 and 5.8

y(T ˜ j − 1) ≤ y (Tj − 1) ≤ x (Tj − 1) ≤ x(T ¯ j − 1) = x(T ˜ j − 1).

147

(5.94)

In view of (5.91) and (5.93), we have ˜ j − 1) = x (Tj ) − (1 − d) x (Tj − 1) x(T ˜ j ) − (1 − d)x(T + δ0 min{ai−1 : i = 1, . . . , n}(4n)−1 e.

(5.95)

By (5.95), ˜ j − 1). x(T ˜ j ) ≥ (1 − d)x(T

(5.96)

Relations (5.92) and (5.95) imply that ˜ j − 1)) + ey(T ˜ j − 1) a(x(T ˜ j ) − (1 − d)x(T ≤ a( x (Tj ) − (1 − d) x (Tj − 1)) + δ0 /4 + e y (Tj − 1) − δ0 ≤ 1. Together with (5.91), (5.94), and (5.96), the relation above implies that Tj Tj −1 ˜ ({x(t)} ˜ t=0 ,{y(t)} t=0 ) is a program. It follows from (5.67), (5.72), (5.85), and (5.91) that x(0) ˜ = x(0) ¯ = x (0) = x0 = x (k) (0).

(5.97)

By (5.85), (5.89), (5.90), and (5.93), x (Tj ) + δ0 (max{ai : i = 1, . . . , n})−1 (4n)−1 e ≥ x (Tj ) + δe ≥ xT(k) . x(T ˜ j) ≥ j (5.98) In view of (5.68), (5.71), (5.89), (5.90), (5.92), and the choice of δ0 (see (5.88)), y (t))| ≤ Δ(2Tj + 2)−1 , t = 0, . . . , Tj + 1, |w(by (k) (t)) − w(b y (Tj − 1))| ≤ Δ(2Tj + 2)−1 . |w(by(T ˜ j − 1)) − w(b

(5.99)

It follows from (5.86), (5.91), and (5.99) that Tj −1 



w(by(t)) ˜ − w(by (k) (t))



t=0

=

Tj −2 

 t=0



 w(by(t)) ¯ − w(by (k) (t)) + w(by(T ˜ j − 1)) − w by (k) (Tj − 1)

148

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function Tj −2

=



[w(by(t)) ¯ − w(b y (t))] +

Tj −2 



t=0

w(b y (t)) − w(by (k) (t))



t=0

+ w(by(T ˜ j − 1)) − w(b y (Tj − 1)) + w(b y (Tj − 1)) − w(by (k) (Tj − 1)) ≥ 2Δ − (Tj − 1)Δ(2Tj + 2)−1 − Δ(2Tj + 2)−1 − Δ(2Ti + 2)−1 ≥ Δ and Tj −1



Tj −1

w(by(t)) ˜ ≥

t=0



w(by (k) (t)) + Δ.

t=0

Combined with (5.97) and (5.98), the inequality above implies that Tj −1

U (x0 , x (k) (Tj ), Tj ) ≥



w(by (k) (t)) + Δ.

t=0

Combined with Proposition 5.13, the relation above implies that U (x0 , k) ≥

k−1 

w(by (k) (t)) + Δ/2.

t=0

This inequality contradicts (5.4). The contradiction we have reached proves that (5.82) holds for T = Tj − 1 and for every natural number j . This implies that (5.82) is valid for every natural number T . Now the validity of Theorem 5.8 follows from (5.81) and (5.82).

5.5 Proof of Proposition 5.7 Fix a number M0 > 0 for which M0 > x(0) + d −1 max{ai−1 : i = 1, . . . , n}.

(5.100)

Lemma 5.3 and (5.100) imply that x(t) ≤ M0 e for all integers t ≥ 0.

(5.101)

Corollary 5.6 and (5.101) imply that there exists M > 0 such that for every nonnegative integer T1 and every integer T2 > T1 , we have

5.6 Proof of Theorem 5.9

149 T 2 −1

[w(by(t)) − μ] ≤ M.

(5.102)

t=T1

Assume that the sequence { of (5.102),

T −1 t=0

lim inf T →∞

T −1

[w(by(t)) − μ]}∞ T =1 is not bounded. In view

[w(by(t)) − μ] = −∞.

(5.103)

t=0

Let Q be a positive number. It follows from (5.103) that there exists an integer T0 ≥ 1 such that T 0 −1

[w(by(t)) − μ] < −Q − M.

t=0

By the inequality above and the choice of M (see (5.102)), for every natural number T > T0 , T −1

T 0 −1

T −1

t=0

t=0

t=T0

[w(by(t)) − μ] =

[w(by(t)) − μ] +

[w(by(t)) − μ]

< −Q − M + M = −Q. Since Q is an arbitrary positive number, we obtain that lim

T →∞

T −1

[w(by(t)) − μ] = −∞.

t=0

This completes the proof of Proposition 5.7.

5.6 Proof of Theorem 5.9 Assume that a program {x(t), y(t)}∞ t=0 satisfies T −1

w(by(t)) = U (x(0), x(T ), T ) for all integers T ≥ 1,

(5.104)

t=0

lim sup by(t) > 0. t→∞

(5.105)

150

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

In view of (5.105), there exist a number Δ > 0 and a strictly increasing sequence of natural numbers {Ti }∞ i=1 for which by(Ti − 1) ≥ Δ for all integers i ≥ 1.

(5.106)

Theorem 5.8 implies that there exist a program { x (t), y (t)}∞ t=0 and a positive number M1 > 0 such that x (0) = x(0),   T2 −1     [w(b y (t)) − (T2 − T1 )μ] ≤ M1   t=T1  for every integer T1 ≥ 0 and every integer T2 > T1 .

(5.107)

(5.108)

Fix a number M0 > x(0) + max{(ai d)−1 : i = 1, . . . , n}.

(5.109)

Lemma 5.3, (5.107), and (5.109) imply that x(t), x (t) ≤ M0 e for all integers t ≥ 0.

(5.110)

By (5.109) and Lemma 5.10, there exists an integer τ0 ≥ 4 such that the following property holds: (P4) If an integer S ≥ τ0 , if a program ({u(t)}St=0 , {v(t)}S−1 t=0 ) satisfies u(0) ≤ M0 e, bv(S − 1) ≥ Δ, n satisfies u ˜ 0 ≤ M0 e, then there exists a program and if u˜ 0 ∈ R+

 S−1 S {u(t)} ˜ , { v(t)} ˜ t=0 t=0 such that u(0) ˜ = u˜ 0 , u(S) ˜ ≥ u(S). Let i ≥ 1 be an integer satisfying Ti > τ0 .  Ti −1 i , {y(t)} Consider the program {x(t)}Tt=T t=Ti −τ0 . Property (P4), (5.106), i −τ0 and (5.110) imply that there exists a program

5.6 Proof of Theorem 5.9

151 Ti −1 i ({x (1) (t)}Tt=T , {y (1) (t)}t=T ) i −τ0 i −τ0

which satisfies x (Ti − τ0 ), x (1) (Ti ) ≥ x(Ti ). x (1) (Ti − τ0 ) =

(5.111)

Set ˜ = y (t), t = 0, . . . , Ti − τ0 − 1, = x (t), t = 0, . . . , Ti − τ0 , y(t)

x(t) ˜ x(t) ˜ =

x (1) (t),

t = Ti − τ0 + 1, . . . , Ti , y(t) ˜ = y (1) (t), t = Ti − τ0 , . . . , Ti − 1. (5.112)

Ti Ti −1 It follows from (5.111) and (5.112) that ({x(t)} ˜ ˜ t=0 , {y(t)} t=0 ) is a program. By (5.107), (5.111), and (5.112), we have

x(0) ˜ = x(0), x(T ˜ i ) ≥ x(Ti ).

(5.113)

In view of (5.104), (5.108), (5.112), and (5.113), we have

0≤

T i −1

w(by(t)) −

t=0

=

T i −1

T i −1

w(by(t)) −

Ti −τ 0 −1 

t=0

≤

T i −1

w(by(t)) ˜

t=0

w(b y (t)) −

w(by (1) (t))

t=Ti −τ0

t=0

w(by(t)) − μ(Ti − τ0 ) + M1 =

t=0

T i −1

T i −1

w(by(t)) − μTi + M1 + μτ0

t=0

and T i −1

w(by(t)) − μTi ≥ −M1 − μτ0 .

t=0

Since the inequality above is true for an arbitrary integer i ≥ 1 for which Ti > τ0 , we obtain that lim sup

T −1

T →∞ t=0

[w(by(t)) − μ] ≥ M1 − μτ0 .

152

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

Together with Proposition 5.7, this implies that the sequence { μ]}∞ t=1 is bounded. This completes the proof of Theorem 5.9.

T −1 t=0

[w(by(t)) −

5.7 The RSS Model with Discounting For every nonnegative integer t, let wt : [0, ∞) → [0, ∞) be a continuous increasing function which represents the preferences of the planner at moment of time t. We suppose that the following assumption holds. Assumption A For every nonnegative integer t ≥ 0, wt (0) = 0, and for every positive number M, lim wt (M) = 0.

t→∞

n and every natural number T define For every point x0 ∈ R+

U (x0 , T ) = sup

T −1 

wt (by(t)) :

−1 ({x(t)}Tt=0 , {y(t)}Tt=0 )

t=0

is a program such that x(0) = x0 .

(5.114)

The next proposition follows immediately from the continuity of wt , t = 0, 1, . . . . n and every integer T ≥ 1, there exists a Proposition 5.17 For every x0 ∈ R+ T −1 program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) = x0 and T −1

wt (by(t)) = U (x0 , T ).

t=0

We prove the following theorem which was obtained in [52]. n , there exists a program {x (t), y (t)}∞ such that Theorem 5.18 For every z ∈ R+ z z t=0 xz (0) = z and the following property holds: For every pair of positive numbers M0 , δ, there exists an integer L(δ) ≥ 1 such n which satisfies z ≤ M e, that for every natural number S ≥ L(δ) and every z ∈ R+ 0 S−1  t=0

wt (byz (t)) ≥ U (z, S) − δ.

5.7 The RSS Model with Discounting

153

n , and let a program {x (t), y (t)}∞ be as guaranteed Corollary 5.19 Let z ∈ R+ z z t=0 by Theorem 5.18. Then for every program {x(t), y(t)}∞ t=0 satisfying x(0) = z, the inequality

lim inf

T −1 

T →∞

wt (byz (t)) −

t=0

T −1

 wt (by(t)) ≥ 0

t=0

holds. Proof Let a positive number M0 satisfy z ≤ M0 e, δ > 0, and let an integer L(δ) ≥ 1 be as guaranteed by Theorem 5.18. Then for every integer S ≥ L(δ) , we have S−1 

wt (by(t)) −

t=0

S−1 

wt (byz (t)) ≤ U (z, S) − (U (S, z) − δ) ≤ δ.

t=0

This completes the proof of Corollary 5.19. Example 5.20 Let w : [0, ∞) → [0, ∞) be a continuous increasing function, w(0) = 0, {ρt }∞ t=0 ⊂ (0, 1), limt→∞ ρt = 0, wt = ρt w, t = 0, 1, . . . . Then Assumption A holds.  Assume that ∞ t=0 ρt = ∞ and that w(s) > 0 for every positive number s. Let n z ∈ R+ be given. Set x(0) = z, y(0) = 0, for every natural number t, y(t) = (2n)−1 min{1, a1−1 , . . . , ai−1 , . . . , an−1 }e, for every nonnegative integer t, x(t + 1) = (1 − d)x(t) + (2n)−1 (a1−1 , . . . , an−1 ). Evidently, {x(t), y(t)}∞ t=0 is a program, and for every natural number T , T 

wt (by(t)) =

t=0

=

 T 



T 

 ρt w be(2n)−1 min{1, a1−1 , . . . , an−1 }

t=1



ρt w be(2n)−1 min{1, a1−1 , . . . , an−1 } → ∞ as T → ∞.

t=1

This implies that U (z, T ) → ∞ as T → ∞.

154

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

5.8 An Auxiliary Result for Theorem 5.18 Fix γ ∈ (0, 1) which satisfies γ < (2n)−1

n 

  bi min 1, a1−1 , . . . , ai−1 , . . . , an−1 .

(5.115)

i=1

In the sequel we assume that the sum over empty set is zero. Lemma 5.21 Let a number M0 satisfy M0 > max{(ai d)−1 : i = 1, . . . , n},

(5.116)

and let δ be a positive number. Then there exists an integer L¯ ≥ 1 such that for ¯ there exists an integer τ ≥ L for which the following assertion every integer L ≥ L, holds: −1 For every natural number T ≥ τ and every program ({x(t)}Tt=0 , {y(t)}Tt=0 ) which satisfies x(0) ≤ M0 e,

T −1

wt (by(t)) = U (x(0), T ),

(5.117)

t=0

the inequality L−1 

wt (by(t)) ≥ U (x(0), L) − δ

(5.118)

t=0

is valid. Proof Lemma 5.10 implies that there exists an integer L0 ≥ 4 such that the following property holds: (P5) If an integer S ≥ L0 , if a program ({u(t)}St=0 , {v(t)}S−1 t=0 ) satisfies u(0) ≤ M0 e, bv(S − 1) ≥ γ , n satisfies u ˜ 0 ≤ M0 e, then there exists a program and if u˜ 0 ∈ R+

 S−1 S {u(t)} ˜ , { v(t)} ˜ t=0 t=0 which satisfies u(0) ˜ = u˜ 0 , u(S) ˜ ≥ u(S).

5.8 An Auxiliary Result for Theorem 5.18

155

Assumption (A) implies that there exists an integer L¯ ≥ 1 such that for every ¯ we have natural number L ≥ L, wL (M0 be) < (δ/8)(4L0 )−1 .

(5.119)

¯ and fix a natural number Assume that an integer L ≥ L, τ ≥ L + L0 + 2.

(5.120)

−1 ) Assume that an integer T ≥ τ and that a program ({x(t)}Tt=0 , {y(t)}Tt=0 satisfies (5.117). We claim that (5.118) holds. L−1 L , {y(t)} Proposition 5.17 implies that there exists a program ({x(t)} ˜ t=0 ˜ t=0 ) satisfying

x(0) ˜ = x(0),

L−1 

wt (by(t)) ˜ = U (x(0), L).

(5.121)

t=0

Lemma 5.3, (5.116), (5.117), and (5.121) imply that x(t) ˜ ≤ M0 e, t = 0, . . . , L, x(t) ≤ M0 e, t = 0, . . . , T .

(5.122)

There are two cases: by(t) < γ , t = L + L0 , . . . , T − 1;

(5.123)

max{by(t) : t = L + L0 , . . . , T − 1} ≥ γ .

(5.124)

T −1 ) as follows: Assume that (5.123) is valid. Set ({x (1) (t)}Tt=0 , {y (1) (t)}t=0

x (1) (t) = x(t), ˜ t = 0, . . . , L, y (1) (t) = y(t), ˜ t = 0, . . . , L − 1, y(L) ˜ = 0, y (1) (t) = (2n)−1 min{1, a1−1 , . . . , ai−1 , . . . , an−1 }e, t = L + 1, . . . , T − 1, x (1) (t + 1) = (1 − d)x (1) (t)+(2n)−1 (a1−1 , . . . , ai−1 , . . . , an−1 ), t=L, . . . , T − 1. (5.125) T −1 It is clear that ({x (1) (t)}Tt=0 , {y (1) (t)}t=0 ) is a program. It follows from (5.114), (5.115), (5.117), (5.119), (5.121), (5.123), (5.125), assumption (A), and the monotonicity of the functions wt , t = 0, 1, . . . that

0≥

T −1 t=0

T −1 T −1

 wt by (1) (t) − U (x(0), T ) = wt (by (1) (t)) − wt (by(t)) t=0

t=0

156

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

=

L−1 

wt (by(t)) ˜ + wL (0)

t=0

+

T −1





wt (2n)−1

t=L+1

−

T −1





n 

bi min{1, a1−1 , . . . , ai−1 , . . . , an−1 }

i=1

wt (by(t)) ≥ U (x(0), L) −

t=0

−

L+L 0 t=L+1

+

T −1

L−1 

wt (by(t)) − wL (M0 be)

t=0

wt (M0 be)



wt (2n)

−1

t=L+1

≥ U (x(0), L) −

 bi ) min{1, a1−1 , . . . , ai−1 , . . . , an−1 }

 − wt (γ )

i=1 L−1 

wt (by(t)) −

L+L 0

wt (M0 be)

t=L

t=0

≥ U (x(0), L) −

(

n 

L−1 

wt (by(t)) − δ

t=0

and L−1 

wt (by(t)) ≥ U (x(0), L) − δ.

t=0

Therefore if (5.123) holds, then (5.118) is true. Assume that (5.124) is valid. Then in view of (5.124), there exists an integer S0 which satisfies L + L0 ≤ S0 − 1 ≤ T − 1, by (S0 − 1) ≥ γ , by (t) < γ for each integer t satisfying L0 + L ≤ t < S0 − 1.

(5.126)

T −1 ). Set Let us define a sequence ({x (2) (t)}Tt=0 , {y (2) (t)}t=0

x (2) (t) = x(t), ˜ t = 0, . . . , L, y (2) (t) = y(t), ˜ t = 0, . . . , L − 1, y (2) (L) = 0, y (2) (t) = (2n)−1 min{1, a1−1 , . . . , ai−1 , . . . , an−1 }e

5.8 An Auxiliary Result for Theorem 5.18

157

if an integer t satisfies L < t ≤ S0 − L0 − 1, x (2) (t + 1) = (1 − d)x (2) (t) + (2n)−1 (a1−1 , . . . , ai−1 , . . . , an−1 ) if an integer t satisfies L ≤ t ≤ S0 − L0 − 1.

(5.127)

S0 −L0 −1 0 −L0 It is easy to see that ({x (2) (t)}St=0 , {y (2) (t)}t=0 ) is a program. In view of (5.117), (5.121), (5.127), and Lemma 5.3,

x(t) ≤ M0 e, t = 0, . . . , T , x (2) (t) ≤ M0 e, t = 0, . . . , S0 − L0 .

(5.128)

By (5.126), (5.128), and property (P5), there exists a program

 S0 −1 (2) 0 {x (2) (t)}St=S , {y (t)} t=S0 −L0 0 −L0 such that x (2) (S0 ) ≥ x(S0 ).

(5.129)

0 0 −1 , {y (2) (t)}St=0 ) is a program. It is clear that ({x (2) (t)}St=0 Set

y (2) (t) = y(t) for all integers t satisfying S0 ≤ t ≤ T − 1, x (2) (t + 1) = (1 − d)x (2) (t) + x(t + 1) − (1 − d)x(t) for all integers t satisfying S0 ≤ t ≤ T − 1.

(5.130)

T −1 It is clear that ({x (2) (t)}Tt=0 , {y (2) (t)}t=0 ) is a program. By (5.121) and (5.127),

x (2) (0) = x(0).

(5.131)

By (5.114), (5.115), (5.117), (5.121), (5.126)–(5.128), (5.130), (5.131), assumption (A), and the monotonicity of the functions wt , t = 0, 1, . . . , we have 0≥

=

T −1

T −1 −1



  T wt by (2) (t) − U (x(0), T ) = wt by (2) (t) − wt (by(t))

t=0

t=0

S 0 −1 t=0

=

L−1  t=0

0 −1

 S wt by (2) (t) − wt (by(t))

t=0

wt (by(t)) ˜ + wL (0) +

S 0 −1 t=L+1

 wt by (2) (t)

t=0

158

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

−

L−1 

wt (by(t)) −

S 0 −1

wt (by(t))

t=L

t=0

≥ U (x(0), L) −

L−1 



wt (by(t)) − wL (M0 be)

t=0

+

S0 −L 0 −1

wt (be(2n)−1 min{1, a1−1 , . . . , ai−1 , . . . , an−1 })

t=L

+

S 0 −1

wt (by

(2)

(t)) −

t=S0 −L0

S0 −L 0 −1

wt (by(t)) −

≥ U (x(0), L) −

wt (by(t))

t=S0 −L0

t=L L−1 

S 0 −1



wt (by(t)) − wL (M0 be)

t=0

+

S0 −L 0 −1 

wt (be(2n)−1 min{1, a1−1 , . . . , ai−1 , . . . , an−1 })



t=L

−

S0 −L 0 −1

wt (by(t)) −

S 0 −1

wt (beM0 )

t=S0 −L0

t=L

≥ U (x(0), L) −

L−1 

wt (by(t)) − wL (M0 be)

t=0

+

 

wt be(2n)−1 min{1, a1−1 , . . . , ai−1 , . . . , an−1 }

−wt (γ ) : t is an integer such that L + L0 ≤ t ≤ S0 − 1 − L0 −

L+L 0 −1 

wt (M0 be) −

S 0 −1



wt (M0 be)

t=S0 −L0

t=L

≥ U (x(0), L) −

L−1 

wt (by(t)) − wL (M0 be)

t=0

−

L+L 0 −1  t=L

wt (M0 be) −

S 0 −1

wt (M0 be).

t=S0 −L0

¯ (5.129), the relation L ≥ L, ¯ and (5.126), the Combined with the choice of L, relations above imply that

5.9 Proof of Theorem 5.18 L−1 

159

wt (by(t)) ≥ U (x(0), L) − (δ/8) − 2L0 (δ/8)(4L0 )−1 ≥ U (x(0), L) − δ/2.

t=0

Hence if (5.124) holds, then (5.118) is valid. Therefore (5.118) is true in both cases. Lemma 5.21 is proved.

5.9 Proof of Theorem 5.18 n and every integer T ≥ 1, there exists Proposition 5.17 implies that for every z ∈ R+ T −1 a program ({xz,T (t)}Tt=0 , {yz,T (t)}t=0 ) which satisfies

xz,T (0) = z,

T −1

wt (byz,T (t)) = U (z, T ).

(5.132)

t=0

Let M0 > max{(ai d)−1 : i = 1, . . . , n}, and let δ be a positive number. Lemma 5.21 implies that there exists an integer Lδ ≥ 1 for which the following property holds: (P6) For every integer L ≥ Lδ , there exists an integer τL ≥ L such that for every n which satisfies z ≤ M e, we have integer T ≥ τL and every point z ∈ R+ 0 L−1 

wt (byz,T (t)) ≥ U (z, L) − δ/4.

t=0 n satisfy z ≤ M e. Lemma 5.3 implies that there exist a strictly increasing Let z ∈ R+ 0 ∞ sequence of natural numbers {Tk }∞ k=1 and a program {xz (t), yz (t)}t=0 such that for every nonnegative integer t, we have

xz,Tk (t) → xz (t), yz,Tk (t) → yz (t) as k → ∞.

(5.133)

Evidently, xz (0) = z. Let an integer L satisfy L ≥ Lδ , and let an integer τL ≥ L be as guaranteed by the property (P6). By (5.133), there exists an integer k ≥ 1 for which Tk ≥ τL , L−1  L−1      wt (byz (t)) − wt (byz,Tk (t)) ≤ δ/4.    t=0

t=0

(5.134)

160

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

Property (P6), (5.134), and the choice of τL imply that L−1 

wt (byz,Tk (t)) ≥ U (z, L) − δ/4.

t=0

Combined with (5.134) the inequality above implies that L−1 

wt (byz (t)) ≥ U (z, L) − δ.

t=0

This completes the proof of Theorem 5.18.

5.10 Weakly Agreeable Programs A program {x ∗ (t), y ∗ (t)}∞ t=0 is called weakly agreeable if for every nonnegative integer t, u(x ∗ (t), x ∗ (t + 1)) = w(by ∗ (t))

(5.135)

and if for every integer T0 ≥ 1 and every positive number , there exists a natural   −1 number T > T0 such that for every program ({x(t)}Tt=0 , {y(t)}Tt=0 ) which satisfies T −1  , {y  (t)}t=0 ) such that x(0) = x ∗ (0), there exists a program ({x  (t)}Tt=0 x  (0) = x(0), x  (t) = x ∗ (t), t = 0, . . . , T0 , T  −1 t=0



w(by (t)) ≥

T  −1

w(by(t)) − .

t=0

The notion of weakly agreeable programs is a weakened version of the notion of agreeable programs which is well-known in the literature [32–34]. We will prove the following three results obtained in [111]. Theorem 5.22 Any weakly agreeable program is good. Theorem 5.23 Any weakly agreeable program is weakly maximal. Theorem 5.24 A program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly agreeable if and only if there exist a strictly increasing sequence of natural numbers {Sk }∞ k=1 and a sequence of Sk Sk −1 (k) (k) programs ({x (t)}t=0 , {y (t)}t=0 ), k = 1, 2, . . . such that x (k) (0) = x ∗ (0), k = 1, 2, . . .

(5.136)

5.11 Proof of Theorem 5.22

U (x ∗ (0), Sk ) −

161 S k −1

 w by (k) (t) → 0 as k → ∞

(5.137)

t=0

and that for all integers t ≥ 0, x ∗ (t) = lim x (k) (t), y ∗ (t) = lim y (k) (t). k→∞

k→∞

(5.138)

n , there exists a weakly Theorem 5.24 easily implies that for every x0 ∈ R+ ∞ agreeable program {x(t), y(t)}t=0 satisfying x(0) = x0 .

5.11 Proof of Theorem 5.22 Assume that a program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly agreeable. Fix a positive number M0 > max{(dai ))−1 : i = 1, . . . , n} + ||x ∗ (0)||.

(5.139)

Lemma 5.3 and (5.139) imply that x ∗ (t) ≤ M0 e for every nonnegative integer t.

(5.140)

In follows from Theorem 5.5 and (5.139) that there exists a positive number M1 such that (M0 , p) − pμ| ≤ M1 for every natural number p. |U

(5.141)

Theorem 5.8 and (5.139) imply that there exist a positive number M2 and a program ∞ which satisfies {x(t), ˜ y(t)} ˜ t=0 x(0) ˜ = x ∗ (0), |

S 2 −1

(5.142)

w(by(t)) ˜ − μ(S2 − S1 )| ≤ M2 for all pairs of integers S1 ≥ 0, S2 > S1 .

t=S1

(5.143) Assume that T0 ≥ 1 is an integer. By the definition of a weakly agreeable program, there exists a natural number T1 > T0 for which the following property holds: T1 −1 1 (P7) for every program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) = x ∗ (0), T T −1 1 1 there exists a program ({x  (t)}t=0 , {y  (t)}t=0 ) satisfying x  (0) = x(0), x  (t) = x ∗ (t), t = 0, . . . , T0 ,

162

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function T 1 −1

w(by  (t)) ≥

t=0

T 1 −1

w(by(t)) − 1.

(5.144)

t=0

T1 −1 1 Proposition 5.1 implies that there exists a program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies

x(0) = x ∗ (0),

T 1 −1

w(by(t)) = U (x ∗ (0), T1 ).

(5.145)

t=0

Property (P7) and (5.145) imply that there exists a program T1 −1 1 ({x  (t)}Tt=0 , {y  (t)}t=0 )

such that (5.144) is valid and T 1 −1

w(by  (t)) ≥

T 1 −1

w(by(t)) − 1.

(5.146)

w(by  (t)) ≥ U (x ∗ (0), T1 ) − 1.

(5.147)

t=0

t=0

By (5.145) and (5.146), T 1 −1 t=0

Lemma 5.3, (5.139), and (5.144) imply that x  (t) ≤ M0 e for all integers t = 0, . . . , T1 .

(5.148)

It follows from (5.135), (5.139), (5.144), and (5.147) that T 0 −1 t=0

0 −1 1 −1 1 −1  ∗  T    T    T   w by (t) ≥ w by (t) = w by (t) − w by  (t)

t=0

t=0

t=T0

T 1 −1     ≥ U x ∗ (0), T1 − 1 − w by  (t) . t=T0

Combined with (5.141)–(5.143) and (5.148), the relation above implies that T 0 −1 t=0

(M0 , T1 − T0 ) w(by ∗ (t)) ≥ U (x ∗ (0), T1 ) − 1 − U

5.12 Auxiliary Results

163

≥ U (x ∗ (0), T1 ) − 1 − M1 − (T1 − T0 )μ ≥

T 1 −1

w(by(t)) ˜ − 1 − μ(T1 − T0 ) − M1 ≥ T0 μ − M2 − M1 − 1

t=0

and T 0 −1

w(by ∗ (t)) ≥ T0 μ − 1 − M1 − M2 .

t=0

Since the inequality above is valid for every integer T0 ≥ 1, Corollary 5.6 implies that the program {x ∗ (t), y ∗ (t)}∞ t=0 is good. This completes the proof of Theorem 5.22.

5.12 Auxiliary Results It is easy to see that the following auxiliary result holds. Proposition 5.25 A program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly agreeable if and only if u(x ∗ (t), x ∗ (t + 1)) = w(by ∗ (t)) for every nonnegative integers t and for every positive number  and every integer T0 ≥ 1, there exist a natural number T > T0 and a program 

  −1 {x  (t)}Tt=0 , {y  (t)}Tt=0 which satisfies x  (t) = x ∗ (t), t = 0, . . . , T0 ,

T  −1

  w by  (t) ≥ U (x ∗ (0), T ) − .

t=0

Proposition 5.26 Let a program {x ∗ (t), y ∗ (t)}∞ t=0 be such that for every positive number  and every integer T0 ≥ 1, there exist a natural number T > T0 and a program 

  −1 {x  (t)}Tt=0 , {y  (t)}Tt=0 satisfying x  (t) = x ∗ (t), t = 0, . . . , T0 , y  (t) = y ∗ (t), t = 0, . . . , T0 − 1,

(5.149)

164

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function T  −1

    w by  (t) ≥ U x ∗ (0), T − .

(5.150)

t=0

Then the program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly agreeable. Proof In order to prove the proposition, it is sufficient to show that u(x ∗ (t), x ∗ (t + 1)) = w(by ∗ (t)) for every nonnegative integer t.

(5.160)

Let T0 ≥ 1 be an integer and let  > 0. Then there exist an integer T > T0 and a program

   −1 {x  (t)}Tt=0 , {y  (t)}Tt=0 such that (5.149) and (5.150) are valid. It follows from (5.149) and (5.150) that for all integers t = 0, . . . , T0 − 1, we have         w by ∗ (t) = w by  (t) ≥ u x  (t), x  (t + 1) −  = u x ∗ (t), x ∗ (t + 1) − . Since  is an arbitrary positive number, we obtain that     w by ∗ (t) = u x ∗ (t), x ∗ (t + 1) for all integers t ∈ {0, . . . , T0 − 1}. Since T0 is an arbitrary natural number, we conclude that (5.166) is true. This completes the proof of Proposition 5.26.

5.13 Proof of Theorem 5.23 Let {x ∗ (t), y ∗ (t)}∞ t=0 be a weakly agreeable program. We claim that it is weakly maximal. Assume the contrary. Then there exist a natural number T0 and a positive number  such that U (x ∗ (0), x ∗ (T ), T0 ) >

T 0 −1

w(by ∗ (t)) + 4.

t=0 T0 −1 0 This implies that there exists a program ({x(t)}Tt=0 , {y(t)}t=0 ) for which

x(0) = x ∗ (0), x(T0 ) ≥ x ∗ (T0 ),

T 0 −1 t=0

w(by(t)) >

T 0 −1 t=0

w(by ∗ (t))+4.

(5.161)

5.14 Proof of Theorem 5.24

165

Proposition 5.25 and (5.135) imply that there exist a natural number T1 > T0 and a program

 T1 −1 1 {x  (t)}Tt=0 , {y  (t)}t=0 which satisfies x  (t) = x ∗ (t), t = 0, . . . , T0 , y  (t) = y ∗ (t), t = 0, . . . , T0 − 1, T 1 −1

    w by  (t) ≥ U x ∗ (0), T1 − .

(5.162)

(5.163)

t=0

For all integers t satisfying T0 ≤ t < T1 , set y(t) = y  (t), x(t + 1) = (1 − d)x(t) + x  (t + 1) − (1 − d)x  (t).

(5.164)

In view of (5.161) and (5.162), x(t) ≥ x  (t) for all t = T0 , . . . , T1 . Evidently,

 T1 −1 1 {x(t)}Tt=0 , {y(t)}t=0 is a program. It follows from (5.161), (5.162), and (5.164) that ∗

U (x (0), T1 ) ≥

T 1 −1

w(by(t)) =

t=0

w(by(t)) +

T 0 −1 t=0

T 1 −1

T 1 −1

  w by  (t)

t=T0

t=0

> 4 +

= 4 +

T 0 −1

1 −1   T   w by ∗ (t) + w by  (t)

t=T0

  w by  (t) ≥ U (x ∗ (0), T1 ) + 3,

t=0

a contradiction. The contradiction we have reached proves Theorem 5.23.

5.14 Proof of Theorem 5.24 Assume that {x ∗ (t), y ∗ (t)}∞ t=0 is a weakly agreeable program. Then for all nonnegative integers t, we have u(x ∗ (t), x ∗ (t + 1)) = w(by ∗ (t)).

(5.165)

166

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

Put S0 = 0. By induction we define a strictly increasing sequence of integers Sk Sk −1 (k) (k) {Sk }∞ k=1 and a sequence of programs ({x (t)}t=0 , {y (t)}t=0 ). Assume that k ≥ 1 is an integer and we defined integers S0 < · · · < Sk−1 . Since the program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly agreeable, by Proposition 5.25 there exist an integer Sk −1 k Sk > k + Sk−1 and a program ({x (k) (t)}St=0 , {y (k) (t)}t=0 ) which satisfies x

(k)

∗

(t) = x (t), t = 0, . . . , k,

S k −1

   w by (k) (t) ≥ U x ∗ (0), Sk − 1/k.

t=0

(5.166)

Evidently, by (5.165), we may assume without loss of generality that y (k) (t) = y ∗ (t), t = 0, . . . , k − 1.

(5.167)

In such a manner, we define a sequence of integers {Sk }∞ k=1 and a sequence of Sk Sk −1 (k) (k) programs ({x (t)}t=0 , {y (t)}t=0 ) such that (5.166) and (5.167) hold for all integers k ≥ 1. Evidently, for all integers t ≥ 0, x ∗ (t) = lim x (k) (t), y ∗ (t) = lim y (k) (t). k→∞

k→∞

Assume now that {Sk }∞ k=1 is strictly increasing sequence of natural numbers and Sk −1 k a sequence of programs ({x (k) (t)}St=0 , {y (k) (t)}t=0 ), k = 1, 2, . . . satisfy (5.136), (5.137), and (5.138). We show that ({x ∗ (t), y ∗ (t)}∞ t=0 is weakly agreeable. Let  be a positive number and T0 ≥ 1 be an integer. Fix M0 > x ∗ (0) + max{(dai )−1 : i = 1, . . . , n}.

(5.168)

By (5.137), we may assume that k −1

   S U x ∗ (0), Sk − w by (k) (t) ≤ 1 for every natural number k.

(5.169)

t=0

Lemma 5.3, (5.136), and (5.168) imply that x (k) (t) ≤ M0 e, t = 0, . . . , Sk , k = 1, 2, . . . , x ∗ (t) ≤ M0 e, t = 0, 1, . . . . (5.170) Theorem 5.8 and (5.168) imply that there exist a program {x(t), y(t)}∞ t=0 and a positive number M1 such that x(0) = x ∗ (0),

5.14 Proof of Theorem 5.24

167

   Q   2 −1   w(by(t)) − μ(Q − Q ) 2 1  ≤ M1 for all integers Q1 ≥ 0, Q2 > Q1 .    t=Q1 (5.171) Theorem 5.5 implies that there exists M2 > 0 such that    U (M0 , p) − pμ ≤ M2 for all integers p ≥ 1.

(5.172)

By (5.169) and (5.171), for every natural number k, we have S k −1

S k −1

   ∗ (k) w by (t) ≥ U x (0), Sk − 1 ≥ w(by(t)) − 1 ≥ μSk − M1 − 1.

t=0

t=0

(5.173) By (5.170), (5.172), and (5.173), for every natural number k and every natural number S < Sk − 1, we have S−1 

 w by (k) (t)

=

Sk −1 t=0

  Sk −1  (k)  w by (k) (t) − t=S w by (t)

t=0

(M0 , Sk − S) ≥ μSk − M1 − 1 − U ≥ μSk − M1 − 1 − M2 − (Sk − S)μ = μS − M1 − 1 − M2 . (5.174) In view of (5.138) and (5.174), for all natural numbers S, we have S−1 

  w by ∗ (t) ≥ μS − M1 − 1 − M2 .

(5.175)

t=0

It follows from (5.8) and (5.175) that there exists a natural number τ > T0 + 4 such that y ∗ (τ ) > 0. Thus there exists an integer j ∈ {1, . . . , n} such that yj∗ (τ ) > 0.

(5.176)

Fix a number δ > 0 such that 4δ

n  (1 + ai ) < yj∗ (τ ),

(5.177)

i=1 n |w(bz1 ) − w(bz2 )| ≤ /4 for each z1 , z2 ∈ R+

satisfying z1 , z2 ≤ M0 e and z1 − z2  ≤ δ

n  (1 + ai ). i=1

(5.178)

168

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

In view of (5.136) and (5.137), there exists an integer k ≥ 1 such that k −1 

 S  w by (k) (t)) ≤ /8, Sk > τ + 4, U x ∗ (0), Sk −

(5.179)

t=0

 τ  τ      ∗ (k) w(by (t)) − w(by (t)) ≤ /8,    t=0

(5.180)

t=0

' ' ' ' ' ∗ ' ' ' 'x (t) − x (k) (t)' , 'y ∗ (t) − y (k) (t)' ≤ δ, t = 0, . . . , τ + 1.

(5.181)

x  (t) = x ∗ (t), t = 0, . . . , τ, y  (t) = y ∗ (t), t = 0, . . . , τ − 1,

(5.182)

Set

 

∗



∗

x (τ + 1) = x (τ + 1) + δe, y (τ ) = y (τ ) − δ

n 

 ai ej .

(5.183)

i=1

By (5.182) and (5.183), we have x  (τ + 1) ≥ (1 − d)x ∗ (τ ) = (1 − d)x  (τ ).

(5.184)

It follows from (5.177) and (5.183) that y  (τ ) ≥ 0.

(5.185)

It follows from (5.182) and (5.183) that   ey  (τ ) + a x  (τ + 1) − (1 − d)x  (τ )  n     ∗ = ey (τ ) − δ ai + a x ∗ (τ + 1) − (1 − d)x ∗ (τ ) + δae i=1

  = ey (τ ) + a x ∗ (τ + 1) − (1 − d)x ∗ (τ ) ≤ 1. ∗

(5.186)

+1 By (5.176), (5.182), and (5.184)–(5.186), ({x  (t)}τt=0 , {y  (t)}τt=0 ) is a program. In view of (5.181) and (5.183), we have

x  (τ + 1) ≥ x (k) (τ + 1).

(5.187)

For all integers t satisfying τ + 1 ≤ t < Sk , set y  (t) = y (k) (t), x  (t + 1) = (1 − d)x  (t) + x (k) (t + 1) − (1 − d)x (k) (t).

(5.188)

5.15 Weakly Maximal Programs

169

It follows from (5.187) and (5.188) that x  (t) ≥ x (k) (t), t = τ + 1, . . . , Sk .

(5.189)

It is easy to see that

 Sk −1 k {x  (t)}St=0 , {y  (t)}t=0 is a program. By (5.170), (5.178), (5.179), (5.180), (5.182), (5.183), and (5.188), we have S k −1

k −1 k −1

  S      S w by  (t) − U x ∗ (0), Sk ≥ w by  (t) − w by (k) (t) − /8

t=0

t=0

=

τ 

t=0

  w by  (t) −

t=0

≥

τ 

τ 

 w by (k) (t) − /8

t=0 τ      w by  (t) − w by ∗ (t) − /4

t=0

t=0

    = w by  (τ ) − w by ∗ (τ ) − /4 ≥ −/2 and S k −1

    w by  (t) − U x ∗ (0), Sk ≥ −/2.

(5.190)

t=0

Therefore for every  > 0 and every integer T0 ≥ 1, we showed the existence of a Sk −1 k , {y  (t)}t=0 ) which satisfies natural number Sk > T0 + 4 and a program ({x  (t)}St=0 y  (t) = y ∗ (t), x  (t) = x ∗ (t), t = 0, . . . , T0 , (5.179), and (5.190). ∞ ∗ Proposition 5.26 implies that ({x ∗ (t)}∞ t=0 , {y (t)}t=0 ) is a weakly agreeable program. Theorem 5.24 is proved.

5.15 Weakly Maximal Programs We prove three theorems obtained in [103].

170

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

We assume for simplicity that w(0) = 0 and begin with the following result which establishes the continuity of the function U (·, ·, T ). This result will be proved in Section 5.16. n , U (x , x˜ , T ) > 0, and Theorem 5.27 Let T > 0 be an integer, x0 , x˜0 ∈ R+ 0 0 d < 1. n is continuous at (x , x˜ ). Then the function (y, z) → U (y, z, T ), y, z ∈ R+ 0 0

The following result will be proved in Section 5.17. Theorem 5.28 Let M0 > max{(dai )−1 : i = 1, . . . , n}. Then there exists M1 > 0 such that for each good weakly maximal program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e and each pair of integers S1 ≥ 0 and S2 > S1 ,    S2 −1     w(by(t)) − μ(S − S ) 2 1  ≤ M1 .    t=S1

(5.191)

By definition for any good program {x(t), y(t)}∞ t=0 , there is a constant M1 > 0 such that (5.191) holds. In view of Theorem 5.28, the constant M1 depends only on the constant M0 , and the inequality (5.191) holds for all programs {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e. The next theorem will be proved in Section 5.18. Theorem 5.29 Let {x (k) (t), y (k) (t)}∞ t=0 , k = 1, 2, . . . be good weakly maximal programs. Assume that for any integer t ≥ 0, there exists x(t) = limk→∞ x (k) (t), (k) y(t) = limk→∞ yt . Then {x(t), y(t)}∞ t=0 is a good weakly maximal program.

5.16 Proof of Theorem 5.27 n satisfy U (x , x˜ , T ) > 0, Lemma 5.30 Let T be a natural number, x0 , x˜0 ∈ R+ 0 0 n satisfying and  > 0. Then there exists δ > 0 such that for each y0 , y˜0 ∈ R+ ||y0 − x0 ||, ||y˜0 − x˜0 || ≤ δ, the following inequality holds:

U (y0 , y˜0 , T ) < U (x0 , x˜0 , T ) + . Proof Let us assume the contrary. Then for each integer k ≥ 1, there exist (k) (k) n such that x0 , x˜0 ∈ R+ ' ' ' ' ' ' (k) ' (k) ' 'x0 − x0 ' ≤ k −1 , 'x˜0 − x˜0 '] ≤ k −1 ,

(5.192)



(k) (k) U x0 , x˜0 , T ≥ U (x0 , x˜0 , T ) + .

(5.193)

5.16 Proof of Theorem 5.27

171

T −1 For every natural number k, there exists a program ({x (k) (t)}Tt=0 , {y (k) (t)}t=0 ) such that (k)

(k)

(k)

x (k) (0) = x0 , xT ≥ x˜0 , T −1

w(by (k) (t)) ≥ U (x0 , x˜0 , T ) − k −1 . (k)

(k)

(5.194)

(5.195)

t=0

By (5.192), (5.193), and Lemma 5.3, the set   x (k) (t) : t = 0, 1, . . . , T , k = 1, 2, . . . is bounded. Extracting a subsequence and re-indexing, we may assume without loss of generality that for t = 0, . . . , T , there exists x(t) = lim x (k) (t) k→∞

(5.196)

and for t = 0, . . . , T − 1, there exists y(t) = lim y (k) (t). k→∞

(5.197)

T −1 It is clear that ({x(t)}Tt=0 , {y(t)}t=0 ) is a program. By (5.192), (5.194), and (5.196), (0)

x(0) = x0 , x(T ) ≥ x˜0 .

(5.198)

In view of (5.193), (5.195), (5.197), and (5.198),

U (x0 , x˜0 , T ) ≥

T −1 t=0

w(by(t)) = lim

k→∞

T −1

 w by (k) (t)

t=0

 (k) (k) = lim sup U x0 , x˜0 , T ≥ U (x0 , x˜0 , T ) + . k→∞

The contradiction we have reached proves the lemma. n be such that Lemma 5.31 Let T be a natural number, x0 , x˜0 ∈ R+

U (x0 , x˜0 , T ) > 0, n d < 1, and let  > 0. Then there exists δ > 0 such that for each y0 , y˜0 ∈ R+ satisfying

172

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

y0 − x0 ||, ||y˜0 − x˜0 || ≤ δ,

(5.199)

the following inequality holds: U (y0 , y˜0 , T ) ≥ U (x0 , x˜0 , T ) − . Proof Since U (x0 , x˜0 , T ) > 0, it follows from Lemma 5.3 and the continuity of w T −1 T , {y(t)} that there exists a program ({x(t)} ¯ t=0 ) such that t=0 ¯ ¯ ) ≥ x˜0 , x(0) ¯ = x0 , x(T T −1

w(by(t)) ¯ = U (x0 , x˜0 , T ) > 0.

(5.200)

t=0

By (5.200), there exists an integer τ1 ∈ [0, T − 1] such that w(by(τ ¯ 1 )) > 0; ¯ = 0. if an integer t satisfies 0 ≤ t < τ1 then w(by(t))

(5.201)

Then max{y¯i (τ1 ) : i = 1, . . . , n} > 0. Fix i1 ∈ {1, . . . , n} such that y¯i1 (τ1 ) > 0.

(5.202)

Choose a positive number Δ such that ¯ 1 )/8, Δ < 4−1 y¯i1 (τ1 ), Δ < (be)−1 by(τ ¯ 1 )) − /8. w(by(τ ¯ 1 ) − Δbe) > w(by(τ

(5.203)

We choose δ > 0 such that for all i = 1, . . . , n, δ < (4n)−1 (1 + ai )−1 Δ(1 − d)T .

(5.204)

n satisfy (5.199). Set Assume that y0 , y˜0 ∈ R+

x(0) = y0 . For each integer t satisfying 0 ≤ t < τ1 , set

(5.205)

5.16 Proof of Theorem 5.27

173

x(t + 1) = (1 − d)x(t) + x(t ¯ + 1) − (1 − d)x(t), ¯ y(t) = 0.

(5.206)

τ1 −1 1 , {y(t)}t=0 ) is a program. By (5.199), It is clear that if τ1 > 0, then ({x(t)}τt=0 (5.200), (5.205), and (5.206),

¯ 1 ) ≤ x(0) − x(0) ¯ = y0 − x0  ≤ δ. x(τ1 ) − x(τ

(5.207)

Note that (5.207) holds if τ1 > 0 and if τ1 = 0. Set yi (τ1 ) = max{y¯i (τ1 ) − δ, 0}, i ∈ {1, . . . , n} \ {i1 }, yi1 (τ1 ) = y¯i1 (τ1 ) − Δ, (5.208)

 x(τ1 + 1) = (1 − d)x(τ1 ) + x(τ ¯ 1 + 1) − (1 − d)x(τ ¯ 1 ) + n−1 a1−1 , . . . , an−1 Δ. (5.209) In view of (5.203), (5.204), and (5.206)–(5.208), 0 ≤ y(τ1 ) ≤ x(τ1 ).

(5.210)

x(τ1 + 1) ≥ (1 − d)x(τ1 ).

(5.211)

Clearly,

By (5.204) and (5.208), a(x(τ1 + 1) − (1 − d)x(τ1 )) + ey(τ1 ) = a(x(τ ¯ 1 + 1) − (1 − d)x(τ ¯ 1 )) + Δ + ey(τ1 ) ≤ a(x(τ ¯ 1 + 1) − (1 − d)x(τ ¯ 1 )) + Δ + ey(τ ¯ 1 ) − Δ ≤ 1. τ1 +1 1 , {y(t)}τt=0 ) is a Together with (5.210) and (5.211), this implies that ({x(t)}t=0 program. It follows from (5.204), (5.207), and (5.209) that for all i ∈ {1, . . . , n},

¯ 1 ) − x(τ ¯ 1 ) + n−1 ai−1 Δ xi (τ1 + 1) ≥ x¯i (τ1 + 1) − x(τ ≥ x¯i (τ1 + 1) − δ + n−1 ai−1 Δ ≥ x¯i (τ1 + 1) + 2−1 n−1 ai−1 Δ and 

¯ 1 + 1) + (2n)−1 Δ a1−1 , . . . , an−1 . x(τ1 + 1) ≥ x(τ By (5.201), (5.203), (5.204), and (5.208),

(5.212)

174

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function τ1 

τ1 

w(by(t)) −

t=0

=w

 n  

t=0



≥ w max

≥w



bi yi (τ1 ) − w

i=1

 n 

w(by(t)) ¯ = w(by(τ1 )) − w(by(τ ¯ 1 ))  n 

 bi y¯i (τ1 )

i=1 n 

bi (y¯i (τ1 ) − Δ), 0

i=1

bi y¯i (τ1 ) − Δ

i=1



n 

−w

 bi

−w

 n 

i=1

 n  i=1

 bi y¯i (τ1 ) 

bi y¯i (τ1 ) > −/8.

(5.213)

i=1

If τ1 + 1 = T , then by (5.199), (5.200), (5.204), and (5.212), ¯ 1 + 1) + δe ≥ x˜0 + δe ≥ y˜0 x(T ) = x(τ1 + 1) ≥ x(τ

(5.214)

and by (5.200), (5.205), (5.213), and (5.214), U (y0 , y˜0 , T ) ≥

τ1  t=0

w(by(t)) ≥

τ1 

w(by(t)) ¯ − /8 = U (x0 , x˜0 , T ) − /8.

t=0

Thus in the case τ1 + 1 = T , the assertion of the lemma holds. Assume that τ1 + 1 < T . For each integer t satisfying τ1 + 1 ≤ t < T , set x(t + 1) = (1 − d)x(t) + x(t ¯ + 1) − (1 − d)x(t), ¯ y(t) = y(t). ¯

(5.215)

−1 ) is a program. By (5.212), In view of (5.212) and (5.215), ({x(t)}Tt=0 , {y(t)}Tt=0 and (5.215),

¯ 1 + 1)) x(T ) − x(T ¯ ) = (1 − d)T −τ1 −1 (x(τ1 + 1) − x(τ ≥ (1 − d)T −τ1 −1 (2n)−1 Δ(a1−1 , . . . , an−1 ) ≥ (1 − d)T (2n)−1 Δ(a1−1 , . . . , an−1 ). Together with (5.199), (5.200), and (5.204), this implies that 

x(T ) ≥ x(T ¯ ) + (1 − d)T (2n)−1 Δ a1−1 , . . . , an−1 ≥ x˜0 + δe ≥ y˜0 . By (5.200), (5.205), (5.213), (5.215), and (5.216) U (y0 , y˜0 , T ) − U (x0 , x˜0 , T ) ≥

T −1 t=0

w(by(t)) −

T −1 t=0

w(by(t)) ¯

(5.216)

5.17 Proof of Theorem 5.28

=

τ1 

175

w(by(t)) −

t=0

τ −1 

w(by(t)) ¯ > −/8.

t=0

Thus the assertion of the lemma holds. Lemma 5.31 is proved. Theorem 5.27 now easily follows from Lemmas 5.30 and 5.31.

5.17 Proof of Theorem 5.28 There is δ > 0 such that w(t) < μ/8 for each t ∈ [0, δ].

(5.217)

Let M > 0 be as guaranteed by Theorem 5.8. By Corollary 5.6, there is M˜ > 0 such that the following property holds: (P8) for each program {x(t), y(t)}∞ t=0 satisfying x(0) ≤ M0 e and each integer T ≥ 1, T −1

˜ [w(by(t)) − μ] ≤ M.

t=0

By Lemma 5.10, there exists a natural number p ≥ 4 such that the following property holds: −1 ) which satisfies (P9) for each integer τ ≥ p, each program ({x(t)}τt=0 , {y(t)}τt=0 x(0) ≤ M0 e, by(τ − 1) ≥ δ, n which satisfies x˜ ≤ M e, there exists a program and each x˜0 ∈ R+ 0 0

 τ −1 τ {x(t)} ¯ , { y(t)} ¯ t=0 t=0 such that x(0) ¯ = x˜0 , x(τ ¯ ) ≥ x(τ ). Set M1 = 2M˜ + M + pμ. Assume that {x(t), y(t)}∞ t=0 is a good program such that

(5.218)

176

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

x(0) ≤ M0 e

(5.219)

and that for all integers T ≥ 1, T −1

w(by(t)) = U (x(0), x(T ), T ).

(5.220)

t=0 ∞ such By the choice of M and Theorem 5.8, there exists a program {x(t), ¯ y(t)} ¯ t=0 that

x(0) ¯ = x(0)

(5.221)

and that for each integer S1 ≥ 0 and each integer S2 > S1 , |

S 2 −1

w(by(t)) ¯ − μ(S2 − S1 )| ≤ M.

(5.222)

t=S1

By (5.222), there exists an increasing sequence of natural numbers{Sk }∞ k=1 such that w(by(Tk − 1)) ≥ μ/2, k = 1, 2, . . .

(5.223)

In view of (5.217) and (5.223), by(Tk − 1) > δ, k = 1, 2, . . .

(5.224)

We may assume without loss of generality that Tk > p for all integers k ≥ 1. Let k ≥ 1 be an integer, and set x(t) ˜ = x(t), ¯ t = 0, . . . , Tk − p, y(t) ˜ = y(t), ¯ t = 0, . . . , Tk − p − 1.

(5.225)

By (5.219), (5.221), (5.224), (5.225), Lemma 5.3, and property (P9) applied to the program

 Tk −1 k {x(t)}Tt=T , , {y(t)} t=T −p −p k k Tk Tk −1 there exists a program ({x(t)} ˜ ˜ t=Tk −p , {y(t)} t=Tk −p ) such that

x(T ˜ k ) ≥ x(Tk ). By (5.220)–(5.222), (5.225), and (5.226),

(5.226)

5.17 Proof of Theorem 5.28 T k −1

177

w(by(t)) = U (x(0), x(Tk ), Tk ) ≥

t=0

T k −1

w(by(t)) ˜

t=0

=

Tk −p−1

w(by(t)) ¯ +

t=0

T k −1

w(by(t)) ˜

t=Tk −p

≥ μ(Tk − p) − M ≥ μ(Tk ) − M − pμ. Thus for each integer k ≥ 1, T k −1

w(by(t)) ≥ μ(Tk ) − M − pμ.

(5.227)

t=0

Assume that integers S1 ≥ 0 and S2 > S1 . Choose a natural number k such that Tk > S2 . By (P8), Lemma 5.3, (5.219), and (5.227), S 2 −1 t=S1

w(by(t)) =

T k −1

w(by(t))

t=0

− −



{t is an integer and 0 ≤ t ≤ S1 − 1 : w(by(t))}

T k −1

w(by(t))

t=S2

˜ − [Tk − S2 ]μ − M˜ = μ(S2 − S1 ) ≥ μTk − M − pμ − [μS1 + M] − M − pμ − 2M˜

(5.228)

(here we assume that the sum over an empty set is zero). In view of (5.219), (P8), and Lemma 5.14, S 2 −1

w(by(t)) ≤ M˜ + (S2 − S1 )μ.

t=S1

Together with (5.218) and (5.228), this implies that    S2 −1     ˜ + M + μp = M1 . w(by(t)) − μ(S − S ) 2 1  ≤ 2M    t=S1 Theorem 5.28 is proved.

178

5 The Robinson–Solow–Srinivasan Model with a Nonconcave Utility Function

5.18 Proof of Theorem 5.29 Clearly, {x(t), y(t)}∞ t=0 is a program. By Theorem 5.28, there is M1 > 0 such that for each integer k ≥ 1 and each pair of integers S1 ≥ 0 and S2 > S1 ,    S   2 −1 (k)  w(by (t)) − μ(S2 − S1 ) ≤ M1 .    t=S1

(5.229)

This implies that for each pair of integers S1 ≥ 0 and S2 > S1 ,    S   2 −1  (w(by(t)) − μ(S2 − S1 ) ≤ M1 .    t=S1

(5.230)

In view of (5.230), the program {x(t), y(t)}∞ t=0 is good. By (5.230) there is a strictly increasing sequence of natural numbers {Tj }∞ j =1 such that w(by(Tj − 1)) > μ/2 for all natural numbers j.

(5.231)

Let j ≥ 1 be an integer. Applying Theorem 5.27, we obtain that Tj −1

 t=0

Tj −1

w(by(t)) = lim

k→∞



w(by (k) (t))

t=0

= lim U (x (k) (0), x (k) (Tj ), Tj ) = U (x(0), x(Tj ), Tj ). k→∞

This implies that {x(t), y(t)}∞ t=0 is weakly maximal. Theorem 5.29 is proved.

Chapter 6

Infinite Horizon Nonautonomous Optimization Problems

In this chapter we study infinite horizon optimal control problems with nonautonomous optimality criterions. The utility functions, which determine the optimality criterion, are nonconcave. The class of models contains, as a particular case, the Robinson–Solow–Srinivasan model. We establish the existence of good programs and optimal programs.

6.1 The Model Description and Main Results 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}.

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi ,

i=1

and let x >> y, x > y, x ≥ y have their usual meaning. n Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x2 its Euclidean norm in R n . We assume that  ·  is a norm in R n . For every mapping a : X → 2Y \ {∅}, where X, Y are nonempty sets, set graph(a) = {(x, y) ∈ X × Y : y ∈ a(x)}. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_6

179

180

6 Infinite Horizon Nonautonomous Optimization Problems

Let K be a nonempty compact subset of R n . Denote by P(K) the set of all nonempty closed subsets of K. For every pair of nonempty sets A, B ⊂ R n , define   H (A, B) = sup sup inf x − y, sup inf x − y . x∈A y∈B

y∈B x∈A

(6.1)

For every nonnegative integer t, let at : K → P(K) be such that graph(at ) is a closed subset of R n × R n . Assume that there exists a number κ ∈ (0, 1) such that for every pair of points x, y ∈ K and every nonnegative integer t, H (at (x), at (y)) ≤ κx − y

(6.2)

and that for every nonnegative integer t, the upper semicontinuous function ut : {(x, x  ) ∈ K × K, x  ∈ at (x)} → [0, ∞) satisfies sup{sup{ut (x, x  ) : (x, x  ) ∈ graph(at )} : t = 0, 1, . . . } < ∞.

(6.3)

A sequence {x(t)}∞ t=0 ⊂ K is called a program if x(t + 1) ∈ at (x(t)) for every nonnegative integer t. 2 Let T1 , T2 be integers such that T1 < T2 . A sequence {x(t)}Tt=T ⊂ K is called a 1 program if x(t + 1) ∈ at (x(t)) for every integer t satisfying T1 ≤ t < T2 . We suppose that the following assumptions hold: (A1) for every positive number δ, there exists a positive number λ such that if an integer t ≥ 0 and if (x, x  ) ∈ graph(at ) satisfies ut (x, x  ) ≥ δ, then there exists z ∈ at (x) satisfying z ≥ x  + λe; (A2) there exist a program { x (t)}∞ t=0 and a positive number Δ such that for every nonnegative integer t; ut ( x (t), x (t + 1)) ≥ Δ (A3) for every nonnegative integer t, every (x, y) ∈ graph(at ), and every x˜ ∈ K which satisfies x˜ ≥ x, there exists y˜ ∈ at (x) ˜ for which ˜ y) ˜ ≥ ut (x, y). y˜ ≥ y, ut (x, In the sequel we assume that supremum of empty set is −∞. For every point x0 ∈ K and every natural number T , define U (x0 , T ) = sup

T −1  t=0

ut (x(t), x(t + 1)) :

6.1 The Model Description and Main Results

181

T −1 {x(t)}t=0

is a program and x(0) = x0 .

(6.4)

Let x0 , x˜0 ∈ K and let T ≥ 1 be an integer. Define U (x0 , x˜0 , T ) = sup

T −1 

T −1 ut (x(t), x(t + 1)) : {x(t)}t=0 is a program such that

t=0



x(0) = x0 , x(T ) ≥ x˜0 .

(6.5)

Let T ≥ 1 be an integer. Define (T ) = sup U

T −1 

ut (x(t), x(t + 1)) :

T −1 {x(t)}t=0

is a program .

(6.6)

t=0

The results presented in this section were obtained in [99]. Upper semicontinuity of ut , t = 0, 1, . . . implies the following two propositions. Proposition 6.1 For every x0 ∈ K and every integer T ≥ 1, there exists a program {x(t)}Tt=0 which satisfies x(0) = x0 and T −1

ut (x(t), x(t + 1)) = U (x0 , T ).

t=0

Proposition 6.2 For every integer T ≥ 1, there exists a program {x(t)}Tt=0 T −1 (T ). satisfying t=0 ut (x(t), x(t + 1)) = U For every x0 ∈ K and every pair of integers T1 < T2 , define U (x0 , T1 , T2 ) = sup

⎧ 2 −1 ⎨T ⎩

ut (x(t), x(t + 1)) :

t=T1

T2 −1 is a program and x(T1 ) = x0 {x(t)}t=T 1

⎫ ⎬ ⎭

.

(6.7)

Upper semicontinuity of ut , t = 0, 1, . . . implies the following result. Proposition 6.3 For every x0 ∈ K and every pair of integers T1 < T2 , there exists 2 such that x(T1 ) = x0 and a program {x(t)}Tt=T 1

182

6 Infinite Horizon Nonautonomous Optimization Problems T 2 −1

ut (x(t), x(t + 1)) = U (x0 , T1 , T2 ).

t=T1

Let x0 , x˜0 ∈ K and let T1 < T2 be integers. Define U (x0 , x˜0 , T1 , T2 ) = sup

⎧ 2 −1 ⎨T ⎩

2 ut (x(t), x(t + 1)) : {x(t)}Tt=T is a program and 1

t=T1

x(T1 ) = x0 , x(T2 ) ≥ x˜0

⎫ ⎬ ⎭

(6.8)

.

Let T1 , T2 be integers such that T1 < T2 . Define (T1 , T2 ) = sup U

⎧ 2 −1 ⎨T ⎩

t=T1

⎫ ⎬

2 ut (x(t), x(t + 1)) : {x(t)}Tt=T is a program . 1 ⎭

(6.9)

We will prove the following theorem which is the main result of this section. Theorem 6.4 There exists a positive number M such that for every x0 ∈ K, ∞ such that x(0) there exists a program {x(t)} ¯ ¯ = x0 and that for every pair of t=0 nonnegative integers T1 , T2 satisfying T1 < T2 , the inequality   T   2 −1   ut (x(t), ¯ x(t ¯ + 1)) − U (T1 , T2 ) ≤ M   t=T1  holds. Moreover, for every natural number T , T −1

ut (x(t), ¯ x(t ¯ + 1)) = U˜ (x(0), ¯ x(T ¯ ), 0, T ),

t=0

if the following properties hold: • for every nonnegative integer t and every (z, z ) ∈ graph(at ) satisfying ut (z, z ) > 0, the function ut is continuous at the point (z, z ); • for every nonnegative integer t and each z, z1 , z2 , z3 ∈ K which satisfy z1 ≤ z2 ≤ z3 and zi ∈ at (z), i = 1, 3, the inclusion z2 ∈ at (z) is valid. ∞ whose existence is guaranteed by Theorem 6.4 in infinite The program {x(t)} ¯ t=0 horizon optimal control is considered as an (approximately) optimal program.

Theorem 6.5 Assume that {x(t)}∞ t=0 is a program; that there exists a positive number M such that for every natural number T ,

6.1 The Model Description and Main Results T −1

183

ut (x(t), x(t + 1)) ≥ U (0, T , x(0), x(T )) − M0 ;

t=0

and that lim sup ut (x(t), x(t + 1)) > 0. t→∞

Then there exists positive number M1 such that for every pair of integers T1 ≥ 0 satisfying T2 > T1 , the inequality    T   2 −1   u (x(t), x(t + 1)) − U (T , T ) t 1 2  ≤ M1    t=T1 holds. Theorem 6.4 is proved in Section 6.6, while Theorem 6.5 is obtained in Section 6.7. Let M > 0 be as guaranteed by Theorem 6.4. ∞ be as guaranteed by Proposition 6.6 Let x0 ∈ K, and let a program {x(t)} ¯ t=0 ∞ Theorem 6.4. Assume that {x(t)}t=0 is a program. Then either the sequence

T −1 

ut (x(t), x(t + 1)) −

t=0

T −1

∞ ut (x(t), ¯ x(t ¯ + 1)) T =1

t=0

is bounded or T −1

ut (x(t), x(t + 1)) −

t=0

T −1

ut (x(t), ¯ x(t ¯ + 1)) → −∞ as T → ∞.

t=0

Proof Assume that the sequence T −1 

ut (x(t), x(t + 1)) −

t=0

T −1

∞ ut (x(t), ¯ x(t ¯ + 1)) T =1

t=0

is not bounded. Then by Theorem 6.4, lim inf T →∞

T −1  t=0

ut (x(t), x(t + 1)) −

T −1

 ut (x(t), ¯ x(t ¯ + 1)) = −∞.

t=0

Let Q > 0. Then there exists a natural number T0 which satisfies

(6.10)

184

6 Infinite Horizon Nonautonomous Optimization Problems T 0 −1

ut (x(t), x(t + 1)) −

t=0

T 0 −1

ut (x(t), ¯ x(t ¯ + 1)) < −Q − M.

(6.11)

t=0

∞ , and Theorem 6.4, for every natural number In view of (6.11), the choice of {x(t)} ¯ t=0 T > T0 , we have T −1

ut (x(t), x(t + 1)) −

t=0

−

T −1

ut (x(t), ¯ x(t ¯ + 1)) =

t=0 T 0 −1 t=0

ut (x(t), ¯ x(t ¯ + 1)) +

T 0 −1

ut (x(t), x(t + 1))

t=0 T −1

ut (x(t), x(t + 1)) −

t=T0

(T0 , T ) − < −Q − M + U

T −1

T −1

ut (x(t), ¯ x(t ¯ + 1))

t=T0

ut (x(t), ¯ x(t ¯ + 1)) < −Q.

t=T0

Since Q is an arbitrary positive number, we conclude that (6.10) holds. This completes the proof of Proposition 6.6. Now assume that ut = u0 and at = a0 , t = 0, 1, . . . . Let a positive number M be as guaranteed by Theorem 6.4, and set u = u0 , a = a0 . The following result will be proved in Section 6.8. (0, p)/p and Theorem 6.7 There exists μ = limp→∞ U (0, p) − μ| ≤ 2M/p for all natural numbers p. |p−1 U

6.2 Upper Semicontinuity of Cost Functions For every nonnegative integer t, let at : K → P(K) be such that graph(at ) is a closed set, and assume that for every nonnegative integer t, an upper semicontinuous n → [0, ∞) satisfies function φt : R+ n } : t = 0, 1, . . . } < ∞. sup{sup{φt (z) : z ∈ (K − K) ∩ R+

(6.12)

For every nonnegative integer t and every point (x, x  ) ∈ graph(at ), set n , x  + z ∈ a(x)}. ut (x, x  ) = sup{φt (z) : z ∈ R+

(6.13)

By (6.12) and (6.13), ut , t = 0, 1, . . . satisfy (6.3). Note that in many models of economic dynamics, cost functions ut , t = 0, 1, . . . are defined by (6.13).

6.2 Upper Semicontinuity of Cost Functions

185

Lemma 6.8 For every nonnegative integer t, the function ut : [0, ∞) is upper semicontinuous.

graph(at ) →

Proof Let t be a nonnegative integer, and let {(x (j ) , y (j ) )}∞ j =1 ⊂ graph(at ) satisfy lim (x (j ) , y (j ) ) = (x, y).

j →∞

(6.14)

We claim that ut (x, y) ≥ lim sup u(x (j ) , y (j ) ). j →∞

Extracting a subsequence and re-indexing if necessary, we may assume without loss of generality that there exists lim u(x (j ) , y (j ) ).

j →∞

n satisfying In view of (6.13), for every natural number j , there exists z(j ) ∈ R+

y (j ) + z(j ) ∈ at (x (j ) ), φt (z(j ) ) ≥ ut (x (j ) , y (j ) ) − 1/j.

(6.15)

Evidently, the sequence {z(j ) }∞ j =1 is bounded. Extracting a subsequence and reindexing, if necessary, we may assume without loss of generality that there exists z = lim z(j ) . j →∞

In view of (6.14)–(6.16), we have z≥0 and (x, y + z) = lim (x (j ) , y (j ) + z(j ) ) ∈ graph(at ). j →∞

Together with (6.14)–(6.16), the relation above implies that ut (x, y) ≥ φt (z) ≥ lim sup φt (z(j ) ) ≥ lim sup[ut (x (j ) , y (j ) ) − 1/j ] j →∞

j →∞

= lim ut (x (j ) , y (j ) ). j →∞

This completes the proof of Lemma 6.8.

(6.16)

186

6 Infinite Horizon Nonautonomous Optimization Problems

6.3 The Nonstationary Robinson–Solow–Srinivasan Model In this section we consider a subclass of the class of infinite horizon optimal control problems considered in Section 6.1. Infinite horizon problems of this subclass correspond to the nonstationary Robinson–Solow–Srinivasan models. For every nonnegative integer t, let (t)

(t)

α (t) = (α1 , . . . , αn(t) ) >> 0, b(t) = (b1 , . . . , bn(t) ) >> 0,

(6.17)

(t)

d (t) = (d1 , . . . , dn(t) ) ∈ ((0, 1])n and for every nonnegative integer t, let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that wt (0) = 0, inf{wt (z) : t = 0, 1, . . . } > 0 for all z > 0

(6.18)

and such that the following assumption holds: (A4) for every positive number , there exists a positive number δ such that for every nonnegative integer t and every z ∈ [0, δ], the inequality wt (z) ≤  is valid. n , define Let t be a nonnegative integer. For every x ∈ R+

 (t) n : yi ≥ (1 − di )xi , i = 1, . . . , n, at (x) = y ∈ R+ n  (t) (t) αi (yi − (1 − di )xi ) ≤ 1 .

(6.19)

i=1

It is easy to see that for every x ∈ R n , at (x) is a nonempty closed bounded subset n and graph(a ) is a closed subset of R n × R n . Assume that of R+ t + + (t)

inf{di

: i = 1, . . . , n, t = 0, 1, . . . } > 0,

inf{eb(t) : t = 0, 1, . . . } > 0, (t)

(6.20) (6.21)

inf{αi : i = 1, . . . , n, t = 0, 1, . . . } > 0,

(6.22)

sup{bi(t) : i = 1, . . . , n, t = 0, 1, . . . } < ∞,

(6.23)

(t)

sup{αi : i = 1, . . . , n, t = 0, 1, . . . } < ∞ and that for every positive number M, we have

(6.24)

6.3 The Nonstationary Robinson–Solow–Srinivasan Model

187

sup{wt (M) : t = 0, 1, . . . } < ∞, inf{wt (M) : t = 0, 1, . . . } > 0.

(6.25)

The constraint mappings at , t = 0, 1, . . . have already been defined. Let us now define the cost functions ut , t = 0, 1, . . . . For every nonnegative integer t and every (x, x  ) ∈ graph(at ), put  

ut (x, x ) = sup wt (b(t) y) : 0 ≤ y ≤ x, ey +

n 

(t) αi (xi

(t) − (1 − di )xi )

≤1 .

(6.26)

i=1

Fix numbers α ∗ , α∗ > 0, d∗ > 0 which satisfy α∗ < αi < α ∗ , d∗ < di , i = 1, . . . , n, t = 0, 1, . . . . (t)

(t)

(6.27)

Lemma 6.9 Let a number M0 > (α∗ d∗ )−1 , let an integer t ≥ 0, and let (x, x  ) ∈ graph(at ) satisfy x ≤ M0 e. Then x  ≤ M0 e. Proof In view of (6.19), n 

αi (xi − (1 − di )xi ) ≤ 1 (t)

(t)

i=1

and by (6.27), for every i = 1, . . . , n, we have xi ≤ (αi )−1 + (1 − di )xi ≤ α∗−1 + (1 − d∗ )xi ≤ α∗−1 + (1 − d∗ )M0 (t)

(t)

≤ d∗ (α∗ d∗ )−1 + (1 − d∗ )M0 ≤ d∗ M0 + (1 − d∗ )M0 = M0 . This completes the proof of Lemma 6.9. Lemma 6.10 Let t be a nonnegative integer. Then the function ut : graph(at ) → [0, ∞) is upper semicontinuous. Moreover, if (x, y) ∈ graph(at ) and ut (x, y) > 0, then ut is continuous at (x, y). Proof Let (j ) (j ) (x, y) ∈ graph(at ), {(x (j ) , y (j ) )}∞ j =1 ⊂ graph(at ), lim (x , y ) = (x, y). j →∞

(6.28) We claim that

188

6 Infinite Horizon Nonautonomous Optimization Problems

ut (x, y) ≥ lim sup ut (x (j ) , y (j ) ). j →∞

Extracting a subsequence and re-indexing, we may assume that there exists limj →∞ ut (x (j ) , y (j ) ). In view of (6.25) and (6.26), for every natural number n which satisfies j , there exists z(j ) ∈ R+ z

(j )

≤x

(j )

n 

(j )

(6.29)

wt (b(t) z(j ) ) ≥ ut (x (j ) , y (j ) ) − 1/j.

(6.30)

, ez

(j )

+

(t)

(j )

αi (yi

(t)

− (1 − di )xi ) ≤ 1,

i=1

Extracting a subsequence and re-indexing, we may assume without loss of generality that there exists z = lim z(j ) .

(6.31)

0 ≤ z ≤ x.

(6.32)

j →∞

By (6.28) and (6.31),

It follows from (6.28), (6.29), and (6.31) that ez +

n 

(t)

αi

i=1

= lim

j →∞

 (t) yi − (1 − di )xi

ez

(j )

+

n 

αi(t)



 (j ) (t) (j ) yi − (1 − di )xi ≤ 1.

i=1

Combined with (6.26) and (6.30)–(6.32), the equation above implies that ut (x, y) ≥ wt (b(t) z) = lim wt (b(t) z(j ) ) = lim ut (x (j ) , y (j ) ). j →∞

j →∞

Hence ut is upper lower semicontinuous. Assume now that (x, y) ∈ graph(at ) satisfies ut (x, y) > 0,

(6.33)

and show that ut is continuous at (x, y). Evidently, it is sufficient to show that ut is lower semicontinuous at (x, y). Assume that

6.3 The Nonstationary Robinson–Solow–Srinivasan Model

189

(x (j ) , y (j ) ) ∈ graph(at ) for all natural numbers j, lim (x (j ) , y (j ) ) = (x, y). j →∞

(6.34) Let  be a positive number. It is sufficient to show that lim inf ut (x (j ) , y (j ) ) ≥ ut (x, y) − . j →∞

n for which It follows from (6.26) and (6.33) that there exists z ∈ R+ n 

(t)

(t)

αi (yi − (1 − di )xi ) ≤ 1,

(6.35)

wt (b(t) z) > 0, wt (b(t) z) > ut (x, y) − /4.

(6.36)

z ≤ x, ez +

i=1

By (6.18) and (6.36), there exists q ∈ {1, . . . , n} satisfying bq(t) zq > 0.

(6.37)

Relations (6.18) and (6.37) imply that there exists γ ∈ (0, 1) which satisfies wt (b(t) γ z) ≥ wt (b(t) z) − /4.

(6.38)

In view of (6.34), (6.35), and (6.37), there exists an integer j0 ≥ 1 such that for every natural number j ≥ j0 , we have γ z ≤ x (j ) , e(γ z) +

n 

(j )

αi(t) (yi

(j )

− (1 − di(t) )xi ) ≤ 1.

(6.39)

i=1

It follows from (6.26), (6.36), (6.38), and (6.39) that for all natural numbers j ≥ j0 , we have ut (x (j ) , y (j ) ) ≥ wt (b(t) γ z) ≥ wt (b(t) z) − /4 > ut (x, y) − /2. This implies that ut is lower semicontinuous at (x, y) and completes the proof of Lemma 6.10. For every vector x = (x1 , . . . , xn ) ∈ R n , put x1 =

n 

|xi |, x∞ = max{|xi | : i = 1, . . . , n}.

(6.40)

i=1

In view of (6.19) and (6.27), for every nonnegative integer t, for every pair of points x, y ∈ K, and for || · || = || · ||p , where p = 1, 2, ∞, we have

190

6 Infinite Horizon Nonautonomous Optimization Problems (t)

(t)

H (at (x), at (y)) ≤ ((1 − di )xi )ni=1 − ((1 − di )yi )ni=1 || ≤ (1 − d∗ )||x − y|| (6.41) (see (6.2)). Proposition 6.11 Let δ be a positive number. Then there exists a positive number λ such that for every nonnegative integer t and every (x, y) ∈ graph(at ) satisfying ut (x, y) ≥ δ, the inclusion y + λe ∈ at (x) is valid. Proof Assumption (A4) implies that there exists a positive number δ0 such that for 1 such that w (ξ ) ≥ δ/2, we have every nonnegative integer t and every ξ ∈ R+ t ξ ≥ δ0 .

(6.42)

b∗ = sup{bi : t = 0, 1, . . . , i = 1, . . . , n}

(6.43)

Put (t)

(see (6.23)). Fix number λ for which λnα ∗ < 2−1 b∗−1 δ0 .

(6.44)

(x, y) ∈ graph(at ), ut (x, y) ≥ δ.

(6.45)

Assume that an integer t ≥ 0,

n which satisfies In view of (6.26) and (6.45), there exists z ∈ R+

0 ≤ z ≤ x, ez +

n 

(t)

(t)

αi (yi − (1 − di )xi ) ≤ 1, wt (b(t) z) ≥ δ/2.

(6.46)

i=1

By (6.46) and the choice of δ0 , we have b(t) z ≥ δ0 .

(6.47)

Relations (6.43) and (6.47) imply that ez =

n  i=1

zi =

n 

(bi )−1 bi zi ≥ b∗−1 b(t) z ≥ b∗−1 δ0 . (t)

(t)

(6.48)

i=1

We claim that y+λe ∈ at (x). Evidently, (see (6.19) and (6.45)) for any i = 1, . . . , n, (t)

yi + λ ≥ yi ≥ (1 − di )xi . In view of (6.27), (6.44), (6.46), and (6.48),

(6.49)

6.3 The Nonstationary Robinson–Solow–Srinivasan Model n 

αi(t) ((y + λe)i − (1 − di(t) )xi ) =

n 

i=1

191

αi(t) (yi − (1 − di(t) )xi ) + λ

n 

i=1

αi(t)

i=1

≤ 1 − ez + λ

n 

αi ≤ 1 − b∗−1 δ0 + λnα ∗ < 1 (t)

i=1

and combined with (6.49), this implies that y +λe ∈ at (x). This completes the proof of Proposition 6.11. Proposition 6.12 There exist a program { x (t)}∞ t=0 and Δ > 0 such that for all integers t ≥ 0. x (t), x (t + 1)) ≥ Δ ut ( Proof Fix numbers λ0 > 0, λ1 > 0 such that λ0 nα ∗ < 1/2, λ1 < λ0 , λ1 n < 1/4.

(6.50)

In view of (6.21), there exists a positive number 0 satisfying eb(t) ≥ 0 , t = 0, 1, . . . .

(6.51)

= inf{wt (λ1 0 ) : t = 0, 1, . . . }. Δ

(6.52)

Set

> 0. Define Relation (6.25) implies that Δ y (t) = λ1 e, t = 0, 1, . . . x (t) = λ0 e, t = 0, 1, . . . ,

(6.53)

It follows from (6.27), (6.50), and (6.53) that for i = 1, . . . , n and t = 0, 1, . . . , we have xi (t) = λ0 di(t) > 0, xi (t + 1) − (1 − di(t) ) n 

=

 n 

(t)

(t)

αi [ xi (t + 1) − (1 − di ) xi (t)]

i=1 (t) (t) αi di

 λ0 ≤ λ0

i=1

n 

αi ≤ λ0 nα ∗ < 1/2 (t)

(6.54)

(6.55)

i=1

and for all t = 0, 1, . . . , we have e y (t) +

n  i=1

αi(t) [ xi (t + 1) − (1 − di(t) ) xi (t)] ≤ λ1 n + 1/2 < 1.

(6.56)

192

6 Infinite Horizon Nonautonomous Optimization Problems

Thus { x (t)}∞ t=0 is a program. In view of (6.26), (6.50)–(6.53), and (6.56), for every nonnegative integer t, we have x (t), x (t + 1)) ≥ wt (b(t) y (t)) ≥ wt (λ1 eb(t) ) ≥ wt (λ1 0 ) ≥ Δ. ut ( This completes the proof of Proposition 6.12. Proposition 6.13 Let t be a nonnegative integer, let (x, y) ∈ graph(at ), and let a n satisfy x˜ ≥ x. Then there exists y˜ ∈ a (x) point x˜ ∈ R+ t ˜ which satisfies y˜ ≥ y and ut (x, ˜ y) ˜ ≥ ut (x, y). n which satisfies Proof In view of (6.26), there exists z ∈ R+

0 ≤ z ≤ x, ez +

n 

(t)

(t)

αi (yi − (1 − di )xi ) ≤ 1, wt (b(t) z) = ut (x, y).

(6.57)

i=1

For all integer i = 1, . . . , n, define y˜i = x˜i (1 − di(t) ) + yi − (1 − di(t) )xi .

(6.58)

It follows from (6.19), (6.57), and (6.58) that for i = 1, . . . , n, we have (t)

y˜i ≥ (1 − di )x˜i , n 

αi(t) (y˜i − (1 − di(t) )x˜i ) =

i=1

n 

αi(t) (yi − (1 − di(t) )xi ) ≤ 1 − ez.

i=1

Thus y˜ ∈ at (x). ˜ By the inequality x˜ ≥ x and (6.58), y˜ ≥ y. It is not difficult to see that ˜ y) ˜ ≥ wt (b(t) z) = ut (x, y). ut (x, Proposition 6.13 is proved. It is not difficult to see that the following result holds. n , x ∈ a (x), i = 1, 3, Proposition 6.14 Let an integer t ≥ 0, x, x1 , x2 , x3 ∈ R+ i t x1 ≤ x2 ≤ x3 . Then x2 ∈ a(xt ).

Thus we have defined the mappings at and the cost functions ut , t = 0, 1, . . . . The control system considered in this section is a special case of the control system studied in Section 6.1. As we have already mentioned before, this control system corresponds to the nonstationary Robinson–Solow–Srinivasan model. Note that this control system satisfies the assumptions posed in Section 6.1 and therefore all the results stated there hold for this system. Indeed, choose M0 > (α∗ d∗ )−1 , and put n : z ≤ M e}. Lemma 6.9 implies that a (K) ⊂ K, t = 0, 1, . . . . K = {z ∈ R+ 0 t

6.4 Auxiliary Results for Theorems 6.4, 6.5, and 6.7

193

Relation (6.2) follows from (6.41). It is clear that (6.3) holds. By Lemma 6.10, ut is upper semicontinuous for every nonnegative integer t. Proposition 6.11 implies (A1). Assumption (A2) follows from Proposition 6.12, and assumption (A3) follows from Proposition 6.13.

6.4 Auxiliary Results for Theorems 6.4, 6.5, and 6.7 Lemma 6.15 Let δ be a positive number. Then there exists an integer T0 ≥ 4 such that for every natural number τ1 ≥ 0, every natural number τ2 ≥ T0 + τ1 , every 2 program {x(t)}τt=τ satisfying 1 uτ2 −1 (x(τ2 − 1), x(τ2 )) ≥ δ,

(6.59)

τ2 ˜ and every x˜0 ∈ K, there exists a program {x(t)} t=τ1 which satisfies

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ Proof In view of assumption (A1), there exists a number λ ∈ (0, 1) such that the following property holds: (P1) For every nonnegative integer t and every (x, x  ) ∈ graph(at ) which satisfies ut (x, x  ) ≥ δ, there exists z ∈ at (x) for which z ≥ x  + λe. Fix a positive number D0 such that z ≤ D0 for all z ∈ K.

(6.60)

There exists a positive number c0 such that z2 ≤ c0 z for all z ∈ K.

(6.61)

Fix an integer T0 ≥ 4 satisfying 2D0 c0 κ T0 < λ

(6.62)

(see (6.2)). 2 Assume that integers τ1 ≥ 0, τ2 ≥ T0 + τ1 , that a program {x(t)}τt=τ 1 satisfies (6.59), and that x˜0 ∈ K. It follows from (6.59) and property (P1) that n such that there exists z ∈ R+ z ∈ aτ2 −1 (x(τ2 − 1)), z ≥ x(τ2 ) + λe. τ2 −1 In view of (6.2), there exists a program {x(t)} ˜ t=τ1 such that

(6.63)

194

6 Infinite Horizon Nonautonomous Optimization Problems

x(τ ˜ 1 ) = x˜0 ,

(6.64)

x(t ˜ + 1) − x(t + 1) ≤ κx(t) ˜ − x(t), t = τ1 , . . . , τ2 − 2. By (6.2) and (6.63), there exists ˜ 2 − 1)) x(τ ˜ 2 ) ∈ aτ2 −1 (x(τ which satisfies ˜ 2 − 1). x(τ ˜ 2 ) − z ≤ κx(τ2 − 1) − x(τ

(6.65)

τ2 Evidently, {x(t)} ˜ t=τ1 is a program. It follows from (6.60), (6.64), and (6.65) that

˜ 1 ) − x(τ1 ) ≤ κ τ2 −τ1 (2D0 ) ≤ κ T0 (2D0 ) x(τ ˜ 2 ) − z ≤ κ τ2 −τ1 x(τ and by (6.61), ||x(τ ˜ 2 ) − z||2 ≤ 2D0 c0 κ T0 . The inequality above implies that for all natural numbers i = 1, . . . , n, we have |x˜i (τ2 ) − zi | ≤ 2D0 c0 κ T0 and in view of (6.62) and (6.63), we have x(τ ˜ 2 ) ≥ z − 2D0 c0 κ T0 e ≥ x(τ2 ) + [λ − 2D0 c0 κ T0 ]e ≥ x(τ2 ). This completes the proof of Lemma 6.15. Fix a number γ > 0 satisfying γ < 1/2 and γ < 4−1 Δ.

(6.66)

Lemma 6.16 Let M1 be a positive number. Then there exist integers L1 , L2 ≥ 4 such that for every pair of integers T1 ≥ 0, T2 ≥ L1 + L2 + T1 , every program T2 −1 {x(t)}t=T satisfying 1 T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M1 ,

(6.67)

t=T1

and every integer τ ∈ [T1 + L1 , T2 − L2 ], the following inequality holds: max{ut (x(t), x(t + 1)) : t = τ, . . . , τ + L2 − 1} ≥ γ .

(6.68)

6.4 Auxiliary Results for Theorems 6.4, 6.5, and 6.7

195

Proof Lemma 6.15 implies that there exists an integer L1 ≥ 4 such that the following property holds: 2 satisfies (P2) If integers S1 ≥ 0, S2 ≥ S1 + L1 , if a program {v(t)}St=S 1

uS2 −1 (v(S2 − 1), v(S2 )) ≥ γ , S2 and if v˜0 ∈ K, then there exists a program {v(t)} ˜ ˜ 1 ) = v˜0 , t=S1 such that v(S v(S ˜ 2 ) ≥ v(S2 ).

Fix a number M2 for which M2 > ut (z, z ) for each integer t ≥ 0 and each (z, z ) ∈ graph(at )

(6.69)

and an integer L2 ≥ 1 which satisfies −1 (M1 + L1 γ + 1) + 16Δ −1 (M1 + M2 + L1 + 2). L2 > 4(L1 + 1) + 16Δ

(6.70)

T2 −1 Assume that integers T1 ≥ 0, T2 ≥ L1 + L2 + T1 , that a program {x(t)}t=T 1 satisfies (6.67), and that an integer τ satisfies

T 1 + L1 ≤ τ ≤ T2 − L2 .

(6.71)

We claim that (6.68) is valid. Assume the contrary. Then ut (x(t), x(t + 1)) < γ , t = τ, . . . , τ + L2 − 1.

(6.72)

There are two cases: ut (x(t), x(t + 1)) < γ , t = τ, . . . , T2 − 1;

(6.73)

max{ut (x(t), x(t + 1)) : t = τ, . . . , T2 − 1} ≥ γ .

(6.74)

Now we define a natural number τ0 as follows. If (6.73) holds, then we put τ0 = T2 . If (6.74) is valid, then in view of (6.72), there exists an integer τ0 ≥ 1 such that τ + L2 ≤ τ0 ≤ T2 − 1,

(6.75)

uτ0 (x(τ0 ), x(τ0 + 1)) ≥ γ ,

(6.76)

ut (x(t), x(t + 1)) < γ , t = τ, . . . , τ0 − 1.

(6.77)

It is not difficult to see that in both cases (6.77) is valid and that in both cases τ0 − τ ≥ L2 .

(6.78)

196

6 Infinite Horizon Nonautonomous Optimization Problems

Assume that (6.73) holds. Assumption (A2), the choice of L1 , property τ +L1 (P2), (6.66), (6.70), and (6.71) imply that there exists a program {x(t)} ˜ t=τ which satisfies x (τ + L1 ). x(τ ˜ ) = x(τ ), x(τ ˜ + L1 ) ≥

(6.79)

x(t) ˜ = x(t), t = T1 , . . . , τ.

(6.80)

Define

In view of (6.79), (6.80), (A3), and (A2), there exists x(t) ˜ ∈ K, t = τ + L1 + T2 1, . . . , T2 such that {x(t)} ˜ is a program, t=T1 x(t) ˜ ≥ x (t) for all integers t = τ + L1 , . . . , T2 , ˜ x(t ˜ + 1)) ≥ ut ( x (t), x (t + 1)), t = τ + L1 , . . . , T2 − 1. ut (x(t),

(6.81) (6.82)

Assumption (A2), (6.66), (6.67), (6.70), (6.71), (6.73), (6.80), and (6.82) imply that M1 ≥ U (x(T1 ), T1 , T2 ) −

T 2 −1

ut (x(t), x(t + 1))

t=T1

≥

T 2 −1

ut (x(t), ˜ x(t ˜ + 1)) −

t=T1

=

T 2 −1

ut (x(t), ˜ x(t ˜ + 1)) −

T 2 −1

ut (x(t), ˜ x(t ˜ + 1)) +

T 2 −1

ut ( x (t), x (t + 1))

t=τ +L1

t=τ T 2 −1

ut (x(t), x(t + 1))

t=τ

τ +L 1 −1 

−

ut (x(t), x(t + 1))

t=T1

t=τ

≥

T 2 −1

ut (x(t), x(t + 1))

t=τ

≥

T 2 −1

ut ( x (t), x (t + 1)) −

t=τ +L1

T 2 −1

ut (x(t), x(t + 1))

t=τ

− (T2 − τ )γ = (T2 − τ − L1 )(Δ − γ ) ≥ (T2 − τ − L1 )Δ −1 (T2 − τ − L1 ) − L1 γ −L1 γ ≥ Δ2 2 − L1 ) − L1 γ ≥ 2−1 Δ(L 2 − L1 γ ≥ 4−1 ΔL

6.4 Auxiliary Results for Theorems 6.4, 6.5, and 6.7

197

and −1 (M1 + L1 γ ). L2 ≤ 8Δ The inequality above contradicts (6.70). The contradiction we have reached proves that (6.73) is not true. Thus (6.74) holds and there exists an integer τ0 ≥ 1 satisfying (6.75)–(6.77). Assumption (A2), the choice of L1 , property (P2), and (6.66) τ +L1 imply that there exists a program {x(t)} ˜ t=τ such that x (τ + L1 ). x(τ ˜ ) = x(τ ), x(τ ˜ + L1 ) ≥

(6.83)

x(t) ˜ = x(t), t = T1 , . . . , τ.

(6.84)

Put

By (A2), (A3), (6.75), (6.76), and (6.83), there exist x(t) ˜ ∈ K, t = τ + 1 + τ0 −L1 L1 , . . . , τ0 − L1 such that {x(t)} ˜ is a program, t=τ +L1 x(t) ˜ ≥ x (t), t = τ + L1 , . . . , τ0 − L1 , ˜ x(t ˜ + 1)) ≥ ut ( x (t), x (t + 1)), t = τ + L1 , . . . , τ0 − L1 − 1. ut (x(t),

(6.85) (6.86)

τ0 −L1 Evidently, {x(t)} ˜ t=T1 is a program. In view of the choice of L1 , property (P2),

τ0 +1 and (6.76), there exist x(t) ˜ ∈ K, t = τ0 −L1 +1, . . . , τ0 +1 such that {x(t)} ˜ t=τ0 −L1 is a program,

x(τ ˜ 0 + 1) ≥ x(τ0 + 1).

(6.87)

τ0 +1 It is clear that {x(t)} ˜ t=T1 is a program. If T2 > τ0 + 1, then relation (6.87) and assumption (A3) imply that there exist x(t) ˜ ∈ K, t = τ0 + 2, . . . , T2 such that T2 {x(t)} ˜ is a program, t=τ0 +1

x(t) ˜ ≥ x(t), t = τ0 + 1, . . . , T2 , ˜ x(t ˜ + 1)) ≥ ut (x(t), x(t + 1)), t = τ0 + 1, . . . , T2 − 1. ut (x(t),

(6.88) (6.89)

It follows from (6.66), (6.67), (6.69)–(6.71), (6.74), (6.77), (6.78), (6.84), (6.86), (6.89), and assumption (A2) that M1 ≥ U (x(T1 ), T1 , T2 ) −

T 2 −1 t=T1

ut (x(t), x(t + 1))

198

6 Infinite Horizon Nonautonomous Optimization Problems

≥

T 2 −1

ut (x(t), ˜ x(t ˜ + 1)) −

t=T1

=

T 2 −1

τ0 

ut (x(t), ˜ x(t ˜ + 1)) −

T 2 −1

ut (x(t), x(t + 1))

t=τ

ut (x(t), ˜ x(t ˜ + 1)) −

t=τ

≥

ut (x(t), x(t + 1))

t=T1

t=τ

≥

T 2 −1

τ0 

ut (x(t), x(t + 1))

t=τ

τ0 −L 1 −1

ut (x(t), ˜ x(t ˜ + 1)) − (τ0 − τ )γ − uτ0 (x(τ0 ), x(τ0 + 1))

t=τ +L1

≥

τ0 −L 1 −1

ut ( x (t), x (t + 1)) − (τ0 − τ )γ − uτ0 (x(τ0 ), x(τ0 + 1))

t=τ +L1

0 − τ − 2L1 ) − (τ0 − τ )γ − M2 = (Δ − γ )(τ0 − τ − 2L1 ) ≥ Δ(τ −2L1 γ − M2 ≥ (Δ/2)(τ 0 − τ − 2L1 ) − 2L1 − M2 −1 − 2L1 − M2 ≥ (Δ/2)(L 2 − 2L1 ) − 2L1 − M2 ≥ 4 L2 Δ

and −1 (M1 + M2 + 2L1 ). L2 ≤ 4(Δ) This inequality contradicts (6.70). The contradiction we have reached proves (6.68) and Lemma 6.16 itself. Lemma 6.17 Let M1 be a positive number. Then there exist integers L¯ 1 , L¯ 2 ≥ 1 and a positive number M2 such that for every pair of integers τ1 ≥ 0, τ2 ≥ L¯ 1 + 2 L¯ 2 + τ1 and every program {x(t)}τt=τ satisfying 1 τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M1 ,

(6.90)

t=τ1

the following assertion holds. If integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1 t=T1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M2 .

(6.91)

6.4 Auxiliary Results for Theorems 6.4, 6.5, and 6.7

199

Proof Let integers L1 , L2 ≥ 4 be as guaranteed by Lemma 6.16. Lemma 6.15 implies that there exists an integer L3 ≥ 4 such that the following property holds: 2 satisfies (P3) If integers S1 ≥ 0, S2 ≥ L3 + S1 , if a program {v(t)}St=S 1

uS2 −1 (v(S2 − 1), v(S2 )) ≥ γ , S2 and if v˜0 ∈ K, then there exists a program {v(t)} ˜ ˜ 1 ) = v˜0 , t=S1 such that v(S v(S ˜ 2 ) ≥ v(S2 ).

Fix a positive a number M0 for which M0 > ut (z, z ) for every nonnegative integer t and every (z, z ) ∈ graph(at ), (6.92) integers L¯ 1 , L¯ 2 ≥ 1, and a number M2 > 0 satisfying L¯ 1 ≥ L1 , L¯ 2 > 2(L1 + L2 + L3 + 1),

(6.93)

M2 > M1 + M0 (L3 + L2 ).

(6.94)

2 Assume that integers τ1 ≥ 0, τ2 ≥ L¯ 1 + L¯ 2 + τ1 , a program {x(t)}τt=τ 1 satisfies (6.90), and integers T1 , T2 satisfy

T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ], L¯ 1 ≤ T2 − T1 .

(6.95)

We claim that (6.91) is true. Proposition 6.3 implies that there exists a program 2 {x (1) (t)}Tt=T which satisfies 1 x (1) (T1 ) = x(T1 ),

T 2 −1

ut (x (1) (t), x (1) (t + 1)) = U (x(T1 ), T1 , T2 ).

(6.96)

t=T1

It follows from (6.93) and (6.95) that T1 + L1 ≤ T1 + L¯ 1 + L3 ≤ T2 + L3 ≤ τ2 − L¯ 2 + L3 ≤ τ2 − 2L2 − L3 .

(6.97)

By the choice of the integers L1 , L2 , Lemma 6.16, (6.90), (6.93), and (6.97), we have max{ut (x(t), x(t + 1)) : t = T2 + L3 , . . . , T2 + L2 + L3 − 1} ≥ γ . Hence there exists an integer τ ∈ [T2 + L3 , . . . , T2 + L3 + L2 − 1] for which uτ (x(τ ), x(τ + 1)) ≥ γ .

(6.98)

200

6 Infinite Horizon Nonautonomous Optimization Problems

+1 Property (P3) and (6.98) imply that there exists a program {x (2) (t)}τt=T such that 2

x (2) (T2 ) = x (1) (T2 ), x (2) (τ + 1) ≥ x(τ + 1).

(6.99)

Define ˜ = x (1) (t), t = T1 + 1, . . . , T2 , x(t) ˜ = x(t), t = τ1 , . . . , T1 , x(t) x(t) ˜ = x (2) (t), t = T2 + 1, . . . , τ + 1.

(6.100)

τ +1 It is easy to see that {x(t)} ˜ t=τ1 is a program. By (6.99) and (6.100), we have

x(τ ˜ + 1) ≥ x(τ + 1).

(6.101)

Assumption (A3) and (6.101) imply that there exist x(t) ˜ ∈ K, t = τ + 2, . . . , τ2 τ2 such that (x(t)} ˜ is a program, t=τ1 x(t) ˜ ≥ x(t), t = τ + 1, . . . , τ2 ,

(6.102)

˜ x(t ˜ + 1)) ≥ ut (x(t), x(t + 1)), t = τ + 1, . . . , τ2 − 1. ut (x(t),

(6.103)

¯ By (6.90), (6.92), (6.94), (6.96), (6.100), (6.103), and the choice of L, M1 ≥ U (x(τ1 ), τ1 , τ2 ) −

τ 2 −1

ut (x(t), x(t + 1))

t=τ1

≥

τ 2 −1

ut (x(t), ˜ x(t ˜ + 1)) −

t=τ1

=

τ 2 −1

≥

ut (x(t), ˜ x(t ˜ + 1)) −

T 2 −1

ut (x(t), ˜ x(t ˜ + 1)) −

ut (x(t), x(t + 1))

τ 

ut (x(t), x(t + 1))

t=T1

ut (x(t), ˜ x(t ˜ + 1)) −

t=T1

≥ U (x(T1 ), T1 , T2 ) −

T 2 −1 t=T1

T 2 −1 t=T1

and

τ 2 −1 t=T1

t=T1

≥

ut (x(t), x(t + 1))

t=τ1

t=T1 τ 

τ 2 −1

ut (x(t), x(t + 1)) −

τ 

ut (x(t), x(t + 1))

t=T2

ut (x(t), x(t + 1)) − (τ − T2 + 1)M0

6.5 Properties of the Function U T 2 −1

201

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M1 − M0 (L3 + L2 )

t=T1

> U (x(T1 ), T1 , T2 ) − M2 . This completes the proof of Lemma 6.17.

6.5 Properties of the Function U It is not difficult to see that the following auxiliary result holds. Proposition 6.18 Let τ1 ≥ 0, τ1 > τ1 be integers, let Δ ≥ 0, T1 , T2 be integers 2 such that τ1 ≤ T1 < T2 ≤ τ2 , and let {x(t)}τt=τ be a program satisfying 1 τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − Δ.

t=τ1

Then T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − Δ.

t=T1

Lemma 6.19 There exist an integer L ≥ 1 and a positive number M1 such that for every pair of points x0 , x˜0 ∈ K and every pair of integers T1 ≥ 0, T2 ≥ T1 + L, the inequality |U (x0 , T1 , T2 ) − U (x˜0 , T1 , T2 )| ≤ M1 is valid. Proof Let integers L1 , L2 ≥ 4 be as guaranteed by Lemma 6.16 with M1 = 1. Lemma 6.15 implies that there exists a natural number L3 ≥ 4 such that the following property holds: 2 satisfies (P4) If integers S1 ≥ 0, S2 ≥ S1 + L3 , if a program {v(t)}St=S 1

uS2 −1 (v(S2 − 1), v(S2 )) ≥ γ , S2 and if v˜0 ∈ K, then there exists a program {v(t)} ˜ ˜ 1 ) = v˜0 , t=S1 such that v(S v(S ˜ 2 ) ≥ v(S2 ).

Fix an integer

202

6 Infinite Horizon Nonautonomous Optimization Problems

L > 2(L1 + L2 + L3 + 1),

(6.104)

a number M0 > ut (z, z ), t = 0, 1, . . . , (z, z ) ∈ graph(at )

(6.105)

and set M1 = M0 (L1 + L2 + L3 ).

(6.106)

Assume that x0 , x˜0 ∈ K and that integers T1 ≥ 0, T2 ≥ T1 + L. Proposition 6.3 2 implies that there exists a program {x(t)}Tt=T which satisfies 1 x(T1 ) = x0 ,

T 2 −1

ut (x(t), x(t + 1)) = U (x0 , T1 , T2 ).

(6.107)

T 1 + L 1 + L 3 < T 1 + L − L2 ≤ T 2 − L 2 .

(6.108)

t=T1

By (6.104), we have

By the choice of L1 , L2 , Lemma 6.16, (6.104), and (6.107), max{ut (x(t), x(t + 1)) : t = L3 + L1 + T1 , . . . , L3 + L1 + L2 + T1 − 1} ≥ γ . Thus there exists an integer τ ∈ {T1 + L1 + L3 , . . . , T1 + L3 + L1 + L2 − 1}

(6.109)

such that uτ (x(τ ), x(τ + 1)) ≥ γ .

(6.110)

Property (P4), the choice of L3 , (6.109), and (6.110) imply that there exists a τ +1 program {x(t)} ˜ t=T1 satisfying ˜ + 1) ≥ x(τ + 1). x(T ˜ 1 ) = x˜0 , x(τ

(6.111)

Assumption (A3) and (6.111) imply that there exist x(t) ˜ ∈ K, t = τ + 2, . . . , T2 T2 such that {x(t)} ˜ is a program, t=τ +1 x(t) ˜ ≥ x(t), t = τ + 1, . . . , T2 , ˜ x(t ˜ + 1)) ≥ ut (x(t), x(t + 1)), t = τ + 1, . . . , T2 − 1. ut (x(t),

(6.112) (6.113)

6.5 Properties of the Function U

203

T2 Evidently, {x(t)} ˜ t=T1 is a program. In view of (6.105)–(6.107), (6.109), (6.111), and (6.113), we have

U (x˜0 , T1 , T2 ) ≥

T 2 −1

ut (x(t), ˜ x(t ˜ + 1)) =

t=T1

⎡

−⎣

T 2 −1

ut (x(t), x(t + 1))

t=T1 T 2 −1

ut (x(t), x(t + 1)) −

t=T1

⎡

≥ U (x0 , T1 , T2 ) − ⎣

T 2 −1

⎤ ut (x(t), ˜ x(t ˜ + 1))⎦

t=T1 τ 

ut (x(t), x(t + 1)) −

t=T1

≥ U (x0 , T1 , T2 ) −

τ 

τ 

⎤ ut (x(t), ˜ x(t ˜ + 1))⎦

t=T1

ut (x(t), x(t + 1)) ≥ U (x0 , T1 , T2 )

t=T1

−(τ − T1 )M0 ≥ U (x0 , T1 , T2 ) − (L1 + L2 + L3 )M0 = U (x0 , T1 , T2 ) − M1 . Therefore we have shown that for each x0 , x˜0 ∈ K and each pair of integers T1 ≥ 0, T2 ≥ T1 + L, U (x˜0 , T1 , T2 ) ≥ U (x0 , T1 , T2 ) − M1 . Lemma 6.19 is proved. Corollary 6.20 There exist a positive number M1 and an integer L ≥ 1 such that for every pair of integers T1 ≥ 0, T2 ≥ T1 + L and every x0 ∈ K, the inequality (T1 , T2 )| ≤ M1 |U (x0 , T1 , T2 ) − U holds. Lemmas 6.16 and 6.17 and Corollary 6.20 imply the following auxiliary result. Lemma 6.21 Let M1 > 0. Then there exist natural numbers L¯ 1 , L¯ 2 and M2 > 0 such that for each pair of  integers τ1 ≥ 0, τ2 ≥ τ1 + L¯ 1 + L¯ 2 and each program τ2 −1 2 {x(t)}τt=τ which satisfies t=τ1 ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M1 , the 1 following assertion holds: If integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1 t=T1

(T1 , T2 ) − M2 . ut (x(t), x(t + 1)) ≥ U

204

6 Infinite Horizon Nonautonomous Optimization Problems

6.6 Proof of Theorem 6.4 Let M1 = 1, and let integers L¯ 1 , L¯ 2 ≥ 1 and a positive number M2 be as guaranteed by Lemma 6.21. Let x0 ∈ K be given. Proposition 6.3 implies that for every integer k ≥ 1, there exists a program {x (k) (t)}kt=0 satisfying x (k) (0) = x0 ,

k−1 

ut (x (k) (t), x (k) (t + 1)) = U (x0 , 0, k).

(6.114)

t=0

By the choice of L¯ 1 , L¯ 2 , M2 , and Lemma 6.21, the following property holds: (i) For every natural number k ≥ L¯ 1 + L¯ 2 and every pair of integers T1 , T2 ∈ T2 −1 (k) (k) [0, k − L¯ 2 ] satisfying L¯ 1 ≤ T2 − T1 , t=T1 ut (x (t), x (t + 1)) ≥ (T1 , T2 ) − M2 . U Evidently, there exists a strictly increasing sequence of natural numbers {kj }∞ j =1 such that for every nonnegative integer t, there exists x(t) ¯ = lim x (kj ) (t). j →∞

(6.115)

∞ is a program. By (6.114) and (6.115), we have It is clear that {x(t)} ¯ t=0

x(0) ¯ = x0 .

(6.116)

In view of (6.115), property (i), and upper semicontinuity of the functions ut , t = 0, 1, . . . , the following property holds: (ii) for each pair of nonnegative integers T1 , T2 which satisfy T2 − T1 ≥ L¯ 1 ,    T   2 −1  (T1 , T2 ) ≤ M2 . ut (x(t), ¯ x(t ¯ + 1)) − U     t=T1

(6.117)

Fix number M0 > 0 such that M0 > ut (z, z ) for every integer t ≥ 0 and every (z, z ) ∈ graph(at ). (6.118) Put M = M2 + M0 L¯ 1 .

(6.119)

Assume that nonnegative integers T1 , T2 satisfy T1 < T2 . If T2 − T1 ≥ L¯ 1 , then property (ii), (6.117), and (6.118) imply

6.6 Proof of Theorem 6.4

205

   T   2 −1   u ( x(t), ¯ x(t ¯ + 1)) − U (T , T ) t 1 2  ≤ M2 ≤ M.    t=T1 If T2 − T1 ≤ L¯ 1 , then in view of (6.118) and (6.119), we have    T   2 −1   ¯ 1 < M. u ( x(t), ¯ x(t ¯ + 1)) − U (T , T ) t 1 2  ≤ (T2 − T1 )M0 ≤ M0 L    t=T1 Thus in both cases    T2 −1     u ( x(t), ¯ x(t ¯ + 1)) − U (T , T ) t 1 2  ≤ M.    t=T1

(6.120)

Assume now that the following properties hold: (iii) for every nonnegative integer t and every (z, z ) ∈ graph(at ) satisfying ut (z, z ) > 0, the function ut is continuous at (z, z ); (iv) if an integer t ≥ 0 and z, z1 , z2 , z3 ∈ K satisfy zi ∈ at (z), i = 1, 3 and z1 ≤ z2 ≤ z3 , then z2 ∈ at (z). In order to complete the proof of the theorem, it is sufficient to show that for every natural number T , we have T −1

ut (x(t), ¯ x(t ¯ + 1)) = U (x(0), x(T ), 0, T ).

(6.121)

t=0

Denote by E the set of all natural numbers τ such that ¯ − 1), x(τ ¯ )) > 0. uτ −1 (x(τ

(6.122)

Assumption (A2) and (6.120) imply that the set E is infinite. By Proposition 6.18, it is sufficient to show that (6.121) is valid for all T = τ − 1, where τ ∈ E. Let τ ∈ E and T = τ − 1. We claim that (6.121) holds. Assume the contrary. Then there exist a program {x(t)}Tt=0 and a number Δ > 0 such that x(0) = x(0), ¯ x(T ) ≥ x(T ¯ ), T −1 t=0

ut (x(t), x(t + 1)) ≥

T −1

ut (x(t), ¯ x(t ¯ + 1)) + 2Δ.

t=0

In view of the inclusion τ ∈ E and the definition of E,

(6.123) (6.124)

206

6 Infinite Horizon Nonautonomous Optimization Problems

uT (x(T ¯ ), x(T ¯ + 1)) = uτ −1 (x(τ ¯ − 1), x(τ ¯ )) > 0.

(6.125)

It follows from (6.125) and (A1) that there exists a number λ0 ∈ (0, 1) and ¯ − 1)) = aT (x(T ¯ )) z0 ∈ aτ −1 (x(τ

(6.126)

¯ ) + λ0 e = x(T ¯ + 1) + λ0 e. z0 ≥ x(τ

(6.127)

such that

There exists a number c0 > 1 such that y ≤ c0 y2 ≤ c02 y for all y ∈ R n .

(6.128)

In view of (6.125), (6.127), and properties (iii) and (iv), we may assume without loss of generality that ¯ − 1), z0 ) − uτ −1 (x(τ ¯ − 1), x(τ ¯ ))| ≤ Δ/4. |uτ −1 (x(τ

(6.129)

Assumption (A3), (6.123), and (6.126) imply that there exists z1 ∈ aT (x(T )) such that ¯ ), z0 ). z1 ≥ z0 , uT (x(T ), z1 ) ≥ uT (x(T

(6.130)

Fix a positive number δ < min{1, λ0 , Δτ −1 }.

(6.131)

∞ (see (6.115)) and upper semicontinuity By the construction of the program {x(t)} ¯ t=0 of ut , t = 0, 1, . . . , there is a natural number k > τ + 4 such that

¯ x (k) (t) − x(t) 2 ≤ δ, t = 0, . . . , τ + 2, ¯ x(t ¯ + 1)) + δ, t = 0, . . . , τ + 2. ut (x (k) (t), x (k) (t + 1)) ≤ ut (x(t),

(6.132) (6.133)

Define x(t) ˜ = x(t), t = 0, . . . , τ − 1.

(6.134)

We claim that z1 ≥ x (k) (τ ). In view of (6.132), ¯ )||2 ≤ δ. ||x (k) (τ ) − x(τ By (6.127), (6.130), (6.131), and (6.135), we have

(6.135)

6.6 Proof of Theorem 6.4

207

x (k) (τ ) ≤ x(τ ¯ ) + δe ≤ x(τ ¯ ) + λ 0 e ≤ z0 ≤ z1 .

(6.136)

Put x(τ ˜ ) = z1 .

(6.137)

τ ˜ Since z1 ∈ aT (xT ) = aτ −1 (x˜τ −1 ), {x(t)} t=0 is a program. In view of (6.116), (6.123), (6.134), (6.136), and (6.137),

˜ ) ≥ x (k) (τ ). x(0) ˜ = x(0) ¯ = x0 , x(τ

(6.138)

By (6.124), (6.129), (6.130), (6.134), (6.137), and the equality T = τ − 1, τ −1 

ut (x(t), ˜ x(t ˜ + 1)) −

t=0

τ −1 

ut (x(t), ¯ x(t ¯ + 1))

t=0

≥

τ −2 

ut (x(t), x(t + 1)) + uτ −1 (x(τ ˜ − 1), x(τ ˜ )) −

t=0

≥

τ −2 

τ −1 

ut (x(t), ¯ x(t ¯ + 1))

t=0

ut (x(t), ¯ x(t ¯ + 1)) + 2Δ + uτ −1 (x(τ − 1), z1 ) −

t=0

τ −1 

ut (x(t), ¯ x(t ¯ + 1))

t=0

≥ 2Δ +

τ −2 

ut (x(t), ¯ x(t ¯ + 1)) + uτ −1 (x(τ ¯ − 1), z0 ) −

t=0

τ −1 

ut (x(t), ¯ x(t ¯ + 1))

t=0

¯ − 1), z0 ) − uτ −1 (x(τ ¯ − 1), x(τ ¯ )) ≥ (3/2)Δ. ≥ 2Δ + uτ −1 (x(τ

(6.139)

It follows from (6.131), (6.133), and (6.139) that τ −1 

ut (x(t), ˜ x(t ˜ + 1)) −

t=0

=

τ −1 

ut (x (k) (t), x (k) (t + 1))

t=0 τ −1 

ut (x(t), ˜ x(t ˜ + 1)) −

t=0

−

τ −1 

ut (x(t), ¯ x(t ¯ + 1)) +

t=0

τ −1 

τ −1 

ut (x(t), ¯ x(t ¯ + 1))

t=0

ut (x (k) (t), x (k) (t + 1)) ≥ (3/2)Δ − δτ ≥ Δ/2.

t=0

In view of (6.138) and (6.140), we have U (x0 , x (k) (τ ), 0, τ ) ≥

τ −1  t=0

ut (x (k) (t), x (k) (t + 1)) + Δ/2.

(6.140)

208

6 Infinite Horizon Nonautonomous Optimization Problems

This inequality contradicts (6.114). The contradiction we have reached proves that (6.121) holds for all T = τ − 1 where τ ∈ E. Theorem 6.4 is proved.

6.7 Proof of Theorem 6.5 In the sequel we assume that the sum over empty set is zero. There exist a positive number Δ and a strictly increasing sequence of natural numbers {τi }∞ i=1 such that τ1 ≥ 4 and that uτi −1 (x(τi−1 ), x(τi )) ≥ Δ for all natural numbers i.

(6.141)

Let a positive number M be as guaranteed by Theorem 6.4. Lemma 6.15 implies that there exists an integer L0 ≥ 4 such that the following property holds: (P5) For every nonnegative integer S1 , every integer S2 ≥ S1 + L0 , every program 2 {v(t)}St=S satisfying 1 uS2 −1 (v(S2 − 1), v(S2 )) ≥ Δ, S2 and every v˜0 ∈ K, there exists a program {v(t)} ˜ ˜ 1) = t=S1 such that v(S v˜0 , v(S ˜ 2 ) ≥ v(S2 ).

Corollary 6.20 and (6.3) imply that there exists a positive number M∗ such that (T1 , T2 )| ≤ M∗ |U (v0 , T1 , T2 ) − U for every v0 ∈ K and every pair of integers T1 < T2 ,

(6.142)

ut (z, z ) ≤ M∗ for every nonnegative integer t, and every (z, z ) ∈ graph(at ). (6.143) Fix a number M1 > L0 M∗ + M0 + 3M.

(6.144)

∞ such that Theorem 6.4 implies that there exists a program {x(t)} ¯ t=0

x(0) ¯ = x(0)

(6.145)

and that for each pair of integers S1 , S2 satisfying S1 < S2 ,    S   2 −1  ut (x(t), ¯ x(t ¯ + 1)) − U (S1 , S2 ) ≤ M.    t=S1

(6.146)

6.7 Proof of Theorem 6.5

209

Assume that T1 , T2 are integers such that 0 ≤ T1 < T2 . We claim that    T2 −1    ut (x(t), x(t + 1)) − U (T1 , T2 ) ≤ M1 .    t=T1

(6.147)

If T2 ≤ T1 + L0 , then this inequality follows from (6.143) and (6.144). Assume that T2 > T1 + L0 . There exists an integer i ≥ 1 such that τi > T2 + 2L0 .

(6.148)

τi By (6.141), (6.148), and (P5), there exists a program {x(t)} ˜ t=τi −L0 such that

¯ i − L0 ), x(τ ˜ i ) ≥ x(τi ). x(τ ˜ i − L0 ) = x(τ

(6.149)

x(t) ˜ = x(t), ¯ t = 0, . . . , τi − L0 − 1.

(6.150)

Define

τi Evidently, {x(t)} ˜ t=0 is a program, and by (6.149), (6.150), and (6.145), τ i −1

ut (x(t), x(t + 1)) ≥

t=0

τ i −1

ut (x(t), ˜ x(t ˜ + 1)) − M0 .

t=0

By (6.143) and (6.151), τ i −1

ut (x(t), x(t + 1)) ≥

t=0

τi −L 0 −1 

ut (x(t), ¯ x(t ¯ + 1)) − M0

t=0

≥

τ i −1

ut (x(t), ¯ x(t ¯ + 1)) − M0 − L0 M∗ .

t=0

Together with (6.146) this implies that −(M0 + L0 M∗ ) ≤

τ i −1 t=0

≤

ut (x(t), x(t + 1)) −

τ i −1 t=0

ut (x(t), ¯ x(t ¯ + 1))

 {ut (x(t), x(t + 1)) : 0 ≤ t < T1 }  − {ut (x(t), ¯ x(t ¯ + 1)) : 0 ≤ t < T1 }

(6.151)

210

6 Infinite Horizon Nonautonomous Optimization Problems

+

T 2 −1

ut (x(t), x(t + 1)) −

t=T1

+

τ i −1

ut (x(t), ¯ x(t ¯ + 1))

t=T1

ut (x(t), x(t + 1)) −

t=T2

≤M+

T 2 −1

τ i −1

ut (x(t), ¯ x(t ¯ + 1))

t=T2

T 2 −1

(T1 , T2 ) − M) + U (T2 , τi ) ut (x(t), x(t + 1)) − (U

t=T1

−

τi 

ut (x(t), ¯ x(t ¯ + 1))

t=T2

≤

T 2 −1

(T1 , T2 ) + 3M ut (x(t), x(t + 1)) − U

t=T1

and combined with (6.144) this implies that T 2 −1

(T1 , T2 ) ≥ −3M − (M0 + L0 M∗ ) > −M1 . ut (x(t), x(t + 1)) − U

t=T1

This completes the proof of Theorem 6.5.

6.8 Proof of Theorem 6.7 ∞ be as guaranteed by Theorem 6.4. Then for Let x0 ∈ K, M1 > 0 and let {x(t)} ¯ t=0 every pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 ,

   T2 −1    ut (x(t), ¯ x(t ¯ + 1)) − U (T1 , T2 ) ≤ M.    t=T1

(6.152)

Fix a positive number Δ such that Δ > u(z, z ) for each (z, z ) ∈ graph(a).

(6.153)

Let p be a natural number. We show that for all sufficiently large natural numbers T ,

6.8 Proof of Theorem 6.7

211

  T −1    −1  u(x(t), ¯ x(t ¯ + 1)) ≤ 2M/p. p U (0, p) − T −1  

(6.154)

t=0

Assume that T ≥ p is a natural number. Then there exist integers q, s such that q ≥ 1, 0 ≤ s < p, T = pq + s.

(6.155)

By (6.155) we have T −1

T −1

⎛ (0, p) = T −1 ⎝ u(x(t), ¯ x(t ¯ + 1)) − p−1 U

t=0

pq−1 

u(x(t), ¯ x(t ¯ + 1))

t=0

⎞  + {u(x(t), ¯ x(t ¯ + 1)) : t is an integer such that pq ≤ t ≤ T − 1}⎠ (0, p) −p−1 U  {u(x(t), ¯ x(t ¯ + 1)) : t is an integer such that pq ≤ t ≤ T − 1} = T −1 +(T

−1

pq)(pq)

−1

q−1 (i+1)p−1  

⎡ = (T −1 pq)(pq)−1 ⎣ ⎤

i=0 q−1  i=0

⎛

(0, p) u(x(t), ¯ x(t ¯ + 1)) − p−1 U

t=ip

(i+1)p−1 

⎝

⎞ (0, p)⎠ u(x(t), ¯ x(t ¯ + 1)) − U

t=ip

(0, p)⎦ − p−1 U (0, p) +q U +T −1



 u(x(t), ¯ x(t ¯ + 1)) : t is an integer such that pq ≤ t ≤ T − 1 . (6.156)

In view of (6.152), (6.153), (6.155), and (6.156),   T −1    −1  (0, p) u(x(t), ¯ x(t ¯ + 1)) − p−1 U T   t=0

(0, p)|q/T − 1/p| ≤ T −1 pΔ + (pq)−1 qM + U (0, p)s(pT )−1 → M/p as T → ∞. ≤ T −1 pΔ + M/p + U Since p is an arbitrary natural number, we conclude that

212

6 Infinite Horizon Nonautonomous Optimization Problems

T −1

T −1

u(x(t), ¯ x(t ¯ + 1))}∞ T =1

t=0

is a Cauchy sequence. Evidently, there exists lim T

T →∞

−1

T −1

u(x(t), ¯ x(t ¯ + 1))

t=0

and for every integer p ≥ 1, we have   T −1    −1  −1 u(x(t), ¯ x(t ¯ + 1)) ≤ 2M/p. p U (0, p) − lim T   T →∞

(6.157)

t=0

Since (6.157) is true for every integer p ≥ 1, we conclude that lim T −1

T →∞

T −1

(0, p)/p. u(x(t), ¯ x(t ¯ + 1)) = lim U p→∞

t=0

(6.158)

Define (0, p)/p. μ = lim U p→∞

(6.159)

It follows from (6.157)–(6.159) that, for every integer p ≥ 1, we have (0, p) − μ| ≤ 2M/p. |p−1 U This completes the proof of Theorem 6.7.

6.9 Overtaking Optimal Programs In this section we study the existence of overtaking optimal solutions for a large class of infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a model proposed by Robinson, Solow, and Srinivasan with nonconcave utility functions representing the preferences of the planner. The results of this section were obtained in [104]. We continue to use the notation and definitions introduced in Section 6.1. In particular, we assume that for every nonnegative integer t, at : K → P(K) is such that graph(at ) is a closed subset of R n × R n .

6.9 Overtaking Optimal Programs

213

We also suppose that there exists κ ∈ (0, 1) such that for every pair of points x, y ∈ K and every nonnegative integer t, we have H (at (x), at (y)) ≤ κx − y

(6.160)

and that for every nonnegative integer t, the upper semicontinuous function ut : graph(at ) → [0, ∞) satisfies lim sup{ut (x, x  ) : (x, x  ) ∈ graph(at )} = 0.

(6.161)

t→∞

In Section 6.1 we introduced assumptions (A1)–(A3). Here we assume that (A3) holds, but we do not assume (A1) and (A2). Namely, we assume: (A3) for every nonnegative integer t, every (x, y) ∈ graph(at ), and every x˜ ∈ K which satisfies x˜ ≥ x, there exists y˜ ∈ at (x) ˜ for which ˜ y) ˜ ≥ ut (x, y). y˜ ≥ y, ut (x, We also suppose that the following assumptions hold: (A5) there exist a positive γ and a sequence of positive numbers {Δt }∞ t=0 such that: (i) for every nonnegative integer s and every point z0 ∈ K, there exists a sequence {x(t)}∞ t=s ⊂ K such that x(s) = z0 , x(t + 1) ∈ at (x(t)) for all integers t ≥ s and that ut (x(t), x(t + 1)) ≥ Δt for every natural number t ≥ s + 1; (ii) for every nonnegative integer t if (x, x  ) ∈ graph(at ) satisfies ut (x, x  ) ≥ Δt , then there exists z ∈ at (x) such that z ≥ x  + γ e; For every point x0 ∈ K and every natural number T , define U (x0 , T ) = sup

T −1 

ut (x(t), x(t + 1)) :

t=0 T −1 {x(t)}t=0

is a program and x(0) = x0 .

(6.162)

Upper semicontinuity of ut , t = 0, 1, . . . implies the following result. Proposition 6.22 For every x0 ∈ K and every integer T ≥ 1, there exists a program {x(t)}Tt=0 such that x(0) = x0 and T −1

ut (x(t), x(t + 1)) = U (x0 , T ).

t=0

We prove the following theorem.

214

6 Infinite Horizon Nonautonomous Optimization Problems

Theorem 6.23 For every z ∈ K, there exists a program {xz (t)}∞ t=0 such that xz (0) = z and the following assertion holds: For every positive number δ, there exists an integer L(δ) ≥ 1 such that for every natural number S ≥ L(δ) and every z ∈ K, S−1 

ut (xz (t), xz (t + 1)) ≥ U (z, S) − δ.

t=0

Theorem 6.23 easily implies the following corollary. Corollary 6.24 Let z ∈ K, and let a program {xz (t)}∞ t=0 be as guaranteed by Theorem 6.23. Then for every program {x(t)}∞ satisfying x(0) = z, t=0 lim sup

T −1 

T →∞

ut (x(t), x(t + 1)) −

t=0

T −1

 ut (xz (t), xz (t + 1)) ≤ 0.

t=0

6.10 A Subclass of Infinite Horizon Problems In this section we consider a subclass of the class of infinite horizon optimal control problems considered in Section 6.9. Infinite horizon problems of this subclass correspond to the nonstationary Robinson–Solow–Srinivasan models. For every nonnegative integer t, let (t)

(t)

α (t) = (α1 , . . . , αn(t) ) >> 0, b(t) = (b1 , . . . , bn(t) ) >> 0,

(6.163)

(t)

d (t) = (d1 , . . . , dn(t) ) ∈ ((0, 1])n and for every nonnegative integer t, let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that wt (0) = 0, lim wt (z) = 0 for all z > 0. t→∞

(6.164)

n , define Let t ≥ 0 be an integer. For every x ∈ R+

 (t)

n at (x) = y ∈ R+ : yi ≥ (1 − di )xi , i = 1, . . . , n, n  i=1

(t) αi (yi

(t) − (1 − di )xi )

≤1 .

(6.165)

6.10 A Subclass of Infinite Horizon Problems

215

It is not difficult to see that for every x ∈ R n , at (x) is a nonempty closed bounded n and graph(a ) is a closed subset of R n × R n . Suppose that subset of R+ t + + (t)

inf{di

: i = 1, . . . , n, t = 0, 1, . . . } > 0, inf{eb(t) : t = 0, 1, . . . } > 0,

(6.166) (6.167)

(t)

(6.168)

(t)

(6.169)

(t)

(6.170)

inf{αi : i = 1, . . . , n, t = 0, 1, . . . } > 0, sup{bi : i = 1, . . . , n, t = 0, 1, . . . } < ∞, sup{αi : i = 1, . . . , n, t = 0, 1, . . . } < ∞.

The constraint mappings at , t = 0, 1, . . . have already been defined. Let us now define the cost functions ut , t = 0, 1, . . . . For every nonnegative integer t and every (x, x  ) ∈ graph(at ), define  

ut (x, x ) = sup wt (b(t) y) : 0 ≤ y ≤ x, ey +

n 

(t) αi (xi

(t) − (1 − di )xi )

≤1 .

(6.171)

i=1

Fix α ∗ , α∗ > 0, d∗ > 0 such that α∗ < αi(t) < α ∗ , d∗ < di(t) , i = 1, . . . , n, t = 0, 1, . . . .

(6.172)

Lemma 6.25 Let a number M0 > (α∗ d∗ )−1 , let an integer t ≥ 0, and let (x, x  ) ∈ graph(at ) satisfy x ≤ M0 e. Then x  ≤ M0 e. For the proof see Lemma 6.9. Lemma 6.26 Let t ≥ 0 be an integer. Then the function ut : graph(at ) → [0, ∞) is upper semicontinuous. For the proof see Lemma 6.10. n Proposition 6.27 Let t ≥ 0 be an integer, let (x, y) ∈ graph(at ), and let x˜ ∈ R+ satisfy x˜ ≥ x. Then there is y˜ ∈ at (x) ˜ such that y˜ ≥ y and ut (x, ˜ y) ˜ ≥ ut (x, y).

For the proof see Proposition 6.13. Set β = inf{b(t) e : t = 0, 1, . . . }, b∗ = sup{bi(t) : i = 1, . . . , n, t = 0, 1, . . . }, . (6.173)

216

6 Infinite Horizon Nonautonomous Optimization Problems

In view of (6.167) and (6.169), β > 0 and b∗ < ∞. Proposition 6.28 There exist γ > 0 and a sequence of positive numbers {Δt }∞ t=0 such that: n , there exists a sequence (i) for every nonnegative integer s and every x0 ∈ R+ n ∞ {x(t)}t=s ⊂ R+ such that x(s) = x0 , x(t + 1) ∈ at (x(t)) for every integer t ≥ s and ut (x(t), x(t + 1)) ≥ Δt for every integer t ≥ s + 1; (ii) for every nonnegative integer t if (x, x  ) ∈ graph(at ) satisfies ut (x, x  ) ≥ Δt , then x  + γ e ∈ at (x).

Proof Choose numbers λ0 , λ1 ∈ (0, 1) for which λ0 n(1 + α ∗ ) < 1 and λ1 < λ0

(6.174)

and set γ = (b∗ )−1 βλ1 n−1 α∗−1 , Δt = wt (βλ1 ), t = 0, 1, . . . .

(6.175)

n . Define Let an integer s ≥ 0 and x0 ∈ R+

x(s) = x0 , y(s) = 0, xi (s + 1) = (1 − di(s) )xi (s) + λ0 for i = 1, . . . , n,

(6.176)

and for every integer t ≥ s + 1, define xi (t + 1) = (1 − di(t) )xi (t) + λ0 , i = 1, . . . , n, y(t) = λ1 e.

(6.177)

It follows from (6.165), (6.172), (6.174), (6.176), and (6.177) that x(t + 1) ∈ at (x(t)) for all integers t ≥ s. By (6.172), (6.174), and (6.177), for all integers t > s, ey(t) +

n 

(t)

(t)

αi [xi (t + 1) − (1 − di )xi (t)]

i=1

=

n 

αi λ0 + λ1 n ≤ α ∗ λ0 n + λ1 n ≤ λ0 n(1 + α ∗ ) < 1. (t)

i=1

Combined with (6.171), (6.173), (6.175), and (6.177), this implies that for all integers t ≥ s + 1, ut (x(t), x(t + 1)) ≥ wt (b(t) λ1 e) ≥ wt (λ1 β) = Δt .

6.10 A Subclass of Infinite Horizon Problems

217

Therefore (i) is true. Let us now prove that the property (ii) is true. Assume that an integer t ≥ 0 and that (x, x  ) ∈ graph(at ), ut (x, x  ) ≥ Δt .

(6.178)

In follows from (6.165), (6.171), and (6.178) that xi ≥ (1 − di )xi , i = 1, . . . , n (t)

(6.179)

n satisfying and there exists y ∈ R+

y ≤ x, ey +

n 

αi(t) (xi − (1 − di(t) )xi ) ≤ 1,

i=1

wt (b(t) y) = ut (x, x  ).

(6.180)

In view of (6.175), (6.178), and (6.180), we have wt (b(t) y) ≥ Δt ≥ wt (βλ1 ), and since the function wt is strictly increasing, relation (6.173) implies that βλ1 ≤ b(t) y ≤ b∗ ey and ey ≥ (b∗ )−1 βλ1 .

(6.181)

We claim that x  + γ e ∈ at (x). In view of (6.179), for i = 1, . . . , n, xi + γ ≥ xi ≥ (1 − di(t) )xi .

(6.182)

It follows from (6.172), (6.175), (6.180), and (6.181) that for integers i = 1, . . . , n, we have n 

αi (xi + γ − (1 − di )xi ) = (t)

(t)

i=1

n 

αi (xi − (1 − di )xi ) + γ (t)

(t)

i=1

≤

n  i=1

n  i=1

αi (xi − (1 − di )xi ) + γ nα ∗ (t)

(t)

(t)

αi

218

6 Infinite Horizon Nonautonomous Optimization Problems

=

n 

αi(t) (xi − (1 − di(t) )xi ) + (b∗ )−1 βλ1

i=1

≤

n 

αi (xi − (1 − di )xi ) + ey ≤ 1. (t)

(t)

i=1

Combined with (6.182) and (6.165), the relation above implies that x  + γ e ∈ at (x). Therefore (ii) holds and Proposition 6.28 holds too. For every x = (x1 , . . . , xn ) ∈ R n , put x1 =

n 

|xi |, x∞ = max{|xi | : i = 1, . . . , n}.

i=1

It is easy to see that for every nonnegative integer t, for every pair of vectors x, y ∈ n , and for  ·  =  ·  , where p = 1, 2, ∞, R+ p (t)

(t)

H (at (x), at (y)) ≤ ((1 − di )xi )ni=1 − ((1 − di )yi )ni=1  ≤ (1 − d∗ )x − y. (6.183) Thus we have defined the mappings at and the cost functions ut , t = 0, 1, . . . . The control system considered in this section is a special case of the control system studied in Section 6.9. As we have already mentioned before, this control system corresponds to the nonstationary Robinson–Solow–Srinivasan model. Note that this control system satisfies the assumptions posed in Section 6.9 and therefore all the results stated there hold for this system. Indeed, fix M0 > (α∗ d∗ )−1 , and set n : z ≤ M0 e}. K = {z ∈ R+

In view of Lemma 6.25, at (K) ⊂ K, t = 0, 1, . . . . Relation (6.160) follows from (6.183). Evidently, (6.161) is true by (6.164). Lemma 6.26 implies that ut is upper semicontinuous for every nonnegative integer t. By Proposition 6.38, assumption (A5) holds. (A3) follows from Proposition 6.27.

6.11 Auxiliary Results for Theorems 6.23

219

6.11 Auxiliary Results for Theorems 6.23 Lemma 6.29 Let δ be a positive number. Then there exists an integer T0 ≥ 4 such 2 that for every integer τ1 ≥ 0, every integer τ2 ≥ T0 + τ1 , every program {x(t)}τt=τ 1 n for which there exists z ∈ R satisfying z ∈ aτ2 −1 (x(τ2 − 1)) and z ≥ x(τ2 ) + δe,

(6.184)

τ2 ˜ and every x˜0 ∈ K, there exists a program {x(t)} t=τ1 such that

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ Proof Fix a positive number D0 for which z ≤ D0 for all z ∈ K.

(6.185)

There exists a positive number c0 such that z2 ≤ c0 z for all z ∈ K.

(6.186)

Fix an integer T0 ≥ 4 such that 8D0 c0 κ T0 < δ.

(6.187)

2 and z ∈ Assume that integers τ1 ≥ 0, τ2 ≥ T0 + τ1 , a that program {x(t)}τt=τ 1 satisfy (6.184), and that x˜0 ∈ K. In view of (6.160), there exists a program τ2 −1 {x(t)} ˜ t=τ1 such that

n R+

x(τ ˜ 1 ) = x˜0 , x(t ˜ + 1) − x(t + 1) ≤ κx(t) ˜ − x(t), t = τ1 , . . . , τ2 − 2.

(6.188)

By (6.160) and (6.184), there exists ˜ 2 − 1)) x(τ ˜ 2 ) ∈ aτ2 −1 (x(τ

(6.189)

˜ 2 − 1). x(τ ˜ 2 ) − z ≤ κx(τ2 − 1) − x(τ

(6.190)

such that

τ2 Evidently, {x(t)} ˜ t=τ1 is a program. It follows from (6.185), (6.189), and (6.190) that

˜ 1 ) − x(τ1 ) ≤ κ τ2 −τ1 (2D0 ) ≤ κ T0 (2D0 ) x(τ ˜ 2 ) − z ≤ κ τ2 −τ1 x(τ

220

6 Infinite Horizon Nonautonomous Optimization Problems

and by (6.186) x(τ ˜ 2 ) − z2 ≤ 2D0 c0 κ T0 . This implies that for all integers i = 1, . . . , n, |x˜i (τ2 ) − zi | ≤ 2D0 c0 κ T0 . Combined with (6.184) and (6.187), this implies that x(τ ˜ 2 ) ≥ z − 2D0 c0 κ T0 e ≥ x(τ2 ). This completes the proof of Lemma 6.29. Lemma 6.30 Let δ be a positive number. Then there exists an integer L¯ ≥ 1 such ¯ there exists an integer τ ≥ L for which the following that for every integer L ≥ L, assertion holds: For every integer T ≥ τ and every program {x(t)}Tt=0 satisfying T −1

ut (x(t), x(t + 1)) = U (x(0), T ),

(6.191)

t=0

the inequality L−1 

ut (x(t), x(t + 1)) ≥ U (x(0), L) − δ

t=0

is valid. Proof Lemma 6.29 implies that there exists an integer L0 ≥ 4 such that the following property holds: 2 (P6) If integers τ1 ≥ 0 and τ2 ≥ L0 + τ1 , if for a program {x(t)}τt=τ there exists 1 n z ∈ R+ which satisfies

z ∈ aτ2 −1 (x(τ2 − 1)) and z ≥ x(τ2 ) + γ e, τ2 ˜ and if x˜0 ∈ K, then there exists a program {x(t)} t=τ1 such that

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ In view of (6.161), there exists an integer L¯ ≥ 1 such that for every natural ¯ we have number t ≥ L, sup{ut (x, x  ) : (x, x  ) ∈ graph(at )} ≤ δ(32L0 )−1 .

(6.192)

6.11 Auxiliary Results for Theorems 6.23

221

¯ and fix a natural number Assume that an integer L ≥ L, τ ≥ L + L0 + 2.

(6.193)

Assume that an integer T ≥ τ and that a program {x(t)}Tt=0 satisfies (6.191). We claim that L−1 

ut (x(t), x(t + 1)) ≥ U (x(0), L) − δ.

t=0 L satisfying Proposition 6.22 implies that there exists a program {x(t)} ˜ t=0 L−1 

ut (x(t), ˜ x(t ˜ + 1)) = U (x(0), L).

(6.194)

ut (x(t), x(t + 1)) < Δt , t = L + L0 , . . . , T − 1;

(6.195)

max{ut (x(t), x(t + 1)) − Δt : t = L + L0 , . . . , T − 1} ≥ 0.

(6.196)

x(0) ˜ = x(0),

t=0

There are two cases:

Assume that (6.195) is valid. Assumption (A5) implies that there exists a program {x (1) (t)}Tt=0 which satisfies x (1) (t) = x(t), ˜ t = 0, . . . , L, ut (x (1) (t), x (1) (t + 1)) ≥ Δt , t = L + 1, . . . , T − 1. It follows from (6.191), (6.192), (6.194), (6.195), and (6.197) that 0≥

T −1

ut (x (1) (t), x (1) (t + 1)) −

T −1

t=0

=

L−1 

ut (x(t), ˜ x(t ˜ + 1)) +

T −1

T −1

ut (x (1) (t), x (1) (t + 1))

t=L

t=0

−

ut (x(t), x(t + 1))

t=0

ut (x(t), x(t + 1))

t=0

≥ U (x(0), L) +

T −1 t=L+1

Δt −

T −1 t=0

ut (x(t), x(t + 1))

(6.197)

222

6 Infinite Horizon Nonautonomous Optimization Problems

≥ U (x(0), L) −

L−1 

ut (x(t), x(t + 1))

t=0

+

T −1

Δt −

L+L 0 −1 

ut (x(t), x(t + 1)) −

t=L

t=L+1

≥ U (x(0), L) −

L−1 

T −1

Δt

t=L+L0

ut (x(t), x(t + 1)) − δ

t=0

and L−1 

ut (x(t), x(t + 1)) ≥ U (x(0), L) − δ.

t=0

Assume that (6.196) is true. Then there exists an integer S0 for which L + L0 ≤ S0 − 1 ≤ T − 1,

(6.198)

uS0 −1 (x(S0 − 1), x(S0 )) ≥ ΔS0 −1 , ut (x(t), x(t + 1)) < Δt for every integer t satisfying L0 + L ≤ t < S0 − 1. (6.199) Assumption (A5) implies that there exists a program

0 −L0 {x (2) (t)}St=0

which satisfies

x (2) (t) = x(t), ˜ t = 0, . . . , L,

(6.200)

ut (x (2) (t), x (2) (t + 1)) ≥ Δt for all integers t satisfying L + 1 ≤ t ≤ S0 − L0 − 1. (6.201) It follows from (6.198) and (A5) that there exists y ∈ R n such that y ∈ aS0 −1 (x(S0 − 1)) and y ≥ x(S0 ) + γ e.

(6.202)

0 such Property (P6) and (6.202) imply that there exists a program {x (2) (t)}St=S 0 −L0 that

x (2) (S0 ) ≥ x(S0 ).

(6.203)

0 is a program. In view of (6.203) and (A3), there exists a Evidently, {x (2) (t)}St=0 program {x (2) (t)}Tt=0 such that

ut (x (2) (t), x (2) (t + 1)) ≥ ut (x(t), x(t + 1)), t = S0 , . . . , T − 1.

(6.204)

6.12 Proof of Theorem 6.23

223

It follows from (6.204), (6.191), (6.192), (6.194), and (6.199)–(6.201) that 0≥

T −1

ut (x (2) (t), x (2) (t + 1)) − U (x(0), T )

t=0

=

T −1

ut (x (2) (t), x (2) (t + 1)) −

t=0

≥

L−1 

ut (x(t), x(t + 1))

t=0

ut (x(t), ˜ x(t ˜ + 1)) −

t=0

+

T −1

L−1 

ut (x(t), x(t + 1))

t=0

S 0 −1

ut (x (2) (t), x (2) (t + 1)) −

t=L

S 0 −1

ut (x(t), x(t + 1))

t=L

≥ U (x(0), L) −

L−1 

S0 −L 0 −1

ut (x(t), x(t + 1)) +

t=L

t=0

−

S0 −L 0 −1

ut (x (2) (t), x (2) (t + 1))

ut (x(t), x(t + 1)) −

S 0 −1

ut (x(t), x(t + 1))

t=S0 −L0

t=L

≥ U (x(0), L) −

L−1 

ut (x(t), x(t + 1))

t=0

−

L0 +L−1

ut (x(t), x(t + 1)) −

ut (x(t), x(t + 1))

t=S0 −L0

t=L

≥ U (x(0), L) −

S 0 −1

L−1 

ut (x(t), x(t + 1)) − δ,

t=0 L−1 

ut (x(t), x(t + 1)) ≥ U (x(0), L) − δ.

t=0

Thus in both cases, the inequality above holds. This completes the proof of Lemma 6.30.

6.12 Proof of Theorem 6.23 Proposition 6.22 implies that for every z ∈ K and every integer T ≥ 1, there exists a program {xz,T (t)}Tt=0 such that

224

6 Infinite Horizon Nonautonomous Optimization Problems

xz,T (0) = z,

T −1

ut (xz,T (t), xz,T (t + 1)) = U (z, T ).

(6.205)

t=0

Let δ be a positive number. Lemma 6.30 implies that there exists an integer Lδ ≥ 1 such that the following property holds: (P7) For every integer L ≥ Lδ , there exists an integer τL ≥ L such that for every natural number T ≥ τL and every z ∈ K, L−1 

ut (xz,T (t), xz,T (t + 1)) ≥ U (z, L) − δ/4.

t=0

Let z ∈ K. There exist a strictly increasing sequence of natural numbers ∞ {Tk }∞ k=1 and a program {xz (t)}t=0 such that for each integer t ≥ 0, we have xz,Tk (t) → xz (t) as k → ∞.

(6.206)

xz (0) = z.

(6.207)

Evidently,

Let an integer L satisfy L ≥ Lδ , and let an integer τL ≥ L be as guaranteed by the property (P7). By (6.206) and upper semicontinuity of the functions ut , t = 0, 1, . . . , there exists an integer k ≥ 1 such that Tk ≥ τL ,

L−1 

ut (xz (t), xz (t +1)) ≥

t=0

L−1 

ut (xz,Tk (t), xz,Tk (t +1))−δ/4.

t=0

In view of (P7), (6.208), and the choice of τL , L−1 

ut (xz,Tk (t), xz,Tk (t + 1)) ≥ U (z, L) − δ/4.

t=0

Combined with (6.208) this implies that L−1  t=0

Theorem 6.23 is proved.

ut (xz (t), xz (t + 1)) ≥ U (z, L) − δ.

(6.208)

Chapter 7

One-Dimensional Robinson–Solow–Srinivasan Model

In this chapter we prove turnpike results for a class of discrete-time optimal control problems. These control problems arise in economic dynamics and describe the nonstationary Robinson–Solow–Srinivasan model. We study the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

7.1 Preliminaries and Main Results 1 ) be the set of real Denote by Card(E) the cardinality of a set E. Let R 1 (R+ Y (nonnegative) numbers. For each mapping a : X → 2 \ {∅}, where X, Y are nonempty sets, put graph(a) = {(x, y) ∈ X × Y : y ∈ a(x)}. For each integer t ≥ 0, let

αt > 0, dt ∈ (0, 1]

(7.1)

and for each integer t ≥ 0, let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that wt (0) = 0 and inf{wt (z) : an integer t ≥ 0} > 0 for all z > 0.

(7.2)

We suppose that the following assumption holds: (A1) For each  > 0 there exists δ > 0 such that wt (δ) ≤  for each integer t ≥ 0.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_7

225

226

7 One-Dimensional Robinson–Solow–Srinivasan Model

We now give a formal description of the model. 1 set Let t ≥ 0 be an integer. For each x ∈ R+   1 at (x) = y ∈ R+ : y ≥ (1 − dt )x and αt (y − (1 − dt )x) ≤ 1 .

(7.3)

1, It is clear that for each x ∈ R+

  at (x) = (1 − dt )x, αt−1 + (1 − dt )x 1 × R 1 . Suppose that and that graph(at ) is a closed subset of R+ +

inf{dt : t = 0, 1, . . . } > 0,

(7.4)

inf{αt : t = 0, 1, . . . } > 0,

(7.5)

sup{αt : t = 0, 1, . . . } < ∞,

(7.6)

sup{wt (M) : t = 0, 1, . . . } < ∞ for each M > 0.

(7.7)

The constraint mappings at , t = 0, 1, . . . have already been defined. Now we define the cost functions ut , t = 0, 1, . . . . For each integer t ≥ 0 and each (x, x  ) ∈ graph(at ), set     ut x, x  = sup{wt (y) : 0 ≤ y ≤ x and y + αt x  − (1 − dt )x ≤ 1}.

(7.8)

Clearly, for each integer t ≥ 0 and each (x, x  ) ∈ graph(at ),     ut x, x  = wt min{x, 1 − αt (x  − (1 − dt )x)} . Choose α ∗ , α∗ , d∗ > 0 such that α∗ < αt < α ∗ , d∗ < dt for all integers t ≥ 0.

(7.9)

Clearly, for each integer t ≥ 0, the function ut : graph(at ) → [0, ∞) is upper semicontinuous. 1 A sequence {x(t)}∞ t=0 ⊂ R+ is called a program if x(t + 1) ∈ at (x(t)) for all 2 ⊂ integers t ≥ 0. Let T1 , T2 be integers such that T1 < T2 . A sequence {x(t)}Tt=T 1 1 R+ is called a program if x(t+1) ∈ at (x(t)) for all integers t satisfying T1 ≤ t < T2 . In the sequel we assume that the supremum over an empty set is −∞. 1 and each pair of integers T < T , set For each x0 ∈ R+ 1 2

7.1 Preliminaries and Main Results

U (x0 , T1 , T2 ) = sup

227

⎧ 2 −1 ⎨T ⎩

2 ut (x(t), x(t + 1)) : {x(t)}Tt=T 1

t=T1

is a program and x(T1 ) = x0

⎫ ⎬ ⎭

(7.10)

.

1 and let T < T be integers. Set Let x0 , x˜0 ∈ R+ 1 2

U (x0 , x˜0 , T1 , T2 ) = sup

⎧ 2 −1 ⎨T ⎩

2 ut (x(t), x(t + 1)) : {x(t)}Tt=T 1

t=T1

is a program such that x(T1 ) = x0 , x(T2 ) ≥ x˜0 }.

(7.11)

Let T1 , T2 be integers such that T1 < T2 . Set

M (T1 , T2 ) = sup U

⎧ 2 −1 ⎨T ⎩

ut (x(t), x(t + 1))

t=T1

2 is a program and x(T1 ) ≤ M : {x(t)}Tt=T 1

⎫ ⎬ ⎭

.

(7.12)

Upper semicontinuity of ut , t = 0, 1, . . . , compactness of sets of admissible programs, and the optimization theorem of Weierstrass imply the following results. 1 and each pair of integers T < T , there exists Proposition 7.1 For each x0 ∈ R+ 1 2 T2 a program {x(t)}t=T1 such that x(T1 ) = x0 and T 2 −1

ut (x(t), x(t + 1)) = U (x0 , T1 , T2 ).

t=T1

Proposition 7.2 For each natural number T and each M > 0, there exists a program {x(t)}Tt=0 such that T −1 t=0

M (0, T ) ut (x(t), x(t + 1)) = U

228

7 One-Dimensional Robinson–Solow–Srinivasan Model

and x(0) ≤ M. Fix M∗ > (α∗ d∗ )−1 + 1.

(7.13)

It is clear that the model considered here is a particular case of the model discussed in Chapter 6 with n = 1 (see Section 6.3). Therefore all the results of Chapter 6 can be applied. Theorem 6.4 and Lemma 6.9 imply the following result. Theorem 7.3 There exists M¯ > 0 such that for each x0 ∈ [0, M∗ ], there exists ∞ such that x(0) a program {x(t)} ¯ ¯ = x0 , for each pair of integers T1 , T2 ≥ 0 t=0 satisfying T1 < T2 ,    T2 −1    ut (x(t), ¯ x(t ¯ + 1)) − UM∗ (T1 , T2 ) ≤ M¯    t=T1 and that for each integer T > 0, T −1

ut (x(t), ¯ x(t ¯ + 1)) = U (x(0), ¯ x(T ¯ ), 0, T ).

t=0

Lemma 6.9 and Proposition 6.6 imply the following result. ∞ be as guaranteed Theorem 7.4 Let x0 ∈ [0, M∗ ] and let a program {x(t)} ¯ t=0 ∞ by Theorem 7.3. Assume that {x(t)}t=0 is a program. Then either the sequence T −1 T −1 ut (x(t), x(t + 1)) − t=0 ut (x(t), ¯ x(t ¯ + 1))}∞ { t=0 T =1 is bounded or T −1

ut (x(t), x(t + 1)) −

t=0

T −1

ut (x(t), ¯ x(t ¯ + 1)) → −∞ as T → ∞.

t=0

Other results of this chapter were obtained in [100]. Let M¯ > 0 be as guaranteed by Theorem 7.3. Fix x∗0 ∈ [0, M∗ ] and let a program {x ∗ (t)}∞ t=0 be as guaranteed by Theorem 7.3. Namely, x ∗ (0) = x∗0 , T −1

ut (x ∗ (t), x ∗ (t + 1)) = U (x ∗ (0), x ∗ (T ), 0, T )

t=0

for each integer T > 0 and

(7.14)

7.1 Preliminaries and Main Results

229

   T   2 −1 ∗ ∗   ¯ u (x (t), x (t + 1)) − U (T , T ) t M∗ 1 2  ≤ M    t=T1

(7.15)

for each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 . For each integer t ≥ 0, set   y ∗ (t) = min x ∗ (t), 1 − αt (x ∗ (t + 1) − (1 − dt )x ∗ (t)) .

(7.16)

We will show that the program {x ∗ (t)}∞ t=0 is the turnpike for the model. A function w : [0, ∞) → R 1 is called strictly concave if for each x, y ∈ [0, ∞) satisfying x = y and each α ∈ (0, 1), w(αx + (1 − α)y) > αw(x) + (1 − α)w(y). The following two results are consequences of the optimization theorem of Weierstrass. Proposition 7.5 Assume that w : [0, ∞) → [0, ∞) is continuous strictly concave function. Let , M > 0. Then there exists δ0 > 0 such that for each x, y ∈ [0, M] satisfying |x − y| ≥ , w(2−1 (x + y)) − 2−1 w(x) − 2−1 w(y) ≥ δ0 . Proposition 7.6 Assume that w : [0, ∞) → [0, ∞) is a strictly increasing continuous function, M > 0, and  ∈ (0, M). Then inf{w(x) − w(y) : x, y ∈ [0, M] and x ≥ y + } > 0. We suppose that the following assumptions hold: (A2) For each , M > 0 there exists 0 > 0 such that for each x, y ∈ (0, M] satisfying |x − y| ≥  and each integer t ≥ 0,

 wt 2−1 (x + y) − 2−1 wt (x) − 2−1 wt (y) ≥ 0 . (A3) For each M > 0 and each  ∈ (0, M], there is 1 > 0 such that for each integer t ≥ 0 and each x, y ∈ [0, M] satisfying x ≥ y + , wt (x) − wt (y) ≥ 1 . (A4) For each M > 0 and each  > 0, there exists δ > 0 such that for each integer t ≥ 0 and each x, y ∈ [0, M] satisfying |x − y| ≤ δ, the inequality |wt (x) − wt (y)| ≤  holds. Note that (A2) is an assumption of uniform concavity of the functions wt , t = 0, 1, . . . , (A3) is an assumption of uniform strict monotonicity of the functions wt , t = 0, 1, . . . , and (A4) is an assumption of uniform equicontinuity of the functions wt , t = 0, 1, . . . .

230

7 One-Dimensional Robinson–Solow–Srinivasan Model

It is easy to see that (A1) follows from (A4) and (7.2). We assume that d ∗ := sup{dt : t = 0, 1, . . . } < 1.

(7.17)

The following theorems describe the structure of optimal program of the model. Theorem 7.7 Let M > 0 and  > 0. Then there exists a natural number Q such 2 that for each pair of integers T1 ≥ 0 and T2 ≥ Q + T1 and each program {x(t)}Tt=T 1 T2 −1 which satisfies x(T1 ) ≤ M∗ , u (x(t), x(t + 1)) ≥ U (x(T ), T , T ) − M, 1 1 2 t=T1 t the following inequality holds:   Card {t ∈ {T1 , . . . , T2 } : |x(t) − x ∗ (t)| > } ≤ Q. Theorem 7.7 is proved in Section 7.5. Theorem 7.8 Let M,  > 0. Then there exist a natural number p and δ > 0 such 2 that for each pair of integers T1 ≥ 0, T2 ≥ 2p + T1 and each program {x(t)}Tt=T 1 T2 −1 satisfying x(T1 ) ≤ M∗ , t=T1 ut (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ, U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 + p, T2 − p]. Theorem 7.9 Let M > 0 and  > 0. Then there exist a natural number p and δ > 0 such that for each pair of integers T1 ≥ 0, T2 ≥ p + T1 and each program 2 which satisfies x(T1 ) ≤ M∗ , |x(T1 ) − x ∗ (T1 )| ≤ δ, {x(t)}Tt=T 1 T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ,

t=T1

U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 , T2 − p]. Theorem 7.8 is proved in Section 7.3, while Theorem 7.9 is proved in Section 7.4. Theorem 7.10 Let  > 0. Then there exist a natural number p and δ > 0 such 2 that for each pair of integers T1 ≥ 0, T2 ≥ 2p + T1 and each program {x(t)}Tt=T 1 T2 −1 which satisfies x(T1 ) ≤ M∗ , t=T1 ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − δ, the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 + p, T2 − p]. Theorem 7.11 Let  > 0. Then there exist a natural number p and δ > 0 such that 2 for each pair of integers T1 ≥ 0, T2 ≥ p + T1 and each program {x(t)}Tt=T which 1 ∗ satisfies x(T1 ) ≤ M∗ , |x(T1 ) − x (T1 )| ≤ δ,

7.2 Auxiliary Results

231

T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − δ

t=T1

the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 , T2 − p]. Theorems 7.10 and 7.11 easily follow from Theorems 7.8 and 7.9, respectively. A program {x(t)}∞ t=0 is called good if the sequence T −1 

ut (x(t), x(t + 1)) −

T −1

t=0

∞ ∗

∗

ut (x (t), x (t + 1)) T =1

t=0

is bounded. In view of Theorem 7.4, if the sequence {x(t)}∞ t=0 is not good, then lim

T −1 

T →∞

ut (x(t), x(t + 1)) −

t=0

T −1

 ∗

∗

ut (x (t), x (t + 1)) = −∞.

t=0

In Section 7.6 we prove the following result. ∗ Theorem 7.12 Assume that a program {x(t)}∞ t=0 is good. Then x(t) − x (t) → 0 as t → ∞. ∞  A program {x(t)}∞ t=0 is called overtaking optimal if for each program {x (t)}t=0  satisfying x (0) = x(0),

lim sup T →∞

T −1 





ut (x (t), x (t + 1)) −

t=0

T −1

 ut (x(t), x(t + 1)) ≤ 0.

t=0

In Section 7.6 we prove the following result. ∞ be as guaranteed by Theorem 7.13 Let x0 ∈ [0, M∗ ] and let a program {x(t)} ¯ t=0 ∞ Theorem 7.3. Then {x(t)} ¯ t=0 is a unique overtaking optimal program with the initial state x0 .

7.2 Auxiliary Results We begin with the result which follows from Lemma 6.9 and Corollary 6.20. Proposition 7.14 There exist M1 > 0 and a natural number L such that for each pair of integers T1 ≥ 0, T2 ≥ T1 + L and each x0 ∈ [0, M∗ ], the inequality M∗ (T1 , T2 )| ≤ M1 holds. |U (x0 , T1 , T2 ) − U The following lemma shows the uniform equicontinuity of the functions ut , t = 0, 1, . . . .

232

7 One-Dimensional Robinson–Solow–Srinivasan Model

Lemma 7.15 Let  > 0 and M ≥ M∗ . Then there exists δ > 0 such that for each integer t ≥ 0 and each     x, x  , y, y  ∈ graph(at )

(7.18)

  x, y ≤ M, |x − y|, x  − y   ≤ δ

(7.19)

satisfying

the inequality |ut (x, x  ) − ut (y, y  )| ≤  holds. Proof By (A4) there is δ > 0 such that for each integer t ≥ 0 and each z, z ∈ [0, M] satisfying |z − z | ≤ 2δ(1 + α ∗ ), the following inequality holds:   wt (z) − wt (z ) ≤ /2.

(7.20)

Let an integer t ≥ 0 and let (x, x  ), (y, y  ) satisfy (7.18) and (7.19). In order to prove the lemma, it is sufficient to show that ut (y, y  ) ≥ ut (x, x  ) − . By (7.8) there is z ≥ 0 such that     z ≤ x, z ≤ 1 − αt x  − (1 − dt )x , ut x, x  = wt (z).

(7.21)

By (7.9), (7.19), and (7.21), y ≥ x − δ ≥ z − δ,     1−αt y  − (1 − dt )y ≥ 1−αt x  + δ − (1 − dt )(x − δ) ≥ z−2α ∗ δ.

(7.22) (7.23)

Put   z = max 0, z − δ(1 + 2α ∗ ) .

(7.24)

In view of (7.22) and (7.24), 0 ≤ z ≤ y.

(7.25)

It follows from (7.3), (7.18), (7.23), and (7.24) that   1 − αt y  − (1 − dt )y ≥ z .

(7.26)

Relations (7.8), (7.18), (7.24), (7.25), and (7.26) imply that     wt z ≤ ut y, y  . By (7.19), (7.21), (7.24), (7.25), and the choice of δ (see (7.20)),

(7.27)

7.2 Auxiliary Results

233

  wt (z ) − wt (z) ≤ /2. Together with (7.21) and (7.27), this implies that     ut y, y  ≥ wt (z) − /2 ≥ ut x, x  − /2. Lemma 7.15 is proved. The next result easily follows from (7.8) and the strict monotonicity of wt , t = 0, 1, . . . . Lemma 7.16 Let t ≥ 0 be an integer, (x, x  ) ∈ graph(at ), and let y ∈ [0, x] satisfy y + αt (x  − (1 − dt )x) ≤ 1. Then wt (y) = ut (x, x  ) if and only if y = min{x, 1 − αt (x  − (1 − dt )x)}. > 0 and a program { By Proposition 6.12, there exist Δ x (t)}∞ t=0 such that x (0) < 1 and that ut ( x (t), x (t + 1)) ≥ Δ for all integers t ≥ 0. Let a positive number γ satisfy γ < 1/2 and γ < 4−1 Δ.

(7.28)

Lemmas 6.9 and 6.16 imply the following result. Lemma 7.17 Let M1 > 0. Then there exist integers L1 , L2 ≥ 4 such that for each 2 satisfying pair of integers T1 ≥ 0, T2 ≥ L1 + L2 + T1 , each program {x(t)}Tt=T 1 x(T1 ) ≤ M∗ ,

T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M1

t=T1

and each integer τ ∈ [T1 + L1 , T2 − L2 ], the following inequality holds: max{ut (x(t), x(t + 1)) : t = τ, . . . , τ + L2 − 1} ≥ γ . Lemmas 6.9 and 6.17 imply the following result. Lemma 7.18 Let M1 > 0. Then there exist natural numbers L¯ 1 , L¯ 2 and M2 > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ L¯ 1 + L¯ 2 + τ1 and each program 2 satisfying {x(t)}τt=τ 1

234

7 One-Dimensional Robinson–Solow–Srinivasan Model

x(τ1 ) ≤ M∗ ,

τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M1

t=τ1

the following assertion holds: If integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M2 .

t=T1

Recall that the positive constant M¯ was fixed in Section 7.1 after the statement of Theorem 7.4. Lemma 7.19 Let  > 0. Then there exist a natural number L0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and each z1 , z2 , z1 , z2 ∈ [0, M∗ ] satisfying   zi − z  ≤ δ, i = 1, 2, i

(7.29)

M∗ (τ1 , τ2 ) − M¯ − 2 U (z1 , z2 , τ1 , τ2 ) ≥ U

(7.30)

the inequality   U z1 , z2 , τ1 , τ2 ≥ U (z1 , z2 , τ1 , τ2 ) −  holds. Proof By Lemma 7.17 there exist natural numbers L1 , L2 ≥ 4 such that the following property holds: 2 (P1) for each pair of integers T1 ≥ 0, T2 ≥ L1 +L2 +T1 , each program {x(t)}Tt=T 1 which satisfies x(0) ≤ M∗ ,

T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M¯ − 4

t=T1

and each integer τ ∈ [T1 + L1 , T2 − L2 ], the following inequality holds: max{ut (x(t), x(t + 1)) : t = τ, . . . , τ + L2 − 1} ≥ γ . In view of (A1), there is γ0 > 0 such that wt (γ0 ) < γ /4 for each integer t ≥ 0. (A4) implies that there is δ1 > 0 such that

(7.31)

7.2 Auxiliary Results

235

δ1 < γ0 /4;

(7.32)

for each integer t ≥ 0 and each y, y  ∈ [0, M∗ ] satisfying |y − y  | ≤ 2δ1 ,   wt (y) − wt (y  ) ≤ 8−1 .

(7.33)

Choose natural numbers L3 > L1 + L2 + 4, L0 > 8 + 4(L1 + L2 + 2L3 ).

(7.34)

By Lemma 7.15, there exists δ2 > 0 such that δ2 α ∗ < γ0 /4

(7.35)

and that for each integer t ≥ 0 and each (x, x  ), (y, y  ) ∈ graph(at ) satisfying x, y ≤ M∗ , |x − x  |, |y − y  | ≤ 2δ2 , the following inequality holds:

     |ut x, x  − ut y, y  | ≤ 8−1  (L3 + L2 + 1)−1 .

(7.36)

  δ = min δ1 , δ2 , (4−1 γ0 )(1 + α ∗ )−1 , 8−1 (1 − d ∗ )L3 +L2 δ1 (1 + α ∗ )−1 .

(7.37)

Set

Assume that integers τ1 ≥ 0, τ2 ≥ τ1 + L0 ,

(7.38)

z1 , z2 , z1 , z2 ∈ [0, M∗ ], |zi − zi | ≤ δ, i = 1, 2,

(7.39)

M∗ (τ1 , τ2 ) − M¯ − 2. U (z1 , z2 , τ1 , τ2 ) ≥ U

(7.40)

2 such that By the continuity of wt , t = 0, 1, . . . , there is a program {x(t)}τt=τ 1

x(τ1 ) = z1 , x(τ2 ) ≥ z2 , τ 2 −1

(7.41)

M∗ (τ1 , τ2 ) − M¯ − 2. ut (x(t), x(t + 1)) = U (z1 , z2 , τ1 , τ2 ) ≥ U

t=τ1

It follows from (P1), (7.34), (7.38), (7.39), and (7.41) that max{ut (x(t), x(t + 1)) : t = L3 + τ1 , . . . , L3 + τ1 + L2 − 1} ≥ γ ,

(7.42)

max{ut (x(t), x(t + 1)) : t = τ2 − L3 − L2 , . . . , τ2 − L3 − 1} ≥ γ .

(7.43)

236

7 One-Dimensional Robinson–Solow–Srinivasan Model

By (7.42) and (7.43), there are integers t1 ∈ [L3 + τ1 , L3 + τ1 + L2 − 1], t2 ∈ [τ2 − L3 − L2 , τ2 − L3 − 1]

(7.44)

such that ut1 (x(t1 ), x(t1 + 1)) ≥ γ , ut2 (x(t2 ), x(t2 + 1)) ≥ γ .

(7.45)

For each integer t ∈ [τ1 , τ2 − 1], set y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )xt )}.

(7.46)

In view of (7.46) and Lemma 7.16 for all integers t ∈ [τ1 , τ2 − 1], ut (x(t), x(t + 1)) = wt (y(t)).

(7.47)

It follows from (7.31), (7.45), (7.46), and the monotonicity of wt , t = 0, 1, . . . that y(t1 ), y(t2 ) > γ0 .

(7.48)

τ2 Now we construct a program {x(t)} ˜ t=τ1 . Set

x(τ ˜ 1 ) = z1

(7.49)

and for all integers t = τ1 , . . . , t1 − 1, put ˜ + x(t + 1) − (1 − dt )x(t). x(t ˜ + 1) = (1 − dt )x(t)

(7.50)

t1 Clearly, {x(t)} ˜ t=τ1 is a program. By (7.39), (7.41), (7.49), and (7.50), for all t = τ1 , , . . . , t1 ,

|x(t) ˜ − x(t)| ≤ |x(τ ˜ 1 ) − x(τ1 )| ≤ δ.

(7.51)

x(t ˜ 1 + 1) = x(t1 + 1) + δ.

(7.52)

Put

In view of (7.46) and (7.48), αt1 (x(t1 + 1) − (1 − dt1 )x(t1 )) ≤ 1 − y(t1 ) ≤ 1 − γ0 . By (7.37) and (7.51)–(7.53), ˜ 1 + 1) − (1 − dt1 )x(t ˜ 1 )) αt1 (x(t

(7.53)

7.2 Auxiliary Results

237

≤ αt1 (x(t1 + 1) − (1 − dt1 )x(t1 ) + 2δ) ≤ 1 − γ0 + 2δα ∗ < 1.

(7.54)

By (7.51) and (7.52), ˜ 1 ) ≥ x(t1 + 1) − (1 − dt1 )x(t1 ) ≥ 0. x(t ˜ 1 + 1) − (1 − dt1 )x(t t1 +1 Therefore {x(t)} ˜ t=τ1 is a program. It follows from (7.37), (7.51), (7.52), the choice of δ2 (see (7.35) and (7.36)), (7.39)–(7.41), (7.49), and Lemma 6.9 that

˜ x(t+1))| ˜ ≤ (8−1 )(L3 +L2 + 1)−1 , t = τ1 , . . . , t1 . |ut (x(t), x(t+1))−ut (x(t), (7.55) In view of (7.44) and (7.55),  t  t1 1      ut (x(t), x(t + 1)) − ut (x(t), ˜ x(t ˜ + 1))    t=τ1

(7.56)

t=τ1

≤ (8−1 )(L3 + L2 + 1)−1 (t1 − τ1 + 1) ≤ 8−1 . For all t = t1 + 2, . . . , t2 set ˜ − 1) − (1 − dt−1 )x(t − 1). x(t) ˜ = x(t) + (1 − dt−1 )x(t

(7.57)

Relations (7.52) and (7.57) imply that x(t) ˜ ≥ x(t), t = t1 + 1, . . . , t2 . Clearly, t2 {x(t)} ˜ t=τ1 is a program. Set ˜ 2 ) − (1 − dt2 )x(t2 ). x(t ˜ 2 + 1) = x(t2 + 1) + (αt2 )−1 δ1 + (1 − dt2 )x(t

(7.58)

By (7.32), (7.46), (7.48), and (7.58), ˜ 2 + 1) − (1 − dt2 )x(t ˜ 2 )) = αt2 (x(t2 + 1) − (1 − dt2 )x(t2 ) + (αt2 )−1 δ1 ) αt2 (x(t ≤ 1 − y(t2 ) + δ1 ≤ 1 − γ0 + δ1 < 1 − γ0 /2.

(7.59)

By (7.58) and (7.59), ˜ 2 )). x(t ˜ 2 + 1) ∈ at2 (x(t

(7.60)

In view of (7.32), (7.48), (7.52), and (7.57), x(t ˜ 2 ) > y(t2 ) − δ1 > γ0 /2. Relation (7.59) implies that

(7.61)

238

7 One-Dimensional Robinson–Solow–Srinivasan Model

y(t2 ) − δ1 ≤ 1 − αt2 (x(t ˜ 2 + 1) − (1 − dt2 )x(t ˜ 2 )).

(7.62)

It follows from (7.8), (7.61), (7.62), the choice of δ1 (see (7.33)), (7.39), (7.41), (7.47), and (7.61) that ˜ 2 ), x(t ˜ 2 + 1)) ≥ wt2 (y(t2 ) − δ1 ) ut2 (x(t ≥ wt2 (y(t2 )) − /8 = ut2 (x(t2 ), x(t2 + 1)) − /8.

(7.63)

For all t = t2 + 1 . . . , τ2 − 1 put ˜ + x(t + 1) − (1 − dt )x(t). x(t ˜ + 1) = (1 − dt )x(t)

(7.64)

By (7.57), (7.58), and (7.64), x(t) ˜ ≥ x(t), t = t2 + 1, . . . , τ2 .

(7.65)

τ2 Now it is easy to see that {x(t)} ˜ t=τ1 is a program. In view of (7.8), (7.64), and (7.65),

˜ x(t ˜ + 1)) ≥ ut (x(t), x(t + 1)), t = t2 + 1, . . . , τ2 − 1. ut (x(t),

(7.66)

Relations (7.17) and (7.64) imply that for all t = t2 + 1, . . . , τ2 − 1, ˜ − x(t)) ≥ (1 − d ∗ )(x(t) ˜ − x(t)). x(t ˜ + 1) − x(t + 1) = (1 − dt )(x(t) Combined with (7.9), (7.17), (7.47), (7.57), and (7.58), this implies that ˜ 2 + 1)−x(t2 + 1)) ≥ (1 − d ∗ )L3 +L2 δ1 (α ∗ )−1 . x(τ ˜ 2 )−x(τ2 ) ≥ (1−d ∗ )τ2 −t2 −1 (x(t (7.67) It follows from (7.37), (7.39), (7.41), and (7.67) that x(τ ˜ 2 ) ≥ x(τ2 ) + (1 − d ∗ )L3 +L2 δ1 (α ∗ )−1 ≥ z2 + (1 − d ∗ )L3 +L2 δ1 (α ∗ )−1 ≥ z2 − δ + (1 − d ∗ )L3 +L2 δ1 (α ∗ )−1 > z2 . By the relation above, (7.41), (7.49), (7.57), the relation x(t) ˜ ≥ x(t), t = t1 + 1, . . . , t2 , (7.8), (7.56), (7.63), and (7.66), U (z1 , z2 , τ1 , τ2 ) ≥

τ 2 −1

ut (x(t), ˜ x(t ˜ + 1)) ≥

t=τ1

⎡

τ 2 −1

+⎣

t=τ1

ut (x(t), ˜ x(t ˜ + 1)) −

τ 2 −1 t=τ1

τ 2 −1 t=τ1

ut (x(t), x(t + 1)) ⎤

ut (x(t), x(t + 1))⎦

7.2 Auxiliary Results

239

≥ U (z1 , z2 , τ1 , τ2 ) +

t1 

ut (x(t), ˜ x(t ˜ + 1)) −

t=τ1

t1 

ut (x(t), x(t + 1))

t=τ1

˜ 2 ), x(t ˜ 2 + 1))−ut2 (x(t2 ), x(t2 + 1)) ≥ U (z1 , z2 , τ1 , τ2 ) − 8−1  − /8. +ut2 (x(t Lemma 7.19 is proved. Lemma 7.20 Let  ∈ (0, 1). Then there exist a natural number L0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and each z1 , z2 ∈ [0, M∗ ] satisfying |z1 − x ∗ (τ1 )|, |z2 − x ∗ (τ2 | ≤ δ

(7.68)

the inequality |U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − U (z1 , z2 , τ1 , τ2 )| ≤  holds. Proof By Lemma 7.19 there exist a natural number L0 ≥ 4 and δ > 0 such that the following property holds: (P2) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and each z1 , z2 , z1 , z2 ∈ [0, M∗ ] satisfying |zi − zi | ≤ δ, i = 1, 2 and M∗ (τ1 , τ2 ) − M¯ − 2 U (z1 , z2 , τ1 , τ2 ) ≥ U the inequality U (z1 , z2 , τ1 , τ2 ) ≥ U (z1 , z2 , τ1 , τ2 ) − /4 holds. Assume that integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and that z1 , z2 ∈ [0, M∗ ] satisfy (7.68). By (7.68), (P2), (7.15), (7.13) U (z1 , z2 , τ1 , τ2 ) ≥ U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − /4.

(7.69)

Together with (7.14) and (7.15), this implies that M∗ (τ1 , τ2 ) − M¯ − 2. U (z1 , z2 , τ1 , τ2 ) ≥ U Property (P2), (7.68), and (7.70) imply that U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) ≥ U (z1 , z2 , τ1 , τ2 ) − /4. Combined with (7.69) this implies that |U (z1 , z2 , τ1 , τ2 ) − U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 )| ≤ /2.

(7.70)

240

7 One-Dimensional Robinson–Solow–Srinivasan Model

Lemma 7.20 is proved. Lemma 7.21 Let  > 0. Then there exist a natural number L0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 , each z1 , z2 ∈ [0, M∗ ], and each 2 program {x(t)}τt=τ satisfying 1

τ 2 −1

|z1 − x ∗ (τ1 )|, |z2 − x ∗ (τ2 )| ≤ δ,

(7.71)

x(τ1 ) = z1 , x(τ2 ) ≥ z2 ,

(7.72)

ut (x(t), x(t + 1)) ≥ U (z1 , z2 , τ1 , τ2 ) − δ

(7.73)

t=τ1

the following inequality holds for all t = τ1 , . . . , τ2 − 1: |y ∗ (t) − min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}| ≤ .

(7.74)

Proof By (A2) there is a positive number 1 such that for each integer t ≥ 0 and each x, y ∈ [0, M∗ ] satisfying |x − y| ≥ /2,

 wt 2−1 (x + y) − 2−1 wt (x) − 2−1 wt (y) ≥ 21 .

(7.75)

Lemma 7.20 implies that there is δ > 0 and a natural number L0 ≥ 4 such that δ < 1 /2; for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and each z1 , z2 ∈ [0, M∗ ] satisfying |z1 − x ∗ (τ1 )|, |z2 − x ∗ (τ2 )| ≤ δ, the following inequality holds:   U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − U (z1 , z2 , τ1 , τ2 ) ≤ 1 /4.

(7.76)

Assume that pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 , z1 , z2 ∈ [0, M∗ ] and that a 2 program {x(t)}τt=τ satisfies (7.71)–(7.73). We show that for all t = τ1 , . . . , τ2 − 1 1 (7.74) holds. Assume the contrary. Then there is an integer s ∈ [τ1 , τ2 − 1] such that   ∗ y (s) − min{x(s), 1 − αs (x(s + 1) − (1 − ds )x(s))} > .

(7.77)

For all t = τ1 , . . . , τ2 − 1 set y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}.

(7.78)

By (7.78) and Lemma 7.16, ut (x(t), x(t + 1)) = wt (y(t)), t = τ1 . . . , τ2 − 1.

(7.79)

7.2 Auxiliary Results

241

2 It is not difficult to see that {2−1 x(t) + 2−1 x ∗ (t)}τt=τ is a program and that 1

 ut 2−1 (x(t) + x ∗ (t)), 2−1 (x(t + 1) + x ∗ (t + 1))

 ≥ wt 2−1 y(t) + 2−1 y ∗ (t) , t = τ1 , . . . , τ2 − 1.

(7.80)

By (7.73) and the choice of δ (see (7.76)), τ 2 −1

ut (x(t), x(t + 1)) ≥ U (z1 , z2 , τ1 , τ2 )−δ ≥ U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 )

t=τ1

− δ − 1 /4.

(7.81)

In view of (7.72), 2−1 x(τ1 ) + 2−1 x ∗ (τ1 ) = (z1 + x ∗ (τ1 ))/2, 2−1 x(τ2 ) + 2−1 x ∗ (τ2 ) ≥ 2−1 (x ∗ (τ2 ) + z2 )). In view of (7.71) and the choice of δ (see (7.76)),   U (2−1 z1 + 2−1 x ∗ (τ1 ), 2−1 z2 + 2−1 x ∗ (τ2 ), τ1 , τ2 )  −U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) ≤ 1 /4.

(7.82)

(7.83)

Relations (7.16), (7.79), (7.80), and Lemma 7.16 imply that for all t = τ1 , . . . , τ2 − 1,

 ut 2−1 (x(t)+x ∗ (t)), 2−1 (x(t + 1)+x ∗ (t + 1)) ≥ 2−1 wt (y(t)) + 2−1 wt (y ∗ (t))   = 2−1 ut (x(t), x(t + 1)) + 2−1 ut x ∗ (t), x ∗ (t + 1) . (7.84) By the choice of 1 (see (7.75)), (7.77), and (7.78), ws (2−1 (y ∗ (s) + y(s))) − ws (y ∗ (s))/2 − ws (y(s))/2 ≥ 21 .

(7.85)

It follows from (7.16), (7.79), (7.80), (7.85), and Lemma 7.16 that

 us 2−1 (x(s) + x ∗ (s)), 2−1 (x(s + 1) + x ∗ (s + 1))   ≥ 2−1 us (x(s), s(s + 1)) + 2−1 us x ∗ (s), x ∗ (s + 1) + 21 .

(7.86)

242

7 One-Dimensional Robinson–Solow–Srinivasan Model

In view of (7.14), (7.72), (7.81), (7.82), (7.84), (7.86), and the relation δ < 1 /2, U (2−1 z1 + 2−1 x ∗ (τ1 ), 2−1 z2 + 2−1 x ∗ (τ2 ), τ1 , τ2 ) ≥

τ 2 −1

ut (2−1 (x(t) + x ∗ (t)), 2−1 (x(t + 1) + x ∗ (t + 1)))

t=τ1

≥ 2−1

τ 2 −1

ut (x(t), x(t + 1)) + 2−1

t=τ1

τ 2 −1

ut (x ∗ (t), x ∗ (t + 1)) + 21

t=τ1

≥ 2−1 U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) + 2−1 U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − δ − 1 /4 + 21 ≥ U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) + 1 . This contradicts (7.83). The contradiction we have reached proves (7.74) for all t = τ1 , . . . , τ2 − 1. Lemma 7.21 is proved. Lemma 7.22 Let  > 0. Then there exist a natural number L0 ≥ 4 and δ > 0 such 2 that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and each program {x(t)}τt=τ 1 which satisfies x(τ1 ) ≤ M∗ ,

τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ

(7.87)

t=τ1

and M∗ (τ1 , τ2 ) − M¯ − 2, U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U

(7.88)

for each integer t ∈ [τ1 , τ2 − L0 ], the following inequality holds: x(t) +  ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)).

(7.89)

Proof By Lemma 7.17, there exist natural numbers L1 , L2 ≥ 4 such that the following property holds: (P3) for each pair of integers T1 ≥ 0, T2 ≥ L1 + L2 + T1 and each program 2 {x(t)}Tt=T which satisfies x(0) ≤ M∗ and 1 T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M¯ − 4

t=T1

and each integer τ ∈ [T1 + L1 , T2 − L2 ], the following inequality holds:

7.2 Auxiliary Results

243

max{ut (x(t), x(t + 1)) : t = τ, . . . , τ + L2 − 1} ≥ γ . By (A1) there is γ0 > 0 such that wt (γ0 ) < γ /4 for each integer t ≥ 0.

(7.90)

Choose γ1 > 0 such that γ1 (d∗ )−1 < γ0 /4

(7.91)

and natural numbers L3 , L0 such that L3 > 4(L1 + L2 ), (M∗ + 1)(1 − d∗ )L3 < γ0 /4, L0 ≥ 4(L1 + L2 + L3 + 4).

(7.92) (7.93)

Choose a positive number γ2 such that γ2 < γ1 /4, γ2 < (1 − d ∗ )L3 +L2 (α ∗ )−1 /8.

(7.94)

By (A3) there is δ ∈ (0, 1) such that the following property holds: is valid (P4) for each integer t ≥ 0 and each z1 , z2 ∈ [0, M∗ ] satisfying z2 ≥ z1 + γ2 min{1, α∗ }, the inequality wt (z2 ) − wt (z1 ) ≥ 4δ. 2 Assume that integers τ1 ≥ 0, τ2 ≥ τ1 +L0 , a program {x(t)}τt=τ satisfies (7.87), 1 (7.88), and an integer t0 ∈ [τ1 , τ2 − L0 ].

(7.95)

We show that (7.89) holds with t = t0 . Assume the contrary. Then x(t0 ) +  < 1 − αt0 (x(t0 + 1) − (1 − dt0 )x(t0 )).

(7.96)

For all t = τ1 , . . . , τ2 − 1 set y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}.

(7.97)

τ2 Now we define a program {x(t)} ˜ t=τ1 . Set

˜ 0 + 1) = x(t0 + 1) + αt−1 . x(t) ˜ = x(t), t = τ1 , . . . , t0 , x(t 0 Clearly, ˜ 0 ). x(t ˜ 0 + 1) ≥ (1 − dt0 )x(t By (7.96)–(7.98),

(7.98)

244

7 One-Dimensional Robinson–Solow–Srinivasan Model

αt0 (x(t ˜ 0 + 1) − (1 − dt0 )x(t ˜ 0 )) = αt0 (x(t0 + 1) − (1 − dt0 )x(t0 )) +  < 1 − x(t0 ) ≤ 1 − y(t0 ).

(7.99)

t0 +1 Therefore {x(t)} ˜ t=τ1 is a program. In view of (7.8) and (7.97)–(7.99) and Lemma 7.16,

˜ 0 ), x(t ˜ 0 + 1)) ≥ wt0 (y(t0 )) = ut0 (x(t0 ), x(t0 + 1)). ut0 (x(t

(7.100)

Property (P3) and relations (7.92), (7.93), and (7.95) imply that max{ut (x(t), x(t + 1)) : t = t0 + 1 + L3 , . . . , t0 + L3 + L2 } ≥ γ .

(7.101)

It follows from (7.101) that there exists an integer t1 such that t0 + 1 + L3 ≤ t1 ≤ t0 + L3 + L2 , ut1 (x(t1 ), x(t1 + 1)) ≥ γ .

(7.102)

By (7.97), Lemma 7.16, (7.90), and (7.102), γ ≤ ut1 (x(t1 ), x(t1 + 1)) = wt1 (y(t1 )) and x(t1 ) ≥ y(t1 ) ≥ γ0 .

(7.103)

We show that there is an integer t2 such that t0 + 1 ≤ t2 ≤ t1 − 1,

(7.104)

x(t2 + 1) − (1 − dt2 )x(t2 ) ≥ γ1 .

(7.105)

Let us assume the contrary. Then for all integers t ∈ [t0 + 1, t1 − 1], x(t + 1) − (1 − dt )x(t) ≤ γ1 .

(7.106)

By (7.13), (7.87), and Lemma 6.9, x(t0 + 1) ≤ M∗ , and in view of (7.106) and (7.9) for all integers t ∈ [t0 + 1, t1 − 1], x(t + 1) ≤ (1 − dt )x(t) + γ1 ≤ (1 − d∗ )x(t) + γ1 . Combined with (7.91), (7.92), and (7.102), this implies that x(t1 ) ≤ M∗ (1 − d∗ )t1 −t0 −1 + γ1

∞ 

(1 − d∗ )i ≤ M∗ (1 − d∗ )L3 + γ1 d∗−1 < γ0 /2.

i=0

This contradicts (7.103). Therefore there is an integer t2 for which (7.104) and (7.105) hold. For each integer t satisfying t0 + 1 ≤ t < t2 , set

7.2 Auxiliary Results

245

x(t ˜ + 1) = (1 − dt )x(t) ˜ + x(t + 1) − (1 − dt )x(t).

(7.107)

By (7.98) and (7.107), x(t) ˜ ≥ x(t), t = t0 + 1, . . . , t2 ,

(7.108)

t2 {x(t)} ˜ t=τ1 is a program, and in view of (7.8) for any integer t satisfying t0 + 1 ≤ t < t2 ,

˜ x(t ˜ + 1)) ≥ ut (x(t), x(t + 1)). ut (x(t),

(7.109)

It follows from (7.17) and (7.107) that for any integer t satisfying t0 + 1 ≤ t < t2 , ˜ − x(t)) ≥ (1 − d ∗ )(x(t) ˜ − x(t)). x(t ˜ + 1) − x(t + 1) = (1 − dt )(x(t)

(7.110)

By (7.92), (7.98), (7.102), (7.104), and (7.110), ˜ 0 + 1) − x(t0 + 1)) x(t ˜ 2 ) − x(t2 ) ≥ (1 − d ∗ )t2 −t0 −1 (x(t ≥ (1 − d ∗ )L3 +L2 −1 αt−1  ≥ (1 − d ∗ )L3 +L2 −1 (α ∗ )−1 . 0

(7.111)

Set ˜ 2 ) + x(t2 + 1) − (1 − dt2 )x(t2 ) − γ2 . x(t ˜ 2 + 1) = (1 − dt2 )x(t

(7.112)

In view of (7.94), (7.105), and (7.112), ˜ 2 ). x(t ˜ 2 + 1) ≥ (1 − dt2 )x(t

(7.113)

It follows from (7.17), (7.94), (7.111), and (7.112) that ˜ 2 ) − x(t2 )) − γ2 x(t ˜ 2 + 1) = x(t2 + 1) + (1 − dt2 )(x(t ≥ x(t2 + 1) + (1 − d ∗ )L3 +L2 (α ∗ )−1  − γ2 > x(t2 + 1). Thus x(t ˜ 2 + 1) > x(t2 + 1).

(7.114)

Relations (7.9), (7.97), and (7.112) imply that ˜ 2 + 1) − (1 − dt2 )x(t ˜ 2 )) 1 − αt2 (x(t = 1 − αt2 (x(t2 + 1) − (1 − dt2 )x(t2 )) + αt2 γ2 ≥ y(t2 ) + αt2 γ2 ≥ y(t2 ) + α∗ γ2 .

(7.115)

246

7 One-Dimensional Robinson–Solow–Srinivasan Model

Relations (7.94), (7.97), and (7.111) imply that x(t ˜ 2 ) ≥ x(t2 ) + γ2 ≥ y(t2 ) + γ2 .

(7.116)

By (7.97), (7.115), (7.116), (P4), and Lemma 7.16, ˜ 2 ), x(t ˜ 2 + 1)) ≥ wt2 (y(t2 ) + γ2 min{1, α∗ }) ut2 (x(t ≥ wt2 (y(t2 )) + 4δ = ut2 (x(t2 ), x(t2 + 1)) + 4δ.

(7.117)

For any integer t satisfying t2 + 1 ≤ t ≤ τ2 − 1, put ˜ + x(t + 1) − (1 − dt )x(t). x(t ˜ + 1) = (1 − dt )x(t)

(7.118)

Relations (7.114) and (7.118) imply that, x(t) ˜ ≥ x(t), t = t2 + 1, . . . , τ2 ,

(7.119)

τ2 {x(t)} ˜ t=τ1 is a program, and in view of (7.8) for all integers t satisfying t2 + 1 ≤ t ≤ τ2 − 1,

˜ x(t ˜ + 1)) ≥ ut (x(t), x(t + 1)). ut (x(t),

(7.120)

It follows from (7.87), (7.98), (7.100), (7.109), (7.117), (7.119), and (7.120) that U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥

τ 2 −1

ut (x(t), ˜ x(t ˜ + 1))

t=τ1

≥

τ 2 −1

ut (x(t), x(t + 1)) + 4δ ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ + 4δ.

t=τ1

The contradiction we have reached proves (7.89) with t = t0 and Lemma 7.22 itself. Corollary 7.23 For each integer t ≥ 0, x ∗ (t) ≥ 1 − αt (x ∗ (t + 1) − (1 − dt )x ∗ (t)). We chose M¯ > 0 in Section 7.1 being as guaranteed by Theorem 7.3. Clearly this ¯ Taking into account constant can be replaced by any other number larger than M. this remark, we obtain the following corollary of Lemma 7.22. Corollary 7.24 Let , M > 0. Then there exist a natural number L0 ≥ 4 and δ > 0 2 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 +L0 and each program {x(t)}τt=τ 1 M∗ (τ1 , τ2 ) − M¯ − M and for which satisfies (7.87) and U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U each integer t ∈ [τ1 , τ2 − L0 ], the inequality (7.89) holds.

7.2 Auxiliary Results

247

Lemma 7.25 Let  > 0. Then there exist δ > 0 and a natural number L0 such 2 that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L0 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ , |x(τi ) − x ∗ (τi )| ≤ δ, i = 1, 2, τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ

(7.121) (7.122)

t=τ1

the following inequality holds: |x(t) − x ∗ (t)| ≤ , t = τ1 , . . . , τ2 − L0 .

(7.123)

Proof Choose a positive number 1 such that (α∗−1 + 1)(81 )(d∗ )−1 < /8.

(7.124)

By Lemma 7.21 there exist a natural number L1 ≥ 4 and δ1 > 0 such that the following property holds: (P5) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L1 , each z1 , z2 ∈ [0, M∗ ], and 2 each program {x(t)}τt=τ satisfying 1 |z1 − x ∗ (τ1 )|, |z2 − x ∗ (τ2 )| ≤ δ1 , x(τ1 ) = z1 , x(τ2 ) ≥ z2 and τ 2 −1

ut (x(t), x(t + 1)) ≥ U (z1 , z2 , τ1 , τ2 )) − δ1

t=τ1

the following inequality holds for all t = τ1 , . . . , τ2 − 1:   ∗ y (t) − min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))} ≤ 1 . By Lemma 7.22 there exist a natural number L2 ≥ 4 and δ2 > 0 such that M∗ (1 − d∗ )L2 < /8

(7.125)

and the following property holds: 2 (P6) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L2 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ , τ 2 −1 t=τ1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ2

248

7 One-Dimensional Robinson–Solow–Srinivasan Model

and M∗ (τ1 , τ2 ) − M¯ − 2 U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U we have that for each integer t ∈ [τ1 , τ2 − L2 ], the inequality x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)) holds. By Lemma 7.20 there exist a natural number L3 ≥ 4 and δ3 > 0 such that the following property holds: (P7) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L3 and each z1 , z2 ∈ [0, M∗ ] satisfying |z1 − x ∗ (τ1 )|, |z2 − x ∗ (τ2 )| ≤ δ3 , the following inequality holds: |U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − U (z1 , z2 , τ1 , τ2 )| ≤ 4−1 . Set δ = min{δ1 , δ2 , δ3 , /8}

(7.126)

L0 > 4(L1 + L2 + L3 ).

(7.127)

and choose a natural number

2 satisfies (7.121) Assume that integers τ1 ≥ 0, τ2 ≥ τ1 +L0 and a program {x(t)}tt=τ 1 and (7.122). For all t = τ1 , . . . , τ2 − 1 set

y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}.

(7.128)

By (P5), (7.121), (7.122), and (7.126)–(7.128), |y ∗ (t) − y(t)| ≤ 1 , t = τ1 , . . . , τ2 − 1.

(7.129)

In view of (P7), (7.14), (7.15), (7.121), (7.126), and (7.127), M∗ (τ1 , τ2 )− M¯ −4−1 . U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − 4−1 ≥ U (7.130) By (P6), (7.121), (7.122), (7.126), (7.127), and (7.130) for each integer t ∈ [τ1 , τ2 − L2 ], x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)).

(7.131)

By (7.128) and (7.131) for each integer t ∈ [τ1 , τ2 − L2 ], |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t)))| ≤ 1 . Corollary 7.23 and (7.16) imply that for all integers t ∈ [τ1 , τ2 − 1],

(7.132)

7.2 Auxiliary Results

249

y ∗ (t) = 1 − αt (x ∗ (t + 1) − (1 − dt )x ∗ (t)).

(7.133)

By (7.129), (7.132), and (7.133) for all integers t = τ1 , . . . , τ2 − L2 , |αt (x ∗ (t + 1) − (1 − dt )x ∗ (t)) − (αt (x(t + 1) − (1 − dt )x(t)))| ≤ |y ∗ (t) − y(t)| + |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t)))| ≤ 1 + 1 . (7.134) Assume that an integer τ satisfies τ2 − L2 ≥ τ > τ1 + L2 .

(7.135)

Then s := τ − L2 − 1 ≥ τ1 . By (7.9), (7.134), and (7.135) for all integers t ∈ [s, s + L2 ],   x(t + 1) − x ∗ (t + 1)| ≤ (1 − dt )|x(t) − x ∗ (t)   + αt−1 αt (x ∗ (t + 1) − (1 − dt )x ∗ (t)) − (αt (x(t + 1) − (1 − dt )x(t)))   ≤ (1 − d∗ ) x(t) − x ∗ (t) + α∗−1 (21 ). (7.136) In view of (7.124)–(7.126), (7.136), and the choice of δ, |x(τ ) − x ∗ (τ )| ≤ M∗ (1 − d∗ )L2 +1 + α∗−1 (21 )

∞  (1 − d∗ )i i=0

= M∗ (1 − d∗ )L2 +1 + α∗−1 (21 )d∗−1 < /4. Thus |x(τ ) − x ∗ (τ )| ≤ /4

(7.137)

for all integers τ satisfying τ1 + L2 < τ ≤ τ2 − L2 . Assume that an integer τ ≤ τ1 + L2 and τ ≥ τ1 .

(7.138)

By (7.9), (7.127), (7.134), and (7.138), for all integers t satisfying τ1 ≤ t < τ ,   x(t + 1) − x ∗ (t + 1)| ≤ (1 − dt )|x(t) − x ∗ (t)   + αt−1 αt (x ∗ (t + 1) − (1 − dt )(x ∗ (t)) − αt (x(t + 1) − (1 − dt )x(t)))

250

7 One-Dimensional Robinson–Solow–Srinivasan Model

  ≤ (1 − d∗ ) x(t) − x ∗ (t) + α∗−1 (21 ). In view of the relation above, (7.121), (7.124), and (7.126), |x(τ )−x ∗ (τ )| ≤ |x(τ1 )−x ∗ (τ1 )|+α∗−1 (21 )

∞  (1−d∗ )i ≤ δ +α∗−1 (21 )d∗−1 < . i=0

Thus |x(τ ) − x ∗ (τ )| <  for all integers τ = τ1 , . . . , τ1 + L2 . Together with (7.137) this implies that |x(τ )−x ∗ (τ )| <  for τ = τ1 , . . . , τ2 −L2 . Lemma 7.25 is proved. Lemma 7.26 Let , M > 0. Then there exists a natural number L0 ≥ 4 such that 2 for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 +L0 and each program {x(t)}τt=τ satisfying 1 x(τ1 ) ≤ M∗ , τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M

(7.139) (7.140)

t=τ1

the following inequality holds: Card({t ∈ {τ1 , . . . , τ2 − 1} : |y ∗ (t) − min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}| > }) ≤ L0 .

(7.141)

Proof By Proposition 7.14 there exist M1 > 0 and a natural number L1 such that for each pair of integers T1 ≥ 0, T2 ≥ T1 + L1 and each x0 ∈ [0, M∗ ], M∗ (T1 , T2 )| ≤ M1 . |U (x0 , T1 , T2 ) − U

(7.142)

By (A2) there is 1 ∈ (0, 1) such that the following property holds: (P8) for each integer t ≥ 0 and each z1 , z2 ∈ [0, M∗ ] satisfying |z1 − z2 | ≥ , wt (2−1 z1 + 2−1 z2 ) − 2−1 wi (z1 ) − 2−1 wt (z2 ) ≥ 1 . Choose a natural number L0 > 1−1 (M¯ + M + M1 ) + 4 + L1 .

(7.143)

2 Assume that integers τ1 ≥ 0, τ2 ≥ τ1 +L0 and a program {x(t)}τt=τ satisfies (7.139) 1 and (7.140). By (7.139) and (7.140)–(7.143),

7.2 Auxiliary Results τ 2 −1

251

M∗ (τ1 , τ2 ) − M1 − M. ut (x(t), x(t + 1)) ≥ U

(7.144)

t=τ1

For t = τ1 , . . . , τ2 − 1 set y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}.

(7.145)

In view of (7.145) and Lemma 7.16, wt (y(t)) = ut (x(t), x(t + 1)), t = τ1 , . . . , τ2 − 1.

(7.146)

Clearly, {2−1 (x(t) + x ∗ (t)) : t = τ1 , . . . , τ2 } is a program, and for all t = τ1 , . . . , τ2 − 1,

 ut 2−1 (x(t) + x ∗ (t)), 2−1 (x(t + 1) + x ∗ (t + 1)) ≥ wt (2−1 (y(t) + y∗ (t))). (7.147) Set E = {t ∈ {τ1 , . . . , τ2 − 1} : |y(t) − y ∗ (t)| > }.

(7.148)

It follows from (7.139), (P8), (7.148), and Lemma 6.9 that for each t ∈ E, wt (2−1 y(t) + 2−1 y ∗ (t)) ≥ 2−1 wt (y(t)) + 2−1 wt (y ∗ (t)) + 1 .

(7.149)

By (7.15), (7.16), (7.139), (7.144), (7.146), (7.147), (7.149), and Lemma 7.16, M∗ (τ1 , τ2 ) ≥ U

τ 2 −1

ut (2−1 (x(t) + x ∗ (t)), 2−1 (x(t + 1) + x ∗ (t + 1)))

t=τ1

≥

τ 2 −1

wt (2−1 (y(t) + y ∗ (t)))

t=τ1 −1

≥2

τ 2 −1 t=τ1

−1

≥2

−1

wt (y(t)) + 2

τ 2 −1

wt (y ∗ (t)) + 1 Card(E)

t=τ1

    M∗ (τ1 , τ2 ) − M¯ + 2−1 U M∗ (τ1 , τ2 ) − M1 − M U

+ 1 Card(E), 1 Card(E) ≤ M¯ + M + M1 and in view of (7.143), Card(E) ≤ 1−1 (M¯ + M + M1 ) < L0 .

252

7 One-Dimensional Robinson–Solow–Srinivasan Model

Lemma 7.26 is proved. Lemma 7.27 Let M,  > 0. Then there exist δ > 0 and a natural number L0 ≥ 4 2 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 +L0 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ ,

τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ,

(7.150)

t=τ1

U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U (x(τ1 ), τ1 , τ2 ) − M

(7.151)

the following inequality holds: Card({t ∈ {τ1 , . . . , τ2 − 1} : |x(t + 1) − (1 − dt )x(t) − (x ∗ (t + 1) − (1 − dt )x ∗ (t))| > }) ≤ L0 . Proof Choose a positive number 1 such that

 1 + α∗−1 (21 ) < .

(7.152)

By Lemma 7.26 there is an integer L1 ≥ 4 such that the following property holds: 2 (P9) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L1 , each program {x(t)}τt=τ 1 τ2 −1 satisfying x(τ1 ) ≤ M∗ , t=τ1 ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − 2 − M, we have Card({t ∈ {τ1 , . . . , τ2 − 1} : |y ∗ (t) − min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}| > 1 } ≤ L1 . By Proposition 7.14 there exist M1 > 0 and a positive number L2 such that for each pair of integers T1 ≥ 0, T2 ≥ T1 + L2 and each x0 ∈ [0, M∗ ], M∗ (T1 , T2 )| ≤ M1 . |U (x0 , T1 , T2 ) − U

(7.153)

By Corollary 7.24 there exist a natural number L3 ≥ 4 and δ ∈ (0, 1/4) such that the following property holds: 2 (P10) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L3 , each program {x(t)}τt=τ 1 τ2 −1 satisfying x(τ1 ) ≤ M∗ , t=τ1 ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ, M∗ (τ1 , τ2 ) − M¯ − 2 − M1 − M U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U and for each integer t ∈ [τ1 , τ2 − L3 ], we have

7.2 Auxiliary Results

253

x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )(x(t)). Choose a natural number L0 > 4(L1 + L2 + L3 ).

(7.154)

2 Assume that integers τ1 ≥ 0, τ2 ≥ τ1 +L0 and a program {x(t)}τt=τ satisfies (7.150) 1 and (7.151). By (7.150)–(7.154),

τ 2 −1

M∗ (τ1 , τ2 ) − M − 1 − M1 . ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M − 1 ≥ U

t=τ1

(7.155) By (P10), (7.150), (7.154), and (7.155) for each integer t ∈ [τ1 , τ2 − L3 ], x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)).

(7.156)

For each integer t ∈ {τ1 , . . . , τ2 − 1}, set y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}.

(7.157)

E1 = {t ∈ {τ1 , . . . , τ2 − 1} : |y ∗ (t) − y(t)| > 1 }.

(7.158)

Set

It follows from (P9), (7.150), (7.151), (7.154), (7.155), (7.157), and (7.158) that Card(E1 ) ≤ L1 .

(7.159)

By Corollary 7.23, (7.12), and (7.156)–(7.158), for each t ∈ [τ1 , τ2 − L3 ] \ E1 ,   αt (x(t + 1) − (1 − dt )x(t)) − αt (x ∗ (t + 1) − (1 − dt )x ∗ (t))     = y ∗ (t) − (1 − αt (x(t + 1) − (1 − dt )x(t))) ≤ y ∗ (t) − y(t) + |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t)))| ≤ 1 + 1 .

(7.160)

and (7.160) for all integers t ∈ [τ1 , τ2 − L3 ] \ E1 ,   x(t + 1) − (1 − dt )x(t) − (x ∗ (t + 1) − (1 − dt )x ∗ (t)) ≤ α −1 (21 ) < . ∗ (7.161) By (7.154), (7.159), and (7.161), Card({t ∈{τ1 , . . . , τ2 − 1} : |x(t + 1) − (1 − dt )x(t) − (x ∗ (t + 1) − (1 − dt )x ∗ (t))| > })

254

7 One-Dimensional Robinson–Solow–Srinivasan Model

≤ Card({τ1 , . . . , τ2 − 1} \ ({τ1 , . . . , τ2 − L3 } \ E1 )) ≤ L3 + 1 + Card(E1 ) ≤ L3 + 1 + L1 < L0 . Lemma 7.27 is proved.

7.3 Proof of Theorem 7.8 By Lemma 7.25 there are δ1 > 0 and a natural number L1 such that the following property holds: 2 (P11) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L1 and each program {x(t)}τt=τ 2 ∗ satisfying |x(τi ) − x (τi )| ≤ δ1 , i = 1, 2, x(τ1 ) ≤ M∗ , τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ1

t=τ1

the inequality |x(t) − x ∗ (t)| ≤  holds for all t = τ1 , . . . , τ2 − L1 . Choose a positive number δ2 < δ1 d∗ /8.

(7.162)

By Lemma 7.27 there exist δ ∈ (0, δ2 ) and a natural number L2 ≥ 4 such that the following property holds: 2 (P12) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L2 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ ,

τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ,

t=τ1

U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U (x(τ1 ), τ2 , τ2 ) − M − 2 we have Card({t ∈ {τ1 , . . . , τ2 − 1} : |x(t + 1) − (1 − dt )x(t) − (x ∗ (t + 1) − (1 − dt )x ∗ (t))| > δ2 }) ≤ L2 . Choose natural numbers L3 ≥ 4 and p such that

7.3 Proof of Theorem 7.8

255

M∗ (1 − d∗ )L3 < δ1 /8,

(7.163)

p > 2(L1 + L2 + 4)(4L3 + 8).

(7.164)

2 satisfies Assume that integers τ1 ≥ 0, τ2 ≥ τ1 + 2p and that a program {x(t)}τt=τ 1

x(τ1 ) ≤ M∗ ,

τ 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ,

(7.165)

t=τ1

U (x(τ1 ), x(τ2 ), τ1 , τ2 ) ≥ U (x(τ1 ), τ1 , τ2 ) − M.

(7.166)

Set E1 = {t ∈ {τ1 , . . . , τ2 − 1} : |x(t + 1) − (1 − dt )x(t) − (x ∗ (t + 1) − (1 − dt )x ∗ (t))| > δ/2}.

(7.167)

By (P12) and (6.164)–(6.167), Card(E1 ) ≤ L2 .

(7.168)

Set E2 = {t ∈ {τ1 , . . . , τ2 } : there is s ∈ E1 such that |t − s| ≤ 2L3 + 2}.

(7.169)

In view of (7.164), (7.168), and (7.169), Card(E2 ) ≤ L2 (4L3 + 5) < p/2.

(7.170)

Relations (7.164) and (7.170) imply that there are S1 , S2 ∈ [τ1 , τ2 ] such that S1 ≤ τ1 + p/2, S2 ≥ τ2 − p/2, S1 , S2 ∈ E2 .

(7.171)

It follows from (7.164), (7.169), and (7.171) that {S1 , . . . , S1 + 2L3 + 2} ∩ E1 = ∅, {S2 − 2L3 − 2, . . . , S2 } ∩ E1 = ∅. By (7.167) and (7.172) for all integers t ∈ {S1 , . . . , S1 + 2L3 + 2} ∪ {S2 − 2L3 − 2, . . . , S2 }, we have |x(t + 1) − (1 − dt )x(t) − (x ∗ (t + 1) − (1 − dt )x ∗ (t))| ≤ δ2

(7.172)

256

7 One-Dimensional Robinson–Solow–Srinivasan Model

and in view of (7.9), |x(t + 1)−x ∗ (t + 1)| ≤ (1 − dt )|x(t) − x ∗ (t)| + δ2 ≤ (1 − d∗ )|x(t)−x ∗ (t)| + δ2 . (7.173) By (7.162), (7.163), (7.165), and (7.173), |x(S1 + 2L3 + 2) − x ∗ (S1 + 2L3 + 2)| ≤ |x(S1 ) − x ∗ (S1 )|(1 − d∗ )2L3 + δ2

∞ 

(1 − d∗ )i ≤ M∗ (1 − d∗ )2L3 + δ2 d∗−1 < δ1 ,

i=0

|x(S2 ) − x ∗ (S2 )| ≤ |x(S2 − 2L3 − 2) − x ∗ (S2 − 2L3 − 2)|(1 − d∗ )2L3 + δ2

∞ 

(1 − d∗ )i ≤ M∗ (1 − d∗ )2L3 + δ2 d∗−1 < δ1 .

i=0

Thus |x(S2 ) − x ∗ (S2 )| < δ1 , |x(S1 + 2L3 + 2) − x ∗ (S1 + 2L3 + 2)| < δ1 .

(7.174)

In view of (7.165), S 2 −1

ut (x(t), x(t + 1)) ≥ U (x(S1 +2L3 + 2), x(S2 ), S1 +2L3 + 2, S2 ) − δ.

t=S1 +2L3 +2

(7.175) By (7.164) and (7.171), S2 − (S1 + 2L3 + 2) ≥ p > L1 .

(7.176)

It follows from (7.174), (7.175), (7.176), and (P11) that |x(t) − x ∗ (t)| ≤  for all t = S1 + 2L3 + 2, . . . , S2 − L1 . Theorem 7.8 is proved.

7.4 Proof of Theorem 7.9 By Lemma 7.25 there exist δ1 ∈ (0, ) and a natural number L1 such that the following property holds: 2 (P13) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L1 and each program {x(t)}τt=τ 1 ∗ satisfying x(τ1 ) ≤ M∗ , |x(τi ) − x (τi )| ≤ δ1 , i = 1, 2, and

7.4 Proof of Theorem 7.9 τ 2 −1

257

ut (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ1

t=τ1

the inequality |x(t) − x ∗ (t)| ≤  holds for all t = τ1 , . . . , τ2 − L1 . By Theorem 7.8 there exist a natural number L2 and δ ∈ (0, δ1 ) such that the following property holds: (P14) for each pair of integers T1 ≥ 0, T2 ≥ T1 + 2L2 and each proT2 −1 2 gram {x(t)}Tt=T which satisfies x(T1 ) ≤ M∗ , t=T1 ut (x(t), x(t + 1)) ≥ 1 U (x(T1 ), x(T2 ), T1 , T2 ) − δ, U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M we have |x(t) − x ∗ (t)| ≤ δ1 , t ∈ [T1 + L2 , T2 − L2 ].

(7.177)

Choose a natural number p > 2(2L2 + 2L1 ).

(7.178)

2 satisfies Assume that integers T1 ≥ 0, T2 ≥ T1 + p and that a program {x(t)}Tt=T 1

x(T1 ) ≤ M∗ , |x(T1 ) − x ∗ (T1 )| ≤ δ, T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ,

(7.179) (7.180)

t=T1

U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M.

(7.181)

By (P14) and (7.178)–(7.181), the inequality (7.177) is true. By (7.177), |x(T1 + L2 + L1 ) − x ∗ (T1 + L2 + L1 )| ≤ δ1 .

(7.182)

In view of (7.179), (7.180), (7.182), and (P13), |x(t) − x ∗ (t)| ≤ , t ∈ {T1 . . . . , T1 + L2 }.

(7.183)

By (7.177) and (7.183), |x(t) − x ∗ (t)| ≤ , t = T1 , . . . , T2 − L2 . Theorem 7.9 is proved.

258

7 One-Dimensional Robinson–Solow–Srinivasan Model

7.5 Proof of Theorem 7.7 By Lemma 7.18 there are M0 > 0 and natural numbers L¯ 1 , L¯ 2 such that the following property holds: 2 (P15) for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 +L¯ 1 +L¯ 2 , each program {x(t)}τt=τ 1 τ2 −1 which satisfies x(τ1 ) ≤ M∗ , t=τ1 ut (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M − 1, and each pair of integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfying L¯ 1 ≤ T2 − T1 , the T2 −1 inequality t=T ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M0 holds. 1 By Theorem 7.8 there exist a natural number p and δ > 0 such that the following property holds: 2 (P16) for each pair of integers T1 ≥ 0, T2 ≥ T1 +2p and each program {x(t)}Tt=T 1 T2 −1 which satisfies x(T1 ) ≤ M∗ , t=T1 ut (x(t), x(t +1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 )− δ, U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M0 − 1 the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 + p, T2 − p]. Choose a natural number Q such that Q > 8(L¯ 1 + L¯ 2 ) + 2(2 + Mδ −1 )[6p + L¯ 1 + 2L¯ 2 + 8].

(7.184)

Assume that integers T1 ≥ 0, T2 ≥ Q + T1

(7.185)

2 satisfies and a program {x(t)}Tt=T 1

x(T1 ) ≤ M∗ ,

T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M.

(7.186)

t=T1

Define by induction a sequence of nonnegative integers T1 = τ0 < τ1 < . . . , . . . . < τq = T2 with an integer q ≥ 1. Put τ0 = T1 . Assume that an integer j ≥ 0, τi , i = 0, . . . , j has been defined and define τj +1 . If τj = T2 , then j = q and the construction is completed. Assume that τj < T2 . There are two cases: T 2 −1

ut (x(t), x(t + 1)) ≥ U (x(τj ), x(T2 ), τj , T2 ) − δ;

(7.187)

ut (x(t), x(t + 1)) < U (x(τj ), x(T2 ), τj , T2 ) − δ.

(7.188)

t=τj T 2 −1 t=τj

7.5 Proof of Theorem 7.7

259

If (7.187) holds, then set τj +1 = T2 , q = j + 1 and the construction is completed. If (7.188) holds, then there is a natural number τj +1 > τj such that τj +1 −1



ut (x(t), x(t + 1)) < U (x(τj ), x(τj +1 ), τj , τj +1 ) − δ;

(7.189)

t=τj

if an integer s satisfies τj < s < τj +1 , then s−1 

ut (x(t), x(t + 1)) ≥ U (x(τj ), x(s), τj , s) − δ.

(7.190)

t=τj

Therefore by induction we defined a sequence of τi . It is not difficult to see that this sequence is finite. Let τq be its last element. We construct by induction a sequence of programs. First we define a program 2 {x (q) }Tt=T . If 1 τq −1



ut (x(t), x(t + 1)) ≥ U (x(τq−1 ), x(τq ), τq−1 , τq ) − δ,

(7.191)

t=τq−1

then set x (q) (t) = x(t), t = T1 , . . . , T2 .

(7.192)

If τq −1



ut (x(t), x(t + 1)) < U (x(τq−1 ), x(τq ), τq−1 , τq ) − δ,

(7.193)

t=τq−1 2 then there exists a program {x (q) (t)}Tt=T such that 1

x (q) (t) = x(t), t = T1 , . . . , τq−1 , x (q) (T2 ) ≥ x(T2 ), τq −1



t=τq−1

τq −1

ut (x

(q)

(t), x

(q)

(t + 1)) ≥



ut (x(t), x(t + 1)) + δ.

(7.194)

t=τq−1

Assume that an integer j ≥ 1 satisfies j ≤ q and that we have already defined a 2 program {x (j ) (t)}Tt=T such that 1 x (j ) (t) = x(t), t = T1 , . . . , τj −1 , x (j ) (T2 ) ≥ x(T2 ),

(7.195)

260

7 One-Dimensional Robinson–Solow–Srinivasan Model

T 2 −1

ut (x (j ) (t), x (j ) (t + 1)) ≥

t=τj −1

T 2 −1

ut (x(t), x(t + 1)) + δ(q − j ).

(7.196)

t=τj −1

Clearly for j = q the assumption is true. 2 Assume that an integer j ≥ 2 and define a program {x (j −1) (t)}Tt=T . Set 1 x (j −1) (t) = x(t), t = T1 , . . . , τj −2 .

(7.197)

By (7.189), τj −1 −1



ut (x(t), x(t + 1)) < U (x(τj −2 ), x(τj −1 ), τj −2 , τj −1 ) − δ.

(7.198)

t=τj −2 τ −1

j By (7.198) there is a program {x (j −1) (t)}t=τ } such that j −2

x (j −1) (τj −1 ) ≥ x(τj −1 ), τj −1 −1



ut (x

(j −1)

(t), x

(j −1)

(7.199)

τj −1 −1

(t + 1)) ≥

t=τj −2



ut (x(t), x(t + 1)) + δ.

(7.200)

t=τj −2 τ −1

j Clearly, {x (j −1) (t)}t=T is a program. For all integers t satisfying τj −1 ≤ t < T2 , 1 put

x (j −1) (t + 1) = (1 − dt )x (j −1) (t) + x (j ) (t + 1) − (1 − dt )x (j ) (t).

(7.201)

2 is a program such that By (7.195), (7.197), (7.199), and (7.201), {x (j −1) (t)}Tt=T 1

x (j −1) (T2 ) ≥ x (j ) (T2 ) ≥ x(T2 ).

(7.202)

By (7.8), (7.195), (7.197), and (7.201), T 2 −1 t=τj −1

ut (x (j −1) (t), x (j −1) (t + 1)) −

T 2 −1

ut (x (j ) (t), x (j ) (t + 1)) ≥ 0.

(7.203)

t=τj −1

2 Therefore {x j −1 (t)}Tt=T is a program, (7.197) and (7.202) hold, and in view of 1 (7.196), (7.200), and (7.203),

7.5 Proof of Theorem 7.7 T 2 −1

261

ut (x (j −1) (t), x (j −1 (t + 1)) −

t=τj −2

T 2 −1

ut (x(t), x(t + 1))

t=τj −2 T 2 −1

≥δ +

ut (x (j ) (t), x (j ) (t + 1))−

t=τj −1

T 2 −1

ut (x(t), x(t + 1)) ≥ δ(q − j + 1).

t=τj −1

Therefore the assumptions posed for j hold also for j − 1, and by induction we 2 construct a sequence of programs {x (j ) (t)}Tt=T , j = q, . . . , 1 satisfying (7.195) and 1 (1) (7.196) for j = q, . . . , 1. In particular, x (T1 ) = x(T1 ), x (1) (T2 ) ≥ x(T2 ), T 2 −1

ut (x (1) (t), x (1) (t + 1)) −

t=T1

T 2 −1

ut (x(t), x(t + 1)) ≥ δ(q − 1)

t=T1

and in view of (7.186), δ(q − 1) ≤ U (x(T1 ), T1 , T2 ) −

T 2 −1

ut (x(t), x(t + 1)) ≤ M

t=T1

and q ≤ Mδ −1 + 1.

(7.204)

Assume that an integer j satisfies 0 ≤ j ≤ q − 1, τj +1 − τj > L¯ 2 + 4 + 2p + L¯ 1 .

(7.205)

If j < q − 1, then (7.190) holds for all integers s satisfying τj < s < τj +1 and in particular τj +1 −2



ut (x(t), x(t + 1)) ≥ U (x(τj ), x(τj +1 − 1), τj , τj +1 − 1) − δ.

(7.206)

t=τj

If j = q − 1, then (7.206) holds or (7.187) is valid. It is easy to see that (7.206) holds in all the cases. By (7.184)–(7.186), (7.205), and (P15), τj +1 −2−L¯ 2



ut (x(t), x(t + 1)) ≥ U (x(τj ), τj , τj +1 − L¯ 2 − 1) − M0 .

t=τj

Relation (7.206) implies that

(7.207)

262

7 One-Dimensional Robinson–Solow–Srinivasan Model

τj +1−2−L¯ 2



ut (x(t), x(t + 1)) ≥ U (x(τj ), x(τj +1 − 1 − L¯ 2 ), τj , τj +1 − L¯ 2 − 1)−δ.

t=τj

(7.208) By (P16), (7.186), (7.205), (7.207), and (7.208), |x(t) − x ∗ (t)| ≤  for all integers t ∈ [τj + p, τj +1 − 1 − p − L¯ 2 ].

(7.209)

Thus (7.209) holds for all integers j satisfying (7.205) and {t ∈ {T1 , . . . , T2 } : |x(t) − x ∗ (t)| > } ⊂ {T1 , . . . , T2 } \ ∪{{τj + p, . . . τj +1 − 1 − p − L¯ 2 } : j is an integer such that (7.205) holds} ⊂ ∪{{τj , . . . , τj +1 } : j is an integer such that 0 ≤ j ≤ q − 1, τj +1 − τj ≤ L¯ 2 + 4 + 2p + L¯ 1 } ∪{{τj , . . . , τj + p − 1} ∪ {τj +1 − p − L¯ 2 , . . . , τj +1 } : j is an integer satisfying (7.205)}. Together with (7.184) and (7.204), this implies that Card({t ∈ {T1 , . . . , T2 } : |x(t) − x ∗ (t)| > }) ≤ q(L¯ 2 + 6 + 2p + L¯ 1 ) + 2q(p + 2 + L¯ 2 + 1) ≤ 2q(8 + 2L¯ 2 + L¯ 1 + 4p) < 2(Mδ −1 + 1)(8 + 2L¯ 2 + 4p + L¯ 1 ) < Q. Theorem 7.7 is proved.

7.6 Proofs of Theorems 7.12 and 7.13 Proof of Theorem 7.12 We may assume that M∗ > x(0). There exist M0 > 0 such that T −1  T −1     ut (x(t), x(t + 1)) − ut (x ∗ (t), x ∗ (t + 1)) ≤ M0    t=0

t=0

for all integers T ≥ 1. Combined with (7.15) this implies that for each integer T ≥ 0, the inequality

7.6 Proofs of Theorems 7.12 and 7.13

263

T −1     M∗ (0, T ) ≤ M¯ + M0 ut (x(t), x(t + 1)) − U    t=0

holds. Now Theorem 7.12 follows from Theorem 7.7. Proof of Theorem 7.13 Since M∗ is an arbitrary positive number satisfying (7.13) and x∗,0 is an arbitrary element of [0, M∗ ], we may assume without loss of generality ∞ ∞ ∞ ∗ that x0 = x∗ . Then {x(t)} ¯ t=0 = {x (t)}t=0 . Let aprogram {x(t)}t=0 satisfy T −1 ∗ x(0) = x∗,0 = x (0). We show that lim supT →∞ [ t=0 ut (x(t), x(t + 1)) − T −1 ∗ ∗ t=0 ut (x (t), x (t + 1))] ≤ 0. By Theorem 7.4, we may assume without loss of generality that the sequence T −1 

ut (x(t), x(t + 1)) −

t=0

T −1

∗

∗

ut (x (t), x (t + 1))

t=0

is bounded. By Theorem 7.12, lim |x(t) − x ∗ | = 0.

(7.210)

x ∗ (t), x(t) ≤ M∗ , t = 0, 1, . . .

(7.211)

t→∞

In view of Lemma 6.9,

By (7.15), (7.210), (7.211), and Lemma 7.19, lim |U (0, T , x ∗ (0), x ∗ (T )) − U (0, T , zx(0), x(T ))| = 0.

(7.212)

T →∞

Relations (7.14) and (7.212) imply that lim sup T →∞

T −1 

ut (x(t), x(t + 1)) −

t=0

T −1

 ∗

∗

ut (x (t), x (t + 1))

t=0

  ≤ lim sup U (0, T , x(0), x(T )) − U (0, T , x ∗ (0), x ∗ (T )) = 0 T →∞

and {x ∗ (t)}∞ t=0 is an overtaking optimal program. Assume now that {x(t)}∞ t=0 is an overtaking optimal program satisfying x(0) = x ∗ (0). Clearly, lim

T →∞

T −1  t=0

ut (x(t), x(t + 1)) −

T −1

 ut (x ∗ (t), x ∗ (t + 1)) = 0.

t=0

Together with (7.15) and Theorem 7.9, this implies the validity of Theorem 7.13.

Chapter 8

Optimal Programs

In this chapter we continue to study the Robinson–Solow–Srinivasan model and compare different optimality criterions. In particular, we are interested in good programs, agreeable and weakly maximal programs.

8.1 Preliminaries 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}.

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi

i=1

and let x >> y, x > y, x ≥ y have their usual meaning. n Let e(i), i = 1, . . . , n be the ith unit vector in R n and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n . Let a = (a1 , . . . , an ) >> 0, b = (b1 , . . . , bn ) >> 0, d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_8

265

266

8 Optimal Programs

Clearly, there exists σ ∈ {1, . . . , n} such that cσ ≥ ci for all i = 1, . . . , n.

(8.1)

We may assume without loss of generality that for each i ∈ {1, . . . , n} cσ = ci if and only if i ≥ σ.

(8.2)

(Note that in Chapters 2 and 3, we assumed that σ = n.) The planner’s preferences are formalized by a continuous, strictly increasing, concave, and differentiable function w : [0, ∞) → R 1 . Let n n × R+ : x  − (1 − d)x ≥ 0 and a(x  − (1 − d)x) ≤ 1}, Ω = {(x, x  ) ∈ R+

(8.3)

for every (x, x  ) ∈ Ω, n : 0 ≤ y ≤ x and ey ≤ 1 − a(x  − (1 − d)x)}. Λ(x, x  ) = {y ∈ R+

(8.4)

and     u x, x  = max w(by) : y ∈ Λ(x, x  ) .

(8.5)

Recall that a sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0 n n × R+ , x(t + 1) ≥ (1 − d)x(t), (x(t), y(t)) ∈ R+

0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(8.6)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences T2 −1 n and for each integer 2 ({x(t)}Tt=T , {y(t)}t=T ) is called a program if x(T2 ) ∈ R+ 1 1 t satisfying T1 ≤ t < T2 relations (8.6) hold. For i = 1, . . . , n set i = w  (cσ ) qi . qi = ai bi /(1 + dai ), p

(8.7)

We have the following important auxiliary result. Lemma 8.1 w(cσ ) ≥ w(by) + p x  − p x for any (x, x  ) ∈ Ω and for any y ∈ Λ(x, x  ). For the proof of this result, see the proof of Lemma 2.2, and note that its proof remains valid without the assumption that cσ > ci for all i ∈ {1, . . . , n} \ {σ }. For any (x, x  ) ∈ Ω and any y ∈ Λ(x, x  ), set δ(x, y, x  ) = p (x − x  ) − (w(by) − w(cσ )).

(8.8)

8.1 Preliminaries

267

By Lemma 8.1,       δ x, y, x  ≥ 0 for each x, x  ∈ Ω and each y ∈ Λ x, x  .

(8.9)

It is easy to see that the following lemma holds. T −1 Lemma 8.2 Let T > 0 be an integer and ({x(t)}Tt=0 , {y(t)}t=0 ) be a program. Then T −1

(w(by(t)) − w(cσ )) = p (x(0) − x(T )) −

t=0

T −1

δ(x(t), y(t), x(t + 1)).

t=0

The model considered here is a particular case of the model studied in Chapter 5. Therefore we can use all the results obtained there. A program {x(t), y(t)}∞ t=0 is called good if there exists M ∈ R 1 such that T  (w(by(t)) − w(cσ )) ≥ M for all integers T ≥ 0. t=0

 A program is called bad if limT →∞ Tt=0 (w(by(t)) − w(cσ )) = −∞. n , there exists good program By Theorem 5.8, for any initial state x0 ∈ R+ ∞ {x(t), y(t)}t=0 such that x(0) = x0 . Lemmas 5.3 and 8.2 and (8.9) imply the following result. Proposition 8.3 A program {x(t), y(t)}∞ t=0 is good if and only if ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

t=0

A program

{x(t), y(t)}∞ t=0

is bad if and only if

∞ 

δ(x(t), y(t), x(t + 1)) = ∞.

t=0

Corollary 8.4 Any program that is not good is bad. n , and define Let x0 ∈ R+

 Δ(x0 ) = inf

∞ 

δ(x(t), y(t), x(t + 1)) ,

(8.10)

t=0

where the infimum is taken over all programs {x(t), y(t)}∞ t=0 with x(0) = x0 . By Theorem 5.8 and Proposition 8.3, Δ(x0 ) < ∞. We now conclude this section by the following result.

268

8 Optimal Programs

n . Then there exists a program {x(t), y(t)}∞ from x Proposition 8.5 Let x0 ∈ R+ 0 t=0 such that ∞ 

δ (x(t), y(t), x(t + 1)) = Δ(x0 ).

t=0

For its proof, see Proposition 2.10.

8.2 Optimality Criteria A program {x ∗ (t), y ∗ (t)}∞ t=0 is called finitely optimal if for each integer T > 0 ∗ ∗ and each program {x(t), y(t)}∞ t=0 satisfying x(0) = x (0) and x(t) = x (t) for all t ≥ T , the following inequality holds: T −1

  w(by(t)) − w(by ∗ (t)) ≤ 0.

t=0

A program {x ∗ (t), y ∗ (t)}∞ t=0 is called weakly optimal if for each program {x(t), y(t)}∞ t=0 satisfying x(0) = x ∗ (0), the following inequality holds: lim inf T →∞

T    w(by(t)) − w(by ∗ (t)) ≤ 0. t=0

A program {x ∗ (t), y ∗ (t)}∞ t=0 is called overtaking optimal if lim sup

T    w(by(t)) − w(by ∗ (t)) ≤ 0

T →∞ t=0

∗ for every program {x(t), y(t)}∞ t=0 satisfying x(0) = x (0). ∞ ∗ ∗ A program {x (t), y (t)}t=0 is called weakly maximal if for each integer T > 0 −1 ) satisfying x(0) = x ∗ (0), x(T ) ≥ x ∗ (T ), and each program ({x(t)}Tt=0 , {y(t)}Tt=0 the following inequality holds: T −1 t=0

  w(by(t)) − w(by ∗ (t)) ≤ 0.

8.3 Four Theorems

269

A program {x ∗ (t), y ∗ (t)}∞ t=0 is called agreeable if for all integers t ≥ 0, u(x ∗ (t), x ∗ (t + 1)) = w(by ∗ (t)) and if for any natural number T0 and any  > 0, there exists an integer T > T0 T −1 such that for any integer T ≥ T and any program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying T −1 T ∗   x(0) = x (0), there exists a program ({x (t)}t=0 , {y (t)}t=0 ) such that x  (0) = x(0), x  (t) = x ∗ (t), t = 0, . . . , T0 , T −1

w(by  (t)) ≥

t=0

T −1

w(by(t)) − .

t=0

n , let For any natural number T and any x0 ∈ R+

U (x0 , T ) = sup

T −1 

w(by(t)) :

T −1 ({x(t)}Tt=0 , {y(t)}t=0 )

is a program from x0 .

t=0

We follow the convention that the supremum of an empty set is negative infinity.

8.3 Four Theorems We now present the results which are proved in this section. All of them were obtained in [54]. Theorem 8.6 Assume that {x(t), y(t)}∞ t=0 is a program such that ∞ 

δ(x(t), y(t), x(t + 1)) = Δ(x(0)).

(8.11)

t=0

Then {x(t), y(t)}∞ t=0 is a weakly maximal program. By Theorem 5.9, if {x(t), y(t)}∞ t=0 is a weakly maximal program and lim sup ||y(t)|| > 0 t→∞

then {x(t), y(t)}∞ t=0 is a good program. Theorem 8.7 Let {x(t), y(t)}∞ t=0 be weakly optimal. Then it is weakly maximal. Theorem 8.8 Any agreeable program is weakly maximal.

270

8 Optimal Programs

Theorem 8.9 Assume that cσ > ci for all i ∈ {1, . . . , n} \ {σ } and that for each good program {x(t), y(t)}∞ t=0 , x, x ), lim (x(t), y(t)) = (

t→∞

where x = (1/(1 + daσ ))eσ . Then for each weakly maximal program {x(t), y(t)}∞ t=0 which satisfies lim sup y(t) > 0 t→∞

the following equality holds: ∞ 

δ(x(t), y(t), x(t + 1)) = Δ(x(0)).

t=0

8.4 Proof of Theorem 8.6 Assume that {x(t), y(t)}∞ t=0 is not a weakly maximal program. Then there exist an τ −1 τ , {y(t)} ¯ integer τ > 0 and a program ({x(t)} ¯ t=0 ) such that t=0 x(0) ¯ = x(0), x(τ ¯ ) ≥ x(τ ), τ −1  t=0

w(by(t)) ¯ >

τ −1 

w(by(t)).

(8.12) (8.13)

t=0

Set z = x(τ ¯ ) − x(τ ).

(8.14)

y(t) ¯ = y(t) for all integers t ≥ τ,

(8.15)

x(t) ¯ = x(t) + (1 − d)t−τ z for all integers t > τ.

(8.16)

Define

∞ is a program. By (8.14) and (8.16), It is not difficult to see that {x(t), ¯ y(t)} ¯ t=0

8.5 Proof of Theorem 8.7

271

lim (x(t) ¯ − x(t)) = 0.

(8.17)

t→∞

It follows from Lemma 8.2, (8.12), and (8.15) that for each integer T > τ , T 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1)) −

t=0

=

T 

δ(x(t), y(t), x(t + 1))

t=0 T  [w(by(t)) − w(by(t))] ¯ +p (x(0) ¯ − x(T ¯ + 1)) − p (x(0) − x(T + 1)) t=0

=

τ −1  [w(by(t)) − w(by(t))] ¯ +p (x(T + 1) − x(T ¯ + 1)). t=0

Combined with (8.13) and (8.17), this relation implies that

lim

T →∞

=

T 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1)) −

t=0

T 

 δ(x(t), y(t), x(t + 1))

t=0

τ −1 

[w(by(t)) − w(by(t))] ¯ < 0.

t=0

In view of this equation and (8.11), ∞ 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1))


τ −1  t=0

w(by(t)).

(8.18)

272

8 Optimal Programs

Define for all integers t ≥ τ , y(t) ¯ = y(t), x(t ¯ + 1) = (1 − d)x(t) ¯ + [x(t + 1) − (1 − d)x(t)].

(8.19)

∞ is a program. By (8.19) for all natural It is not difficult to see that {x(t), ¯ y(t)} ¯ t=0 numbers T > τ , T 

w(by(t)) ¯ −

T 

t=0

w(by(t)) =

t=0

τ −1 

w(by(t)) ¯ −

t=0

τ −1 

w(by(t)).

t=0

This equality and (8.18) imply that lim

T 

T →∞

=

w(by(t)) ¯ −

t=0 τ −1 

T 

 w(by(t))

t=0

w(by(t)) ¯ −

t=0

τ −1 

w(by(t)) > 0.

t=0

Thus the program {x(t), y(t)}∞ t=0 is not weakly optimal. The contradiction we have reached proves the theorem.

8.6 Proof of Theorem 8.8 Assume the contrary. Then there is an agreeable program {x ∗ (t), y ∗ (t)}∞ t=0 which is not weakly maximal. Then there exist a natural number T0 and a program T0 T0 −1 ¯ ({x(t)} ¯ t=0 , {y(t)} t=0 ) such that ¯ 0 ) ≥ x ∗ (T0 ), x(0) ¯ = x ∗ (0), x(T T 0 −1 t=0

w(by(t)) ¯ >

T 0 −1

w(by ∗ (t)) =

t=0

T 0 −1

u(x ∗ (t), x ∗ (t + 1)).

(8.20) (8.21)

t=0

Put  = 4−1 [

T 0 −1 t=0

w(by(t)) ¯ −

T 0 −1

w(by ∗ (t))].

(8.22)

t=0

Let a natural number T > T0 be as guaranteed by the definition of an agreeable   −1 program. Clearly, there exists a program ({x(t)}Tt=0 , {y(t)}Tt=0 ) such that

8.6 Proof of Theorem 8.8

273

x(0) = x ∗ (0),

T  −1

w(by(t)) = U (x ∗ (0), T ).

(8.23)

t=0

By the choice of T , (8.23) and the agreeability of {x ∗ (t), y ∗ (t)}∞ t=0 , there is a T T −1   program ({x (t)}t=0 , {y (t)}t=0 ) such that x  (0) = x(0), x  (t) = x ∗ (t), t = 0, . . . , T0 , T  −1

(8.24)

Tq −1



w(by (t)) ≥

t=0



w(by(t)) − .

(8.25)

t=0

For all integers t satisfying T0 ≤ t < T , set ¯ + 1) = (1 − d)x(t) ¯ + x  (t + 1) − (1 − d)x  (t). y(t) ¯ = y  (t), x(t By (8.20), (8.24), and (8.26), x(t) ¯ ≥ x  (t)), t = T0 , . . . , T T T −1 and ({x(t)} ¯ ¯ t=0 ) is a program. t=0 , {y(t)} By (8.20)–(8.26),

U (x ∗ (0), T ) ≥

T  −1

w(by(t)) ¯ =

t=0

=

T 0 −1

T 0 −1

∗

T 0 −1

w(by (t)) + 4 +

T 0 −1 t=0

≥

T  −1

w(by(t)) ¯

T  −1

w(by  (t))

t=T0

u(x ∗ (t), x ∗ (t + 1)) + 4 +

T  −1

w(by  (t))

t=T0

t=0

≥

T  −1 t=T0

t=0

t=0

=

w(by(t)) ¯ +

w(by  (t)) +

T  −1

w(by  (t)) + 4

t=T0

w(by(t)) + 3 = U (x ∗ (0), T ) + 3,

t=0

a contradiction. The contradiction we have reached proves Theorem 8.8.

(8.26)

274

8 Optimal Programs

8.7 Proof of Theorem 8.9 Let {x(t), (t)}∞ t=0 be a weakly maximal program such that lim sup ||y(t)|| > 0.

(8.27)

t→∞

We will show that ∞ 

δ(x(t), y(t), x(t + 1)) = Δ(x(0)).

t=0

Let us assume the contrary. Then ∞ 

δ(x(t), y(t), x(t + 1)) > Δ(x(0)).

(8.28)

t=0

By (8.27), the program {x(t), y(t)}∞ t=0 is good. Therefore x, x ). lim (x(t), y(t)) = (

t→∞

(8.29)

∞ such that In view of Proposition 8.5, there is a good program {x(t), ¯ y(t)} ¯ t=0

x(0) ¯ = x(0),

∞ 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1)) = Δ(x(0)).

(8.30)

t=0

Choose a positive number  such that 8
2 such that x(t) ¯ − x , x(t) − x  < δ for all integers t ≥ T0 .

(8.33)

By (8.30) and (8.31), there exists an integer τ > T0 such that τ −1 

δ(x(t), y(t), x(t + 1)) −

t=0 −1

≥2

∞ 

τ −1 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1))

t=0



δ(x(t), y(t), x(t + 1)) − Δ(x(0)) > 4.

(8.34)

t=0

In view of (8.33) and (P1), there exist n , x¯ ≥ x(τ + 1) y¯ ∈ R+

(8.35)

such that (x(τ ¯ ), x) ¯ ∈ Ω, y¯ ∈ Λ(x(τ ¯ ), x), ¯ δ(x(τ ¯ ), y, ¯ x) ¯ ≤ /8, x¯ − x  ≤ (4(1 +  p ))−1 .

(8.36)

Define x(t) ˜ = x(t), ¯ t = 0, . . . , τ, x(τ ˜ + 1) = x, ¯ y(t) ˜ = y(t), ¯ t = 0, . . . , τ − 1, y(τ ˜ ) = y. ¯

(8.37)

It is not difficult to see that τ +1 τ ˜ ({x(t)} ˜ t=0 ) t=0 , {y(t)}

is a program. By (8.27) and (8.30), x(0) ˜ = x(0) ¯ = x(0).

(8.38)

276

8 Optimal Programs

Relations (8.35) and (8.27) imply that x(τ ˜ + 1) ≥ x(τ + 1).

(8.39)

It follows from (8.31), (8.34), and (8.36)–(8.38) that τ 

δ(x(t), y(t), x(t + 1)) −

t=0

≥

τ 

δ(x(t), ˜ y(t), ˜ x(t ˜ + 1))

t=0 τ −1 

δ(x(t), y(t), x(t + 1)) −

t=0

τ −1 

δ(x(t), ¯ y(t), ¯ x(t ¯ + 1))

t=0

− δ(x(τ ˜ ), y(τ ˜ ), x(τ ˜ + 1)) 

∞  −1 δ(x(t), y(t), x(t + 1)) − Δx(0) − δ(x(τ ¯ ), y, ¯ x) ¯ ≥2

−1

≥2

−1

≥4

t=0 ∞ 

 δ(x(t), y(t), x(t + 1)) − Δ(x(0)) − /8

t=0 ∞ 

 δ(x(t), y(t), x(t + 1)) − Δ(x(0)) .

(8.40)

t=0

By Lemma 8.2, (8.31), (8.33), (8.36)–(8.38), (8.40), and the choice of δ, τ  [w(by(t)) − w(by(t))] ˜ t=0

=p (x(0) − x(τ + 1)) −

τ 

− p (x(0) ˜ − x(τ ˜ + 1)) −

−1

≤ −4

−1

≤ −4

δ(x(t), y(t), x(t + 1))

t=0 τ 

 δ(x(t), ˜ y(t), ˜ x(t ˜ + 1))

t=0 ∞ 

  δ(x(t), y(t), x(t + 1)) − Δ(x(0)) + p x(τ ˜ + 1) − x(τ + 1)

t=0 ∞ 



 δ(x(t), y(t), x(t + 1)) − Δ(x(0))

t=0

+ || p||||x¯ − x || + || p|||| x − x(τ + 1)||

8.8 Maximal Programs

−1

≤ −4

∞ 

277

 δ(x(t), y(t), x(t + 1))−Δ(x(0)) + /8 + /4 ≤ −2 +  < 0.

t=0

By the relation above, (8.38) and (8.39), {x(t), y(t)}∞ t=0 is not weakly maximal. The contradiction we have reached proves the theorem.

8.8 Maximal Programs The following optimality criterion was introduced in [10]. A program {x(t), y(t)}∞ t=0 is called maximal if there exist no program 

x  (t), y  (t)

∞ t=0

and a natural number S such that x  (0) = x(0),

S−1 

[w(by  (t)) − w(by(t))] > 0,

t=0

w(by  (t)) ≥ w(by(t)) for all integers t ≥ S + 1. Clearly any overtaking optimal program is maximal. We assume that {i ∈ {1, . . . , n} : cσ = cu } = {σ }.

(8.41)

x = (1 + daσ )−1 eσ .

(8.42)

Set

We prove the following result which was obtained in [115]. Theorem 8.10 Assume that for each good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

Then any maximal program is overtaking optimal. For its proof we use the following result which follows from Theorems 3.9 and 8.9.

278

8 Optimal Programs

Proposition 8.11 Assume that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let a program {x(t), y(t)}∞ t=0 be weakly maximal and satisfy lim sup y(t) > 0. t→∞

Then the program {x(t), y(t)}∞ t=0 is overtaking optimal. Proof of Theorem 8.10 Assume that a program {x(t), y(t)}∞ t=0 is maximal. We show that it is overtaking optimal. Clearly, {x(t), y(t)}∞ t=0 is weakly maximal. By Proposition 8.11, in order to complete the proof, it is sufficient to show that lim sup y(t) > 0, t→∞

Assume the contrary. Then lim y(t) = 0.

t→∞

(8.43)

∞ such that There exists a good program {x(t), ¯ y(t)} ¯ t=0

x(0) ¯ = x(0).

(8.44)

¯ = x. lim y(t)

(8.45)

Clearly, t→∞

By (8.43) and (8.45), there are  > 0 and a natural number T0 such that w(by(t)) ¯ ≥ w(by(t)) +  for all integers t ≥ T0 .

(8.46)

By (8.46), there is a natural number T1 > T0 such that T 1 −1 t=0

w(by(t)) ¯ >

T 1 −1

w(by(t)).

t=0

Together with (8.44) and (8.46), this implies that {x(t), y(t)}∞ t=0 is not maximal, a contradiction. The contradiction we have reached proves the theorem.

8.9 One-Dimensional Model

279

8.9 One-Dimensional Model In this section we consider a particular case of the model considered in Section 8.1 with n = 1. Let a > 0 and d ∈ (0, 1]. Recall that a sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0 1 1 × R+ , (x(t), y(t)) ∈ R+

x(t + 1) ≥ (1 − d)x(t), 0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + y(t) ≤ 1.

(8.47)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences

 T2 −1 2 {x(t)}Tt=T , {y(t)} t=T 1 1 1 and if for each integer t satisfying T ≤ t < T is called a program if x(T2 ) ∈ R+ 1 2 relations (8.47) hold. Let w : [0, ∞) → [0, ∞) be a continuous strictly increasing function which represents the preferences of the planner. Note that we do not assume the concavity of w. In the sequel we assume that supremum of empty set is −∞. 1 and let T be a natural number. Set Let x0 , x˜0 ∈ R+

U (x0 , x˜0 , T ) = sup{

T −1

T −1 w(y(t)) : ({x(t)}Tt=0 , {y(t)}t=0 )

(8.48)

t=0

is a program such that x(0) = x0 , x(T ) ≥ x˜0 }. A program {x(t), y(t)}∞ t=0 is called weakly maximal if for all integers T > 0, T −1

w(y(t)) = U (x(0), x(T ), T ).

t=0

Theorems 5.8 and 5.9 and relation (5.8) imply the following result. 1 there exists a weakly maximal program Theorem 8.12 For any x0 ∈ R+

{x(t), y(t)}∞ t=0 such that

(8.49)

280

8 Optimal Programs

x(0) = x0 , lim sup y(t) > 0 t→∞

and lim inf T −1 T →∞

T −1

w(y(t)) > w(0).

t=0

Recall that a program {x∗ (t), y∗ (t)}∞ t=0 is called maximal if there is not a program {x(t), y(t)}∞ t=0 for which there exists a natural number s such that x(0) = x∗ (0), s−1 

w(y(t)) >

t=0

s−1 

w(y∗ (t))

t=0

and w(y(t)) ≥ w(y∗ (t)) for all integers t ≥ s. We prove the following theorem obtained in [105]. Theorem 8.13 A program {x(t), y(t)}∞ t=0 is weakly maximal and satisfies lim sup y(t) > 0

(8.50)

t→∞

if and only if it is maximal. Proof It is not difficult to see that if the program {x(t), y(t)}∞ t=0 is maximal, then it is weakly maximal and satisfies (8.50). Assume that the program {x(t), y(t)}∞ t=0 is weakly maximal and satisfies (8.50). We show that it is maximal. ∞ and an integer Assume the contrary. Then there exist a program {x(t), ¯ y(t)} ¯ t=0 s > 1 such that x(0) ¯ = x(0), s−1  t=0

w(y(t)) ¯ >

s−1 

w(y(t))

(8.51) (8.52)

t=0

and w(y(t)) ¯ ≥ w(y(t)) for all integers t ≥ s.

(8.53)

8.9 One-Dimensional Model

281

Since the program {x(t), y(t)}∞ t=0 is weakly maximal, Equations (8.51)–(8.53) imply that x(t) > x(t) ¯ for all integers t ≥ s.

(8.54)

Since the function w is strictly increasing, Equation (8.53) implies that y(t) ¯ ≥ y(t) for all integers t ≥ s.

(8.55)

We define a program {x ∗ (t), y ∗ (t)}∞ t=0 . Set ¯ t = 0, . . . , s, x ∗ (t) = x(t),

(8.56)

¯ t = 0, . . . , s − 1 y ∗ (t) = y(t), and for all integers t ≥ s, put y ∗ (t) = y(t),

(8.57)

x ∗ (t + 1) = (1 − d)x ∗ (t) + a −1 (1 − y ∗ (t)). ∗ We show that {x ∗ (t), y ∗ (t)}∞ t=0 is a program. By (8.57) y (t) ∈ [0, 1] for all integers t ≥ s and

x ∗ (t + 1) ≥ (1 − d)x ∗ (t) for all integers t ≥ s.

(8.58)

x ∗ (t) ≥ 0 for all integers t ≥ s.

(8.59)

This implies that

In view of (8.57) for all integers t ≥ s, a(x ∗ (t + 1) − (1 − d)x ∗ (t)) + y ∗ (t) = 1.

(8.60)

By definition, for all integers t ≥ 0 x(t + 1) ≤ (1 − d)x(t) + a −1 (1 − y(t))

(8.61)

and ¯ x(t ¯ + 1) ≤ (1 − d)x(t) ¯ + a −1 (1 − y(t)). It follows from (8.57), (8.55), and (8.56) ¯ ≤ x(s) ¯ = x ∗ (s). y ∗ (s) = y(s) ≤ y(s)

(8.62)

282

8 Optimal Programs

By induction we show that for all integers t ≥ s, ¯ x ∗ (t) ≥ x(t).

(8.63)

It follows from (8.56) that (8.63) is true for t = s. Assume that an integer t ≥ s and (8.63) holds. By (8.57), (8.63), (8.55), and (8.61), ¯ + a −1 (1 − y(t)) x ∗ (t + 1) ≥ (1 − d)x(t) ≥ (1 − d)x(t) ¯ + a −1 (1 − y(t)) ¯ ≥ x(t ¯ + 1). Therefore we have shown by induction that (8.63) holds for all integers t ≥ s. It follows from (8.55), (8.57), and (8.63) that for all integers t ≥ s ¯ ≤ x(t) ¯ ≤ x ∗ (t). y ∗ (t) = y(t) ≤ y(t)

(8.64)

By (8.56)–(8.60) and (8.64), imply that {x ∗ (t), y ∗ (t)}∞ t=0 is a program. In view of (8.61) and (8.57), for all integers t ≥ s, x ∗ (t + 1) − x(t + 1) = (1 − d)x ∗ (t) + a −1 (1 − y ∗ (t)) − x(t + 1) ≥ (1 − d)x ∗ (t) + a −1 (1 − y(t)) − (1 − d)x(t) − a −1 (1 − y(t)) = (1 − d)(x ∗ (t) − x(t)). This implies that lim inf(x ∗ (t) − x(t)) ≥ 0. t→∞

(8.65)

Put Δ :=

s−1 

w(y(t)) ¯ −

t=0

s−1 

w(y(t)).

(8.66)

t=0

In view of (8.66) and (8.52), Δ > 0.

(8.67)

By Lemma 5.3, there is M0 > 0 such that x(t), x(t), ¯ x ∗ (t) ≤ M0 for all integers t ≥ 0. By (8.64) there is 0 > 0 such that

(8.68)

8.9 One-Dimensional Model

283

lim sup y(t) > 40 .

(8.69)

|w(z1 ) − w(z2 )| ≤ Δ/16 for all z1 , z2 ∈ [0, 1]

(8.70)

t→∞

There is 1 > 0 such that

such that |z1 − z2 | ≤ 21 . Put  = min{0 , 1 }(1 + a)−1 .

(8.71)

By (8.65) there is a natural number p0 > s + 2 such that x(t) − x ∗ (t) ≤ /2 for all integers t ≥ p0 .

(8.72)

By (8.69), there is an integer p > p0 + 2 such that y(p) > 40 .

(8.73)

Put x(t) ˜ = x ∗ (t), t = 0, . . . , p, y(t) ˜ = y ∗ (t), t = 0, . . . , p − 1, x(p ˜ + 1) = x ∗ (p + 1) + , y(p) ˜ = y ∗ (p) − a.

(8.74)

By (8.74), x(p ˜ + 1) − (1 − d)x(p) ˜ = x ∗ (p + 1) +  − (1 − d)x ∗ (p) ≥ .

(8.75)

It follows from (8.74), (8.57), (8.73), and (8.71) that y(p) ˜ ≥ 40 − a > 0.

(8.76)

By (8.74), a(x(p ˜ + 1) − (1 − d)x(p)) ˜ + y(p) ˜ ≤ a(x ∗ (p + 1) − (1 − d)x ∗ (p)) + a + y ∗ (p) − a ≤ 1. p+1

p

(8.77)

˜ In view of (8.74)–(8.77), ({x(t)} ˜ t=0 , {y(t)} t=0 ) is a program. By (8.74) and (8.72),

284

8 Optimal Programs

x(p ˜ + 1) = x ∗ (p + 1) +  ≥ x(p + 1) − /2 + .

(8.78)

Equations (8.74), (8.56) and (8.51) imply that x(0) ˜ = x ∗ (0) = x(0).

(8.79)

By (8.74), (8.76), (8.70), (8.71), and (8.66), |w(y(p)) ˜ − w(y ∗ (p))| ≤ Δ/16.

(8.80)

It follows from (8.52), (8.56), (8.57), (8.66), (8.67), (8.74), and (8.80) that p 

w(y(t)) ˜ −

t=0

=

p 

w(y(t))

t=0 p−1 

w(y ∗ (t)) −

t=0

≥

p−1 

s−1 

w(y ∗ (t)) −

s−1  t=0

p−1 

w(y(t)) − Δ/16

t=0

w(y ∗ (t)) −

t=0

=

w(y(t)) + w(y(p)) ˜ − w(y ∗ (p))

t=0

t=0

=

p−1 

s−1 

w(y(t)) − Δ/16

t=0

w(y(t)) ¯ −

s−1 

w(y(t)) − Δ/16 > Δ/2 > 0.

(8.81)

t=0

Equations (8.78), (8.79), and (8.81) contradict the weak maximality of the program {x(t), y(t)}∞ t=0 . The contradiction we have reached proves the theorem.

Chapter 9

Turnpike Phenomenon for the RSS Model with Nonconcave Utility Functions

In this chapter we study the turnpike properties for the Robinson–Solow–Srinivasan model. To have these properties means that the approximate solutions of the problems are essentially independent of the choice of an interval and endpoint conditions. The utility functions, which determine the optimality criterion, are nonconcave. We show that the turnpike properties hold and that they are stable under perturbations of an objective function. Moreover, we consider a class of RSS models which is identified with a complete metric space of utility functions. Using the Baire category approach, we show that the turnpike phenomenon holds for most of the models. All the results of this chapter are new.

9.1 Preliminaries and Main Results 1 ) be the set of real (nonnegative) numbers, and let R n be the nLet R 1 (R+ dimensional Euclidean space with nonnegative orthant n R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}.

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n 

xi yi

i=1

and let x >> y, x > y, x ≥ y have their usual meaning. n Let e(i), i = 1, . . . , n be the ith unit vector in R n and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_9

285

286

9 Turnpike for the RSS Model with Nonconcave Utility Functions

norm in R n . For every x ∈ R n set x∞ = max{|xi | : i = 1, . . . , n}. Let a = (a1 , . . . , an ) >> 0, b = (b1 , . . . , bn ) >> 0 be vectors of R n , d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. We assume the following: There exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci .

(9.1)

Recall that a sequence {x(t), y(t)}∞ t=0 is called a program if for each integer t ≥ 0 n n × R+ , x(t + 1) ≥ (1 − d)x(t), (x(t), y(t)) ∈ R+

0 ≤ y(t) ≤ x(t), a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(9.2)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences

 T2 −1 2 {x(t)}Tt=T , {y(t)}t=T 1 1 n and for each integer t satisfying T ≤ t < T is called a program if x(T2 ) ∈ R+ 1 2 relations (9.2) are valid. Define n n × R+ : x  − (1 − d)x ≥ 0 Ω = {(x, x  ) ∈ R+

and a(x  − (1 − d)x) ≤ 1}

(9.3)

n given by and a correspondence Λ : Ω → R+ n : 0 ≤ y ≤ x and ey ≤ 1 − a(x  − (1 − d)x)}, (x, x  ) ∈ Ω. Λ(x, x  ) = {y ∈ R+

(9.4) Assume that w : [0, ∞) → is a continuous strictly increasing function which represents the preferences of the planner. For every (x, x  ) ∈ Ω set R1

u(x, x  ) = max{w(by) : y ∈ Λ(x, x  )}. n and each integer T ≥ 1, set For each x0 ∈ R+

(9.5)

9.1 Preliminaries and Main Results

U (x0 , T ) = sup

287

T −1 

w(by(t)) :



T −1 {x(t)}Tt=0 , {y(t)}t=0

t=0



is a program such that x(0) = x0 .

(9.6)

n and let T be a natural number. Define Let x0 , x˜0 ∈ R+

U (x0 , x˜0 , T ) = sup

T −1 

w(by(t)) :



T −1 {x(t)}Tt=0 , {y(t)}t=0

t=0



is a program such that x(0) = x0 , x(T ) ≥ x˜0 .

(9.7)

(Here we suppose that a supremum over empty set is −∞.) Let M0 be a positive number and let T ≥ 1 be an integer. Set (M0 , T ) = sup U

T −1 

w(by(t)) :

t=0



 T −1 T {x(t)}t=0 , {y(t)}t=0 is a program such that x(0) ≤ M0 e . (9.8) By Theorem 5.4, for each pair of numbers M1 , M2 > max{(dai )−1 : i = 1, . . . , n} we have (Mi , p)/p, i = 1, 2. μ := μ(w) := lim U p→∞

n satisfy (x, x) ∈ Ω. Then Let x ∈ R+

x ≥ (1 − d)x, a(x − (1 − d)x) ≤ 1, d

n  i=1

and

ai xi = dax ≤ 1

(9.9)

288

9 Turnpike for the RSS Model with Nonconcave Utility Functions

xi ≤ (dai )−1 , i = 1, . . . , n. Thus n n : (x, x) ∈ Ω} ⊂ {x ∈ R+ : xi ≤ (dai )−1 , i = 1, . . . , n}. {x ∈ R+

(9.10)

Clearly, n , (x, x) ∈ Ω} ≤ μ(w). sup{w(by) : y ∈ Λ(x, x), x ∈ R+

(9.11)

Suppose that n , (x, x) ∈ Ω} = μ(w). sup{w(by) : y ∈ Λ(x, x), x ∈ R+

(9.12)

n By (9.12), there exist x(w), y(w) ∈ R+

(x(w), x(w)) ∈ Ω, y(w) ∈ Λ(x(w), x(w)) such that w(by(w)) = μ(w). Since the function w is strictly monotone, it is not difficult to see that the following lemma holds. Lemma 9.1 n {y ∈ ∪{Λ(x, x) : x ∈ R+ , (x, x) ∈ Ω}, w(by) = μ(w)}

is the set of all n , (x, x) ∈ Ω} y ∈ ∪{Λ(x, x) : x ∈ R+

such that by ≥ bz for all n , (x, x) ∈ Ω}}. z ∈ ∪{Λ(x, x) : x ∈ R+ 1 , imply Lemma 9.1 and the results of Chapter 2, applied with w(t) = t, t ∈ R+ the following theorem.

9.1 Preliminaries and Main Results

289

Theorem 9.2 n n × R+ : (x, x) ∈ Ω, y ∈ Λ(x, x), w(by) = μ(w)} {(x, y) ∈ R+

= {((1 + daσ )−1 e(σ ), (1 + daσ )−1 e(σ ))}. Set y = x. x = (1 + daσ )−1 e(σ ),

(9.13)

Recall that a program {x(t), y(t)}∞ t=0 is (w)-good (or good if the function w is understood) if the sequence T −1 

∞ [w(by(t)) − μ] T =1

t=0

is bounded. A program {x(t), y(t)}∞ t=0 is (w)-bad (or bad if the function w is understood) if lim

T −1

T →∞

[w(by(t)) − μ] = −∞.

t=0

By Proposition 5.7, any program that is not good is bad. In this chapter we prove the following results. Theorem 9.3 Assume that for each good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

(9.14)

Let M > max{(ai d)−1 : i = 1, . . . , n},  be positive number and Γ ∈ (0, 1). Then there exist a natural number L and n a positive number γ such that for each integer T > 2L + 1, each z0 , z1 ∈ R+ T −1 T −1 satisfying z0 ≤ Me and az1 ≤ Γ d , and each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1 t=0

there are integers τ1 , τ2 such that

w(by(t)) ≥ U (z0 , z1 , T ) − γ ,

290

9 Turnpike for the RSS Model with Nonconcave Utility Functions

τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1 and x(τ2 ) − x  ≤ . x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 9.4 Assume that for each good program {x(t), y(t)}∞ t=0 , Equation (9.14) holds. Let M > max{(ai d)−1 : i = 1, . . . , n} and  be a positive number. Then there exist a natural number L and a positive n satisfying z ≤ Me, number γ such that for each integer T > 2L + 1, each z0 ∈ R+ 0 −1 which satisfies and each program {x(t)}Tt=0 , {y(t)}Tt=0

x(0) = z0 ,

T −1

w(by(t)) ≥ U (z0 , T ) − γ ,

t=0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1 and x(τ2 ) − x  ≤ . x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Clearly, Theorem 9.4 follows from Theorem 9.3 applied with z1 = 0. Theorem 9.5 Assume that for each good program {x(t), y(t)}∞ t=0 , Equation (9.14) holds and M0 > max{(ai d)−1 : i = 1, . . . , n}. Let M1 ,  be positive numbers and Γ ∈ (0, 1). Then there exists a natural number n satisfying z ≤ Me and L such that for each integer T > L, each z0 , z1 ∈ R+ 0 T −1 az1 ≤ Γ d −1 , and each program ({x(t)}Tt=0 , {y(t)}t=0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1

w(by(t)) ≥ U (z0 , z1 , T ) − M1 ,

t=0

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ L.

9.2 Auxiliary Results

291

Theorem 9.6 Assume that for each good program {x(t), y(t)}∞ t=0 , Equation (9.14) holds and M0 > max{(ai d)−1 : i = 1, . . . , n}. Let M1 ,  be positive numbers. Then there exists a natural number L such that n for

each integer T ≥L, each z0 ∈ R+ satisfying z0 ≤ Me, and each program T −1 {x(t)}Tt=0 , {y(t)}t=0 which satisfies T −1

x(0) = z0 ,

w(by(t)) ≥ U (z0 , T ) − M1 ,

t=0

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ L. Theorem 9.6 follows from Theorem 9.5 applied with z1 = 0. Theorem 9.7 Assume that for each good program {x(t), y(t)}∞ t=0 , Equation (9.14) holds and M0 > max{(ai d)−1 : i = 1, . . . , n}. L such that for Let M1 ,  be positive numbers. Then there

exists a natural number  T −1 T each integer T > L and each program {x(t)}t=0 , {y(t)}t=0 which satisfies x(0) ≤ M0 e, T −1

w(by(t)) ≥ T w(b y ) − M1 ,

t=0

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ L.

9.2 Auxiliary Results It is not difficult to see that the following result holds. Proposition 9.8 For each natural number T , U ( x, x , T ) = T μ(w).

292

9 Turnpike for the RSS Model with Nonconcave Utility Functions

Assume that for each good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

(9.15)

Lemma 9.9 Let M there is a natural number τ such that 

0 , M1 > 0,  > 0. Then −1 satisfying for each program {x(t)}τt=0 , {y(t)}τt=0 x(0) ≤ M0 e, τ −1 

w(by(t)) ≥ τ μ(w) − M1

t=0

there is an integer t ∈ [0, τ − 1] such that y(t) − x , x(t) − x  ≤ . Proof We may assume that M0 > max{(ai d)−1 : i = 1, . . . , n}.

(9.16)

Lemma 5.3 and (9.16) imply that if (x, x  ) ∈ Ω and x ≤ M0 e, then x  ≤ M0 e.

(9.17)

Assume that

the lemma does nothold. Then for each natural number k, there exists (k) (k) a program {xt }kt=0 , {yt }k−1 t=0 such that x (k) (0) ≤ M0 e, k−1 

w(by (k) (t)) ≥ kμ(w) − M1 ,

(9.18) (9.19)

t=0

max{yt(k) − x  + xt(k) − x  : t = 0, . . . , k − 1} > .

(9.20)

In view of (9.17) and (9.18), for each natural number k, x (k) (t) ≤ M0 e, t = 0, . . . , k.

(9.21)

By Corollary 5.6, there is M2 > 0 such that the following property holds:

 T −1 (i) for each natural number T and each program {u(t)}Tt=0 , {v(t)}t=0 satisfying u(0) ≤ M0 e, the following inequality holds:

9.2 Auxiliary Results

293 T −1

(w(bv(t)) − μ(w)) ≤ M2 .

t=0

By property (i), (9.19), and (9.21), for each integer k ≥ 1 and each integer T ∈ {0, . . . , k}, T −1 t=0

w(by (k) (t))=

T −1

w(by (k) (t))−



{w(by (k) (t)) : t ∈ {0, . . . , k − 1}, t ≥ T }

t=0

≥ T μ(w) − M1 − M2 .

(9.22)

By extracting subsequence and using diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {kp }∞ p=1 such that for each integer t ≥ 0, there exist x(t) = lim x (kp ) (t), y(t) = lim y (kp ) (t). p→∞

p→∞

(9.23)

Clearly, {x(t), y(t)}∞ t=0 is a program. By (9.22) and (9.23), for all natural numbers s, T −1

(w(by(t)) − μ(w)) ≥ −M1 − M2

t=0

and {x(t), y(t)}∞ t=0 is a good program. By (9.15), there exists a natural number t0 such that for all integers t ≥ t0 , x(t) − x  ≤ /8, y(t) − x  ≤ /8.

(9.24)

By (9.23), there exists an integer p ≥ 1 such that kp > t0 + 4, x(t0 ) − x (kp ) (t0 ) ≤ /8, y(t0 ) − y (kp ) (t0 ) ≤ /8.

(9.25)

In view of (9.24) and (9.25), x  ≤ /4, y (kp ) (t0 ) − y  ≤ /4. x (kp ) (t0 ) − This contradicts (9.20). The contradiction we have reached proves Lemma 9.9. Lemma 9.10 Let  ∈ (0, 1). Then there existsδ > 0 such that for each integer −1 T ≥ 2 and each program {x(t)}Tt=0 , {y(t)}Tt=0 satisfying x(0) − x  ≤ δ, x(T ) − x  ≤ δ,

294

9 Turnpike for the RSS Model with Nonconcave Utility Functions

U (x(0), x(T ), T ) ≤

T −1

w(by(t)) + δ

t=0

the inequalities x(t) − x ∞ ≤ , t = 0, . . . , T , y(t) − x ∞ ≤ , t = 0, . . . , T − 1 hold. Proof By Proposition 3.13 and continuity of w, for every natural number k, there exists δk ∈ (0, 4−k )

(9.26)

such that the following property holds: n satisfying (ii) for each x, x  ∈ R+

x  ≤ δk x − x , x  − n such that there exist x¯ ≥ x  , y ∈ R+

(x, x) ¯ ∈ Ω, y ∈ Λ(x, x), ¯ y − x  ≤ 4−k , x¯ − x  ≤ 4−k , |w(by) − w(b x )| ≤ 4−k . We may assume without loss of generality that δl+1 < δk for all integers k ≥ 1. Assume that the lemma does not hold. Then for each integerk ≥ 1, there exist Tk −1 k such that , {y (k) (t)}t=0 an integer Tk ≥ 2 and a program {x (k) (t)}Tt=0 x (k) (0) − x  ≤ δk , x (k) (Tk ) − x  ≤ δk , T k −1

w(by (k) (t)) ≥ U (x (k) (0), x (k) (Tk ), Tk ) − δk

t=0

and x ∞ : t = 0, . . . , Tk }, max{max{x (k) (t) −

(9.27) (9.28)

9.2 Auxiliary Results

295

max{y (k) (t) − y ∞ : t = 0, . . . , Tk − 1}} > .

(9.29)

Let k ≥ 1 be an integer. Property (ii) and (9.27) imply that there exist n such that x˜ (k) (1) ∈ R+ x˜ (k) (1) ≥ x,

(9.30)

(x (k) (0), x˜ (k) (1)) ∈ Ω

(9.31)

y˜ (k) (0) ∈ Λ(x (k) (0), x˜ (k) (1))

(9.32)

x  ≤ 4−k , x˜ (k) (1) − x  ≤ 4−k , y˜ (k) −

(9.33)

x )| ≤ 4−k . |w(by˜ (k) (0)) − w(b

(9.34)

and

such that

n such that Property (ii) and (9.27) imply that there exist x˜ (k) (Tk ) ∈ R+

x˜ (k) (Tk ) ≥ x (k) (Tk ),

(9.35)

( x , x˜ (k) (Tk )) ∈ Ωk

(9.36)

x , x˜ (k) (Tk )), y˜ (k) (Tk − 1) ∈ Λ(

(9.37)

such that

and

such that y  ≤ 4−k , x˜ (k) (Tk ) − x  ≤ 4−k , y˜ (k) (Tk − 1) −

(9.38)

x )| ≤ 4−k . |w(by˜ (k) (Tk − 1)) − w(b

(9.39)

x¯ (k) (0) = x (k) (0), x¯ (k) (1) = x˜ (k) (1),

(9.40)

Set

x + (1 − d)t (x˜ (k) (1) − x), ¯ t = 1, . . . , Tk − 1, x¯ (k) (t) =

(9.41)

y, y¯ (k) (0) = y˜ (k) (0), y¯ (k) (t) = ˜ k − 1), t ∈ {1, . . . , Tk − 1} \ {Tk − 1}, y¯ (k) (Tk − 1) = y(T

(9.42)

296

9 Turnpike for the RSS Model with Nonconcave Utility Functions

x¯ (k) (Tk ) = x˜ (k) (Tk ) + (1 − d)Tk (x˜ (k) (1) − x ).

(9.43)

In view of (9.30)–(9.32), (9.36), (9.37), and (9.40)–(9.43), 

Tk −1 k {x¯ (k) (t)}Tt=0 , {y¯ (k) (t)}t=0 is a program. It follows from (9.28), (9.34), (9.35), (9.39), (9.40), (9.42), and (9.43) that T k −1

w(by (k) (t)) ≥ U (x (k) (0), x (k) (Tk ), Tk ) − δk

t=0

≥ −δk +

T k −1

w(by¯ (k) (t)) ≥ Tk w(b y ) − 2 · 4−k .

(9.44)

t=0 n such that Property (ii) and (9.27) imply that there exist z(k) , ξ (k) ∈ R+

(x (k) (Tk ), x (k+1) (0) + z(k) ) ∈ Ω,

(9.45)

x  ≤ 4−k , x (k+1) (0) + z(k) −

(9.46)

ξ (k) ∈ Λ(x (k) (Tk ), x (k+1) (0) + z(k) ),

(9.47)

y  ≤ 4−k , ξ (k) −

(9.48)

x )| ≤ 4−k . |w(bξ (k) ) − w(b

(9.49)

By induction we construct a program ({x(t), y(t)}∞ t=0 ). Set x(t) = x (1) (t), t = 0, . . . , T1 , y(t) = y (1) (t), t = 0, . . . , T1 − 1, x(T1 + 1) = x (2) (0) + z(1) , y(T1 ) = ξ (1) , for all t = 1, . . . , T2 , x(T1 + t + 1) = x (2) (t) + (1 − t)t z(1) and for all t = 0, . . . , T2 − 1, y(T1 + t + 1) = y (2) (t). 

T2 +T1 +1 2 +T1 is a program. Set Clearly, {x(t)}t=0 , {y(t)}Tt=0 z˜ (0) = 0.

(9.50) (9.51)

9.2 Auxiliary Results

297

Assume that k ≥ 2 is an integer and we defined a program x(t), t = 0, . . . ,

k 

(Ti + 1) − 1, y(t), t = 0, . . . ,

i=1

k  (Ti + 1) − 2 i=1

n such that (9.50) holds and for each integer p ∈ [2, k] there exists z˜ (p−1) ∈ R+ such that

z˜ (p−1) = z(p−1) + (1 − d)Tp−1 +1 z˜ (p−2) , ⎛ ⎞ p−1  y ⎝ (Ti + 1) − 1⎠ = ξ (p−1) , i=1

⎛

p−1 

x⎝

(9.52) (9.53)

⎞

(Ti + 1)⎠ = x (p) (0) + z˜ (p−1) ,

(9.54)

i=1

for all integers t = 1, . . . , Tp , ⎛

p−1 

x⎝

⎞ (Ti + 1) + t ⎠ = x (p) (t) + (1 − d)t z˜ (p−1) ,

i=1

⎞

⎛

p−1 

y⎝

(9.55)

(Ti + 1) + t − 1⎠ = y (p) (t − 1).

(9.56)

i=1

It is not difficult to see that for k = 2 our assumption holds. By (9.45), (x (k) (Tk ), x (k+1) (0) + z(k) ) ∈ Ω, ξ (k) ∈ Λ(x (k) (Tk ), x (k+1) (0) + z(k) ), (9.57) (k)

ξ (k) ≤ xTk ,

(9.58)

a(x (k+1) (0) + z(k) − (1 − d)x (k) (Tk )) ≤ 1,

(9.59)

a(x (k+1) (0) + z(k) − (1 − d)x (k) (Tk )) + eξ (k) ≤ 1.

(9.60)

In view of (9.55), x

 k 

 (Ti + 1) − 1 = x (k) (Tk ) + (1 − d)Tk z˜ (k−1) .

i=1

By (9.57) and (9.60),

(9.61)

298

9 Turnpike for the RSS Model with Nonconcave Utility Functions

a(x

(k+1)

(0) + z

(k)

Tk +1 (k−1)

− (1 − d)

z˜

 k   ) − (1 − d)x (Ti + 1) − 1 + eξ (k) ≤ 1. i=1

(9.62)

It follows from (9.57)–(9.62) that     k  (k+1) (k) Tk +1 (k−1) (Ti + 1) − 1 , x (0) + z + (1 − d) z˜ ∈ Ω, x i=1

ξ (k)

(9.63)   k    (k+1) (k) Tk +1 (k−1) ∈Λ x (Ti + 1) − 1 , x (0) + z + (1 − d) z˜ . i=1

(9.64) Set 

 k  y (Ti + 1) − 1 = ξ (k) ,

(9.65)

i=1

 k   x (Ti + 1) = x (k+1) (0) + z(k) + (1 − d)Tk +1 z˜ (k−1) ,

(9.66)

i=1

for all t = 1, . . . , Tk+1 ,   k  (Ti + 1) + t = x (k+1) (t) + (1 − d)t z(k) + (1 − d)Tk +1+t z˜ (k−1) , x i=1

(9.67) y

 k 

 (Ti + 1) + t − 1 = y (k+1) (t − 1).

(9.68)

i=1

By (9.60) and (9.63)–(9.67), x(t), t = 0, . . . ,

k+l 

(Ti + 1) − 1, y(t), t = 0, . . . ,

i=1

k+1  (Ti + 1) − 2 i=1

is a program. Thus, the assumption made for k also holds for k + 1 with z˜ (k) = z(k) + (1 − d)Tk +1 z˜ (k−1) .

9.2 Auxiliary Results

299

Therefore by induction we constructed a program {x(t), y(t)}∞ t=0 such that (9.50) holds and (9.52)–(9.56) hold for all integers p ≥ 2. By (9.44), (9.50), and (9.58), T 1 −1

w(by(t)) ≥ T1 w(b y ) − 2−1 

(9.69)

t=0

and for each integers k ≥ 2, k

i=1 (Ti +1)−2



 t= k−1 i=1 (Ti +1)

w(by(t)) =

T k −1

w(by (k) (t)) ≥ Tk w(b y ) − 2 · 4−k .

(9.70)

t=0

In view of (9.49) and (9.53), for each integer k ≥ 1, 

 k   w by (Ti + 1) − 1 = w(bξ (k) ) ≥ w(b x ) − 4−k .

(9.71)

i=1

By (9.70) and (9.71), lim sup

T 

T →∞ t=0

(w(by(t)) − w(b y )) > −∞

and {x(t), y(t)}∞ t=0 ) is a good program. Therefore x  = 0, lim y(t) − x  = 0. lim x(t) −

t→∞

t→∞

This implies that there exists an integer k0 ≥ 1 such that for each integer t ≥ k0 i=1 (Ti + 1) − 1), we have x(t) − x  ≤ /8, y(t) − x  ≤ /8.

(9.72)

By (9.26), (9.27), and (9.46), for each integer i ≥ 1, z(i)  ≤ x (i+1) (0) + z(i) − x (i+1) (0) x  +  x − x (i+1) (0) ≤ x (i+1) (0) + z(i) − ≤ δi+1 + 4−i  ≤ 2 · 4−i  z(i) ≤ 2 · 4−i e for all integers i ≥ 1. In view of (9.52) and (9.73), for all integers p ≥ 1,

(9.73)

300

9 Turnpike for the RSS Model with Nonconcave Utility Functions

z˜ (p) ≤

p 

z(i) ≤ 2

i=1

∞ 

4−i e = 2−1 (4/3)e = (2/3)e.

(9.74)

i=1

Let k > k0 be an integer. It follows from (9.54), (9.55), and (9.74) that for all t = 0, . . . , Tk , k−1   0≤x (Ti + 1) + t − x (k) (t) ≤ (2/3)e.

(9.75)

i=1

By (9.56), y

(k)

(t) = y

 k 

 (Ti + 1) + t , t = 0, . . . , Tk − 1.

(9.76)

i=1

In view of (9.72) and (9.76), x  ≤ /8, t = 0, . . . , Tk − 1. y (k) (t) −

(9.77)

Relations (9.72) and (9.75) imply that for all t = 0, . . . , Tk , x (k) (t) − x ∞ ≤ x (k) (t) − x

 k 

 (Ti + 1) + t ∞

i=1



k  + x (Ti + 1) + t

 − x ∞ ≤ (2/3) + /8 < .

i=1

This contradicts (9.29). The contradiction we have reached proves Lemma 9.10. Lemma 9.11 Let M0 > max{(ai d)−1 : i = 1, . . . , n}, M1 > 0,  > 0. Then there is a natural number τ such that for each integer T ≥ τ , T −1 each program ({x(t)}Tt=0 , {y(t)}t=0 ) satisfying x(0) ≤ M0 e, T −1

w(by(t)) ≥ T w(b x ) − M1

t=0

and each S ∈ {0, . . . , T − τ } there is an integer t ∈ [S, S + τ − 1] such that

9.2 Auxiliary Results

301

y(t) − x , x(t) − x  ≤ . Proof Lemma 5.3 implies that if (x, x  ) ∈ Ω and x ≤ M0 e, then x  ≤ M0 e.

(9.78)

In view of Corollary 5.6, there exists M2 > 0 such that the following property holds:

 T −1 (a) for each integer T ≥ 1, each program {u(t)}Tt=0 , {v(t)}t=0 satisfying u(0) ≤ M0 e, we have T −1

w(bv(t)) ≤ T μ + M2 .

t=0

Lemma  exists a natural number τ such that for each

9.9 implies that there τ −1 τ program {x(t)}t=0 , {y(t)}t=0 satisfying x(0) ≤ M0 e, τ −1 

w(by(t)) ≥ τ w(b x ) − M1 − 2M2

(9.79) (9.80)

t=0

there is an integer t ∈ [0, τ − 1] such that y(t) − x , x(t) − x  ≤ . Assume that an integer T satisfying

≥ τ,



T −1 is a program {x(t)}Tt=0 , {y(t)}t=0

x(0) ≤ M0 e, T −1

w(by(t)) ≥ T μ(w) − M1

t=0

and S ∈ {0, . . . , T − τ }. By (9.78),

(9.81)

302

9 Turnpike for the RSS Model with Nonconcave Utility Functions

x(t) ≤ M0 e, t = 0, . . . , T . It follows from the relations above and property (a) that  

{w(by(t)) − w(b y ) : t ∈ {0, . . . , S} \ {S}} ≤ M2 ,

{w(by(t)) − w(b y ) : t ∈ {S + τ, . . . , T } \ {T }} ≤ M2 ,

S+τ −1

(w(by(t)) − w(b y )) =

t=S

T −1

(w(by(t)) − w(b y ))

t=0

 − {w(by(t)) − w(b y ) : t ∈ {0, . . . , S} \ {S}}  − {w(by(t)) − w(b y ) : t ∈ {S + τ, . . . , T } \ {T }} ≥ −M1 − 2M2 . By the relations above and the choice of τ (see (9.80) and (9.81)), there exists t ∈ {S, . . . , S + τ − 1} for which x(t) − x  ≤ , y(t) − x  ≤ . Lemma 9.11 is proved. By the definition, da x + e x ≤ 1. This implies that b y > 0, e y > 0, and a x < d −1 . There exists Γ∗ ∈ (0, 1) such that a x < Γ∗ d −1 .

(9.82)

n and each natural Proposition 9.12 There is m > 0 such that for each z ∈ R+ number T ,

U (z, T ) ≥ T w(b x ) − m. Proof By (9.82) and Proposition 4.5, there exists a natural number k∗ such that the following property holds:

9.2 Auxiliary Results

303



n there is a program {x(t)}k∗ , {y(t)}k∗ −1 such that x(0) = (b) for each z0 ∈ R+ t=0 t=0 x. z0 , x(k∗ ) ≥ Set m = k∗ (w(b y ) − w(0)).

(9.83)

n . Property (b) implies that there exists a program Let z ∈ R+ ∗ ∗ −1 , {y(t)}kt=0 ) ({x(t)}kt=0

such that x. x(0) = z, x(k∗ ) ≥ For all integers t ≥ k∗ define y(t) = y, x(t + 1) = (1 − d)x(t) + d x. It is not difficult to see that {x(t), y(t)}∞ t=0 is a program. For each natural number T , we have U (z, T ) ≥

T −1

w(by(t)).

t=0

By (9.83), for each natural number T , U (z, T ) − T w(b y ) ≥ (w(0) − w(b y ))k∗ ≥ −m. Proposition 9.12 is proved. Proposition 9.13 Let Γ ∈ (0, 1). Then there exists m > 0 such that for each z0 ∈ n , each z ∈ R n satisfying az ≤ Γ d −1 , and each natural number T , R+ 1 1 + x ) − m. U (z0 , z1 , T ) ≥ T w(b

(9.84)

Proof By (9.82) and Proposition 4.5, there exists a natural number k∗ such that the following property holds: 

n there is a program {x(t)}k∗ , {y(t)}k∗ −1 such that x(0) = (c) for each z0 ∈ R+ t=0 t=0 z0 , x(k∗ ) ≥ x. By Proposition 4.5, there exists a natural number k1 such that the following property holds:

304

9 Turnpike for the RSS Model with Nonconcave Utility Functions

n n −1 (d) for

each z0 ∈ R+ and each  z1 ∈ R+ satisfying az1 ≤ Γ d , there is a program k1 −1 1 {x(t)}kt=0 such that x(0) = z0 , x(k1 ) ≥ z1 . , {y(t)}t=0 Set

m = (k∗ + k1 )(w(b y ) − w(0)),

(9.85)

k0 = k1 + k∗ .

(9.86)

n , z ∈ R n satisfies az ≤ Γ d −1 and that T is a Assume that z0 ∈ R+ 1 1 + natural number. We show that (9.84) holds. In view of (9.85) and (9.86), we may consider only the case

T > k0 .

 T −1 Properties (c) and (d) imply that there exists a program {x(t)}Tt=0 , {y(t)}t=0 such that x(0) = z0 , x(k∗ ) ≥ x, for all integers t = k∗ , . . . , T − k1 − 1, y(t) = y, x(t + 1) = (1 − d)x(t) + d x, x(T ) ≥ z1 . It is not difficult to see that y ) ≥ (w(0) − w(b y ))(k∗ + k1 ) ≥ −m. U (z0 , z1 , T ) − T w(b Proposition 9.13 is proved.

9.3 Proof of Theorem 9.3 By Lemma 9.10, there exists γ ∈ (0, min{1, /2}) such that the following property holds:



(i) for each integer S ≥ 2 and each program {x(t)}St=0 , {y(t)}S−1 t=0 satisfying

9.3 Proof of Theorem 9.3

305

x(0) − x  ≤ γ , x(S) − x ≤ γ , U (x(0), x(S), S) ≤

S−1 

w(by(t)) + γ

t=0

the inequalities x(t) − x  ≤ , t = 0, . . . , S, y(t) − x  ≤ , t = 0, . . . , S − 1 hold. Proposition 9.13 implies that there exists m0 > 0 such that the following property holds: n , each z ∈ R n satisfying az ≤ Γ d −1 , and each natural (ii) for each z0 ∈ R+ 1 1 + number T , x ) − m0 . U (z0 , z1 , T ) ≥ T w(b In view of Corollary 5.6, there exists m1 > 0 such that the following property holds: 

T −1 satisfying (iii) for each integer T ≥ 1 and each program {u(t)}Tt=0 , {v(t)}t=0 x(0) ≤ Me we have T −1

w(by(t)) ≤ m1 + T w(b y ).

t=0

By Lemma 9.11, there is a natural number L such that the following property holds: 

−1 satisfying (iv) for each integer τ ≥ L, each program {x(t)}τt=0 , {y(t)}τt=0 x(0) ≤ M0 e, τ −1 

w(by(t)) ≥ τ w(b x ) − m0 − 2m1 − 1

t=0

and each S ∈ {0, . . . , τ − L}, there is an integer t ∈ [S, S + L − 1] such that y(t) − x , x(t) − x ≤ γ .

306

9 Turnpike for the RSS Model with Nonconcave Utility Functions

n, Assume that an integer T > 2L + 1, z0 , z1 ∈ R+

z0 ≤ Me, az1 ≤ Γ d −1

(9.87)



T −1 satisfies and that a {x(t)}Tt=0 , {y(t)}t=0 x(0) = z0 , x(T ) ≥ z1 , T −1

w(by(t)) ≥ U (z0 , z1 , T ) − γ .

(9.88) (9.89)

t=0

Lemma 5.3, (9.87), and (9.88) imply that x(t) ≤ Me, t = 0, . . . , T .

(9.90)

By property (ii) and (9.87)–(9.89), T −1

w(by(t)) ≥ T w(b y ) − m0 − 1.

(9.91)

t=0

Let integers S1 , S2 satisfy 0 ≤ S1 < S2 ≤ T . Property (iii) and (9.91) imply that S 2 −1

(w(by(t)) − w(b y ))

t=S1

=

T −1

(w(by(t)) − w(b y ))

t=0

− −

 

{w(by(t)) − w(b y ) : t ∈ {0, . . . , S1 } \ {S1 }} {w(by(t)) − w(b y ) : t ∈ {S2 , . . . , T } \ {T }}

≥ −m0 − 1 − 2m1 .

(9.92)

By (9.90), (9.92), and property (iv), there exist τ1 ∈ {0, . . . , L}, τ2 ∈ {T − L, . . . , T } such that x  ≤ γ , i = 1, 2. x(τi ) −

(9.93)

9.4 Proofs of Theorems 9.5 and 9.7

307

If x(0) − x  ≤ γ , we may assume that τ1 = 0, and if x(T ) − x  ≤ γ , we may assume that τ2 = T . Property (i), (9.88), (9.89), and (9.93) imply that x(t) − x  ≤ , t = τ1 , . . . , τ2 , y(t) − x  ≤ , t = τ1 , . . . , τ2 − 1 Theorem 9.3 is proved.

9.4 Proofs of Theorems 9.5 and 9.7 Proof of Theorem 9.7 By Lemma 9.10, there exists γ ∈ (0, min{1, /2}) such that the following property holds:



(i) for each integer S ≥ 2 and each program {x(t)}St=0 , {y(t)}S−1 t=0 satisfying x(0) − x  ≤ γ , x(S) − x ≤ γ , U (x(0), x(S), S) ≤

S−1 

w(by(t)) + γ

t=0

the inequalities x(t) − x  ≤ , t = 0, . . . , S, y(t) − x  ≤ , t = 0, . . . , S − 1 hold. In view of Corollary 5.6, there exists m1 > 0 such that the following property holds: 

T −1 satisfying (ii) for each integer T ≥ 1 and each program {u(t)}Tt=0 , {v(t)}t=0 u(0) ≤ Me we have T −1 t=0

w(by(t)) ≤ m1 + T w(b y ).

308

9 Turnpike for the RSS Model with Nonconcave Utility Functions

By Lemma 9.11, there is a natural number L0 ≥ 4 such that the following property holds: 

−1 satisfying (iii) for each integer τ ≥ L0 , each program {x(t)}τt=0 , {y(t)}τt=0 x(0) ≤ M0 e, τ −1 

w(by(t)) ≥ τ w(b x ) − M1 − 2m1 − 1

t=0

and each S ∈ {0, . . . , τ − L0 }, there is an integer t ∈ [S, . . . , S + L0 − 1] such that y(t) − x , x(t) − x ≤ γ . Choose a natural number L > 4(L0 + 4) + (2L0 + 2)(γ −1 (M1 + m1 ) + 2).

(9.94)



−1 satisfies Assume that an integer T ≥ L and that a program {x(t)}Tt=0 , {y(t)}Tt=0 x(0) ≤ M0 e, T −1

w(by(t)) ≥ T w(b y ) − M1 .

(9.95) (9.96)

t=0

Lemma 5.3 and (9.95) imply that x(t) ≤ M0 e, t = 0, . . . , T .

(9.97)

t0 = 0.

(9.98)

Set

By induction, using property (iii) and (9.94)–(9.96), we can construct a finite q sequence of natural numbers {ti }i=1 where q ≥ 2 is an integer, such that tq = T and that for each i ∈ {0, . . . , q − 2}, ti+1 − ti ∈ [L0 , 2L0 ],

(9.99)

tq − tq−1 < 2L0 ,

(9.100)

9.4 Proofs of Theorems 9.5 and 9.7

309

x(ti ) − x  ≤ γ , i = 1, . . . , q − 1.

(9.101)



t1 t1 −1 It is clear that there exists a program {x(t)} ˜ , { y(t)} ˜ t=0 t=0 such that x(0) ˜ = x(0), x(t ˜ 1 ) ≥ x(t1 ), t 1 −1

(9.102)

w(by(t)) ˜ = U (x(0), x(t1 ), t1 ).

(9.103)

t=0

Assume that k ∈ {1, . . . , q − 1} and we defined a program

 tk tk −1 {x(t)} ˜ ˜ t=0 , {y(t)} t=0 such that (9.102) and (9.103) hold, for all i = 1, . . . , k, x(t ˜ i ) ≥ x(ti ),

(9.104)

and for all i = 0 . . . , k − 1, ti+1 −1



ti+1 −1

w(by(t)) ˜ = U (x(t ˜ i ), x(ti+1 ), ti+1 − ti ) ≥

t=ti



w(by(t)).

(9.105)

t=ti

In view of (9.102) and (9.103), our assumption holds for k = 1. By (9.104), x(t ˜ k ) ≥ x(tk ). It is not difficult to see that 

 tk+1 tk+1 −1 ˜ k ) − x(tk )) t=t , {y(t)}t=t x(t) + (1 − d)t−tk (x(t k k

is a program and there exists a program

 tk+1 tk+1 −1 {x(t)} ˜ , { y(t)} ˜ t=tk t=tk such that x(t ˜ k+1 ) ≥ x(tk+1 ), and

(9.106)

310

9 Turnpike for the RSS Model with Nonconcave Utility Functions tk+1 −1



tk+1 −1

w(by(t)) ˜ = U (x(t ˜ k ), x(tk+1 ), tk+1 − tk ) ≥

t=tk



w(by(t)).

(9.107)

t=tk

In view of (9.106) and (9.107), the assumption made for

k holds for k + 1.  T −1 T , {y(t)} such Thus by induction, we have constructed a program {x(t)} ˜ t=0 t=0 ˜ that x(0) ˜ = x(0), x(T ˜ ) ≥ x(T ),

(9.108)

x(t ˜ i ) ≥ x(ti ), i = 1, . . . , q,

(9.109)

and for all i = 0, . . . , q − 1, ti+1 −1



ti+1 −1

w(by(t)) ˜ = U (x(t ˜ i ), x(ti+1 ), ti+1 − ti ) ≥

t=ti



w(by(t)).

(9.110)

t=ti

Proposition 4.9 and (9.108)–(9.10) imply that for all i = 0, . . . , q − 1, ti+1 −1



w(by(t)) ˜ = U (x(t ˜ i ), x(ti+1 ), ti+1 − ti )

t=ti

≥ U (x(ti ), x(ti+1 ), ti+1 − ti ).

(9.111)

By property (ii) and (9.95), T −1

w(by(t)) ˜ ≤ T w(b y ) + m1 .

(9.112)

t=0

In view of (9.96) and (9.112), T −1

(w(by(t)) ˜ − w(by(t))) ˜

t=0

y ) + M1 = m 1 + M 1 . ≤ T w( y ) + m1 − T w(b By (9.111) and (9.113), M1 + m 1 ≥

T −1 t=0

(w(by(t)) ˜ − w(by(t)))

(9.113)

9.4 Proofs of Theorems 9.5 and 9.7

=

q−1 

311

U (x(t ˜ i ), x(ti+1 ), ti+1 − ti ) −

i=0

≥

q−1 

T −1

w(by(t))

t=0 ti+1 −1

(U (x(ti ), x(ti+1 ), ti+1 − ti ) −



w(by(t)))

t=ti

i=0

≥ γ Card({i ∈ {0, . . . , q − 1} : ti+1 −1

U (x(ti ), x(ti+1 ), ti+1 − ti ) −



w(by(t)) ≥ γ }.

(9.114)

t=ti

Set ti+1 −1

E = {i ∈ {0, . . . , q − 1} : U (x(ti ), x(ti+1 ), ti+1 − ti ) −



w(by(t)) ≥ γ }.

t=ti

(9.115) By (9.114) and (9.115), Carrd(E) ≤ γ −1 (M1 + m1 ).

(9.116)

Property (iv), (9.101), (9.110), and (9.115) imply that if i ∈ {1, . . . , q − 1} \ {q − 1} and i ∈ E, then x(t) − x  ≤ , t = ti , . . . , ti+1 , y(t) − x  ≤ , t = ti , . . . , ti+1 − 1. This implies that {t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ⊂ {0, . . . , t1 } ∪ {tq−1 , . . . , T } ∪ {{ti , . . . , ti+1 } : i ∈ E}. Together with (9.94), (9.99), (9.100), and (9.116), this implies that Card{t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ 2(2L0 + 1) + Card(E)(2L0 + 1) + Card(E) ≤ Card(E)(2L0 + 2) + 2(2L0 + 1) ≤ (γ −1 (M1 + m1 ) + 2)(2L0 + 2) < L. Theorem 9.7 is proved.

312

9 Turnpike for the RSS Model with Nonconcave Utility Functions

9.5 Generalizations of the Turnpike Results Assume that for each good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

(9.117)

Theorem 9.14 Let M > max{(ai d)−1 : i = 1, . . . , n},  be a positive number and Γ ∈ (0, 1). Then there exist a natural number L n and a positive number γ such that for each integer T > 2L, each z0 , z1 ∈ R+  T −1 satisfying z0 ≤ Me and az1 ≤ Γ d −1 , and each program {x(t)}Tt=0 , {y(t)}t=0 which satisfies x(0) = z0 , x(T ) ≥ z1 , τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), L) − γ for all τ ∈ {0, . . . , T − L}

t=τ

and T −1

w(by(t)) ≥ U (x(T − L), z1 , L) − γ

t=T −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x  ≤  for all t = τ1 , . . . , τ2 , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 9.15 Let M > max{(ai d)−1 : i = 1, . . . , n} and  be a positive number. Then there exist a natural number L and a positive n satisfying z ≤ Me, number γ such that for each integer T > 2L, each z0 ∈ R+ 0 T −1 T and each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies

9.6 Proof of Theorem 9.14

313

x(0) = z0 , τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), L) − γ for all τ ∈ {0, . . . , T − L}

t=τ

and T −1

w(by(t)) ≥ U (x(T − L), L) − γ ,

t=T −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x  ≤  for all t = τ1 , . . . , τ2 , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 9.15 follows from Theorem 9.14 applied with z2 = 0.

9.6 Proof of Theorem 9.14 Recall (see (9.82)) that Γ∗ ∈ (0, 1) satisfies a x < Γ∗ d −1 .

(9.118)

Γ∗ < Γ.

(9.119)

We may assume that

By Lemma 9.10, (9.118), and (9.119), there exists γ ∈ (0, min{1, /2}) such that a( x + γ e) < Γ d −1 and the following property holds:

(9.120)



(i) for each integer S ≥ 2 and each program {x(t)}St=0 , {y(t)}S−1 t=0 satisfying

314

9 Turnpike for the RSS Model with Nonconcave Utility Functions

x(0) − x  ≤ γ , x(S) − x ≤ γ , U (x(0), x(S), S) ≤

S−1 

w(by(t)) + γ

t=0

the inequalities x(t) − x  ≤ , t = 0, . . . , S, y(t) − x  ≤ , t = 0, . . . , S − 1 are valid. By Proposition 9.13, there exists m0 > 0 such that the following property holds: n , each z ∈ R n satisfying az ≤ Γ d −1 , and each natural (ii) for each z0 ∈ R+ 1 1 + number T , x ) − m0 . U (z0 , z1 , T ) ≥ T w(b By Lemma 9.11, there is a natural number L0 such that the following property holds: 

−1 satisfying (iii) for each integer τ ≥ L0 , each program {x(t)}τt=0 , {y(t)}τt=0 x(0) ≤ M0 e, τ −1 

w(by(t)) ≥ τ w(b x ) − m0 − 1

t=0

and each S ∈ {0, . . . , τ − L0 }, there is an integer t ∈ [S, . . . , S + L0 − 1] such that y(t) − x , x(t) − x ≤ γ . Choose a natural number L > 8(L0 + 1).

(9.121)

n satisfies Assume that an integer T > 2L, z0 , z1 ∈ R+

z0 ≤ Me, az1 ≤ Γ d −1 

T −1 satisfies and that a program {x(t)}Tt=0 , {y(t)}t=0

(9.122)

9.6 Proof of Theorem 9.14

315

x(0) = z0 , x(T ) ≥ z1 , τ +L−1 

(9.123)

w(by(t)) ≥ U (x(τ ), x(τ + L), L) − γ for all τ ∈ {0, . . . , T − L}

t=τ

(9.124) for all τ ∈ {0, . . . , T − L} and T −1

w(by(t)) ≥ U (x(T − L), z1 , L) − γ .

(9.125)

t=T −L

Lemma 5.3, (9.122), and (9.123) imply that x(t) ≤ M0 e, t = 0, . . . , T .

(9.126)



T −1 Consider the program {x(t)}Tt=T −L , {y(t)}t=T −L . Property (ii), (9.122), (9.123), and (9.125) imply that T −1

w(by(t)) ≥ U (x(T − L), z1 , L) − 1 ≥ Lw(b y ) − m0 − 1.

(9.127)

t=T −L

Property (iii), (9.121), (9.126), and (9.127) imply that t0 ∈ {T − L0 , . . . , T }, t1 ∈ {T − 4L0 , . . . , T − 3L0 }

(9.128)

such that x , y(ti ) − x  ≤ γ , i = 0, 1. x(ti ) −

(9.129)

If x(T ) − x  ≤ γ , then we may assume that t0 = T . Assume that k ≥ 1 is an integer and that we defined a strictly decreasing sequence of natural numbers ti , i = 0, . . . , k such that ti+1 < ti , i = 0, . . . , k − 1, for all i = 0, . . . , k − 1, L0 ≤ ti − ti+1 ≤ 4L0 ,

(9.130)

x(ti ) − x  ≤ γ , i = 0, . . . , k,

(9.131)

tk ≥ 4L0 + 4.

(9.132)

316

9 Turnpike for the RSS Model with Nonconcave Utility Functions

(Clearly, for k = 1 our assumption holds.) It follows from (9.120) and (9.131) that for i = 0, . . . , k, x + γ e) < Γ d −1 . ax(ti ) ≤ a(

(9.133)

Property (iii), (9.132) and (9.133) imply that y ) − m0 . U (x(tk − 4L0 − 4), x(tk ), 4L0 + 4) > (4L0 + 4)w(b

(9.134)

Relations (9.121) and (9.124) imply that t k −1

w(by(t)) ≥ (4L0 + 4)w(b y ) − m0 − 1.

(9.135)

tk > 6L0 + 6;

(9.136)

tk ≤ 6L0 + 6, x(0) − x > γ ;

(9.137)

tk ≤ 6L0 + 6, x(0) − x ≤ γ .

(9.138)

t=tk −4L0 −4

There are two cases:

If (9.137) holds, then the construction is completed. If (9.138) is true, then we set tk+1 = 0, and the construction is completed too. Assume that (9.136) holds. Property (iii), (9.135), and (9.136) imply that tk+1 ∈ {tk − 2L0 , . . . , tk − L0 }

(9.139)

x ≤ γ . x(tk+1 ) −

(9.140)

such that

In view of (9.136) and (9.139), tk+1 > 6L0 + 6 − 2L0 > 4L0 + 4. Thus the assumption we made for k also holds for k + 1. Therefore by induction (see (9.128)–(9.132)), we constructed the finite strictly decreasing sequence of nonnegative integers ti , i = 0, . . . , q such that x  ≤ γ , then t0 = T , T ≥ t0 ≥ T − L0 and if x(T ) − for all i = 0, . . . , q, we have x ≤ γ , x(ti ) −

(9.141)

9.7 Stability Results

317

for all i = 0, . . . , q − 1, L0 ≤ ti − ti+1 ≤ 6L0 + 6,

(9.142)

tq ≤ 6L0 + 6,

(9.143)

if x(0) − x  ≤ γ , then tq = 0. Let i ∈ {1, . . . , q}. By (9.121), (9.124), (9.141), and (9.142), ti−1 −1



w(by(t)) ≥ U (x(ti ), x(ti−1 ), ti−1 − ti ) + γ .

t=ti

Property (i), (9.141), (9.142), and the inequality above imply that x(t) − x  ≤ , t = ti , . . . , ti−1 , y(t) − x  ≤ , t = ti , . . . , ti−1 − 1. This implies that x(t) − x  ≤ , t ∈ [tq , t0 ], y(t) − x  ≤ , t ∈ [tq , t0 − 1]. Theorem 9.14 is proved.

9.7 Stability Results n → R 1 , define For every positive number M and every function φ : R+ n such that and 0 ≤ z ≤ Me}. φM = sup{|φ(z)| : z ∈ R+

(9.144)

n → R 1 , i = T , . . . , T − 1 be Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and wi : R+ 1 2 n n, bounded on bounded subsets of R+ functions. For every pair of points z0 , z1 ∈ R+ define ⎧ 2 −1 ⎨T 

T2 −1 = sup , z , z wt (y(t)) : U {wt }t=T 0 1 1 ⎩ t=T1

⎫ ⎬

 T2 T2 −1 {x(t)}t=T1 , {y(t)}t=T1 is a program such that x(T1 ) = z0 , x(T2 ) ≥ z1 , ⎭ (9.145)

318

9 Turnpike for the RSS Model with Nonconcave Utility Functions





T2 −1 U {wt }t=T , z0 = sup 1



T2 −1 2 {x(t)}Tt=T , {y(t)}t=T 1 1



⎧ 2 −1 ⎨T ⎩

wt (y(t)) :

t=T1

is a program such that x(T1 ) = z0

⎫ ⎬ ⎭

.

(9.146)

n → R 1 is a bounded on If M0 > 0 and wt = w, t = T1 , . . . , T2 − 1, where w : R+ n bounded subsets of R+ function, set



T2 −1 , , z , z U (w, z0 , z1 , T1 , T2 ) = U {wt }t=T 0 1 1 

T2 −1 U (w, z0 , T1 , T2 ) = U {wt }t=T , z0 , 1 U (w, M0 , T ) = sup

T −1 

w(y(t)) :

t=0

 −1 {x(t)}Tt=0 , {y(t)}Tt=0 is a program such that x(0) ≤ M0 e

(9.147)

for every natural number T . (Here we assume that supremum over empty set is −∞.) It is not difficult to see that the following result holds. n → R1, Lemma 9.16 Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and wi : R+ n i = T1 , . . . , T2 − 1 be bounded on bounded subsets of R+ upper semicontinuous functions. Then the following assertions hold. 

n , there exists a program {x(t)}T2 , {y(t)}T2 −1 such 1. For every point z0 ∈ R+ T1 t=T1 that

x(T1 ) = z0 ,

T 2 −1

T2 −1 wt (y(t)) = U ({wt }t=T , z0 ). 1

t=T1 n such that U ({w }T2 −1 , z , z ) is finite, there 2. For every pair of points z0 , z1 ∈ R+ t t=T1 0 1 T2 −1 2 exists a program ({x(t)}Tt=T , {y(t)}t=T ) such that x(0) = z0 , x(T2 ) ≥ z1 and 1 1 T 2 −1

T2 −1 wt (y(t)) = U ({wt }t=T , z0 , z1 ). 1

t=T1

The following stability results hold.

9.7 Stability Results

319

Theorem 9.17 Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ˜ such that for each n satisfying z ≤ Me and az ≤ Γ d −1 , each finite integer T > 2L, each z0 , z1 ∈ R+ 0 1 n sequence of functions wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on n and such that bounded subsets of R+ wi − w(b(·))M ≤ γ˜

 T −1 for every integer i ∈ {0, . . . , T −1} and every program {x(t)}Tt=0 , {y(t)}t=0 such that x(0) = z0 , x(T ) ≥ z1 , τ +L−1 

τ +L−1 wt (y(t)) ≥ U ({wt }t=τ , x(τ ), x(τ + L)) − γ˜

t=τ

for every τ ∈ {0, . . . , T − L} and T −1 t=T −L

T −1 wt (y(t)) ≥ U ({wt }t=T −L , x(T − L), z1 ) − γ˜ ,

there exist integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ˜ , then τ2 = T . Moreover if |x(0) − x  ≤ γ˜ , then τ1 = 0, and if x(T ) − Proof Theorem 9.17 follows easily form Theorem 9.14. Namely, let a natural number L and γ > 0 be as guaranteed by Theorem 9.14. Set γ˜ = γ (4−1 (L + 1))−1 . Now it easy to see that the assertion of Theorem 9.17 holds. Theorem 9.17 applied with z1 = 0 implies the following result. Theorem 9.18 Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , each z0 ∈ R+ 0 i + n and such that i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ˜

320

9 Turnpike for the RSS Model with Nonconcave Utility Functions



T −1 which for each i ∈ {0, . . . , T − 1} and each program {x(t)}Tt=0 , {y(t)}t=0 satisfies x(0) = z0 , τ +L−1 

τ +L−1 wt (y(t)) ≥ U ({wt }t=τ , x(τ ), x(τ + L)) − γ˜ ,

t=τ

for each integer τ ∈ {0, . . . , T − L} and T −1 t=T −L

T −1 wt (y(t)) ≥ U ({wt }t=T −L , x(T − L)) − γ˜

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ , then τ2 = T . Moreover if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − The following result is proved in Section 9.8. Theorem 9.19 Let M0 > max{(ai d)−1 : i = 1, . . . , n}, M1 > 0,  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ such that n satisfying z ≤ Me and az ≤ Γ d −1 , for each integer T > L, each z0 , z1 ∈ R+ 0 1 n → R 1 , i = 0, . . . , T − 1 which are each finite sequence of functions wi : R+ n and such that bounded on bounded subsets of R+ wi − w(b(·))M0 ≤ γ T −1 for each i ∈ {0, . . . , T − 1} and each program ({x(t)}Tt=0 , {y(t)}t=0 ) such that

x(0) = z0 , x(T ) ≥ z1 , T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 , z1 ) − M1

t=0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L. Theorem 9.19, applied with z1 = 0, implies the following result.

9.8 Proof of Theorem 9.19

321

Theorem 9.20 Let M0 > max{(ai d)−1 : i = 1, . . . , n}, M1 > 0 and  > 0. Then there exist a natural number L and a positive number γ such that for each n satisfying z ≤ Me, each finite sequence of functions integer T > L, each z0 ∈ R+ 0 n n 1 wi : R+ → R , i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·))M0 ≤ γ 

T −1 which for each i ∈ {0, . . . , T − 1} and each program {x(t)}Tt=0 , {y(t)}t=0 satisfies x(0) = z0 , T −1

T −1 wt (y(t)) ≥ U ({wt }t=0 , z0 ) − M1

t=0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L.

9.8 Proof of Theorem 9.19 Recall (see (9.82)) that Γ∗ ∈ (0, 1) satisfies a x < Γ∗ d −1 .

(9.148)

Γ∗ < Γ.

(9.149)

We may assume that

By Lemma 9.10, (9.148), and (9.149), there exists γ0 ∈ (0, min{1, /2}) such that a( x + γ e) < Γ d −1 and that the following property holds:

(9.150)



(i) for each integer S ≥ 2 and each program {x(t)}St=0 , {y(t)}S−1 t=0 satisfying

322

9 Turnpike for the RSS Model with Nonconcave Utility Functions

x(0) − x  ≤ γ0 , x(S) − x  ≤ γ0 , U (x(0), x(S), S) ≤

S−1 

w(by(t)) + 8γ0

t=0

the inequalities x(t) − x  ≤ , t = 0, . . . , S, y(t) − x  ≤ , t = 0, . . . , S − 1 are valid. By Proposition 9.13, there exists m0 > 0 such that the following property holds: n , each z ∈ R n satisfying az ≤ Γ d −1 , and each natural (ii) for each z0 ∈ R+ 1 1 + number T , x ) − m0 . U (z0 , z1 , T ) ≥ T w(b By Lemma 9.11, there is a natural number L0 > 4 such that the following property holds: 

−1 satisfying (iii) for each integer τ ≥ L0 , each program {x(t)}τt=0 , {y(t)}τt=0 x(0) ≤ M0 e, τ −1 

w(by(t)) ≥ τ w(b x ) − 2M1 − m0 − 1

t=0

and each S ∈ {0, . . . , τ − L0 }, there is an integer t ∈ [S, . . . , S + L0 − 1] such that y(t) − x , x(t) − x  ≤ γ0 . Choose a natural number L > 8(L0 + 1) + 2L0 M1 γ0−1

(9.151)

and γ ∈ (0, γ0 /2) such that γ ≤ (2L + 2)−1 γ0 .

(9.152)

n satisfies Assume that an integer T > L, z0 , z1 ∈ R+

z0 ≤ Me, az1 ≤ Γ d −1 ,

(9.153)

9.8 Proof of Theorem 9.19

323

n → R 1 , i = 0, . . . , T − 1 are bounded on bounded subsets of R n functions wi : R+ + such that

wi − w(b(·))M0 ≤ γ , i = 0, . . . , T − 1

(9.154)



T −1 satisfies and that a program {x(t)}Tt=0 , {y(t)}t=0 x(0) = z0 , x(T ) ≥ z1 , T −1

−1 wt (y(t)) ≥ U ({wt }Tt=0 , z0 , z1 ) − M1 .

(9.155) (9.156)

t=0

Lemma 5.3, (9.153), and (9.155) imply that x(t) ≤ M0 e, t = 0, . . . , T .

(9.157)

By (9.150), (9.152), (9.154), and (9.157), ' T −1 ' T −1 '  ' ' ' w(by(t)) − wt (y(t))' ≤ Lγ ≤ 2−1 , ' ' ' t=T −L

(9.158)

t=T −L



T −1 −1 |U {wt }t=T −L , z0 , z1 − U (w(b(·)), z0 , z1 , T − L, T )| ≤ Lγ ≤ 2 . (9.159) Property (iii), (9.153), (9.156), and (9.158) imply that T −1

w(by(t)) ≥ U (w(b(·)), z0 , z1 , T − L, T ) − M1 − 1

t=T −L

≥ Lw(b y ) − m0 − M1 − 1.

(9.160)

Property (iii), (9.151), (9.157), and (9.160) imply that there exist t0 ∈ {T − L0 , . . . , T }, t1 ∈ {T − 4L0 , . . . , T − 3L0 }

(9.161)

such that x  ≤ γ0 , i = 0, 1. x(ti ) −

(9.162)

Assume that k ≥ 1 is an integer and that we defined a strictly decreasing sequence of natural numbers ti , i = 0, . . . , k such that ti+1 < ti , i = 0, . . . , k − 1,

324

9 Turnpike for the RSS Model with Nonconcave Utility Functions

for all i = 0, . . . , k − 1, L0 ≤ ti − ti+1 ≤ 4L0 ,

(9.163)

x(ti ) − x  ≤ γ0 , i = 0, . . . , k,

(9.164)

tk ≥ 4L0 + 4.

(9.165)

(Clearly, for k = 1 our assumption holds.) It follows from (9.150) and (9.164) that x + γ0 e) < Γ d −1 . ax(tk ) ≤ a(

(9.166)

Property (ii), (9.165), and (9.166) imply that y ) − m0 . U (w(b(·)), x(tk − 4L0 − 4), x(tk ), tk − 4L0 − 4, tk ) ≥ (4L0 + 4)w(b (9.167) Relations (9.151), (9.154), (9.156), (9.157), (9.165), and (9.167) imply that t k −1

w(by(t)) ≥

t=tk −4L0 −4

t k −1

wt (y(t)) − Lγ

t=tk −4L0 −4

tk −1 ≥ U ({wt }t=t , x(tk − 4L0 − 4), x(tk )) − M1 − Lγ k −4L0 −4

≥ U (w(b(·)), x(tk − 4L0 − 4), x(tk ), tk − 4L0 − 4, tk ) − 2γ L − M1 ≥ U (w(b(·)), x(tk − 4L0 − 4), x(tk ), tk − 4L0 − 4, tk ) − 1 − M1 ≥ (4L0 + 4)w(b y ) − m0 − M1 − 1.

(9.168)

Property (iii), (9.157), and (9.168) imply that there exist tk+1 ∈ {tk − 2L0 , . . . , tk − L0 }

(9.169)

x  ≤ γ0 . x(tk+1 ) −

(9.170)

tk+1 ≥ 2L0 + 4.

(9.171)

such that

By (9.165) and (9.169),

If tk+1 < 4L0 + 4,

9.8 Proof of Theorem 9.19

325

then the construction is completed. Otherwise the assumption made for k also holds for k + 1. Therefore by induction we constructed the finite strictly decreasing sequence of natural numbers ti , i = 0, . . . , q such that T ≥ t0 ≥ T − L0 , ti+1 < ti , i = 0, . . . , q − 1, L0 ≤ ti − ti+1 ≤ 4L0 , i = 0, . . . , q − 1,

(9.172)

x(ti ) − x  ≤ γ0 , i = 0, . . . , q,

(9.173)

4L0 + 4 > tq ≥ 2L0 + 4. Set E = {i ∈ {1, . . . , q} : −1

 ti−1  ti−1 −1 U {wt }t=t , x(t ), x(t ) − wt (y(t)) ≥ 4γ0 }. i i−1 i

(9.174)

t=ti T −1 T , {y(t)} By induction we can construct a program ({x(t)} ˜ t=0 ) such that t=0 ˜

˜ = y(t), t = 0, . . . , tq − 1, x(t) ˜ = x(t), t = 0, . . . , tq , y(t)

(9.175)

for i = q, . . . , 1, x(t ˜ i ) ≥ x(ti ),

(9.176)

if i ∈ {1, . . . , q} \ E, then ˜ i ) = y(ti ), t = ti , . . . , ti−1 − 1, x(t ˜ i ) ≥ x(ti ), t = ti , . . . , ti−1 , y(t

(9.177)

if i ∈ E, then ti−1 −1



t

−1

i−1 wt (y(t)) ˜ ≥ U ({wt }t=t i

, x(t ˜ i ), x(ti−1 )) − γ0 ,

(9.178)

t=ti

y(t) ˜ = y(t), t ∈ {t0 , . . . , T } \ {T },

(9.179)

x(T ˜ ) ≥ x(T ).

(9.180)

Proposition 4.9, (9.156), and (9.174)–(9.180) imply that

326

9 Turnpike for the RSS Model with Nonconcave Utility Functions

T −1 M1 ≥ U ({wt }t=0 , z0 , z1 ) −

T −1

wt (y(t))

t=0

≥

T −1

wt (y(t)) ˜ −

t=0

≥

⎧ −1   ⎨ti−1 ⎩

T −1

wt (y(t))

t=0 ti−1 −1

wt (y(t)) ˜ −

t=ti



wt (y(t)) : i ∈ E

t=ti

⎫ ⎬ ⎭

⎫ ⎧ ti−1 −1 ⎬  ⎨

 ti−1 −1 ≥ , x(t ˜ i ), x(ti−1 ) − γ0 − wt (y(t)) : i ∈ E U {wt }t=t i ⎭ ⎩ t=ti

⎫ ⎧ ti−1 −1 ⎬  ⎨

 ti−1 −1 ≥ , x(t ), x(t ) − γ − w (y(t)) : i ∈ E U {wt }t=t i i−1 0 t i ⎭ ⎩ t=ti

≥ 3γ0 Card(E) and Card(E) ≤ 3−1 γ0−1 M1 .

(9.181)

i ∈ {1, . . . , q} \ E.

(9.182)

x  ≤ γ0 , x(ti−1 ) − x  ≤ γ0 . x(ti ) −

(9.183)

Let

In view of (9.173),

Property (i), (9.152), (9.154), (9.155), (9.172), (9.174), and (9.182) imply that ti−1 −1



t

−1

i−1 wt (y(t)) ≥ U ({wt }t=t i

, x(ti ), x(ti−1 )) − 4γ0

t=ti

≥ U (w(b(·)), x(ti ), x(ti−1 ), ti , ti−1 ) − 4γ0 − 4γ L0 ≥ U (w(b(·)), x(ti ), x(ti−1 ), ti , ti−1 ) − 6γ0 . By (9.183) and (9.184), x(t) − x  ≤ , t = ti , . . . , ti−1 , y(t) − x  ≤ , t = ti , . . . , ti−1 − 1.

(9.184)

9.9 An Auxiliary Result

327

By the relation above, {t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ⊂ {0, . . . , tq } ∪ {t0 , . . . , T } ∪ {{ti , . . . , ti−1 } : i ∈ E}. Together with (9.157), (9.172), (9.174), and (9.181), this implies that Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ 4L0 + 4 + L0 + 4L0 Card(E) ≤ 4L0 Card(E) + 5L0 + 4 ≤ 2L0 M1 γ0−1 + 5L0 + 4 < L. This completes the proof of Theorem 9.19.

9.9 An Auxiliary Result Proposition 9.21 Let M0 > 0, d + aσ−1 = 2,

(9.185)

n n ({u(t), v(t)}∞ t=−∞ ⊂ R+ × R+ and for all integers t.

u(t) ≤ M0 e, (u(t), u(t + 1)) ∈ Ω, v(t) ∈ Λ(u(t), u(t + 1)), bv(t) = bσ (1 + daσ )−1 .

(9.186) (9.187) (9.188)

Then for all integers t, x. u(t) = v(t) = (1 + daσ )−1 eσ = Proof Consider the RSS model with the function w(t) = t, t ∈ [0, ∞). By Theorem 3.1, for all (w)-good programs {x(t), y(t)}∞ t=0 , x lim x(t) = lim y(t) = (1 + daσ )−1 eσ =

t→∞

t→∞

(9.189)

and the results of Chapters 3 and 4 can be applied to the RSS model with the function

328

9 Turnpike for the RSS Model with Nonconcave Utility Functions

w(t) = t, t ∈ [0, ∞). For i = 1, . . . , n set qi = ai bi /(1 + dai ). p i = For any (x, x  ) ∈ Ω and any y ∈ Λ(x, x  ), set (see (3.12)) (x − x  ) − (w(by) − w(b x )). δ(x, y, x  ) = p

(9.190)

Lemma 2.2 implies that δ(x, y, x  ) ≥ 0 for every (x, x  ) ∈ Ω and every y ∈ Λ(x, x  ).

(9.191)

By Lemma 3.8 and (9.185)–(9.187), in order to prove the proposition, it is sufficient to show that δ(u(t), v(t), u(t + 1)) = 0 for all integers t. In view of (9.188), for each pair of integers T2 > T1 , T 2 −1

(bv(t) − b x ) = 0.

t=T1

Combined with Lemma 4.11, this implies that the following property holds: ∞ (a) for each  > 0 there exist sequences of integers {tk }∞ k=1 , {sk }k=1 such that

tk > 0, sk < 0 for all integers k ≥ 1, lim tk = ∞, lim sk = −∞,

k→∞

k→∞

x , u(sk ) − x  ≤ , k = 1, 2, . . . . u(tk ) −

(9.192) (9.193)

∞ Let  > 0 and sequences of integers {tk }∞ k=1 , {sk }k=1 be as guaranteed by property (a). Let k ≥ 1 be an integer. By (9.188), (9.190), and (9.193), t k −1

δ(u(t), v(t), u(t + 1))

t=sk

=

t k −1 t=sk

p (u(t) − u(t + 1)) = p ((u(sk ) − u(tk )) ≤ 2 p .

9.10 Perturbations

329

Since  is an arbitrary positive number and k is an arbitrary natural number, we have ∞ 

δ(u(t), v(t), u(t + 1)) = 0.

t=−∞

Proposition 9.21 is proved.

9.10 Perturbations Let w : [0, ∞) → [0, ∞) be a continuous strictly increasing function satisfying w(0) > 0.

(9.194)

d + aσ−1 = 2

(9.195)

Assume that

and that n , (x, x) ∈ Ω} = μ(w). sup{w(by) : y ∈ Λ(x, x), x ∈ R+

(9.196)

Theorem 9.2 implies that   n n × R+ : (x, x) ∈ Ω, y ∈ Λ(x, x), w(by) = μ(w) (x, y) ∈ R+   (9.197) = (1 + daσ )−1 e(σ ), (1 + daσ )−1 e(σ ) . Let λ ∈ (0, 1) satisfy w(0) > bσ (1 + daσ )−1 (1 − λ).

(9.198)

Define a function wλ : [0, ∞) → [0, ∞) as follows: wλ (t) = w(t) + (t − bσ (1 + daσ )−1 )(1 − λ) for all t ∈ [0, bσ (1 + daσ )−1 ] and

(9.199)

330

9 Turnpike for the RSS Model with Nonconcave Utility Functions

wλ (t) = w(bσ (1 + daσ )−1 ) + λ(w(t) − w(bσ (1 + daσ )−1 )) = λw(t) + (1 − λ)w(bσ (1 + daσ )−1 )

(9.200)

for all t > bσ (1 + daσ )−1 . It follows from (9.198)–(9.200) that wλ is a continuous strictly increasing function, wλ (0) > 0, wλ (t) ≤ w(t), t ∈ [0, ∞),

(9.201)

μ(wλ ) ≤ μ(w).

(9.202)

sup{wλ (by) : y ∈ Λ(x, x), x ∈

n R+ ,

(x, x) ∈ Ω}

n ≤ sup{w(by) : y ∈ Λ(x, x), x ∈ R+ , (x, x) ∈ Ω}.

(9.203)

In view of (9.199), wλ (bσ (1 + daσ )−1 ) = w(bσ (1 + daσ )−1 ).

(9.204)

By (9.196), (9.197), and (9.202)–(9.204), μ(wλ ) = μ(w) = w(bσ (1 + daσ )−1 ).

(9.205)

Theorem 9.2 and (9.205) imply that n n × R+ : (x, x) ∈ Ω, y ∈ Λ(x, x), wλ (by) = μ(wλ )} {(x, y) ∈ R+

= {(1 + daσ )−1 e(σ ), (1 + daσ )−1 e(σ )}.

(9.206)

Theorem 9.22 Let {x(t), y(t)}∞ t=0 be a (wλ )-good program. Then x, x ). lim (x(t), y(t)) = ((1 + daσ )−1 e(σ ), (1 + daσ )−1 e(σ )) = (

t→∞

Proof Lemma 5.3 implies that there exists M0 > 0 such that x(t) ≤ M0 for all integers t ≥ 0.

(9.207)

Assume that {tk }∞ k=1 is an increasing sequence of natural numbers such that there exists y0 , x1 ) = lim (x(tk ), y(tk ), x(tk + 1)). ( x0 , k→∞

In order to complete the proof, it is sufficient to show that

(9.208)

9.10 Perturbations

331

x0 = y0 = x1 = x = (1 + daσ )−1 e(σ ).

(9.209)

b y0 = bσ (1 + daσ )−1 .

(9.210)

First, we show that

Since the program {x(t), y(t)}∞ t=0 is (wλ )-good, we have that the sequence T −1 

∞ (wλ (by(t)) − μ(wλ ))

is bounded.

(9.211)

T =1

t=0

If lim

T −1

T →∞

(w(by(t)) − μ(w)) = −∞,

t=0

then in view of (9.202) and (9.205), lim

T −1

T →∞

(wλ (by(t)) − μ(wλ )) = −∞.

t=0

This contradicts (9.211). The contradiction we have reached proves that the sequence T −1  t=0

∞ (w(by(t)) − μ(w)) T =1

is bounded too. Together with (9.205) and (9.211), this implies that there exists M1 > 0 such that for each natural number T , T −1      (wλ (by(t)) − μ(wλ )) ≤ M1 ,   

(9.212)

t=0

T −1      (w(by(t)) − μ(w) ≤ M1 .   

(9.213)

t=0

Assume that b y0 > bσ (1 + daσ )−1 . Fix

(9.214)

332

9 Turnpike for the RSS Model with Nonconcave Utility Functions

Δ ∈ (bσ (1 + daσ )−1 , b y0 ).

(9.215)

In view of (9.208) and (9.215), we may assume without loss of generality that for all natural numbers k, by(tk ) > Δ.

(9.216)

By (9.200), (9.215), and (9.216), there exists δ > 0 such that for each natural number k, w(by(tk )) ≥ wλ (by(tk )) + δ.

(9.217)

It follows from (9.213) and (9.217) that for each natural number k, tk  (w(by(t)) − μ(w)) M1 ≥ t=0

=

k  (w(by(ti )) − μ(w)) i=1

+ ≥



{w(by(t)) − μ(w) : t ∈ {0, . . . , tk } \ {t1 , . . . , tk }}

k  (w(by(ti )) − wλ (by(ti ))) i=1

+

k  (wλ (by(ti )) − μ(wλ )) i=1

+



≥ δk +

{wλ (by(t)) − μ(wλ ) : t ∈ {0, . . . , tk } \ {t1 , . . . , tk }}

tk 

(wλ (by(t)) − μ(wλ )) ≥ δk − M1

t=0

and 2M1 ≥ δk → ∞ as k → ∞. The contradiction we have reached proves that b y0 ≤ bσ (1 + daσ )−1 . Assume that

(9.218)

9.10 Perturbations

333

b y0 < bσ (1 + daσ )−1 .

(9.219)

Δ ∈ (b y0 , bσ (1 + daσ )−1 ).

(9.220)

Fix

By (9.208) and (9.220), we may assume without loss of generality that for each natural number k, by(tk ) < Δ.

(9.221)

By (9.199), (9.220), and (9.221), there exists δ > 0 such that for each natural number k, w(by(tk )) ≥ wλ (by(tk )) + δ.

(9.222)

It follows from (9.201), (9.212), (9.213), and (9.222) that for each natural number k, M1 ≥

tk  (w(by(t)) − μ(w)) t=0

=

k  (w(by(ti )) − μ(w)) i=1

+ ≥



{w(by(t)) − μ(w) : t ∈ {0, . . . , tk } \ {t1 , . . . , tk }}

k  (w(by(ti )) − wλ (by(ti ))) i=1

+

k  (wλ (by(ti )) − μ(wλ )) i=1

+



≥ δk +

{wλ (by(t)) − (μ(wλ )) : t ∈ {0, . . . , tk } \ {t1 , . . . , tk }}

tk  (wλ (by(t)) − μ(wλ )) ≥ δk − M1 t=0

and 2M1 ≥ δk → ∞

334

9 Turnpike for the RSS Model with Nonconcave Utility Functions

as k → ∞. The contradiction we have reached proves that b y0 ≥ bσ (1 + daσ )−1 . Together with (9.218) this implies that b y0 = bσ (1 + daσ )−1 . This implies that lim by(t) = bσ (1 + daσ )−1 .

t→∞

(9.223)

In view of (9.213), for each pair of integers T2 > T1 ≥ 0,    T2 −1     (w (by(t)) − μ(w )) λ λ  ≤ 2M1 .    t=T1

(9.224)

For each integer k ≥ 1, define {x (k) (t), y (k) (t)}∞ t=−tk by x (k) (t) = x(t + tk ), y (k) (t) = y(t + tk ).

(9.225)

Extracting subsequences and using (9.207) and the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {kp }∞ p=1 such that for each integer t, there exist ut = lim x (kp ) (t), vt = lim y (kp ) (t). p→∞

p→∞

(9.226)

By (9.208), (9.225), and (9.226), u0 = x0 , u1 = x1 , v0 = y0 ,

(9.227)

(ut , ut+1 ) ∈ Ω, vt ∈ Λ(ut , ut+1 ).

(9.228)

for each integer t,

It follows from (9.223), (9.225), and (9.226) that for every integer t, bvt = bσ (1 + daσ )−1 . Proposition 9.21, (9.228), and (9.229) imply that for all integers t, x. ut = vt = (1 + daσ )−1 e(σ ) =

(9.229)

9.11 Generic Results

335

Together with (9.227) this implies that x1 = y0 = (1 + daσ )−1 e(σ ). x0 = Theorem 9.22 is proved.

9.11 Generic Results Denote by A the set of all continuous increasing functions w : [0, ∞) → [0, ∞). For every pair of positive numbers , M, set U (, M) = {(w1 , w2 ) ∈ A : |w1 (z) − w2 (z)| ≤  for all z ∈ [0, M]}.

(9.230)

We equip the set A with the uniformity determined by the base U (, M), where , M > 0. It is not difficult to see that the uniform space A is metrizable (by a metric ρA ) and complete. n . Set Let w ∈ A and x ∈ R+  μ(w, x) = sup lim sup T −1 T →∞

{x(t), y(t)}∞ t=0

T −1

w(by(t)) :

t=0

is a program such that x(0) = x

(9.231)

and n }. μ(w) = sup{μ(w, x) : x ∈ R+

Let M0 be a positive number and let T ≥ 1 be an integer. Set U (w, M0 , T ) = sup

T −1 

w(by(t)) :

t=0



 T −1 T {x(t)}t=0 , {y(t)}t=0 is a program such that x(0) ≤ M0 e . (9.232) Let M∗ > max{(ai d)−1 : i = 1, . . . , n}.

(9.233)

336

9 Turnpike for the RSS Model with Nonconcave Utility Functions

Lemma 5.3 and (9.233) imply that if (x, x  ) ∈ Ω and x ≤ M∗ e, then x  ≤ M∗ e.

(9.234)

n . Set Let x ∈ R+ n a(x) = {x  ∈ R+ : (x, x  ) ∈ Ω} n = {x ∈ R+ : x  ≥ (1 − d)x, a(x  − (1 − d)x) ≤ 1} n = (1 − d)x + {u ∈ R+ : au ≤ 1}.

(9.235)

n , its Hausdorff distance is defined by For each pair of nonempty sets A, B ⊂ R+





H (A, B) = max sup inf x − y, sup inf x − y . x∈A y∈B

y∈B x∈A

(9.236)

n, In view of (9.235) and (9.236), for all x, y ∈ R+

H (a(x), a(y)) ≤ (1 − d)x − y.

(9.237)

Relations (9.234) and (9.237) imply the following result. Proposition 9.23 Let {x(t), y(t)}∞ t=0 be a program. Then for all sufficiently large natural numbers t, x(t) ≤ (M∗ + 1)e. Proposition 9.23 implies the following result. Proposition 9.24 For every w ∈ A, n μ(w) = sup{μ(w, x) : x ∈ R+ and x ≤ (M∗ + 1)e}.

Suppose that d + aσ−1 = 2.

(9.238)

Denote by A0 the set of all w ∈ A such that μ(w) = w(bσ (1 + daσ )−1 ). Proposition 9.25 A0 is a closed set in (A,ρA ). Proof Let {wk }∞ k=1 ⊂ A0 , w ∈ A and

(9.239)

9.11 Generic Results

337

lim ρA (wk , w) = 0.

k→∞

(9.240)

Proposition 9.24 and (9.240) imply that w(bσ (1 + daσ )−1 ) = lim wk (bσ (1 + daσ )−1 ). k→∞

(9.241)

It follows from (9.230), (9.231), and (9.240) that for every natural number k, |μ(wk ) − μ(w)| n × | sup{μ(wk , x) : x ∈ R+ and x ≤ (M∗ + 1)e} n − sup{μ(w, x) : x ∈ R+ and x ≤ (M∗ + 1)e}| n ≤ sup{|μ(w, x) − μ(wk , x)| : x ∈ R+ and x ≤ (M∗ + 1)e} n ≤ sup{|w(bz) − wk (bz)| : z ∈ R+ and z ≤ (M∗ + 1)e} → 0

as k → ∞. This implies that lim μ(wk ) = μ(w).

k→∞

Together with (9.239) and (9.241), this implies that μ(w) = w(bσ (1 + daσ )−1 ). Thus w ∈ A0 . Proposition 9.25 is proved. We consider the complete metric space (A0 , ρA ). Clearly, the function w∗ (t) = t, t ∈ [0, ∞) belongs to A0 . It is not difficult to see that the following two propositions hold. Proposition 9.26 Let w ∈ A0 and α ∈ (0, 1]. Then the function αw(z) + (1 − α), z ∈ [0, ∞) belongs to A0 . Proposition 9.27 Let w ∈ A0 and α ∈ (0, 1). Then the function αw(z) + (1 − α)w(b x )z(b x )−1 , z ∈ [0, ∞) belongs to A0 . Proposition 9.27 implies the following result.

(9.242)

338

9 Turnpike for the RSS Model with Nonconcave Utility Functions

Proposition 9.28 There exists a set B ⊂ A0 which is a countable intersection of open everywhere dense subsets of (A0 , ρA ) such that each w ∈ B is strictly increasing. Let B ⊂ A0 be as guaranteed by Proposition 9.28. Proposition 9.28 and Theorem 9.22 imply the following result. Proposition 9.29 There exists a set B0 ⊂ B which is everywhere dense in (A0 , ρA ) such that for each w ∈ B0 and each (w)-good program {x(t), y(t)}∞ t=0 , lim x(t) = lim y(t) = (1 + daσ )−1 e(σ ).

t→∞

t→∞

Theorem 9.30 There exists a set G ⊂ A0 which is a countable intersection of open everywhere dense sets in (A0 , ρA ) such that G ⊂ B and for each w ∈ G and each (w)-good program {x(t), y(t)}∞ t=0 , lim x(t) = lim y(t) = (1 + daσ )−1 e(σ ).

t→∞

t→∞

Proof Let the set B0 be as guaranteed by Proposition 9.29. Assume that u ∈ B0 and q0 , q1 ≥ 1 be integers. By Theorem 9.19, there exist a natural number L(u, q0 , q1 ) and an open neighborhood U ((u, q0 , q1 ) of u in A0 such that the following property holds: (a) for

each integer T > L(u, q0 , q1 ), each w ∈ U ((u, q0 , q1 ), and each program T −1 which satisfies {x(t)}Tt=0 , {y(t)}t=0 x(0) ≤ q0 e, T −1

w(by(t)) ≥ U (w, q0 , T ) − q1

t=0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − (1 + daσ )−1 e(σ ), y(t) − (1 + daσ )−1 e(σ )} > q1−1 n}) ≤ L(u, q0 , q1 ). Set ∞ G = B ∩ (∩∞ q0 =1 ∩q1 =1 ∪{U ((u, q0 , q1 ) : u ∈ B0 }).

(9.243)

Clearly, G is a countable intersection of open everywhere dense sets in A0 . Let w ∈ G. The inclusion G ⊂ B implies that w is strictly increasing.

9.11 Generic Results

339

Let a program {x(t), y(t)}∞ t=0 be (w)-good, and let an integer q0 > max{(dai )−1 : i = 1, . . . , n}

(9.244)

x(0) ≤ q0 e.

(9.245)

be such that

Lemma 5.3, (9.244), and (9.245) imply that x(t) ≤ q0 e, t = 0, 1, . . . .

(9.246)

By Theorem 5.5, there exists an integer q1 > 0 such that for each integer p ≥ 1, U (w, q0 p) ≤

p−1 

w(by(t)) + q1 .

(9.247)

t=0

Let  > 0. Choose a natural number q2 such that 4q2−1 < , q2 > 4q1 .

(9.248)

In view of (9.243), there exists u ∈ B0

(9.249)

w ∈ U (u, q0 , q2 ).

(9.250)

such that

Property (a), (9.246), and (9.248)–(9.250) imply that for all integers T > L(u, q0 , q2 ), Card({t ∈ {0, . . . , T − 1} : max{x(t) − (1 + daσ )−1 e(σ ), y(t) − (1 + daσ )−1 e(σ )} > }) ≤ L(u, q0 , q2 ). Therefore Card({t ∈ {0, 1, . . . } : max{x(t) − (1 + daσ )−1 e(σ ), y(t) − (1 + daσ )−1 e(σ )} > }) ≤ L(u, q0 , q2 ).

340

9 Turnpike for the RSS Model with Nonconcave Utility Functions

This implies that x. lim x(t) = lim y(t) =

t→∞

Theorem 9.30 is proved.

t→∞

Chapter 10

An Autonomous One-Dimensional Model

The one-dimensional Robinson–Solow–Srinivasan model was studied by T. Mitra and M. Ali Khan in [39–45]. In this chapter we discuss the results obtained in [39] which was the starting point of their research. In [39] the value-loss approach of Radner–Gale–McKenzie was used in order to show a multiplicity of optimal programs under certain conditions, and a theory of undiscounted dynamic programming was used to derive properties of the optimal policy correspondence.

10.1 The Model Let a > 0, d ∈ (0, 1) and 2 Ω = {(x, x  ) ∈ R+ : x  ≥ (1 − d)x and a(x  − (1 − d)x) ≤ 1}.

For every (x, x  ) ∈ Ω define 1 : y ≤ x and y ≤ 1 − a(x  − (1 − d)x)} Λ(x, x  ) = {y ∈ R+

and u(x, x  ) = sup{y ∈ Λ(x, x  )}. 1 . A sequence {x(t), y(t)}∞ is called a feasible program from x if Let x0 ∈ R+ 0 t=0

x(0) = x0 and for all nonnegative integers t,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_10

341

342

10 An Autonomous One-Dimensional Model

(x(t), x(t + 1)) ∈ Ω and y(t) ∈ Λ(x(t), x(t + 1)). A sequence {x(t), y(t)}∞ t=0 is called a program from x0 if x(0) = x0 and for all nonnegative integers t, (x(t), x(t + 1)) ∈ Ω and y(t) = u(x(t), x(t + 1)). A sequence {x(t), y(t)}∞ t=0 is called a program if it is a program from x(0). Let {x(t), y(t)}∞ be a program. We associate with it a gross investment sequence t=0 {z(t + 1)}∞ and a consumption sequence {c(t + 1)}∞ t=0 t=0 defined by z(t + 1) = x(t + 1) − (1 − d)x(t) and c(t + 1) = y(t) for all integers t ≥ 0. It is not difficult to see that every program {x(t), y(t)}∞ t=0 is bounded by M(x(0)) := max{x(0), x}, ¯ where x¯ = (ad)−1 is the maximum sustainable capital stock. A program {x(t), y(t)}∞ t=0 is called stationary if (x(t), x(t + 1)) = (x(t + 1), x(t + 2)) for all nonnegative integers t. For a stationary program {x(t), y(t)}∞ t=0 , we have x(t) ≤ x¯ for all nonnegative integers t. A program {x ∗ (t), y ∗ (t)}∞ t=0 is called weakly optimal if lim inf T →∞

T  [u(x(t), x(t + 1)) − u(x ∗ (t), x ∗ (t + 1))] ≤ 0 t=0

∗ for every program {x(t), y(t)}∞ t=0 such that x(0) = x (0)). A stationary weakly optimal program is a program which is stationary and weakly optimal.

10.2 A Golden Rule

343

A program {x(t), y(t)}∞ t=0 is full-employment if for all integers t ≥ 0, y(t) + a(x(t + 1) − (1 − d)x(t)) = 1. It is not difficult to see that the following result holds (see Proposition 1 of [39]. Proposition 10.1 (1) For every program {x(t), y(t)}∞ t=0 , there exists a full-employment program {x  (t), y  (t)}∞ such that x(0) = x  (0)) and y  (t) = y(t) for all integers t ≥ 0. t=0 ∞ (2) If {x(t), y(t)}t=0 is a program from x and x  > x, then there exists a full  employment program {x  (t), y  (t)}∞ t=0 from x such that y (t) ≥ y(t) for all  integers t ≥ 0 and y (t) > y(t) for some integer t ≥ 0. (3) If {x(t), y(t)}∞ t=0 is a program from x which is not a full-employment program, then there is a full-employment program {x  t), y  (t)}∞ t=0 from x such that y  (t) ≥ y(t) for all integers t ≥ 0 and y  (t) > y(t) for some integer t ≥ 0.

10.2 A Golden Rule 1 as a golden rule if ( We define a stock x ∈ R+ x, x ) ∈ Ω and

u( x, x ) ≥ u(x, x  ) for all (x, x  ) ∈ Ω such that x  ≥ x. The following result is proved in [39]. Its proof is given in Section 10.6. Proposition 10.2 (1) The pair ( x, p ) = ((1 + ad)−1 , a(1 + ad)−1 ) satisfies ( x, x) ∈ Ω and x  − p x for all (x, x  ) ∈ Ω. u( x, x ) ≥ u(x, x  ) + p

(10.1)

(2) x is a unique golden-rule stock. We refer to p , given by Proposition 10.2, as the golden price and to the pair ( x, p ) as the golden rule. The value loss (relative to the golden rule) from operating at (x, x  ) ∈ Ω and y = u(x, x  ) is defined by

344

10 An Autonomous One-Dimensional Model

δ(x, x  , y) = u( x, x ) − [u(x, x  ) + p x  − p x] ≥ 0.

(10.2)

We split the value loss in (10.2) as follows: d(x − y) ≥ 0, α(x, x  , y) = p β(x, x  , y) = ( p/a)(1 − y − a(x  − (1 − d)x)) ≥ 0, δ(x, x  , y) = α(x, x  , y) + β(x, x  , y).

(10.3)

Let {x(t), y(t)}∞ t=0 be a program. Then for all integers t ≥ 0, (x(t), x(t + 1)) ∈ Ω and y(t) = u(x(t), x(t + 1)). It follows from (10.2) and (10.3) that for all integers t ≥ 0, u( x, x ) = u(x(t), x(t + 1)) + p x(t + 1) − p x(t) + α(t) + β(t) = u(x(t), x(t + 1)) + p x(t + 1) − p x(t) + β(t)

(10.4)

where α(t) = α(x(t), x(t + 1), y(t)), β(t) = β(x(t), x(t + 1), y(t)), δ(t) = δ(x(t), x(t + 1), y(t)) for every integer t ≥ 0. This yields a useful identity, relating the sum of value losses to the sum of utility differences from the golden-rule utility level, along any program {x(t), y(t)}∞ t=0 and any integer T ≥ 0: T  [u(x(t), x(t + 1)) − u( x, x )] t=0

=p x(0) − p x(T + 1) −

T 

δ(t).

(SV L)

t=0

We can eliminate the variable y(t) = u(x(t), x(t + 1)) from (10.4), by using (10.3), to obtain for all integers t ≥ 0, x(t + 1) = a −1 − ξ x(t) + Ax(t) − B(t) where ξ = a −1 − (1 − d)

10.2 A Golden Rule

345

and A(t) = (1/ad p )α(t), B(t) = (1/ p )β(t) for all integers t ≥ 0. On measuring capital stocks relative to the golden-rule stock, X(t) = x(t) − x , and on noting that x, x = a −1 − ξ we obtain that X(t + 1) − ξ X(t) + A(t) − B(t) for all integers t ≥ 0.

(10.5)

In this chapter we assume that ξ > 0. Recall that a program {x(t), y(t)}∞ t=0 is good if there is a number G such that for all integers T ≥ 1, T 

[u(x(t), x(t + 1)) − u( x, x )] ≥ G.

t=0

The following result is proved in [39]. Its proof is given in Section 10.6. Theorem 10.3 n there exists a good program {x(t), y(t)}∞ satisfying (1) For every x0 ∈ R+ t=0 x(0) = x0 . ∞ (2) If {x(t), y(t)}∞ t=0 δ(t) < ∞. t=0 is a good program, then (3) If ξ = 1 and {x(t), y(t)}∞ is a good program, then t=0

(x(t), y(t)) → ( x, x ) as t → ∞ and T  [u(x(t), x(t + 1)) − u( x, x )] t=0

=p x(0) − p x−

∞  t=0

δ(t).

(U S)

346

10 An Autonomous One-Dimensional Model

(4) Suppose that ξ = 1, {x(t), y(t)}∞ t=0 is a good program and that x˜ is a limit point of the sequence {x(t)}∞ . Then either x(t) converges to x˜ for all odd periods t=0 and to 2 x − x˜ for all even periods or x(t) converges to x˜ for all even periods and to 2 x − x˜ for all odd periods. Note that Assertions 1–3 of Theorem 10.3 were proved in Chapters 2 and 3 for the multidimensional model (see Propositions 2.8 and 2.9 and Theorem 3.2) while Assertion 4 does not have a multidimensional analog.

10.3 Optimality and Value-Loss Minimization The following result plays an important role in the study in [39]. It follows from Propositions 2.10 and 2.11. Proposition 10.4 (1) If {x(t), y(t)}∞ t=0 is a program such that ∞ 

δ(t) ≤

t=0

∞ 

δ  (t)

(V LM)

t=0

∞  for every program {x  (t), y  (t)}∞ t=0 satisfying x(0) = x (0), then {x(t), y(t)}t=0 is weakly optimal. 1 there exists a program {x(t), y(t)}∞ from x such that (2) For every x ∈ R+ t=0 ∞  t=0

δ(t) ≤

∞ 

δ  (t)

t=0

for every program {x  (t), y  (t)}∞ t=0 from x. 1 there exists a weakly optimal program from x. (3) For every x ∈ R+ The next auxiliary result obtained [39] is proved in Section 10.6. Lemma 10.5 (1) For every full-employment program {x(t), y(t)}∞ t=0 and for every integer T > 1, T  [α(t)(−ξ )−t ] = adξ [ pX(0) − p (X(T + 1)(−ξ )−T −1 )].

(10.6)

t=0

(2) Every weakly optimal program is a full-employment program and satisfies (10.6).

10.4 Optimality Does Not Imply Value-Loss Minimization

347

10.4 Optimality Does Not Imply Value-Loss Minimization The following two results obtained [39] are proved in Section 10.6. Proposition 10.6 Let {x(t), y(t)}∞ t=0 be a weakly optimal program. Then it is good. If ξ = 1, then (x(t) − x) = 0 lim p

t→∞

(10.7)

and ∞ 

δ≤

t=0

∞ 

δ  (t)

(V LM)

t=0

 for every program {x  (t), y  (t)}∞ t=0 satisfying x (0) = x(0). Moreover, T  [u(x(t), x(t + 1)) − u( x, x )] t=0

=p x(0) − p x−

∞ 

δ(t).

(U S)

t=0

If ξ = 1, (US) of Proposition 10.6 can be rewritten as T  [u(x(t), x(t + 1)) − u( x, x )] = p x(0) − p x − δ(x)

(10.8)

t=0

where δ(x) =

∞ 

δ(t).

t=0

By Proposition 10.6, δ(x) is well-defined and depends only on x. Theorem 10.7 Assume that ξ = 1 and x0 ∈ ( x , 1). (1) The full-employment program {x(t), y(t)}∞ x for t=0 from x0 satisfying x(t) = all integers t ≥ 1 is weakly optimal from x0 .  (2) The full-employment program {x  (t), y  (t)}∞ t=0 from x0 satisfying x (t) = x0 for  x − x0 for all odd t is weakly optimal from x0 . all even t and x (t) = 2 (3) There is a continuum of weakly optimal programs from x0 .

348

10 An Autonomous One-Dimensional Model

10.5 Optimal Policy Function 1 to subsets of The optimal policy correspondence is a correspondence h from R+ 1 1 R+ such that, given any x ∈ R+ , there is a weakly optimal program {x(t), y(t)}∞ t=0 from x such that x(1) ∈ h(x). 1 → R 1 such that for every x ∈ R 1 , Define a function V : R+ + +

V (x) =

T  [u(x(t), x(t + 1)) − u( x, x )]

(10.9)

t=0

where {x(t), y(t)}∞ t=0 is any weakly optimal program from x. Clearly, V (x) is independent of the choice of the weakly optimal program (see (10.8)). It follows from (US) of Theorem 10.3(3), (10.8), and (10.9) that for every good program {x  (t), y  (t)}∞ t=0 from x, V (x) ≥

T 

[u(x  (t), x  (t + 1)) − u( x, x )].

(10.10)

t=0 1 define For every x ∈ R+ 1 : (x, x  ) ∈ Ω}. Ω(x) = {x  ∈ R+

The next result obtained in [39] is proved in Section 10.6. Proposition 10.8 1 and (1) The value function V is a concave and strictly increasing function on R+ continuous on (0, ∞) satisfying V ( x ) = 0. (2) V satisfies the equation

V (x) = max{u(x, x  ) − u( x, x ) + V (x  ) : x  ∈ Ω(x)} 1. for all x ∈ R+ (3) {x(t), y(t)}∞ t=0 is a weakly optimal program if and only if for every integer t ≥ 0,

V (x(t)) = u(x(t), x(t + 1)) − u( x, x ) + V (x(t + 1)). Since any program starting from a point in X = [0, 1/ad] remains in X, we assume that X is our space of states and confirm our solution of the optimal policy correspondence to this domain. When ξ = 1 we can use Proposition 10.8 in order to describe the optimal policy correspondence for certain subsets of X. More precisely, set

10.6 Proofs

349

k = x (1 − d)−1 , A = [0, x ], B = ( x , k), C = (k, 1/ad].

(10.11)

Proposition 10.8 implies the following results. Corollary 10.9 Assume that ξ = 1. Then the optimal policy correspondence h : 1 satisfies X → R+ h(x) = a −1 − ξ x, x ∈ A, h(x) = (1 − d)x, x ∈ C.

(10.12)

Corollary 10.10 1 (1) Assume that ξ > 1. Then the optimal policy correspondence h : X → R+ satisfies

h(x) = x , x ∈ ( x , k).

(10.13)

1 (2) Assume that ξ < 1. Then the optimal policy correspondence h : X → R+ satisfies

x , 1), h(x) = a −1 − ξ x, x ∈ ( h(x) = (1 − d)x, x ∈ [1, k).

(10.14)

Corollary 10.11 Assume that ξ = 1. Then the optimal policy correspondence h : 1 satisfies X → R+ a −1 − ξ x ∈ h(x), x ∈ A [a −1 − ξ x, x ] ⊂ h(x), x ∈ ( x , 1) [(1 − d)x, x ] ⊂ h(x), x ∈ [1, k), (1 − d)x ∈ h(x), x ∈ C.

10.6 Proofs Proof of Proposition 10.2 (1) It is not difficult to see that ( x, x ) ∈ Ω and u( x, x) = x. For every (x, x  ) ∈ Ω and every y ∈ Λ(x, x  ), set

(10.15)

350

10 An Autonomous One-Dimensional Model

α(x, x  , y) = d p (x − y) and p/a)(1 − y − a(x  − (1 − d)x)). β(x, x  , y) = ( Clearly, α(x, x  , y) ≥ 0 and β(x, x  , y) ≥ 0. We have x = y + p (x  − (1 − d)x) − d p x y+p x  − p = (1 + ad)−1 y + p (x  − (1 − d)x) − α(x, x  , y) = (1 + ad)−1 (y + a(x  − (1 − d)x)) − α(x, x  , y) = x (y + a(x − (1 − d)x)) − α(x, x  , y) = x − x (1 − y − a(x  − (1 − d)x)) − α(x, x  , y) = x − β(x, x  , y) − α(x, x  , y).

(10.16)

Equation (10.16) implies the validity of Proposition 10.2(1). (2) Assume that x is a golden-rule stock. Then (x, x) ∈ Ω and x ∈ Λ(x, x). In view of (10.16), β(x, x, y) = α(x, x, y) = 0 and x = y = x . Proposition 10.2(2) is proved. Proof of Theorem 10.3 (1) Define y(0) = 0 and y(t + 1) = (1 − d)y(t) + d x for all integers t ≥ 0. Clearly, {y(t)}∞ t=0 is a monotonically non-decreasing sequence which converges to x as t → ∞. Define

10.6 Proofs

351

z(t + 1) = d x for all integers t ≥ 0. 1 be given. Define x(0) = x and Let x ∈ R+ x(t + 1) = (1 − d)x(t) + z(t + 1) for all integers t ≥ 0. It is easy to see that {x(t), y(t)}∞ t=0 is a program and that for all integers t ≥ 0, y(t) − x = (1 − d)t (y(0) − x ). This implies that the program {x(t), y(t)}∞ t=0 is good. (2) In view of (SVL), for every integer T ≥ 0, we have T  [u(x(t), x(t + 1)) − u( x, x )] t=0

=p x(0) − p x−

∞ 

δ(t).

t=0

Since the program {x(t), y(t)}∞ t=0 is good, there exists a number G such that for all integers T ≥ 0, T  [u(x(t), x(t + 1)) − u( x, x )] ≥ G. t=0

Hence T 

δ(t) ≤ p x(0) − G

t=0

 for all integers T ≥ 0. This implies that ∞ t=0 δ(t) < ∞. (3) Since the program {x(t), y(t)}∞ is good, it follows from assertion 2 and (10.3) t=0 that ∞  t=0

A(t) < ∞,

∞ 

B(t) < ∞.

t=0

Let  > 0 be given. There exists a natural number τ such that

352

10 An Autonomous One-Dimensional Model ∞ 

A(t) +

t=τ

∞ 

B(t) < /3.

(10.17)

t=τ

Set D = 2M(x). Since x(t) ≤ M(x) for all integers t ≥ 0 and x ≤ x¯ ≤ M(x), we conclude that |X(t)| ≤ D for all integers t ≥ 0. The equation 0 < ξ = a −1 − (1 − d) implies that we have two possible cases: (a) 0 < ξ < 1; (b) ξ > 1. Hence, there exists a natural number ν such that min{Dξ ν , Dξ −ν } < /3.

(10.18)

Let a natural number T ≥ τ be given. Then for all natural numbers n ≥ ν, it follows from (10.5) that X(T + n) = (−ξ )n X(T ) +

n−1 

(−ξ )n−1−s A(T + s) −

s=0

n−1  (−ξ )n−1−s B(T + s).

(10.19)

s=0

In case (a), by (10.17)–(10.19), we have |X(T + n)| < . Consider case (b). We divide (10.19) by (−ξ )n and obtain (−ξ )−n X(T + n) = X(T ) +

n−1 n−1   (−ξ )−1−s A(T + s) − (−ξ )−1−s B(T + s). s=0

(10.20)

s=0

In view of (10.17), (10.18), and (10.20), |X(T )| < . Since T ≥ τ and n ≥ ν are arbitrary, we conclude that in both cases |x(t) − x | = |X(t)| → 0 as t → ∞. Proposition 10.1 implies that y(t) = 1 − a(x(t + 1) − (1 − d)x(t))

10.6 Proofs

353

for all integers t ≥ 0 and y(t) → 1 − a( x − (1 − d) x) = x as t → ∞. It follows from assertion 2 and the relation x(t) → x as t → ∞ that the right-hand side of (SVL) has a limit as T → ∞ and the left-hand side of (SVL) also has a limit as T → ∞. Taking limits in (SVL), by letting T → ∞, we obtain (US). (4) Let  > 0 be given. As in the proof of (3), choose a natural number τ such that (10.17) holds. For every integer T ≥ τ , we sum (10.5) from T to T + s, except that for s odd, use the equation in (10.5) with a negative sign and conclude that −/3 ≤ −

∞ 

A(t) −

t=T

∞ 

B(t)

t=T

≤ X(T ) + (−1)s X(T + s + 1) ≤

∞  t=T

A(t) +

∞ 

B(t) ≤ /3.

(10.21)

t=T

Let x˜ be an arbitrary limit point of the sequence {x(t)}∞ t=0 . Set X˜ = x˜ − x. ∞ ˜ There exists a subsequence {tr }∞ r=1 such that {X(tr )}r=1 converges to X. There is μ such that for all integers r ≥ μ, we have

tr ≥ τ, X˜ − /3 ≤ X(tr ) ≤ X˜ + /3. By (10.21) for integers r ≥ μ and integers s ≥ 0, X˜ − 2/3 ≤ −X(tr ) − /3 ≤ (−1)s X(tr + s + 1) ≤ −X(tr ) + /3 ≤ −X˜ + 2/3. Thus, for even positive integers s, we obtain that −X˜ − 2/3 ≤ X(tμ + s + 1) ≤ −X˜ + 2/3,

354

10 An Autonomous One-Dimensional Model

so that X(tμ + s) converges to −X˜ for all odd integers s. For odd natural numbers s, we obtain that −X˜ − 2/3 ≤ −X(tr + s + 1) ≤ −X˜ + 2/3. This implies that −X˜ − 2/3 ≤ X(tr + s + 1) ≤ X˜ + 2/3 and that X(tμ + s) converges to X˜ for all odd integers s. If tμ is odd, then X(t) converges to −X˜ for even integers and X(t) converging to X˜ for odd integers. If tμ is even, then X(t) converges to −X˜ for odd integers and X(t) converging to X˜ for even integers. This completes the proof of Theorem 10.3. Proof of Lemma 10.5 Let a sequence {x(t), y(t)}∞ t=0 be a full-employment program. Then B(t) = 0 for all integers t ≥ 0. In view of (10.5), A(t)ξ −t = X(t + 1)ξ −t + ξ X(t)ξ −t for all integers t ≥ 0.

(10.22)

Let a natural number T be given. We sum (10.22) from t = 0 to t = T , except that for t odd, use (10.22) and obtain T 

A(t)(−ξ )−t = ξ(X(0) − X(T + 1)(−ξ )−T −1 ).

(10.23)

t=0

Equation (10.6) follows from (10.23) and the equality A(t)α(t)/ pad, t = 0, 1, 2, . . . . (2) Let {x(t), y(t)}∞ t=0 be a weakly optimal program. By Assertion (3) of Proposition 10.1, it is a full-employment program. In view of Assertion (1), it satisfies (10.6). Lemma 10.5 is proved. Proof of Proposition 10.6 Let {x(t), y(t)}∞ t=0 be a weakly optimal program from x and {x  (t), y  (t)}∞ be a program from x. By (10.4), for every natural number T , t=0 T −1

[u(x  (t), x  (t + 1)) − u( x, x )]

t=0 



x (T ) − =p x (0) − p

T −1 t=0

and

δ  (t)

(10.24)

10.6 Proofs

355 T −1

[u(x  (t), x  (t + 1)) − u(x(t), x(t + 1))]

t=0 

=p x(T ) − p x (T ) −

T −1



δ (t) +

T −1

t=0

δ(t).

(10.25)

t=0

∞ Assertion 1 of Theorem 10.3 implies that there exists a good program {x(t), ¯ y(t)} ¯ t=0 ∞   ¯ satisfying y (t)}∞ t=0 δ(t) < ∞. Using this program in place of {x (t), t=0  ∞ in (10.25) and the weak optimality of {x(t), y(t)}∞ t=0 δ(t) < t=0 , we conclude that ∞   ∞. Using {x(t), y(t)}∞ t=0 in place of {x (t), y (t)}t=0 in (10.24), we obtain that ∞ {x(t), y(t)}t=0 is good. Now Assertion (3) of Theorem 10.3 implies (10.7). Since the program {x(t), y(t)}∞ t=0 is good, we have ∞ 

δ(t) < ∞.

t=0

∞  then If {x  (t), y  (t)}∞ t=0 δ (t) = ∞ and (VLM) holds. If t=0 is not good,  ∞  (t) < ∞, by Theorem 10.3, {x  (t), y  (t)}∞ is good, then δ t=0 t=0 x(t) = p x lim p

t→∞

and x  (t) = p x. lim p

t→∞

Thus the right-hand side of (10.25) has a limit as T → ∞. Therefore the left-hand side of (10.25) has a limit as T → ∞ which is equal to ∞  t=0

δ(t) −

∞ 

δ  (t).

t=0

Now the weak optimality of {x(t), y(t)}∞ t=0 implies (VLM). Theorem 10.3 implies (US). Proposition 10.6 is proved. Proof of Theorem 10.7 The equality ξ = 1 implies that x = 1/2a, p = 1/2, 1/a = 1 + (1 − d) > 1, a ∈ (0, 1), ad ∈ (0, 1). (1) Since the program {x(t), y(t)}∞ t=0 is full-employment, we conclude that for all integers t ≥ 0,

356

10 An Autonomous One-Dimensional Model

β = 0, δ(t) = α(t). It is easy to see that for all integers t ≥ 1, X(t) = 0, δ(t) = 0. Lemma 10.5 and equality ξ = 1 imply that for every natural number T , T 

δ(t) = δ(0) = α(0) =

t=0

T 

α(t)(−ξ )−t = ad p X(0).

(10.26)

t=0

If the program {x(t), y(t)}∞ t=0 is not weakly optimal, then there exist a positive number θ , an integer N, and a program {x  (t), y  (t)}∞ t=0 from x0 such that for every integer T > N, we have θ≤

T −1

[u(x  (t), x  (t + 1)) − u(x(t), x(t + 1))].

(10.27)

t=0

In view of Proposition 10.1, we may assume without loss of generality that {x  (t), y  (t)}∞ t=0 is a full-employment program from x0 . Therefore for all integers t ≥ 0, β  (t) = 0, δ  (t) = α  (t). Lemma 10.5 and the equality ξ = 1 imply for every natural number T , T 

δ  (t) ≥

t=0

T 

α  (t)(−1)t

t=0

= ad p X(0) − ad p X (T + 1)(−1)T +1 .

(10.28)

By (10.26) and (10.28), T  t=0

δ(t) −

T 

δ  (t) ≤ ad p X (T )(−1)T .

(10.29)

t=0

It follows from (10.25), (10.27), and (10.29) that for every integer T > N, X (T )(−1)T ≥ θ. − p X (T ) + ad p

(10.30)

In view of (10.27), the program {x  (t), y  (t)}∞ t=0 is good. By (10.30), for odd integer T > N,

10.6 Proofs

357

−X (T ) ≥ θ/ p(1 + ad) = θ/a. Theorem 10.3 implies that for even T > N, X (T ) converges to a positive number as T → ∞. On the other hand, for even T > N, by (10.30), p X (T )(ad − 1) ≥ θ, p (1 − ad) < 0, X (T ) ≤ −θ/ a contradiction. Thus {x(t), y(t)}∞ t=0 is a weakly optimal program. (2) It is easy to see that for the program {x  (t), y  (t)}∞ t=0 for every integer t ≥ 0, α  (t) = β  (t) = 0 and (VLM) holds. Proposition 10.4 implies that it is weakly optimal. (3) Since the function h is convex, there is a continuum of weakly optimal programs from x0 . This completes the proof of Theorem 10.7. Proof of Proposition 10.8 The golden-rule program defined by ( x (t), y (t)) = ( x, x ), t = 0, 1, . . . has zero value loss in each period, and it is weakly optimal in view of Proposition 10.4. It follows from (10.10) that V ( x ) = 0.  If {x(t), y(t)}∞ t=0 is a weakly optimal program from x and x > x, then we set

x  (0) = x  , x  (t + 1) = x  (t) + z(t + 1) for every integer t ≥ 0 and y  (t) = y(t) for every integer t ≥ 0, where {z(t + 1)}∞ t=0 is the investment sequence associated with the program {x(t), y(t)}∞ . It is not difficult to see that x  (t) > x(t) for every integer t ≥ 0 t=0 ∞   1. and {x (t), y (t)}t=0 is a program from x  . Therefore, V is non-decreasing on R+ 1 . If z(t + 1) = 0 for all integers Let us show that V is strictly increasing on R+ t ≥ 0, then x(t) → 0 as t → ∞, a contradiction. Therefore z(t + 1) > 0 for some integer t ≥ 0. Let T be the first period for which z(T + 1) > 0. If x  > x, we define x  (0) = x  ,

358

10 An Autonomous One-Dimensional Model

x  (t + 1) = x  (t) + z(t + 1) if t = T , x  (T + 1) = x  (T ) + z (T + 1), where z (T + 1) = z(T + 1) − , 0 <  < z(T + 1) and  is sufficiently close to zero so that x  (T + 1) > x(T + 1). Set y  (t) = y(t) if t = T , y  (T ) = y(T ) + . It is not difficult to see that x  (t) > x(t) for all integers t ≥ 0 and that  {x  (t), y  (t)}∞ t=0 is a program from x . Hence, the function V is strictly increasing 1 on R+ . Since the set Ω is convex and the function u is concave, we conclude that the 1 and continuous on (0, ∞). value function is concave on R+ (2) Assume that {x(t), y(t)}∞ t=0 is a weakly optimal program from x. By definition, V (x) = [u(x, x(1)) − u( x, x )] +

∞  [u(x(t), x(t + 1)) − u( x, x )]. t=1

For all integers t ≥ 0, define (x  (t), y  (t)) = (x(t + 1), y(t + 1)). It is easy to see that {x  (t), y  (t)}∞ t=0 is a good program and that ∞  [u(x(t), x(t + 1)) − u( x, x )] t=1

=

∞  [u(x  (t), x  (t + 1)) − u( x, x )] ≤ V (x(1)) t=0

and

10.6 Proofs

359

V (x) ≤ [u(x, x(1)) − u( x, x )] + V (x(1)).

(10.31)

Assume that (x, x  ) ∈ Ω and that {x  (t), y  (t)}∞ t=0 is a weakly optimal program from x  . Define (x(0), y(0)) = (x, max Λ(x, x  )), (x(t), y(t)) = (x  (t − 1), y  (t − 1)) for all integers t ≥ 1. It is not difficult to see that {x(t), y(t)}∞ t=0 is a good program from x. By (10.9) and (10.10), we have 

x, x )] + V (x) ≥ [u(x, x ) − u(

∞ 

[u(x(t), x(t + 1)) − u( x, x )]

t=1

x, x )] + = [u(x, x  ) − u(

∞  [u(x  (t − 1), x  (t)) − u( x, x )] t=1

x, x )] + = [u(x, x  ) − u(

∞  [u(x  (t), x  (t + 1)) − u( x, x )] t=0



x, x )] + V (x  ). = [u(x, x ) − u(

(10.32)

Clearly, if {x(t), y(t)}∞ t=0 is a weakly optimal program, then equality holds in (10.31). This implies that V (x(1)) =

∞  [u(x(t), x(t + 1)) − u( x, x )], t=1

for all integers t ≥ 0, V (x(t)) = [u(x(t), x(t + 1)) − u( x, x )] + V (x(t + 1))

(10.33)

and that x, x )] + V (x  ) : (x, x  ) ∈ Ω}. V (x) = max{[u(x, x  ) − u( (3) We already proved one half of (iii) in (10.33). Let us now prove the other half. Assume that a program {x(t), y(t)}∞ t=0 satisfies for every integer t ≥ 0, V (x(t)) = [u(x(t), x(t + 1)) − u( x, x )] + V (x(t + 1)). Then, for every integer T ≥ 0,

360

10 An Autonomous One-Dimensional Model

V (x(0)) =

T  [u(x(t), x(t + 1)) − u( x, x )] + V (x(T + 1)).

(10.34)

t=0

By monotonicity of V on X and the inequality x(t) ≤ M(x(0)), t = 0, 1, . . . , we define m = V (M(x(0))) and obtain that V (x(t)) ≤ V (M(x(0))) = m for all integers t ≥ 0. Together with (10.34) this implies that T 

[u(x(t), x(t + 1)) − u( x, x )] = V (x(0)) − V (x(T + 1)) ≥ V (x(0)) − m.

t=0

Therefore the program {x(t), y(t)}∞ t=0 is good and x(t) → x as t → ∞. It follows from the continuity of V that V (x(t)) → V ( x ) = 0 as t → ∞. Together with (10.34) this implies that lim

T →∞

T  [u(x(t), x(t + 1)) − u( x, x )] t=0

exists, is finite, and satisfies V (x) =

T 

[u(x(t), x(t + 1)) − u( x, x )].

t=0

By (10.8), (10.9), (US), and Theorem 10.3, δ(x(0)) =

∞ 

δ(t)

t=0

and {x(t), y(t)}∞ t=0 is a weakly optimal program. Proposition 10.8 is proved.

10.6 Proofs

361

Proof of Corollary 10.9 In view of (10.9) and concavity of V , V (x) − V ( x) ≤ p (x − x) for all x > x, x) ≤ p V+ ( and x) ≤ p for all x > x. V+ (x) ≤ V+ ( Consider the case with x ∈ C. Then for (x, x  ) ∈ Ω we have x¯ := (1 − d)x ≤ x  . Assume, contrary to the claim of the corollary, that there exists x  > x¯ such that x  ∈ h(x). Then x, x > V (x  ) − V (x) ¯ ≤ V+ (x)(x ¯  − x) ¯ ≤p (x  − x) ¯ < a(x  − x). ¯ Together with Proposition 10.8(3), this implies that ¯ + V (x) ¯ u(x, x  ) + V (x  ) = 1 − a(x  − (1 − d)x) + (V (x  ) − V (x)) < 1 − a(x  − x) ¯ + a(x  − x) ¯ + V (x) ¯ = u(x, x) ¯ + V (x) ¯ ≤ V (x) + u( x, x ). Clearly, (10.35) contradicts Proposition 10.8(2). Consider the case x ∈ A. Set x¯ = a −1 − ξ x. We have x¯ ≥ x. Assume, contrary to the corollary, that there exists x  ∈ h(x)

(10.35)

362

10 An Autonomous One-Dimensional Model

satisfying ¯ x  = x. ¯ then for y ∈ Λ(x, x  ), If x  < x, a(x  − (1 − d)x) + y < a(x¯ − (1 − d)x) + x = 1, so labor is not fully employed, a contradiction to Lemma 10.5(2). Assume that x  > x. ¯ Then x, x  > x¯ ≥ V (x  ) − V (x) ¯ ≤ V+ (x)(x ¯  − x) ¯ ≤p (x  − x). ¯ Together with Proposition 10.8(3), this implies that ¯ + V (x) ¯ u(x, x  ) + V (x  ) = 1 − a(x  − (1 − d)x) + (V (x  ) − V (x)) ≤ 1 − a(x¯ − (1 − d)x) − a(x  − x) ¯ +p (x  − x) ¯ + V (x) ¯ < x + V (x) ¯ = u(x, x) ¯ + V (x) ¯ ≤ V (x) + u( x, x ).

(10.36)

Clearly, (10.36) contradicts Proposition 10.8(2). Corollary 10.9 is proved. Proof of Corollary 10.10 (1) Set x(0) ¯ = x, x(t) ¯ = x for all integers t ≥ 0. Clearly, (x(t), ¯ x(t ¯ + 1)) ∈ Ω for all integers t ≥ 0. Set y(t) ¯ = u(x(t), ¯ x(t ¯ + 1)) for all integers t ≥ 0. ∞ is a full-employment program, for all It is easy to see that {x(t), ¯ y(t)} ¯ t=0 integers t ≥ 0,

¯ = 0, δ(t) ¯ = α(t) β(t) ¯

10.6 Proofs

363

and for all integers t ≥ 1, ¯ ¯ = 0. X(t) = 0, δ(t) Lemma 10.5 implies that ∞ 

¯ = δ(0) ¯ δ(t) = α(0) ¯

t=0

=

∞ 

¯ α(t)(−ξ ¯ )−t = adξ p X(0).

(10.37)

t=0

Assume that {x  (t), y  (t)}∞ t=0 is a weakly optimal program satisfying x(0) = x. Then {x  (t), y  (t)}∞ is a full-employment program. Lemma 10.5 and the t=0 inequality ξ > 1 imply that ∞ 

δ  (t) ≥

t=0

∞ 

α  (t) ≥

t=0

≥

∞ 

∞ 

α  (t)ξ −t

t=0

α  (t)(−ξ )−t = adξ p X(0).

(10.38)

t=0

Proposition 10.6 implies that ∞ 

δ  (t) ≤

t=0

∞ 

δ(t)

(10.39)

t=0

for every program {x(t), y(t)} satisfying x(0) = x. In view of (10.37) and (10.38), ∞ 

δ  (t) ≥

t=0

∞ 

¯ δ(t).

t=0

Thus ∞ 

δ  (t) =

t=0

Together with (10.39) this implies that

∞  t=0

¯ δ(t).

364

10 An Autonomous One-Dimensional Model ∞ 

¯ ≤ δ(t)

t=0

∞ 

δ(t)

t=0

for every program {x(t), y(t)} satisfying x(0) = x. Proposition 10.4 implies that the program {x(t), ¯ y(t)} ¯ is weakly optimal. Since ∞  t=0

δ  (t) =

∞ 

¯ δ(t)

t=0

we have equality in all the inequalities of (10.38). Together with (10.5) this implies that for all integers t ≥ 1, α  (t) = 0, X (t + 1) = (−ξ )X (t). Since x  (t) ≤ M(x) for all integers t ≥ 0 and ξ > 1, we conclude that X (1) = 0, x  (1) = x and that x = h(x). (2) Consider the case x ∈ ( x , 1). Set x(0) = x, x(t + 1) = −ξ(x(t) − x) + x for all integers t ≥ 0. It is easy to see that (x(t), x(t + 1)) ∈ Ω for all integers t ≥ 0. Set y(t) = u(x(t), x(t + 1)) for all integers t ≥ 0. Then {x(t), y(t)}∞ t=0 is a full-employment program with δ(t) = 0 for all integers t ≥ 0 in view of (10.5). Proposition 10.4 implies that the program {x(t), y(t)}∞ t=0 is weakly optimal and x(1) = a −1 − ξ x ∈ h(x). Proposition 10.6 implies that if {x  (t), y  (t)}∞ t=0 is a weakly optimal program satisfying x  (0) = x, then δ  (t) = 0 for all integers t ≥ 0 and in view of (10.5), x) + x = a −1 − ξ x. x  (1) = −ξ(x − Thus h(x) = a −1 − ξ x.

10.6 Proofs

365

Consider the case x ∈ [1, k). Set x(0) = x, x(1) = (1 − d)x, x(t + 1) = −ξ(x(t) − x) + x for all integers t ≥ 1. It is easy to see that (x(t), x(t + 1)) ∈ Ω for all integers t ≥ 0. Set y(t) = u(x(t), x(t + 1)) for all integers t ≥ 0. Then {x(t), y(t)}∞ t=0 is a full-employment program with y(0) = 1, δ(t) = 0 for all integers t ≥ 1 in view of (10.5). If {x  (t), y  (t)}∞ t=0 is a program satisfying x  (0) = x, then x  (1) ≥ (1 − d)x, y  (1) ≤ 1 = y(t), δ  (0) ≥ α  (0) ≥ α(0) = δ(0). Proposition 10.4 implies that the program {x(t), y(t)}∞ t=0 is weakly optimal and x(1) = (1 − d)x ∈ h(x). Proposition 10.6 implies that if {x  (t), y  (t)}∞ t=0 is a weakly optimal program satisfying x  (0) = x, then δ  (0) = δ(0), δ  (t) = 0 for all integers t ≥ 1. Thus x  (1) = x(1) = (1 − d)x and h(x) = (1 − d)x. Corollary 10.10 is proved. Proof of Corollary 10.11 Consider the case x ∈ ( x , 1). Following the proof of Theorem 10.7, we obtain that [a −1 − ξ x, x ] ⊂ h(x).

366

10 An Autonomous One-Dimensional Model

Consider the case x ∈ [1, k). Following the proof of Corollary 10.10(2), we obtain that (1 − d)x ∈ h(x). Following the proof of Theorem 10.7(1), we obtain that x ∈ h(x). Since the function h is convex, we conclude that [(1 − d)x, x ] ⊂ h(x). Consider the case x ∈ C. Let T ≥ 0 be the smallest integer such that x¯ = (1 − d)T x < 1. Then T ≥ 1 and (1 − d)T x ≥ (1 − d). Set x(t) = (1 − d)t x, t = 0, . . . , T , x(t + 1) = 2 x − x(t) for all integers t ≥ T . It is easy to see that (x(t), x(t + 1)) ∈ Ω for all integers t ≥ 0. Set y(t) = u(x(t), x(t + 1)) for all integers t ≥ 0. Then {x(t), y(t)}∞ t=0 is a full-employment program with δ(t) = 0 for all integers  t ≥ T . If {x  (t), y  (t)}∞ t=0 is a program satisfying x (0) = x, then for all integers t = 0, . . . , T − 1 x  (t) ≥ x(t), y  (t) ≤ 1 = y(t), δ  (t) ≥ α  (t) ≥ α(t) = δ(t). Proposition 10.4 implies that the program {x(t), y(t)}∞ t=0 is weakly optimal and x(1) = (1 − d)x ∈ h(x). Consider the case x ∈ A. Set x¯ = a −1 − ξ x.

10.6 Proofs

367

Then x¯ ≥ x . If x¯ ∈ ( x , 1), define the program {x(t), y(t)}∞ t=0 as in the proof of Corollary 10.10(2), dealing with x ∈ ( x , 1). If x¯ ∈ [1, k), define the program {x(t), y(t)}∞ t=0 as in the proof of Corollary 10.10(2), dealing with x ∈ ([1, k). If x¯ ∈ C, define the program {x(t), y(t)}∞ t=0 as in the above paragraph. Then, in each case, consider the program {x  (t), y  (t)}∞ t=0 defined by (x  (0), y  (0)) = (x, x) and (x  (t), y  (t)) = (x(t − 1), y(t − 1)) for all integers t ≥ 1. It is not difficult to see that in each case this defines a weakly optimal program. Thus a −1 − ξ x ∈ h(x) in each case.

Chapter 11

The Continuous-Time Robinson–Solow–Srinivasan Model

In this chapter we study the continuous-time Robinson–Solow–Srinivasan model. We establish a convergence of good programs to the golden-rule stock, show the existence of overtaking optimal programs and analyze their convergence to the golden-rule stock, and consider some properties of good programs. We are also interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals.

11.1 Infinite Horizon Problems 1 ) be the set of real (nonnegative) numbers, and let R n be a finiteLet R 1 (R+ n = {x ∈ R n : x ≥ dimensional Euclidean space with nonnegative orthant R+ i  n 0, i = 1, . . . , n}. For any x, y ∈ R , let the inner product xy = ni=1 xi yi , and x y, x > y, x ≥ y have their usual meaning. Let e(i), i = 1, . . . , n be the ith n all of whose coordinates are unity. For unit vector in R n and e be an element of R+ n any x ∈ R , let ||x|| denote the Euclidean norm of x. Denote by mes(E) the Lebesgue measure of a Lebesgue measurable set E ⊂ R 1 . Let a = (a1 , . . . , an ) 0, b = (b1 , . . . , bn ) 0, b1 ≥ b2 · · · ≥ bn , d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. We suppose:

There exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci .

(11.1)

We now give a formal description of our technological structure. Set   n × R n : z + dx ≥ 0 and a(z + dx) ≤ 1 . Ω = (x, z) ∈ R+ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6_11

(11.2) 369

370

11 The Continuous-Time Robinson–Solow–Srinivasan Model

For every point (x, z) ∈ Ω, set   n : y ≤ x and ey ≤ 1 − a(z + dx) . Λ(x, z) = y ∈ R+

(11.3)

Let I be either [0, ∞) or [0, T ] with a positive number T . A pair of functions (x(·), y(·)) is called a program if x : I → R n is an absolutely continuous (a.c.) function on any finite subinterval of I , y : I → R n is a Lebesgue measurable function and if (x(t), x  (t)) ∈ Ω for almost every t ∈ I, 

y(t) ∈ Λ(x(t), x (t)) for almost every t ∈ I.

(11.4) (11.5)

In the sequel if I = [0, T ], then the program (x(·), y(·)) is denoted by (x(t), y(t))Tt=0 , and if I = [0, ∞), then the program (x(·), y(·)) is denoted by (x(t), y(t))∞ t=0 . Let w : [0, ∞) → [0, ∞) be a continuous strictly increasing concave and differentiable function which represents the preferences of the planner. For every point (x, z) ∈ Ω, set u(x, z) = max{w(by) : y ∈ Λ(x, z)}. n such that a point ( A golden-rule stock is a vector x ∈ R+ x , 0) is a solution to the problem: maximize u(x, z) subject to (i) z ≥ 0; (ii) (x, z) ∈ Ω. By Theorem 2.3, there exists a unique golden-rule stock

x = (1/(1 + daσ ))e(σ ).

(11.6)

It is easy see that x is a solution of the problem w(by) → max, y ∈ Λ( x , 0). Put y = x.

(11.7)

i = w  (b x ) qi . qi = ai bi (1 + dai )−1 , p

(11.8)

ξσ = 1 − d − 1/aσ .

(11.9)

For all integers i = 1, . . . , n, put

Set

11.1 Infinite Horizon Problems

371

By Lemma 2.2, w(b x ) ≥ w(by) + p z

(11.10)

for every (x, z) ∈ Ω and for every y ∈ Λ(x, z). We will prove the following three propositions obtained in [101]. Proposition 11.1 Let m0 be a positive number. Then there exists a positive number m1 such that for every positive number T and every program (x(t), y(t))Tt=0 which satisfies x(0) ≤ m0 e, the inequality x(t) ≤ m1 e is valid for every number t ∈ [0, T ]. We use the following notion of good programs. 1 A program (x(t), y(t))∞ t=0 is called good if there exists M ∈ R such that +

T

(w(by(t)) − w(b y ))dt ≥ M for all T ≥ 0.

0

A program is called bad if +

T

lim

T →∞ 0

(w(by(t)) − w(b y ))dt = −∞.

Proposition 11.2 Any program (x(t), y(t))∞ t=0 that is not good is bad. Proposition 11.3 For every point x0 (x(t), y(t))∞ t=0 satisfying x(0) = x0 .

n , there exists a good program ∈ R+

In the sequel we use a notion of an overtaking optimal program. ∞ A program (x(t), ˜ y(t)) ˜ t=0 is overtaking optimal if for every program ∞ (x(t), y(t))t=0 satisfying x(0) = x(0), ˜ the inequality ,+

T

lim sup T →∞

+

T

w(by(t))dt −

0

w(by(t))dt ˜ ≤0

0

holds. We prove the following two theorems obtained in [101]. Theorem 11.4 Assume that a program (x(t), y(t))∞ t=0 is good. Then x. (i) lim x(t) = t→∞

(ii) Let  ∈ (0, 1) and L > 1. Then there exists a positive number T0 such that for every number T ≥ T0 , mes([T , T + L] : y(t) − x  > ) ≤ .

372

11 The Continuous-Time Robinson–Solow–Srinivasan Model

n there exists an overtaking optimal program Theorem 11.5 For every x0 ∈ R+ ∞ (x(t), y(t))t=0 such that x(0) = x.

11.2 Proofs of Propositions 11.1–11.3 Proof of Proposition 11.1 Fix a number m1 > 0 such that m1 > 4m0 , min{ai : i = 1, . . . , n}m1 > 8m0

n 

ai ,

(11.11)

i=1

d min{ai : i = 1, . . . , n}m1 > 16.

(11.12)

Assume that T is a positive number and that (x(t), y(t))Tt=0 is a program which satisfies x(0) ≤ m0 e.

(11.13)

We claim that the inequality x(t) ≤ m1 e is valid for all numbers t ∈ [0, T ]. Assume the contrary. Then there exists a number t0 ∈ (0, T ] such that n . m1 e − x(t0 ) ∈ R+

(11.14)

a(x(t0 )) ≥ min{ai : i = 1, . . . , n}m1 .

(11.15)

In view of (11.14),

Equations (11.11), (11.13), and (11.15) imply that there exists a number t1 for which t1 ∈ (0, t0 ), a(x(t1 )) = 4−1 min{ai : i = 1, . . . , n}m1 ,

(11.16)

a(x(t)) ≥ 4−1 min{ai : i = 1, . . . , n}m1 for all t ∈ [t1 , t0 ].

(11.17)

Since (x(t), y(t))Tt=0 is a program for almost every t ∈ [0, T ], we have x  (t) + dx(t) ≥ 0, a(x  (t) + dx(t)) ≤ 1.

(11.18)

In view of (11.12), (11.17), and (11.18), for almost every t ∈ [t1 , t0 ], we have

11.2 Proofs of Propositions 11.1–11.3

373

ax  (t) ≤ 1 − adx(t) ≤ 1 − 4−1 d min{ai : i = 1, . . . , n}m1 < −1. Therefore the function ax(·) is decreasing on [t1 , t0 ]. On the other hand, it follows from (11.15) and (11.16) that ax(t0 ) > ax(t1 ). The contradiction we have reached completes the Proof of Proposition 11.1. For every point (x, z) ∈ Ω and every y ∈ Λ(x, z), define δ(x, y, z) = w(b y ) − w(by) − p z.

(11.19)

It follows from (11.10) and (11.19) that δ(x, y, z) ≥ 0 for every (x, z) ∈ Ω and every y ∈ Λ(x, z).

(11.20)

It is not difficult to prove the following result. Lemma 11.6 Let T be a positive number and let (x(t), y(t))Tt=0 be a program. Then +

T

+

T

(w(by(t)) − w(b y ))dt = −

0

δ(x(t), y(t), x  (t))dt − p (x(T ) − x(0)).

0

Proposition 11.2 now easily follows from Proposition 11.1 and Lemma 11.6. n be given. Fix a number h satisfying Proof of Proposition 11.3 Let x0 ∈ R+

h + (aσ d)−1 = (x0 )σ .

(11.21)

There exists a number T0 > 4 for which he−dT0 + (aσ d)−1 > (aσ d + 1)−1 .

(11.22)

Set xi (t) = (x0 )i e−dt , t ∈ [0, T0 ], i ∈ {1, . . . , n} \ {σ },

(11.23)

xσ (t) = he−dt + (aσ d)−1 , t ∈ [0, T0 ],

(11.24)

y(t) = 0, t ∈ [0, T0 ].

(11.25)

0 is a program In view of (11.4), (11.5), (11.21), and (11.23)–(11.25), (x(t), y(t))Tt=0 and

x(0) = x0 . By (11.22) and (11.24), we have

(11.26)

374

11 The Continuous-Time Robinson–Solow–Srinivasan Model

xσ (T0 ) > (aσ d + 1)−1 = xσ .

(11.27)

Define x(t), y(t) for all numbers t > T0 as follows: yi (t) = 0, xi (t) = (x0 )i e−dt for all t ∈ (T0 , ∞) and all i ∈ {1, . . . , n} \ {σ }, xσ , xσ (t) = e−d(t−T0 ) (xσ (T0 ) − (1 + daσ )−1 ) yσ (t) = + (1 + daσ )−1 , t ∈ (T0 , ∞). It is not difficult to see that (x(t), y(t))∞ t=0 is a good program. This completes the Proof of Proposition 11.3. Lemma 11.6 and Proposition 11.1 imply the following result. Proposition 11.7 A program (x(t), y(t))∞ t=0 is good if and only if +

∞

δ(x(t), y(t), x  (t))dt := lim

+

T →∞ 0

0

T

δ(x(t), y(t), x  (t))dt

is finite.

11.3 Auxiliary Results Proposition 11.8 Let T > 0, m0 > 0, and let {(x (i) (t), y (i) (t))Tt=0 }∞ i=1 be a sequence of programs satisfying x (i) (0) ≤ m0 e for all integers i ≥ 0.

(11.28)

Then there exist a program (x(t), y(t))Tt=0 and a strictly increasing sequence of natural numbers {ik }∞ k=1 such that x (ik ) (t) → x(t) as k → ∞ uniformly on [0, T ], 

x (ik ) → x  as k → ∞ weakly in L2 ([0, T ]; R n ),

(11.29)

y (ik ) → y as k → ∞ weakly in L2 ([0, T ]; R n ).

(11.31)

(11.30)

Proof In view of Proposition 11.1 and (11.28), there exists a positive number m1 such that x (i) (t) ≤ m1 e for all t ∈ [0, T ] and for all integers i ≥ 1. By (11.3), (11.5), and (11.32),

(11.32)

11.3 Auxiliary Results

375

y (i) (t) ≤ m1 e for all t ∈ [0, T ] and for all integers i ≥ 1.

(11.33)

By (11.2), (11.4), and (11.32), for all integers i ≥ 1 and for almost every t ∈ [0, T ], we have  (i)  (t) ≥ −dx (i) (t) ≥ −dm1 e, x   a x (i) (t) ≤ 1 − adx (i) (t) ≤ 1.

(11.34) (11.35)

In view of (11.35), there exists a positive number m2 such that for all integers i ≥ 1 and almost every t ∈ [0, T ], we have (x (i) ) (t) ≤ m2 .

(11.36)

Note that the space L2 ([0, T ]); R n ) is Hilbert. Equations (11.33) and (11.36) imply (i) ∞ 2 n that {(x (i) ) }∞ t=1 , {y }i=1 are bounded sequences in L ([0, T ]; R ). Therefore there ∞ exist a strictly increasing sequence of natural numbers {ik }k=1 , y ∈ L2 ([0, T ]; R n ) and u ∈ L2 ([0, T ]; R n ) such that  (i )  x k → u as k → ∞ in L2 ([0, T ]; R n ),

(11.37)

y (ik ) → y as k → ∞ in L2 ([0, T ]; R n )

(11.38)

in the weak topology, and there exists lim x (ik ) (0).

(11.39)

k→∞

For every number τ ∈ [0, T ], define +

τ

x(τ ) = lim x (ik ) (0) + k→∞

u(t)dt.

(11.40)

0

It follows from (11.37) and (11.40) that relation (11.29) holds. In order to complete the proof of the proposition, it is sufficient to show that (x(t), y(t))Tt=0 is a program. Let  ∈ (0, 1) be given. Fix number 0 > 0 such that  20 2n + 2 +

n 

 ai

< .

(11.41)

i=1

In view of (11.29), there exists an integer k0 ≥ 1 such that for every natural number k ≥ k0 , we have

376

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x (ik ) (t) − x(t) ≤ 0 , t ∈ [0, T ].

(11.42)

By (11.37), (11.38), and (11.40), (x  , y) is a limit point of the convex hull of the set 

   x (ik ) , y (ik ) : k is an integer and k ≥ k0

in the norm topology of the space L2 ([0, T ]; R n ) × L2 ([0, T ]; R n ). Therefore there exist an integer k1 > k0 and nonnegative numbers αk , k = k0 , . . . , k1 such that k1 

αk = 1,

(11.43)

k=k0

+

T

0

 αk x (ik ) (t) − x  (t)2 dt < 04 ,

k1 



(11.44)

k=k0

+

T



0

k1 

αk y (ik ) (t) − y(t)2 dt < 04 .

(11.45)

k=k0

Define E1 =

⎧ ⎨ ⎩

E2 =

k1 

t ∈ [0, T ] : 

⎧ ⎨ ⎩

k=k0

t ∈ [0, T ] : 

⎫ ⎬  (i )  αk x k (t) − x  (t) ≤ 0 , ⎭

k1 

αk y (ik ) (t) − y(t) ≤ 0

k=k0

(11.46)

⎫ ⎬ ⎭

.

(11.47)

It follows from (11.41) and (11.44)–(11.47) that + 02 mes([0, T ] \ E1 ) ≤

T



0

  αk x (ik ) (t) − x  (t)2 dt < 04 ,

(11.48)

k=k0

+ 02 mes([0, T ] \ E2 ) ≤

k1 

T 0



k1 

αk y (ik ) (t) − y(t)2 dt < 04

(11.49)

k=k0

and mes([0, T ] \ E1 ), mes([0, T ] \ E2 ) < 02 < 0 , mes([0, T ] \ (E1 ∩ E2 )) < 20 < .

(11.50)

11.3 Auxiliary Results

377

Equations (11.2)–(11.5) imply that for almost every t ∈ E1 ∩ E2 and all integers k = k0 , . . . , k1 , we have

   i   x k (t) + dx (ik ) (t) ≥ 0, a x (ik ) (t) + dx (ik ) (t) ≤ 1, 0 ≤ y (ik ) (t) ≤ x (ik ) (t),

   ey (ik ) (t) ≤ 1 − a x (ik ) (t) + dx (ik ) (t) .

(11.51) (11.52) (11.53)

In view of (11.42), (11.43), (11.46), and (11.51), for almost every t ∈ E1 ∩ E2 , we have x  (t) + dx(t) ≥

k1 

  αk x (ik ) (t) − 0 e

k=k0

⎛

+d⎝

k1 

⎞ αk (x ik )(t) − 0 e⎠ ≥ −(1 + d)0 e.

(11.54)

k=k0

It follows from (11.42), (11.44), and (11.51) that for almost every t ∈ E1 ∩ E2 , we have ⎡ ⎛ ⎞⎤ k1 k1      αk x (ik ) (t) + 0 e + d ⎝ αk x (ik ) (t) + 0 e⎠⎦ a(x  (t) + dx(t)) ≤ a ⎣ k=k0

=

k1 

k=k0

    αk a x (ik ) (t) + dx (ik ) (t) + a(0 e + d0 e)

k=k0

≤ 1 + 0 (1 + d)

n 

(11.55)

ai .

i=1

By (11.42), (11.43), (11.46), (11.47), and (11.52), for almost every t ∈ E1 ∩ E2 , we have −0 e ≤ −0 e +

k1  k=k0

≤ 0 e +

k1 

αk y (ik ) (t) ≤ y(t) ≤ 0 e +

k1 

αk y (ik ) (t)

k=k0

αk x (ik ) (t) ≤ 0 e + x(t) + 0 e = x(t) + 20 e.

(11.56)

k=k0

In view of (11.2), (11.4), (11.42), (11.43), (11.46), (11.47), and (11.53), for almost every t ∈ E1 ∩ E2 ,

378

11 The Continuous-Time Robinson–Solow–Srinivasan Model

⎛ ey(t) ≤ e ⎝

k1 

⎞ αk y (ik ) (t) + 0 e⎠ = 0 n +

k=k0

≤ 0 n +

k1 

k1 

αk ey (ik ) (t)

k=k0

    αk 1 − a x (ik ) (t) + dx (ik ) (t)

k=k0

= 0 n + 1 − a

k1 

k1    αk x (ik ) (t) − da αk x (ik ) (t) ≤ 0 n + 1

k=k0

k=k0



− a(x (t) − 0 e) − da(x(t) − 0 e) ≤ 1 + 0 n + 0 ae + d0 ae − a(x  (t) + dx(t)).

(11.57)

It follows from (11.41) and (11.54)–(11.57) that for almost every t ∈ E1 ∩ E2 , x  (t) + dx(t) ≥ −(1 + d)0 e ≥ −e, n    ai ≤ 1 + , a x  (t) + dx(t) ≤ 1 + 0 (1 + d)

(11.58) (11.59)

i=1

−e < −0 e ≤ y(t) ≤ x(t) + 20 e ≤ x(t) + e,   ey(t) ≤ 1 − a x  (t) + dx(t) + 0 (n + ae(1 + d))

(11.60)



≤ 1 − a(x (t) + dx(t)) + . Since  is an arbitrary number from the open interval (0, 1), it follows from (11.50) that for every integer k ≥ 1, there exists a Lebesgue measurable set Fk ⊂ [0, T ] such that mes([0, T ] \ Fk ) ≤ 2−k

(11.61)

and that for almost every t ∈ Fk , the following inequalities hold: x  (t) + dx(t) ≥ −2−k e,

(11.62)

a(x  (t) + dx(t)) ≤ 1 + 2−k ,

(11.63)

−2−k e ≤ y(t) ≤ x(t) + 2−k e,

(11.64)

ey(t) ≤ 1 − a(x  (t) + dx(t)) + 2−k e.

(11.65)

∞ F = ∪∞ j =1 ∩k=j Fk .

(11.66)

Define

11.3 Auxiliary Results

379

By (11.61) and (11.66), mes([0, T ] \ F ) = 0. In view of (11.62)–(11.66), for almost every t ∈ F , x  (t) + dx(t) ≥ 0, a(x  (t) + dx(t)) ≤ 1, 0 ≤ y(t) ≤ x(t), ey(t) ≤ 1 − a(x  (t) + dx(t)). These inequalities imply that (x(t), y(t))Tt=0 is a program. Proposition 11.8 is proved. Proposition 11.9 Let T > 0, m0 > 0, {(x (i) (t), y (i) (t))Tt=0 }∞ i=1 be a sequence of programs satisfying x (i) (0) ≤ m0 e for all integers i ≥ 0

(11.67)

and let (x(t), y(t))Tt=0 be a program such that x (i) (t) → x(t) as i → ∞ uniformly on [0, T ],  (i)  → x  as i → ∞ weakly in L2 ([0, T ]; R n ), x

(11.68)

y (i) → y as i → ∞ weakly in L2 ([0, T ]; R n ).

(11.70)

(11.69)

Then +

T

+ w(by(t))dt ≥ lim sup

0

i→∞

T

 w by (i) (t) dt.

0

Proof Proposition 11.1 and (11.67) imply that there exists a positive number m1 such that x (i) (t) ≤ m1 e for all t ∈ [0, T ] and all natural numbers i.

(11.71)

By (11.3), (11.5), and (11.71), y (i) (t) ≤ m1 e for almost all t ∈ [0, T ] and all natural numbers i. In view of (11.72), we have +

T

 w by (i) (t) dt ≤ T w(m1 be) for all natural numbers i.

0

We may assume without loss of generality that there exists

(11.72)

380

11 The Continuous-Time Robinson–Solow–Srinivasan Model

+

T

lim

i→∞ 0

  w by (i) (t) dt.

Let  ∈ (0, 1) be given. Fix a number 0 > 0 for which 0 < , 20 (w(m1 be) + 4) < /8,

(11.73)

|w(by) − w(by  )| ≤ (8(T + 1))−1

(11.74)

for each y, y  satisfying 0 ≤ y, y  ≤ m1 e and y − y   ≤ 20 . There exists an integer k0 ≥ 1 such that for every natural number k ≥ k0 , we have +

T

|

  w by (k) (t) dt − lim

+

T

i→∞ 0

0

  w by (i) (t) dt| ≤ 0 .

(11.75)

Equations (11.69) and (11.70) imply that (x  , y) is a limit point of the convex hull of the set      x (i) , y (i) : i is an integer and i ≥ k0 in the norm topology of the space L2 ([0, T ]; R n )×L2 ([0, T ]; R n ). This implies that there exist a natural number k1 > k0 and nonnegative numbers αk , k = k0 , . . . , k1 such that k1 

αk = 1,

(11.76)

k=k0

+

T 0



k1 

αk y (k) (t) − y(t)2 dt < 04 .

(11.77)

k=k0

Define E=

⎧ ⎨ ⎩

t ∈ [0, T ] : 

k1 

αk y (k) (t) − y(t) ≤ 0

k=k0

⎫ ⎬ ⎭

.

In view of (11.77) and (11.78), we have + 02 mes([0, T ] \ E) ≤ and

T 0



k1  k=k0

αk y (k) (t) − y(t)2 dt < 04

(11.78)

11.3 Auxiliary Results

381

mes([0, T ] \ E) < 02 .

(11.79)

By (11.72) and (11.79), we have for all integers k = k0 , . . . , k1 +

T

|

  w by (k) (t) dt −

+

0

  w by (k) (t) dt|

E

≤ mes([0, T ] \ E)w(m1 be) < 02 w(m1 be).

(11.80)

In view of (11.3), (11.5), (11.62), and (11.71), we have 0 ≤ y(t) ≤ x(t) ≤ m1 e for a.e. t ∈ [0, T ].

(11.81)

It follows from (11.79) and (11.81) that +

T

|

+ w(by(t))dt −

w(by(t))dt|

0

E

≤ mes([0, T ] \ E)w(m1 be) < 02 w(m1 be).

(11.82)

By (11.74), (11.78), and (11.81), for almost every t ∈ E, we have ⎛ ⎛ w(by(t)) ≥ w ⎝b ⎝

k1 

⎞⎞ αk y (k) (t)⎠⎠ − (8(T + 1))−1

k=k0

≥

k1 

  αk w by (k) (t) − (8(T + 1))−1 .

(11.83)

k=k0

Equations (11.73), (11.75), (11.79), (11.82), and (11.83) imply that +

T

+ w(by(t))dt ≥

0

E

≥

w(by(t))dt − 02 |w(0)|

−02 w(0) +

k1 

+ αk

k=k0

≥

−02 w(0) − /8 +

  w by (k) (t) dt − /8

E k1 

k=k0

+ αk

T 0

+

≥ −202 w(m1 be) − /8 + lim + ≥ lim

i→∞ 0

  w by (k) (t) dt − 02 w(m1 be)

i→∞ 0

T

  w by (i) (t) dt − .

T

  w by (i) (t) dt − 0 (11.84)

382

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Since  is an arbitrary number from the interval (0, 1), this completes the proof of Proposition 11.9. n be given. Set Let x0 ∈ R+

+

∞

Δ(x0 ) = inf 0

δ(x(t), y(t), x  (t))dt : (x(t), y(t))∞ t=0

is program and x(0) = x0 } .

(11.85)

Propositions 11.3 and 11.7 imply that Δ(x0 ) is well-defined and finite. n . Then there exists a program (x(t), y(t))∞ which Proposition 11.10 Let x0 ∈ R+ t=0 satisfies x(0) = x0 and

+

∞

δ(x(t), y(t), x  (t))dt = Δ(x0 ).

0

Proof For every integer k ≥ 1, there exists a program (x (k) (t), y (k) (t))∞ t=0 such that +

∞

x (k) (0) = x0 ,

   δ x (k) (t), y (k) (t), x (k) (t) dt ≤ Δ(x0 ) + 1/k.

(11.86)

0

In view of Proposition 11.2, extracting a subsequence and re-indexing and using a diagonalization process, we may assume without loss of generality that there exists a program (x(t), y(t))∞ t=0 such that for every integer T ≥ 1, we have x (k) (t) → x(t) as k → ∞ uniformly on [0, T ],  (k)  → x  as k → ∞ weakly in L2 ([0, T ]; R n ), x

(11.88)

y (k) → y as k → ∞ weakly in L2 ([0, T ]; R n ).

(11.89)

(11.87)

Let T ≥ 1 be an integer. Proposition 11.9 and (11.87)–(11.89) imply that +

T

+

T

w(by(t))dt ≥ lim sup

0

k→∞

  w by (k) (t) dt.

0

Lemma 11.6, (11.86), (11.87), and (11.90) imply that +

T

δ(x(t), y(t), x  (t))dt

0

+

T

=− 0

(w(by(t)) − w(b y ))dt + p (x(0) − x(T ))

(11.90)

11.4 Proofs of Theorems 11.4 and 11.5

+

T

≤ − lim sup 0

k→∞

,+

T

= lim inf k→∞

+

= lim inf k→∞

+

≤ lim

k→∞ 0

383

    w by (k) (t) dt + T w(b y ) + lim p x (k) (0) − x (k) (T ) k→∞





w(b y ) − w by (k) (t) dt + p x (k) (0) − x (k) (T )

0 T

0 ∞

   δ x (k) (t), y (k) (t), x (k) (t) dt

   δ x (k) (t), y (k) (t), x (k) (t) dt = Δ(x0 ).

Since the relation above is valid for every integer T ≥ 1, we conclude that +

∞

δ(x(t), y(t), x  (t))dt ≤ Δ(x0 ).

0

This completes the Proof of Proposition 11.10.

11.4 Proofs of Theorems 11.4 and 11.5 The next auxiliary result easily follows from Lemma 2.13. Lemma 11.11 The von Neumann facet {(x, z) ∈ Ω : there is y ∈ Λ(x, z) such that δ(x, y, z) = 0} is a subset of {(x, z) ∈ Ω : xi = zi = 0 for all i ∈ {1, . . . , n} \ {σ }, zσ = (1/aσ ) + (ξσ − 1)xσ } with the equality if the function w is linear. If the function w is strictly concave, then the facet is the singleton {( x , 0)}. Lemma 11.12 Let {(x(t), y(t))}∞ t=0 be a good program and Ti > 2i for all natural numbers i. Let   (i) x (t), y (i) (t) = (x(t + Ti ), y(t + Ti )), t ∈ [−i, i]

(11.91)

for every integer i ≥ 1. Then there exist a strictly increasing sequence of natural numbers {ik }∞ k=1 , a locally a.c. function x˜ : R 1 → R n , and a Lebesgue measurable function y˜ : R 1 → R n such that for each natural number j ,

384

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x (ik ) (t) → x(t) ˜ as i → ∞ uniformly on [−j, j ],  (i )  x k → x˜  as k → ∞ weakly in L2 ([−j, j ]; R n ),

(11.92)

y (ik ) → y˜ as k → ∞ weakly in L2 ([−j, j ]; R n ).

(11.94)

(11.93)

Moreover, 0 ≤ y(t) ˜ ≤ x(t) ˜ for a.e. t ∈ R 1 , x˜  (t) + d x(t) ˜ ≥ 0 for a.e. t ∈ R 1 ,   ey(t) ˜ + a x˜  (t) + d x(t) ˜ ≤ 1 for a.e. t ∈ R 1

(11.95) (11.96) (11.97)

and   δ x(t), ˜ y(t), ˜ x˜  (t) = 0 for a.e. t ∈ R 1 .

(11.98)

Proof By Propositions 11.1 and 11.8 and a diagonalization process, there exist a strictly increasing sequence of natural numbers {ik }∞ k=1 , a locally a.c. function x˜ : 1 n R → R , and a Lebesgue measurable function y˜ : R 1 → R n such that (11.92)– (11.97) are valid for all integers j ≥ 1. By (11.19), (11.92)–(11.97), Proposition 11.9, Lemma 11.6, (11.91), and Proposition 11.7, for every integer j ≥ 1, we have +

j −j

δ(x(t), ˜ y(t), ˜ x˜  (t))dt

=

+

  w(b y ) − w(by(t)) ˜ −p (x˜  (t)) dt

j −j

+

=−

j −j

  w(by(t))dt ˜ + 2j w(b y) + p x(j ˜ ) − x(−j ˜ ) +

≤ − lim sup k→∞

+

≤ lim inf k→∞

j

−j

+ = lim inf k→∞

+ = lim

j

−j

j −j

k→∞

      w(b y ) − w by (ik ) (t) − p x (ik ) (t) dt

   δ x (ik ) (t), y (ik ) (t), x (ik ) (t) dt

Tk +j

k→∞ Tk −j

    w by (ik ) (t) dt + 2j w(b y ) − lim p x (ik ) (j ) − x (ik ) (−j )

δ(x(t), y(t), x  (t))dt = 0.

Since this relation holds for every integer j ≥ 1, we conclude that

11.4 Proofs of Theorems 11.4 and 11.5

385

δ(x(t), y(t), x  (t)) = 0 for a.e. t ∈ R 1 . This completes the proof of Lemma 11.12. Lemma 11.13 Let x : R 1 → R n be a locally a.c. function and y : R 1 → R n be a Lebesgue measurable function such that (x(t), x  (t)) ∈ Ω for a.e. t ∈ R 1 , y(t) ∈ Λ(x(t), x  (t)) for a.e. t ∈ R 1 ,   sup ||x(t)|| : t ∈ R 1 < ∞,   δ x(t), y(t), x  (t) = 0 for a.e. t ∈ R 1 .

(11.99) (11.100) (11.101) (11.102)

x for almost every t ∈ R 1 . Then x(t) = x , t ∈ R 1 , y(t) = Proof Lemma 11.11, (11.99), (11.100), and (11.102) imply that for almost every t ∈ R 1 , we have xi (t) = xi (t) = 0 for all i ∈ {1, . . . , n} \ {σ },

(11.103)

xσ (t) = aσ−1 + (ξσ − 1)xσ (t). In view of (11.103), xσ (t) = ce(ξσ −1)t − (aσ (ξσ − 1))−1 , t ∈ R 1

(11.104)

where c is a constant. Note that in view of (11.9), ξσ = 1. By (11.101) and (11.104), c = 0. Combined with (11.9) and (11.104), this implies that x(t) = x for all t ∈ R 1 .

(11.105)

It follows from (11.19), (11.102), and (11.105) that for almost every t ∈ R 1 , w(by(t)) = w(b x ). Since w is strictly increasing, it follows from this equality, (11.100), and (11.105) that y(t) = x for almost every t ∈ R 1 . This completes the proof of Lemma 11.13.

386

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Proof of Theorem 11.4 By Proposition 11.1, there exists a positive number m1 such that x(t) ≤ m1 e for all t ≥ 0. First we show that x(t) → x as t → ∞.

(11.106)

Assume that {Ti }∞ i=1 is a strictly increasing sequence of positive numbers and that h = lim x(Ti ). i→∞

(11.107)

In order to prove (11.106), it is sufficient to show that h = x . We may assume without loss of generality that Ti > 2i for all integers i ≥ 1. Let x (i) , y (i) be defined by (11.91) for all natural numbers i ≥ 1. Lemma 11.12 implies that there exist a strictly increasing sequence of natural numbers {ik }∞ k=1 , a locally a.c. function x˜ : R 1 → R n , and a Lebesgue measurable function y˜ : R 1 → R n such that (11.92)–(11.98) are valid. In view of (11.91), (11.92), and (11.107), we have ˜ h = lim x(Tik ) = lim x (ik ) (0) = x(0). k→∞

k→∞

(11.108)

By (11.92)–(11.98), the bondedness of {x(t) : t ∈ [0, ∞)}, (11.92) x(t) ˜ = x for all t ∈ R 1 and in view of (11.108) h = x(0) ˜ = x. x. Therefore we have shown that limt→∞ x(t) = Now we prove (ii). We may assume that  < (8(daσ + 1))−1 . Fix a positive number γ such that    w(b x) > w b x − n−1 e(σ ) + 4γ

(11.109)

and a positive number 0 such that 0 <  and γ −1 L2 0 < .

(11.110)

There exists a number 1 ∈ (0, ) such that 1 n < γ and 21 < 0 /n,

(11.111)

11.4 Proofs of Theorems 11.4 and 11.5

w(b( x + 1 e)) − w(b x ) ≤ 0 ,  (2)   (1)  n − w bz | ≤ γ for each z(1) , z(2) ∈ R+ |w bz

387

(11.112) (11.113)

such that xσ + 1)e, z(1) − z(2)  ≤ 41 n. z(1) , z(2) ≤ 4( Since the program (x(t), y(t))∞ t=0 is good, we conclude that +

∞

δ(x(t), y(t), x  (t))dt < ∞.

0

Combined with assertion (i), the inequality above implies that there exists a positive number T0 such that +

∞

δ(x(t), y(t), x  (t))dt < 1 ,

(11.114)

2(1 +  p)x(t) − x  ≤ 1 for all t ≥ T0 .

(11.115)

T0

Assume that T ≥ T0 . Then by (11.19), (11.20), and (11.114), we have +

T +L

1 > +

δ(x(t), y(t), x  (t))dt

T T +L

=

[w(b y ) − w(by(t))]dt − p (x(T + L) − x(T )).

T

Combined with (11.115) this implies that +

T +L

[w(b y ) − w(by(t))]dt < 21 .

(11.116)

T

Define E1 = {t ∈ [T , T + L] :  x − y(t) ≤ }, E2 = [T , T + L] \ E1 .

(11.117)

In view of (11.112) and (11.115), for a.e. t ∈ [T , T + L], we have 0 ≤ y(t) ≤ x(t) ≤ x + 1 e, w(by(t)) ≤ w(b( x + 1 e)) ≤ w(b y ) + 0 .

(11.118) (11.119)

Let t ∈ E2 be given. Equation (11.117) implies that there exists j ∈ {1, . . . , n} such that

388

11 The Continuous-Time Robinson–Solow–Srinivasan Model

| xj − yj | > /n. If i ∈ {1, . . . , n} \ {σ }, then in view of (11.111) and (11.118), we have | xi − yi (t)| = |0 − yi (t)| ≤ 1 < /n. Therefore j = σ and | xσ − yσ (t)| > /n. Combined with (11.118) this implies that xσ − /n for a.e. t ∈ E2 . yσ (t) <

(11.120)

By (11.118) and (11.120), for a.e. t ∈ E2 , y(t) − yσ (t)e(σ ) ≤ 1 n, and in view of the choice of 1 (see (11.113)), (11.109), (11.118), and (11.120), we have xσ − /n)e(σ )) + γ w(by(t)) ≤ γ + w(byσ (t)e(σ )) < w(b( < γ + w(b y ) − 4γ < w(b y ) − 3γ .

(11.121)

It follows from (11.116), (11.117), (11.119), and (11.121) that +

T +L

21 > +

[w(b y ) − w(by(t))]dt

T

+ [w(b y ) − w(by(t))]dt +

= E1

[w(b y ) − w(by(t))]dt E2

≥ −0 L + 3γ mes(E2 ). Combined with (11.110) and (11.111), this relation implies that mes(E2 ) ≤ (3γ )−1 (21 + L0 ) ≤ (3γ )−1 (2L2 0 ) ≤ γ −1 L2 0 < . Assertion (ii) is proved. This completes the Proof of Theorem 11.4. n be given. Proposition 11.10 implies that there Proof of Theorem 11.5 Let x0 ∈ R+ ∞ exists a program (x(t), ˜ y(t)) ˜ t=0 such that

+

∞

x(0) ˜ = x0 , 0

δ(x(t), ˜ y(t), ˜ x˜  (t))dt = Δ(x0 ).

(11.122)

11.5 Turnpike Results

389

∞ is a good program. We Proposition 11.7 and (11.122) imply that (x(t), ˜ y(t)) ˜ t=0 ∞ claim that (x(t), ˜ y(t)) ˜ t=0 is an overtaking optimal program. Let (x(t), y(t))∞ t=0 be a program such that

x(0) = x0 . If (x(t), y(t))∞ t=0 is bad, then +

T

[w(by(t)) ˜ − w(by(t))]dt

0

+

T

=

+

T

[w(by(t)) ˜ − w(b y )]dt +

0

[w(b y ) − w(by(t))]dt → ∞ as T → ∞.

0

Assume that the program (x(t), y(t))∞ t=0 is good. Theorem 11.4 implies that x(t), x(t) ˜ → x as t → ∞. Together with (11.19), Proposition 11.7, and (11.122), this implies that +

T

lim sup T →∞



0

 w(by(t)) − w(by(t)) ˜ dt +

= lim sup T →∞

+ =

∞

T

  δ(x(t), ˜ y(t), ˜ x˜  (t)) − δ(x(t), y(t), x  (t)) − p (x  (t) − x˜  (t)) dt

0 

+

∞

δ(x(t), ˜ y(t), ˜ x˜ (t))dt −

0

δ(x(t), y(t), x  (t))dt

0

˜ ) + x(0)) ˜ −p lim (x(T ) − x(0) − x(T + =

T →∞

∞

δ(x(t), ˜ y(t), ˜ x˜  (t))dt −

0

+

∞

δ(x(t), y(t), x  (t))dt ≤ 0.

0

Theorem 11.5 is proved.

11.5 Turnpike Results n and T > 0 be given. Define Let z ∈ R+

+ U (z, T ) = sup

T

w(by(t))dt :

0

 (x(t), y(t))Tt=0 is a program such that x(0) = z .

(11.123)

390

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Proposition 11.1, Theorem 11.5, (11.3), (11.5), and (11.123) imply that U (z, T ) is a finite number. n and let 0 ≤ T < T . Define Let x0 , x1 ∈ R+ 1 2 + U (x0 , x1 , T1 , T2 ) = sup

T2

w(by(t))dt :

T1 2 is a program such that (x(t), y(t))Tt=T 1

(11.124)

x(T1 ) = x0 , x(T2 ) ≥ x1 } . Here we assume that supremum over empty set is −∞. In view of Proposition 11.1, (11.3), (11.5), and (11.124), U (x0 , x1 , T1 , T2 ) < ∞. It is not difficult to see that for n and every positive number T , U (z, T ) = U (z, 0, 0, T ). every point z ∈ R+ We prove the following two theorems, obtained in [110], which describe the structure of approximate optimal solutions of optimal control problems on sufficiently large intervals. Theorem 11.14 Let M, , L be positive numbers and let Γ ∈ (0, 1). Then there n exist T∗ > 0 and a positive number γ such that for each T > 2T∗ , each z0 , z1 ∈ R+ T −1 satisfying z0 ≤ Me and az1 ≤ Γ d , and each program (x(t), y(t))t=0 which satisfies +

T

x(0) = z0 , x(T ) ≥ z1 ,

w(by(t))dt ≥ U (z0 , z1 , 0, T ) − γ

0

there are numbers τ1 , τ2 such that τ1 ∈ [0, T∗ ], τ2 ∈ [T − T∗ , T ], x(t) − x  ≤  for all t ∈ [τ1 , τ2 ] and that for each number S satisfying τ1 ≤ S ≤ τ2 − L, mes({t ∈ [S, S + L] : y(t) − x  > }) ≤ . x  ≤ γ , then τ2 = T . Moreover, if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − Theorem 11.15 Let M0 , M1 ,  be positive numbers, L > 1, and let Γ ∈ (0, 1). Then there exist T∗ > L, a natural number Q, and l > 0 such that for each T > n satisfying z ≤ Me and az ≤ Γ d −1 , and each program T∗ , each z0 , z1 ∈ R+ 0 1 T (x(t), y(t))t=0 which satisfies + x(0) = z0 , x(T ) ≥ z1 ,

T

w(by(t))dt ≥ U (z0 , z1 , 0, T ) − M1

0

there exists a finite sequence of closed intervals [Si , Si ], i = 1, . . . , q such that q ≤ Q, Si − Si ≤ l, i = 1, . . . , q, Si ≤ Si+1 for each integer i satisfying 1 ≤ i ≤ q − 1,

11.6 Auxiliary Results

391

x(t) − x  ≤ , t ∈ [0, T ] \ ∪i=1 [Si , Si ] q

and if S ∈ [0, T − L] satisfies [S, S + L] ⊂ [Si , Si+1 ] with 1 ≤ i < q, then mes({t ∈ [S, S + L] : y(t) − x  > }) ≤ .

11.6 Auxiliary Results Lemma 11.16 Let Γ ∈ (0, 1). Then there exists a number k(Γ ) > 0 such that n and each z ∈ R n satisfying az ≤ Γ d −1 , there is a program for each z0 ∈ R+ 1 1 + ∞ (x(t), y(t))t=0 such that x(0) = z0 and x(t) ≥ z1 for all t ≥ k(Γ ). Proof There exists a positive number k(Γ ) such that 1 − e−dk(Γ ) > Γ.

(11.125)

n and z ∈ R n satisfies Assume that z0 ∈ R+ 1 +

az1 ≤ Γ d −1 .

(11.126)

z2 = Γ −1 z1 ,

(11.127)

Set

x(t) = e−dt (z0 − z2 ) + z2 , y(t) = 0, t ∈ [0, ∞),

(11.128)

In view of (11.128), for all t ≥ 0, we have x(t) ≥ 0, x  (t) + dx(t) = −de−dt (z0 − z2 ) + de−dt (z0 − z2 ) + dz2 = dz2 .

(11.129) (11.130)

By (11.126), (11.127), and (11.130), for all t ≥ 0, x  (t) + dx(t) ≥ 0,

  a(x  (t) + dx(t)) = adz2 = ad Γ −1 z1 ≤ 1. Combined with (11.2), (11.5), (11.128), and (11.129), these inequalities imply that (x(t), y(t))∞ t=0 is a program. In view (11.128), we have

392

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x(0) = z0 .

(11.131)

By (11.125)–(11.128), for all t ≥ k(Γ ),       x(t) ≥ 1 − e−dt z2 ≥ 1 − e−dk(Γ ) z2 ≥ Γ −1 1 − e−dk(γ ) z1 ≥ z1 . This completes the proof of Lemma 11.16. In the sequel with each Γ ∈ (0, 1), we associate a number k(Γ ) > 0 for which the assertion of Lemma 11.16 holds. n and Lemma 11.17 There exists a positive number m such that for every z ∈ R+ every positive number T ,

U (z, T ) ≥ T w(b x ) − m.

(11.132)

Proof In view of (11.6), we have xσ = aσ (1 + daσ )−1 < d −1 . a x = aσ

(11.133)

It follows from (11.133) that there exists a number Γ ∈ (0, 1) such that a x ≤ Γ d −1 .

(11.134)

Fix a number m > k(Γ )[|w(0)| + |w(k(Γ ))| + |w(b x )|]. n be given. In view of (11.134), the choice of k(Γ ), and Lemma 11.17, Let z ∈ R+ k(Γ ) there exists a program (x(t), y(t))t=0 such that

x(0) = z, x(k(Γ )) ≥ x.

(11.135)

Let T > 0 be given. We claim that (11.132) is valid. There are cases T ≤ k(Γ )

(11.136)

T > k(Γ ).

(11.137)

and

Assume that (11.136) is valid. Then by (11.135), (11.136), and the choice of m, we have

11.6 Auxiliary Results

+

393 T

U (z, T ) ≥

w(by(t))dt ≥ T w(0) ≥ T (−|w(0)|) ≥ −k(Γ )|w(0)|

0

= T w(b x ) + [−k(Γ )|w(0)| − T w(b x )] ≥ T w(b x ) − k(Γ )|w(0)| − k(Γ )|w(b x )| ≥ T w(b x) − m and (11.132) is true. Assume that (11.137) is valid. For all numbers t > k(Γ ), define x(t) = x + e−d(t−k(Γ )) (x(k(Γ )) − x ), y(t) = x.

(11.138)

In view of (11.135) and (11.138), for all t ∈ (k(Γ ), ∞), we have 0 ≤ y(t) ≤ x(t).

(11.139)

By (11.1), (11.138), for all t ∈ (k(Γ ), ∞), x = d(1 + daσ )−1 eσ , x  (t) + dx(t) = d 



a(x (t) + dx(t)) ≤ 1, (x(t), x (t)) ∈ Ω.

(11.140) (11.141)

It follows from (11.138), (11.140), for all t ∈ (k(Γ ), ∞) x + (1 + daσ )−1 a(x  (t) + dx(t)) + ey(t) = ad = aσ d(1 + daσ )−1 + (1 + daσ )−1 = 1 and together with (11.5), (11.139), and (11.141), this implies that y(t) ∈ Λ(x(t), x  (t)). Hence we have shown that (x(t), y(t))∞ t=0 is a program. In view of (11.135), (11.137), (11.138), and the choice of m, +

T

U (z, T ) ≥ 0

+ w(by(t))dt =

k(Γ )

w(by(t))dt + (T − k(Γ ))w(b x)

0

≥ T w(b x ) + k(Γ )(w(0) − w(b x )) ≥ T w(b x ) − m. Hence (11.132) is valid. This completes the proof of Lemma 11.17. Lemma 11.18 Let Γ ∈ (0, 1). Then there exists a positive number m such that for n , every z ∈ R n satisfying az ≤ Γ d −1 , and every T > k(Γ ), every z0 ∈ R+ 1 1 + U (z0 , z1 , 0, T ) ≥ T w(b x ) − m.

394

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Proof Lemma 11.17 implies that there exists a positive number m0 such that n and every positive number T . U (z, T ) ≥ T w(b x ) − m0 for every z ∈ R+ (11.142)

Set m = m0 + 1 + k(Γ )(w(b x ) − w(0)).

(11.143)

Assume that n , az1 ≤ Γ d −1 , T > k(Γ ). z 0 , z 1 ∈ R+

(11.144)

In view of the choice of m0 (see (11.142)) and (11.144), there exists a program −k(Γ ) (x(t), y(t))Tt=0 such that x(0) = z0 ,

+

T −k(Γ )

w(by(t))dt ≥ U (z0 , T − k(Γ )) − 1 ≥ (T − k(Γ ))w(b x ) − m0 − 1.

0

(11.145) It follows from the choice of k(Γ ), Lemma 11.16, and (11.144) that there exists a program (x(t), y(t))Tt=T −k(Γ ) such that x(T ) ≥ z1 .

(11.146)

It is clear that (x(t), y(t))Tt=0 is a program. By (11.143), (11.145), and (11.146), we have + U (z0 , z1 , 0, T ) ≥ 0

T

+

T −k(Γ )

w(by(t))dt =

+ w(by(t))dt +

0

T T −k(Γ )

w(by(t))dt

≥ (T − k(Γ ))w(b x ) − m0 − 1 + k(Γ )w(0) = T w(b x ) − k(Γ )(w(b x ) − w(0)) − m0 − 1 = T w(b x ) − m. This completes the proof of Lemma 11.18.

11.6 Auxiliary Results

395

Lemma 11.19 Let m0 be a positive number. Then there exists a positive number m2 such that for every positive number T and every program (x(t), y(t))Tt=0 ) which satisfies x(0) ≤ m0 e, the inequality +

T

[w(by(t)) − w(b x )]dt ≤ m2

0

is valid. Proof Proposition 11.1 implies that there exists a positive number m1 such that for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ m0 e, the inequality x(t) ≤ m1 e for all t ∈ [0, T ].

(11.147)

p m1 n. m2 > 2

(11.148)

Fix

Assume that T > 0 and that a program (x(t), y(t))Tt=0 satisfies x(0) ≤ m0 e. Then inequality (11.147) is valid. Lemma 11.6, (11.20), (11.147), and (11.148) imply that +

T

(w(by(t)) − w(b x ))dt ≤ − p (x(T ) − x(0)) ≤ 2|| p||nm1 ≤ m2 .

0

This completes the proof of Lemma 11.19. It is not difficult to see that the following auxiliary result holds. Lemma 11.20 Assume that nonnegative numbers T1 , T2 satisfy T1 < T2 , 2 (x(t), y(t))Tt=T 1

n . Then (x(t) + e−d(t−T1 ) u, y(t))T2 ) is also a is a program and that u ∈ R+ t=T1 program.

In order to prove Lemma 11.20, it is sufficient to note that for a.e. t ∈ [T1 , T2 ],



 x(t) + e−d(t−T1 ) u + d x(t) + e−d(t−T1 ) u = x  (t) + dx(t).

Lemma 11.20 implies the following result. n , and let Lemma 11.21 Let 0 ≤ T1 < T2 , M > 0, x0 , x1 ∈ R+ 2 (x(t), y(t))Tt=T 1

396

11 The Continuous-Time Robinson–Solow–Srinivasan Model

be a program such that + x(T1 ) = x0 , x(T2 ) ≥ x1 ,

T2

w(by(t))dt ≥ U (x0 , x1 , T1 , T2 ) − M.

T1

Then for each pair of numbers S1 , S2 satisfying T1 ≤ S1 < S2 ≤ T2 the following inequality holds: +

S2

w(by(t))dt ≥ U (x(S1 ), x(S2 ), S1 , S2 ) − M.

S1

Lemma 11.22 Let  be a positive number. Then there exists a positive number δ n satisfying such that for every pair of points z, z ∈ R+ x ≤ δ z − x , z −

(11.149)

and every T ∈ [2−1 , 2], there exists a program (x(t), y(t))Tt=0 such that x(0) = z, x(T ) ≥ z , x(t) − x , |y(t) − x  ≤ , t ∈ [0, T ], x  (t) ≤ , t ∈ [0, T ]. Proof We may assume without loss of generality that  < (1 + daσ )−1 .

(11.150)

ai < (/16), 16δn < .

(11.151)

Fix δ > 0 such that δ

n  i=1

n satisfies (11.149). Assume that T ∈ [2−1 , 2] and that a pair of points z, z ∈ R+ For all numbers t ∈ [0, T ], set

  y(t) = (1 + daσ )−1 −  e(σ ).

(11.152)

y(t) − x  ≤ , t ∈ [0, T ].

(11.153)

Evidently,

Set

11.6 Auxiliary Results

397

ξ = 4δe and define x(t) = z + tξ, t ∈ [0, T ].

(11.154)

In view of (11.149)–(11.154) and the choice of ξ , 0 ≤ y(t) ≤ z ≤ x(t) for all t ∈ [0, T ].

(11.155)

By (11.154) and the choice of ξ , for all t ∈ [0, T ], we have x  (t) + dx(t) = ξ + d(z + tξ ) ≥ 0.

(11.156)

It follows from (11.149) and (11.156) that for all t ∈ [0, T ], a(x  (t) + dx(t)) = adz + (1 + dt)aξ ≤ ad x + δd

n 

ai + (1 + 2d)aξ.

i=1

Combined with (11.6) and (11.151)–(11.153), this implies that for all t ∈ [0, T ], ey(t) + a(x  (t) + dx(t)) ≤ (1 + daσ )−1 −  + aσ d(1 + daσ )−1 + δ

n 

ai + 3aξ

i=1

=1−+δ

n 

ai + 3aξ ≤ 1.

i=1

By the relation above, (11.115), (11.152) (11.154), and (11.156), (x(t), y(t))Tt=0 is a program. By (11.149), (11.153), (11.154), and the choice of ξ , x + δe ≥ z. x(0) = z, x(T ) = z + T ξ ≥ x − δe + 2−1 ξ ≥ In view of (11.149), (11.151), and (11.154), for all t ∈ [0, T ], x(t) − x  ≤ z − x  + T ξ | ≤ δ + 2ξ  ≤ δ + 8δn ≤ 16δn < . Lemma 11.22 is proved. Lemma 11.23 Let m0 , m1 ,  be positive numbers. Then there exists an integer τ ≥ 1 such that for every program (x(t), y(t))τt=0 satisfying + x(0) ≤ m0 e, 0

τ

w(by(t))dt ≥ τ w(b x ) − m1

398

11 The Continuous-Time Robinson–Solow–Srinivasan Model

there exists a number t ∈ [0, τ ] such that x(t) − x  ≤ . Proof Assume the contrary. Then for every integer k ≥ 1, there exists a program (x (k) (t), y (k) (t))kt=0 such that + x

(k)

k

(0) ≤ m0 e,

w(by (k) (t))dt ≥ kw(b x ) − m1 ,

0

x (k) (t) − x  >  for all numbers t ∈ [0, k].

(11.157)

By (11.157) and Proposition 11.1, there exists a number m2 > m0 such that for every integer k ≥ 1, x (k) (t) ≤ m2 e, t ∈ [0, k].

(11.158)

Lemma 11.19 implies that there exists a positive number m3 such that for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ m2 e we have +

T

(w(by(t)) − w(b x ))dt ≤ m3 .

(11.159)

0

Let k ≥ 1 be an integer and let a number s satisfy 0 < s < k. By (11.158) and the choice of m3 (see (11.159)), +

k

  w(by (k) (t)) − w(b x ) dt ≤ m3 .

s

Together with (11.158) the relation above implies that +

s

+   w(by (k) (t)) − w(b x ) dt =

0

  w(by (k) (t)) − w(b x ) dt

k

0

+

−

k

  w(by (k) (t)) − w(b x ) dt ≥ −m1 − m3 .

s

Therefore for every integer k ≥ 1 and every s ∈ (0, k), + 0

s

  w(by (k) (t)) − w(b x ) dt ≥ −m1 − m3 .

(11.160)

11.6 Auxiliary Results

399

By extracting a subsequence and using (11.158), Proposition 11.8, and diagonalization process, we obtain that there exist a strictly increasing sequence of natural ∞ ∗ ∗ numbers {kj }∞ j =1 and a program (x (t), y (t))t=0 such that for every integer q ≥ 1, we have x (kj ) (t) → x ∗ (t) as j → ∞ uniformly on [0, q],  (k )  x j (t) → (x ∗ ) (t) as j → ∞ weakly in L2 ([0, q]; R n ),

(11.161)

y (kj ) → y ∗ as j → ∞ weakly in L2 ([0, q]; R n ).

(11.163)

(11.162)

By (11.158) and (11.161), x ∗ (t) ≤ m2 e for all t ≥ 0.

(11.164)

It follows from (11.158), (11.160)–(11.163), and Proposition 11.9 that for all integers q ≥ 1, +

q

w(by ∗ (t))dt ≥ lim sup j →∞

0

+

q

  w by kj (t) dt ≥ qw(b x )) − m3 − m1 .

0

Combined with Proposition 11.3, the relation above implies that (x ∗ (t), y ∗ (t))∞ t=0 is a good program. Theorem 11.4 implies that lim x ∗ (t) = x.

t→∞

(11.165)

On the other hand, it follows from (11.157) and (11.161) that x || ≥ , t ∈ [0, ∞). ||x ∗ (t) − This contradicts (11.165). The contradiction we have reached completes the proof of Lemma 11.23. Lemma 11.24 Let  be a positive number. Then there exists a positive number γ such that for every number T > 2 and every program (x(t), y(t))Tt=0 satisfying +

x(0) − x  ≤ γ , x(T ) − x ≤ γ , T

w(by(t))dt ≥ U (x(0), x(T ), 0, T ) − γ

0

the following inequality holds:

(11.166) (11.167)

400

11 The Continuous-Time Robinson–Solow–Srinivasan Model

+

T

δ(x(t), y(t), x  (t))dt ≤ .

(11.168)

0

Proof Fix a number 0 > 0 such that p  + 1)−1 , 0 < (/18)( n if y ∈ R+ and y − x  ≤ 0 , then |w(b x ) − w(by)| < /16.

(11.169) (11.170)

Lemma 11.22 implies that there exists a number γ ∈ (0, 0 ) such that the following property holds: n which satisfy (P1) for every T ∈ [2−1 , 2] and every pair of points z, z ∈ R+ x ≤ γ z − x , z − there exists a program (u(t), v(t))Tt=0 such that x , v(t) − x  ≤ 0 , t ∈ [0, T ], u(0) = z, u(T ) ≥ z , u(t) − u (t) ≤ 0 , t ∈ [0, T ]. Assume that T > 2 and that a program (x(t), y(t))Tt=0 satisfies (11.166) and (11.167). In view of (11.166) and property (P1), there exist programs 

u(1) (t), v (1) (t)

1 t=0

 T and u(2) (t), v (2) (t) t=T −1

such that x , u(1) (t) − x , v (1) (t) − x  ≤ 0 , t ∈ [0, 1], u(1) (0) = x(0), u(1) (1) ≥ (11.171)  (1)  (t) ≤ 0 , t ∈ [0, 1],  u x , u(2) (T ) ≥ x(T ), u(2) (t) − x , v (2) (t) − x  ≤ 0 , u(2) (T − 1) =  (2)  (t)|| ≤ 0 , t ∈ [T − 1, T ]. (11.172) || u T Define a program (x(t), ¯ y(t)) ¯ t=0 as follows. Set

x(t) ¯ = u(1) (t), y(t) ¯ = v (1) (t), t ∈ [0, 1], (11.173)   x , y(t) ¯ = x , t ∈ (1, T − 1]. x(t) ¯ = x + e−d(t−1) u(1) (1) − T −1 In view of Lemma 11.20, (11.171), and (11.173), (x(t), ¯ y(t)) ¯ t=0 is a program. By (11.171) and (11.173), we have

11.6 Auxiliary Results

401

x(T ¯ − 1) ≥ x = u(2) (T − 1).

(11.174)

For t ∈ (T − 1, T ] set   ¯ − 1) − u(2) (T − 1) , y(t) ¯ = v (2) (t). x(t) ¯ = u(2) (t) + e−d(t−(T −1)) x(T (11.175) T Lemma 11.20, (11.174), and (11.175) imply that (x(t), ¯ y(t)) ¯ t=0 is a program. By (11.172)–(11.175) we have

x(0) ¯ = x(0), x(T ¯ ) ≥ u(2) (T ) ≥ x(T ).

(11.176)

It follows from (11.167) and (11.176) that +

T

−γ ≤

+

T

w(by(t))dt −

0

w(by(t))dt. ¯

(11.177)

0

Equations (11.171)–(11.173) and (11.175) imply that x x(T ¯ ) − x  ≤ x(T ¯ ) − u(2) (T ) + u(2) (T ) − ¯ − 1) − x  + 0 ≤ x(T ¯ − 1) − u(2) (T − 1) + 0 = x(T ≤ u(1) (1) − x  + 0 ≤ 20 . In view of Lemma 11.6 and (11.177), +

T

−γ ≤ −

+

T

(w(by(t)) ¯ − w(b y ))dt −

0

δ(x(t), y(t), x  (t))dt − p (x(T ) − x(0)).

0

Combined with (11.166), (11.169)–(11.173), and (11.175), the relation above implies that +

T

δ(x(t), y(t), x  (t))dt ≤ γ −

0

+

T

(w(by(t)) ¯ − w(b x ))dt +  px(T ) − x(0)

0

+ ≤γ − + −

1

(w(by(t)) ¯ − w(b x ))dt

0 T T −1

(w(by(t)) ¯ − w(b x ))dt +  p20

≤ γ + /16 + /16 + /8 < . This completes the proof of Lemma 11.24.

402

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Lemma 11.25 Let  > 0 and τ0 > 0. Then there exist γ > 0 and T0 > τ0 such that for every number T ≥ T0 and every program (x(t), y(t))Tt=0 which satisfies + x(0) − x , x(T ) − x ≤ γ ,

T

δ(x(t), y(t), x  (t))dt ≤ γ

0

the following properties hold: x(t) − x  ≤  for all t ∈ [0, T ]; for every S ∈ [0, T − τ0 ], x  > }) ≤ . mes({t ∈ [S, S + τ0 ] : y(t) − Proof We may assume that  < 1. Fix 0 ∈ (0, 16−1 ).

(11.178)

In view of Lemma 11.22 and the continuity of the function δ(·, ·, ·), there exists a sequence of positive numbers {γq }∞ q=1 such that γq ≤ 4−1 γq−1 for all natural numbers q ≥ 2, −q

γq ≤ 4

0 for all natural numbers q.

(11.179) (11.180)

and that for every natural number q, the following property holds: n satisfying (P2) for every pair of points z, z ∈ R+ x  ≤ γq z − x , z − there exists a program (x(t), y(t))1t=0 such that x , y(t) − x x(0) = z, x(1) ≥ z , x(t) − ≤ 4−q 0 , t ∈ [0, 1], x  (t) ≤ 4−q 0 , t ∈ [0, 1],

+

1

δ(x(t), y(t), x  (t))dt ≤ 4−q 0 .

0

Assume that the lemma does not hold. Then for every integer q ≥ 1, there exist Tq ≥ τ0 + q T

q such that and a program (x (q) (t), y (q) (t))t=0

(11.181)

11.6 Auxiliary Results

+ 0

403

x (q) (0) − x , x (q) (Tq ) − x  ≤ γq , Tq

(11.182)

  δ x (q) (t), y (q) (t), (x (q) ) (t) dt ≤ γq

and that at least one of the following properties holds:   x  : t ∈ [0, Tq ] > ; sup x (q) (t) −

(11.183)

(P3) there exists S ∈ [0, Tq − τ0 ] such that   x  >  > . mes t ∈ [S, S + τ0 ] : y (q) (t) −

(11.184)

Extracting a subsequence and re-indexing, we may assume without loss of generality that one of the following cases holds: Equation (11.183) is valid for all integers q ≥ 1; (P3) holds for all integers q ≥ 1. Property (P2), (11.79), and (11.182) imply that for every integer q ≥ 1, there exists a program (u(q (t), v (q) (t))1t=0 such that u(q) (0) = x (q) (Tq ), u(q) (1) ≥ x (q+1) (0),

(11.185)

x , v (q) (t) − x  ≤ 4−q 0 , t ∈ [0, 1], u(q) (t) −    u(q) (t) ≤ 4−q 0 , t ∈ [0, 1], + 1

 

δ u(q) (t), v (q) (t), u(q) (t) dt ≤ 4−q 0 .

(11.186)

(11.187)

0

We construct a program (x(t), ¯ y(t) ¯ ∞ t=0 by induction. Put x(t) ¯ = x (1) (t), y(t) ¯ = y (1) (t), t ∈ [0, T1 ].

(11.188)

Assume that q is a natural number and that we have already defined a program q

i=1 (x(t), ¯ y(t)) ¯ t=0

Ti +q−1

such that x¯

 q 

 Ti + q − 1 ≥ x (q) (Tq ).

(11.189)

i=1

q q (Evidently, for q = 1 our assumption holds.) For t ∈ ( i=1 Ti +q −1, i=1 Ti +q] set

404

11 The Continuous-Time Robinson–Solow–Srinivasan Model

 x(t) ¯ =u

t−

(q)

 q 

 + e−d(t−(

Ti + q − 1

q

i=1 Ti +q−1))

i=1

 q    (q) × x¯ Ti + q − 1 − x (Tq ) , 

i=1

y(t) ¯ = v (q) t −

 q 

(11.190)

 Ti + q − 1

.

i=1

It follows from (11.185), (11.189), (11.190), and Lemma 11.20 that q

i=1 (x(t), ¯ y(t)) ¯ t=0

Ti +q

is a program, x¯

 q 

 Ti + q

=u

(q)

(1) + e

−d

 

 q  (q) Ti + q − 1 − x (Tq ) x¯

i=1

i=1

≥ u(q) (1) ≥ x (q+1) (0). q

For every number t ∈ (

i=1 Ti

+ q,

 x(t) ¯ =x

(q+1)

t−

q+1

 q 

i=1

(11.191)

Ti + q], define  + e−d(t−(

Ti + q

q

i=1 Ti +q))

i=1

 q    (q+1) × x¯ Ti + q − x (0) , i=1

 y(t) ¯ =y

(q+1)

t−

 q 

 Ti + q

(11.192)

.

i=1 q+1

It follows from (11.191), (11.192), and Lemma 11.20 that (x(t), ¯ y(t)) ¯ 0 program and that ⎛ ⎞ q+1  x¯ ⎝ Ti + q ⎠ ≥ x (q+1) (Tq+1 ).

i=1

Ti +q

is a

(11.193)

i=0 ∞ has been constructed by induction. Hence the program (x(t), ¯ y(t)) ¯ t=0 It follows from (11.190), (11.186) (with q = 1), and (11.188) that for every t ∈ [T1 , T1 + 1],

11.6 Auxiliary Results

405

x(t) ¯ − x  ≤ x(t) ¯ − u(1) (t − T1 ) + u(1) (t − T1 ) − x ≤ x(T ¯ 1 ) − x (1) (T1 ) + 4−1 0 = 4−1 0 .

(11.194)

We show by induction that for every integer q ≥ 1, we have  x¯ t +

q 



 T i + q − 1 − Tq

− x (q) (t) ≤ 2

i=1

x(t) ¯ − x ≤

q 

 4−i 0 , t ∈ [0, Tq ],

i=1 q 

2 · 4−i 0 , t ∈

q 

i=1

Ti + q − 1,

i=1

q 

 Ti + q .

(11.195) (11.196)

i=1

In view of (11.188) and (11.194), Equations (11.195) and (11.196) are valid for q = 1. Assume that q ≥ 1 is an integer and that (11.195) and (11.196) are true. For every number t ∈ [0, Tq+1 ], it follows from (11.179), (11.180), (11.182), (11.192), and (11.196) that  x¯ t +

q 

 Ti + q − x

(q+1)

(t) ≤ x¯

i=1

 q 

 Ti + q − x (q+1) (0)

i=1

≤ x¯

 q 

 Ti + q − x  +  x − x (q+1) (0)

i=1

≤ x¯

 q 

 Ti + q − x  + 4−q−1 0

i=1

≤

q 

2 · 4−i 0 + 4−q−1 0 .

(11.197)

i=1

In view of (11.186) and (11.190) (which holds for every integer q ≥ 1), (11.196), q+1 q+1 and (11.197), for every number t ∈ [ i=1 Ti + q, i=1 Ti + q + 1], ⎛

⎛

x(t) ¯ − x  ≤ x(t) ¯ − u(q+1) ⎝t − ⎝

q+1  i=1

⎞⎞ Ti + q ⎠ ⎠ 

⎞⎞ ⎛ q+1  Ti + q ⎠ ⎠ − x + u(q+1) ⎝t − ⎝ ⎛

i=1

406

11 The Continuous-Time Robinson–Solow–Srinivasan Model

⎛ ≤ x¯ ⎝

q+1 

⎞ Ti + q ⎠ − x (q+1) (Tq+1 ) + 4−q−1 0

i=1

≤ x¯

 q 

 −q−1

Ti + q − x + 2 · 4

0 ≤

i=1

q+1 

2 · 4−i 0 .

i=1

Thus we have shown by induction that (11.195) and (11.196) are true for every integer q ≥ 1. ∞ is a good program. In view of Proposition 11.2, in We show that (x(t), ¯ y(t)) ¯ t=0 order to meet this goal, it is sufficient to show that +

T

(w(by(t)) ¯ − w(b y ))dt

0

does not tend to −∞ as T → ∞. Lemma 11.6, (11.180), and (11.182) imply that for every integer q ≥ 1, +

Tq

  (q)   w by (t) − w(b y ) dt ≥ −

0

+

Tq

    δ x (q) (t), y (q) (t), x (q) (t) dt

0

  −  p x (q) (0) − x (q) (Tq ) ≥ −γq − 2 p γq = −γq (1 + 2 p) p )0 ≥ −4−q (1 + 2

(11.198)

and in view of Lemma 11.6 and (11.186), we have +

1

   w bv (q) (t) − w(b y ) dt ≥ −

0

+

1

    δ u(q) (t), v (q) (t), u(q) (t) dt

0

  −  p u(q) (0) − u(q) (1) p ≥ −4−q 0 (1 + 2 p ). ≥ −4−q 0 − 2 · 4−q (0 ) (11.199)

∞ By (11.198), (11.199), and the construction of the program (x(t), ¯ y(t)) ¯ t=0 (see (11.188)–(11.192)), for every natural number q, q

+

i=1 Ti +q−1

0

(w(by(t)) ¯ − w(b x ))dt ≥

q 

−4−i 0 (2 + 4 p ) ≥ −(2 + 4 p )20 .

i=1

∞ is a good program. By Theorem 11.4, Thus (x(t), ¯ y(t)) ¯ t=0

11.7 Proof of Theorem 11.4

407

lim x(t) ¯ = x

t→∞

and there exists a positive number S0 such that x(t) ¯ − x  ≤ 0 for every t ≥ S0 .

(11.200)

It follows from (11.195), (11.200), and (11.178), which is true for every natural number q, that x (q) (t) − x  ≤ , t ∈ [0, Tq ] for all sufficiently large natural numbers q. (11.201) In view of Theorem 11.4, there exists a positive number S1 such that for every number T ≥ S1 , we have ¯ − x  > }) ≤ . mes([T , T + τ0 ] : y(t)

(11.202)

By (11.202), (11.192) (which is true for every integer q ≥ 1), and (11.181), for all sufficiently large natural numbers q and for all numbers S ∈ [0, Tq − τ0 ], we have x  > } ≤ . mes{t ∈ [S, S + τ0 ] : y (q) (t) − This contradicts (P3), while (11.201) contradicts (11.183). The contradiction we have reached proves Lemma 11.25.

11.7 Proof of Theorem 11.4 Proposition 11.1 implies that there exists a positive number M1 such that for every T > 0 and every program (x(t), y(t))Tt=0 which satisfies x(0) ≤ Me, the following inequality is valid: x(t) ≤ M1 e for all t ∈ [0, T ].

(11.203)

Lemma 11.18 implies that there exists a positive number M2 such that for every n , every z ∈ R n satisfying az ≤ Γ d −1 , and every T > k(Γ ), we have z 0 ∈ R+ 1 1 + x ) − M2 . U (z0 , z1 , 0, T ) ≥ T w(b

(11.204)

Lemma 11.19 implies that there exists a positive number M3 such that for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ M1 e, the following inequality is valid:

408

11 The Continuous-Time Robinson–Solow–Srinivasan Model

+

T

[w(by(t)) − w(b y )]dt ≤ M3 .

(11.205)

0

Lemma 11.25 implies that there exist numbers 1 > 0, L1 > L such that for every number T ≥ L1 and every program (x(t), y(t))Tt=0 which satisfies x  ≤ 1 , x(0) − x  ≤ 1 , x(T ) − + T δ(x(t), y(t), x  (t))dt ≤ 1 0

the following properties hold: x(t) − x  ≤  for all t ∈ [0, T ];

(11.206)

for every S ∈ [0, T − L], mes({t ∈ [S, S + L] : ||y(t) − x  > }) ≤ .

(11.207)

Lemma 11.24 implies that there exists a number γ ∈ (0, min{1, , 1 })

(11.208)

such that for every T > 2 and every program (x(t), y(t))Tt=0 satisfying +

x(0) − x  ≤ γ , x(T ) − x ≤ γ , T

w(by(t))dt ≥ U (x(0), x(T ), 0, T ) − γ

(11.209)

0

the following inequality is valid: +

T

δ(x(t), y(t), x  (t))dt ≤ 1 .

(11.210)

0

Lemma 11.23 implies that there exists an integer L2 ≥ 1 such that for every program 2 (x(t), y(t))L t=0

which satisfies +

L2

x(0) ≤ M1 e,

w(by(t))dt ≥ L2 w(b x ) − M2 − M3 − 1

0

there exists a number t ∈ [0, L2 ] for which

(11.211)

11.7 Proof of Theorem 11.4

409

x(t) − x ≤ γ .

(11.212)

Define p nM2 ). l = 2L2 + 2L1 + L, Q > 41−1 (M3 + M2 + 2

(11.213)

Set T∗ = L2 + L1 + k(Γ ) + 2 + Ql.

(11.214)

Assume that n , z0 ≤ Me, az1 ≤ Γ d −1 T > 2T∗ , z0 , z1 ∈ R+

(11.215)

and that a program (x(t), y(t))Tt=0 satisfies +

T

x(0) = z0 , x(T ) ≥ z1 ,

w(by(t))dt ≥ U (z0 , z1 , 0, T ) − γ .

(11.216)

0

By (11.215) and (11.216), relation (11.203) is valid. In view of (11.208), (11.214)– (11.216), and the choice of M2 (see (11.204)), we have +

T

w(by(t))dt ≥ U (z0 , z1 , 0, T ) − γ ≥ T w(b x ) − M2 − 1.

(11.217)

0

By the choice of M3 (see (11.205)) and (11.203), +

T

+

T −L2

[w(by(t)) − w(b x )]dt ≤ M3 ,

[w(by(t)) − w(b x )]dt ≤ M3 .

0

L2

(11.218) Equations (11.217) and (11.218) imply that + +

L2

[w(by(t)) − w(b x )]dt ≥ −M2 − 1 − M3 ,

(11.219)

0 T

T −L2

[w(by(t)) − w(b x )]dt ≥ −M2 − 1 − M3 .

(11.220)

By (11.203), (11.219), (11.220), and the choice of L2 (see (11.211) and (11.212)), there exist τ1 ∈ [0, L2 ], τ2 ∈ [T − L2 , T ] such that

(11.221)

410

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x(τi ) − x  ≤ γ , i = 1, 2.

(11.222)

If x(0) − x || ≤ γ , then we set τ1 = 0, and if x(T ) − x  ≤ γ , then we set τ2 = T . In view of (11.216) and Lemma 11.21, we have +

τ2

w(by(t))dt ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − γ .

(11.223)

τ1

It follows from (11.222), (11.223), and the choice of γ (see (11.208)–(11.210)) that +

τ2

δ(x(t), y(t), x  (t))dt ≤ 1 .

(11.224)

τ1

By (11.215), (11.221), (11.222), (11.224), and the choice of 1 , L2 (see (11.206)– (11.208)), x(t) − x  ≤ , t ∈ [τ1 , τ2 ] and if a number S satisfies τ1 ≤ S ≤ τ2 − L, then mes({t ∈ [S, S + L] : y(t) − x  > }) ≤ . This completes the proof of Theorem 11.4.

11.8 Proof of Theorem 11.5 We may assume that  < 1/4. Proposition 11.1 implies that there exists a positive number M2 such that for every T > 0 and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ M0 e, the following inequality is valid: x(t) ≤ M2 e for all t ∈ [0, T ].

(11.225)

Lemma 11.18 implies that there exists a positive number M3 such that for every n , every z ∈ R n satisfying az ≤ Γ d −1 , and every number T > k(Γ ), we z0 ∈ R+ 1 1 + have x ) − M3 . U (z0 , z1 , 0, T ) ≥ T w(b

(11.226)

In view of Lemma 11.25, there exist 1 ∈ (0, ), L1 > L such that for every number T ≥ L1 and every program (x(t), y(t))Tt=0 satisfying x  ≤ 1 , x(0) − x  ≤ 1 , x(T ) −

(11.227)

11.8 Proof of Theorem 11.5

411

+

T

δ(x(t), y(t), x  (t))dt ≤ 21 ,

0

the inequality x(t) − x  ≤ , t ∈ [0, T ]

(11.228)

is true and for every S ∈ [0, T − L], we have mes({t ∈ [S, S + L] : y(t) − x  > }) ≤ .

(11.229)

Lemma 11.23 implies that there exists an integer L2 ≥ 1 such that for every program 2 (x(t), y(t))L t=0

satisfying + x(0) ≤ M2 e,

L2

w(by(t))dt ≥ L2 w(b x ) − M1 − M3 − 2 − 4 p nM2

0

(11.230) there exists t ∈ [0, L2 ] such that x(t) − x  ≤ 1 .

(11.231)

Fix p nM2 ), l = 2L2 + 2L1 + 8, a natural number Q > 41−1 (M3 + M1 + M2 + 2 (11.232) T∗ > 8L + 8L1 + 8L2 + k(Γ ). Assume that n , z0 ≤ Me, az1 ≤ Γ d −1 T > T ∗ , z 0 , z 1 ∈ R+

(11.233)

and that a program (x(t), y(t))Tt=0 satisfies +

T

x(0) = z0 , x(T ) ≥ z1 ,

w(by(t))dt ≥ U (z0 , z1 , 0, T ) − M1 .

(11.234)

0

By (11.233), (11.234), and the choice of M2 , relation (11.225) is valid. In view of (11.232)–(11.234) and the choice of M3 (see (11.226)),

412

11 The Continuous-Time Robinson–Solow–Srinivasan Model

+

T

w(by(t))dt ≥ T w(b x ) − M3 − M1 .

(11.235)

0

Lemma 11.6 and (11.235) imply that +

T

δ(x(t), y(t), x  (t))dt

0

+

T

=

(w(b x ) − w(by(t))dt + p (x(0) − x(T )) ≤ M3 + M1 + 2 p nM2 .

0

(11.236) q

It is easy to see that there exists a finite sequence of numbers {Ti }i=0 such that T0 = 0, Ti < Ti+1 for each integer i satisfying 0 ≤ i < q, Tq = T , for every integer i satisfying 0 ≤ i < q +

Ti+1

δ(x(t), y(t), x  (t))dt = 1 ,

(11.237)

δ(x(t), y(t), x  (t))dt ≤ 1 .

(11.238)

Ti

+

Tq

Tq−1

In view of (11.236)–(11.238), p nM2 q1 ≤ M3 + M1 + 2 and p nM2 ). q ≤ 1−1 (M3 + M1 + 2

(11.239)

Lemma 11.6, (11.225), (11.237), and (11.238) imply that for every i ∈ {0, . . . , q − 1} and every pair of numbers S1 , S2 ∈ [Ti , Ti+1 ] satisfying S1 < S2 , +

S2

(w(by(t)) − w(b x ))dt

S1

+

=−

S2

δ(x(t), y(t), x  (t))dt + p (x(S1 ) − x(S2 )) ≥ −1 − 2 p nM2 .

S1

(11.240) Define J = {i ∈ {0, . . . , q − 1} : Ti+1 − Ti ≥ 2L2 + 2L1 }.

(11.241)

11.9 Stability of the Turnpike Phenomenon

413

Let i ∈ J . In view of (11.241), the choice of L2 (see (11.230) and (11.231)), (11.225), and (11.240), there exist numbers ti1 , ti2 such that x  ≤ 1 , j = 1, 2. ti1 ∈ [Ti , L2 + Ti ], ti2 ∈ [Ti+1 − L2 , Ti+1 ], x(tij ) − (11.242) It follows from (11.237), (11.238), (11.241), (11.242), and the choice of 1 , L1 (see (11.227)–(11.229)) that x(t) − x  ≤ , t ∈ [ti1 , ti2 ]

(11.243)

and if S ∈ [ti1 , ti2 − L], then mes({t ∈ [S, S + L] : y(t) − x  > }) ≤ .

(11.244)

Set A = {[Ti , Ti+1 ] : i ∈ {0, . . . , q − 1} \ J } ∪ {[Ti , ti1 ], [ti2 , Ti+1 ] : i ∈ J }. (11.245) Evidently, the length of all the intervals belonging to A does not exceed 2L2 + 2L1 < l. In view of (11.232), (11.239), and (11.245), the number of elements of A does not exceed 4q ≤ 41−1 (M3 + M1 + 2 p nM2 ) ≤ Q. The inequalities above and (11.243) imply the validity of Theorem 11.5.

11.9 Stability of the Turnpike Phenomenon In this chapter we prove the following turnpike result obtained in [116]. Theorem 11.26 Let M, , L1 be positive numbers and Γ ∈ (0, 1). Then there exist a positive number L and a positive number γ such that for each T > 2L, n satisfying z ≤ Me and az ≤ Γ d −1 , and each program each z0 , z1 ∈ R+ 0 1 T (x(t), y(t))t=0 which satisfies + τ

x(0) = z0 , x(T ) ≥ z1 , τ +L

w(by(t))dt ≥ U (x(τ ), x(τ + L), 0, L) − γ for all τ ∈ [0, T − L]

414

11 The Continuous-Time Robinson–Solow–Srinivasan Model

and +

T

T −L

w(by(t))dt ≥ U (x(T − L), z1 , 0, L) − γ

there are real numbers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x  ≤  for all t ∈ [τ1 , τ2 ] and that for each number S satisfying τ1 ≤ S ≤ τ2 − L1 , x  > }) ≤ . mes({t ∈ [S, S + L1 ] : y(t) − x  ≤ γ , then τ2 = T . Moreover, if x(0) − x  ≤ γ , then τ1 = 0, and if x(T ) − n → R 1 is called L × BLet −∞ < T1 < T2 < ∞. A function φ : [T1 , T2 ] × R+ measurable if it is measurable with respect to the σ -algebra generated by products of Lebesgue subsets of [T1 , T2 ] and Borel subsets of R n . n → R 1 , set For each M > 0 and each function φ : R+ n and z ≤ Me}. φM = sup{|φ(z)| : z ∈ R+

(11.246)

n → R 1 be an Let numbers T1 , T2 satisfy 0 ≤ T1 < T2 , and let W : [T1 , T2 ] × R+ n. L × B-measurable function which is bounded on bounded subsets of [T1 , T2 ] × R+ n For each z0 , z1 ∈ R+ set

+ U (z0 , z1 , T1 , T2 , W ) = sup

T2

W (t, y(t))dt :

T1

 2 (x(t), y(t))Tt=T , is a program such that x(T ) = z , x(T ) ≥ z 1 0 2 1 1 +

U (z0 , T1 , T2 , W ) = sup

T2

(11.247)

W (t, y(t))dt :

T1

 2 (x(t), y(t))Tt=T is a program such that x(T ) = z 1 0 . 1

(11.248)

(Here we assume that supremum over the empty set is −∞.) Theorem 11.27 Let M, L1 ,  be positive numbers and Γ ∈ (0, 1). Then there exist M0 > M, L > 0, δ > 0 such that for each T1 ≥ 0, each T2 > T1 + 2L, each n which satisfy z ≤ Me and az ≤ Γ d −1 , each L × B-measurable z0 , z 1 ∈ R + 0 1 n → R 1 which is bounded on bounded subsets of function W : [T1 , T2 ] × R+ n and such that for almost every t ∈ [T , T ], [T1 , T2 ] × R+ 1 2 W (t, ·) − w(b(·))M0 ≤ δ

11.9 Stability of the Turnpike Phenomenon

415

2 and each program (x(t), y(t))Tt=T which satisfies 1

+

x(T1 ) = z0 , x(T2 ) ≥ z1 , τ +L

W (t, y(t))dt ≥ U (x(τ ), x(τ + L), τ, τ + L, W ) − δ

τ

for all τ ∈ [T1 , T2 − L] and +

T2

T2 −L

W (t, y(t))dt ≥ U (x(T2 − L), z1 , T2 − L, T2 , W ) − δ

there are real numbers τ1 , τ2 such that τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ], x(t) − x  ≤  for all t ∈ [τ1 , τ2 ] and that for each number S satisfying τ1 ≤ S ≤ τ2 − L1 , x  > }) ≤ . mes({t ∈ [S, S + L1 ] : y(t) − x  ≤ δ, then τ1 = T1 , and if x(T2 ) − x  ≤ δ, then τ2 = T2 . Moreover, if x(T1 ) − Proof Theorem 11.27 follows easily from Theorem 11.26. Namely, let L > 0 and γ > 0 be as guaranteed by Theorem 11.26. Proposition 11.1 implies that there exists a positive number M0 > 0 such that for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ Me, the inequality x(t) ≤ M0 e is true for all numbers t ∈ [0, T ]. Set  −1 δ = γ 4−1 (2L + 1) . Now it is easy to see that the assertion of Theorem 11.27 holds. Theorem 11.27 and Lemma 11.21 imply the following result. Theorem 11.28 Let M, L1 ,  be positive numbers and Γ ∈ (0, 1). Then there exist M0 > M, L > 0, δ > 0 such that for each T1 ≥ 0, each T2 > T1 + 2L, each n which satisfy z ≤ Me and az ≤ Γ d −1 , each L × B- measurable z0 , z 1 ∈ R + 0 1 n → R 1 which is bounded on bounded subsets of function W : [T1 , T2 ] × R+ n and such that for almost every t ∈ [T , T ], [T1 , T2 ] × R+ 1 2 W (t, ·) − w(b(·))M0 ≤ δ 2 and each program (x(t), y(t))Tt=T which satisfies 1

416

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x(T1 ) = z0 , x(T2 ) ≥ z1 , + T2 W (t, y(t))dt ≥ U (z0 , z1 , T1 , T2 , W ) − δ T1

there are real numbers τ1 , τ2 such that τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ], x(t) − x  ≤  for all t ∈ [τ1 , τ2 ] and that for each number S satisfying τ1 ≤ S ≤ τ2 − L1 , x  > }) ≤ . mes({t ∈ [S, S + L1 ] : y(t) − x  ≤ δ, then τ1 = T1 , and if x(T2 ) − x  ≤ δ, then τ2 = T2 . Moreover, if x(T1 ) − The following theorem is proved in Section 11.12. It was obtained in [116]. Theorem 11.29 Let M, M1 , L1 ,  be positive numbers and Γ ∈ (0, 1). Then there exist M∗ > 0, l > 0, δ > 0, a natural number Q such that for each T1 ≥ 0, each n which satisfy z ≤ Me and az ≤ Γ d −1 , each T2 > T1 + Ql, each z0 , z1 ∈ R+ 0 1 n → R 1 which is bounded on bounded L×B-measurable function W : [T1 , T2 ]×R+ n and such that for almost every t ∈ [T , T ], subsets of [T1 , T2 ] × R+ 1 2 W (t, ·) − w(b(·))M∗ ≤ δ 2 which satisfies and each program (x(t), y(t))Tt=T 1

+

x(T1 ) = z0 , x(T2 ) ≥ z1 , T2

W (t, y(t))dt ≥ U (z0 , z1 , T1 , T2 , W ) − M1

T1

there exist a natural number q ≤ Q and monotone increasing finite sequences q

q

{ai }i=1 , {bi }i=1 ⊂ [T1 , T2 ] such that 0 ≤ bi − ai ≤ l, i = 1, . . . , q, bi ≤ ai+1 for each integer i satisfying 1 ≤ i ≤ q − 1, x(t) − x  ≤ , t ∈ and if a number S satisfies

q [T1 , T2 ] \ ∪i=1 [ai , bi ]

(11.249) (11.250) (11.251)

11.10 Discount Case

417 q

[S, S + L1 ] ⊂ [T1 , T2 ] \ ∪i=1 [ai , bi ],

(11.252)

then x  > }) < . mes({t ∈ [S, S + L1 ] : y(t) −

(11.253)

11.10 Discount Case Theorem 11.27 and Proposition 11.1 imply the following result. Theorem 11.30 Let M, L1 ,  be positive numbers and Γ ∈ (0, 1). Then there exist M0 > 0, L > 0, γ > 0, λ > 1 such that for each T1 ≥ 0, each T2 > T1 + 2L, each n which satisfy z ≤ Me and az ≤ Γ d −1 , each L × B-measurable z0 , z1 ∈ R+ 0 1 n → R 1 which is bounded on bounded subsets of function W : [T1 , T2 ] × R+ n and such that for almost every t ∈ [T , T ], [T1 , T2 ] × R+ 1 2 W (t, ·) − w(b(·))M0 ≤ γ , each Lebesgue measurable function α : [T1 , T2 ] → (0, 1] such that for each t1 , t2 ∈ [T1 , T2 ] satisfying |t2 − t1 | ≤ L the inequality α(t1 )α(t2 )−1 ≤ λ holds and each 2 which satisfies program (x(t), y(t))Tt=T 1 +

x(T1 ) = z0 , x(T2 ) ≥ z1 , τ +L

α(t)W (t, y(t))dt ≥ U (x(τ ), x(τ + L), τ, τ + L, αW ) − γ α(τ )

τ

for all τ ∈ [T1 , T2 − L] and +

T2 T2 −L

α(t)W (t, y(t))dt ≥ U (x(T2 − L), z1 , T2 − L, T2 , αW ) − γ α(T2 − L)

there are real numbers τ1 , τ2 such that τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ], x(t) − x  ≤  for all t ∈ [τ1 , τ2 ] and that for each number S satisfying τ1 ≤ S ≤ τ2 − L1 , x  > }) ≤ . mes({t ∈ [S, S + L1 ] : y(t) − Moreover, if x(T1 ) − x  ≤ δ, then τ1 = T1 , and if x(T2 ) − x  ≤ δ, then τ2 = T2 .

418

11 The Continuous-Time Robinson–Solow–Srinivasan Model

11.11 Proof of Theorem 11.26 Proposition 11.1 implies that there exists a positive number M0 such that the following property holds: (P1) for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ Me, the inequality x(t) ≤ M0 e is valid for all t ∈ [0, T ]. Note that   a x = aσ 1 + daσ−1 < d −1 . Therefore we may assume without loss of generality that a x < Γ d −1 , sup{ay : y ∈ R n and y − x  ≤ } < Γ d −1 .

(11.254) (11.255)

Theorem 11.4 implies that there exist L0 > L1 and a positive number γ such that the following property holds: n satisfying z ≤ M e, (P2) for every T > 2L0 , every pair of points z0 , z1 ∈ R+ 0 0 T −1 az1 ≤ Γ d , and every program (x(t), y(t))t=0 which satisfies x(0) = z0 , x(T ) ≥ z1 ,

+

T

w(by(t))dt ≥ U (z0 , z1 , 0, T ) − γ

(11.256) (11.257)

0

there exist τ1 , τ2 such that τ1 ∈ [0, L0 ], τ2 ∈ [T − L0 , T ],

(11.258)

x(t) − x  ≤  for all t ∈ [τ1 , τ2 ],

(11.259)

for every number S satisfying τ1 ≤ S ≤ τ2 − L1 , x  > 4−1 }) ≤ 4−1  mes({t ∈ [S, S + L1 ] : y(t) −

(11.260)

and that x  ≤ γ , then τ2 = T . if x(0) − x  ≤ γ , then τ1 = 0 and if x(T ) − (11.261) Set L = 8L0 + 4.

(11.262)

11.11 Proof of Theorem 11.26

419

n satisfy Let z0 , z1 ∈ R+

z0 ≤ Me, az1 ≤ Γ d −1

(11.263)

and let a number T > 2L. Assume that a program (x(t), y(t))Tt=0 satisfies (11.251), for every τ ∈ [0, T − L], we have +

τ +L

w(by(t))dt ≥ U (x(τ ), x(τ + L), 0, L) − γ

(11.264)

w(by(t))dt ≥ U (x(T − L), z1 , 0, L) − γ .

(11.265)

τ

and that +

T

T −L

We claim that there exist numbers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ],

(11.266)

Equation (11.259) is true, for every number S satisfying τ1 ≤ S ≤ τ2 − L1 , we have x  > }) ≤  mes({t ∈ [S, S + L1 ] : y(t) − and that (11.261) is valid. In view of (11.256), (11.263), and property (P1), we have x(t) ≤ M0 e, t ∈ [0, T ].

(11.267)

Consider the program (x(t), y(t))Tt=T −L . In view of property (P2), applied to this program, (11.256), (11.262), (11.263), (11.265), and (11.267), there exist τ˜1 , τ˜2 such that τ˜1 ∈ [T − L, T − L + L0 ], τ˜2 ∈ [T − L0 , T ], x(t) − x  ≤ , t ∈ [τ˜1 , τ˜2 ],

(11.268) (11.269)

for every number S satisfying τ˜1 ≤ S ≤ τ˜2 − L1 , (11.260) is valid and that x  ≤ γ , then τ˜2 = T . if x(T − L) − x  ≤ γ , then τ˜1 = T − L and if x(T ) − (11.270) Suppose that {t ∈ [0, τ˜1 ] : x(t) − x  > } = ∅.

(11.271)

420

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Define x  > }. t0 = sup{t ∈ [0, τ˜1 ] : x(t) −

(11.272)

It is not difficult to see that x || = . t0 > 0, t0 ≤ τ˜1 , ||x(t0 ) −

(11.273)

t0 ≤ 2L0 .

(11.274)

We claim that

Assume the contrary. Then t0 > 2L0 and there exists a number t1 such that x  > . 0 ≤ t1 < t0 , t1 > t0 − L0 > L0 , x(t1 ) −

(11.275)

0 +L0 . In view of (11.275), Consider the program (x(t), y(t))tt=t 1 −L0

(t0 + L0 ) − (t1 − L0 ) = t0 − t1 + 2L0 > 2L0 .

(11.276)

By (11.267), x(t1 − L0 ) ≤ M0 e.

(11.277)

Equations (11.262), (11.268), and (11.272) imply that t0 < t0 + L0 ≤ τ˜1 + L0 ≤ T − L + 2L0 ≤ T − L0 ≤ τ˜2 .

(11.278)

It follows from (11.269), (11.272), and (11.278) that x  ≤ . x(t0 + L0 ) −

(11.279)

ax(t0 + L0 ) < Γ d −1 .

(11.280)

By (11.265) and (11.279),

By (11.262), (11.264), (11.267), (11.269), (11.275), (11.276), (11.280), 0 +L0 , we have Lemma 11.21, and (P2) applied to the program (x(t), y(t))tt=t 1 −L0 x(t) − x  ≤  for all t ∈ [t1 , t0 ]

11.11 Proof of Theorem 11.26

421

and that x  ≤ . x(t1 ) − This contradicts (11.275). The contradiction we have reached proves that t0 ≤ 2L0 . Then in view of (11.268)–(11.270), (11.272), and (11.273), 0 ≤ t0 ≤ 2L0 , τ˜2 ∈ [T − L0 , T ],

(11.281)

x(t) − x  ≤  for all t ∈ [t0 , τ˜2 ],

(11.282)

if x(T ) − x  ≤ γ , then τ˜2 = T .

(11.283)

x(0) − x ≤ γ ;

(11.284)

x(0) − x > γ .

(11.285)

τ1 = 2L0 , τ2 = τ˜2 .

(11.286)

There are two cases:

If (11.285) is valid, then we put

0 Assume that (11.284) is true and consider the program (x(t), y(t))3L t=0 . In view of (11.262), (11.267), (11.281), and (11.282),

x  ≤ . x(0) ≤ M0 e, x(3L0 ) −

(11.287)

By (11.255) and (11.287), ax(3L0 ) < Γ d −1 .

(11.288)

Equations (11.264), (11.284), (11.287), (11.288), Lemma 11.21, and property (P2) 0 applied to the program (x(t), y(t))3L t=0 imply that x(t) − x  ≤  for all t ∈ [0, 2L0 ]. Combined with (11.281) and (11.282), the inequality above implies that x(t) − x  ≤ , t ∈ [0, τ˜2 ].

(11.289)

τ1 = 0, τ2 = τ˜2 .

(11.290)

Put

In view of (11.289) and (11.290), we have

422

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x(t) − x  ≤  for all t ∈ [τ1 , τ2 ].

(11.291)

It follows from (11.181)–(11.283), (11.286), (11.290), and (11.291) that in both cases, we have defined τ1 , τ2 ∈ [0, T ] such that: in the case of (11.284) τ1 = 0

(11.292)

τ1 = 2L0 ;

(11.293)

τ2 = τ˜2 ≥ T − L0 ,

(11.294)

if x(T ) − x  ≤ γ , then τ2 = T ,

(11.295)

x(t) − x  ≤ , t ∈ [τ1 , τ2 ].

(11.296)

and in the case of (11.285)

Assume that a real number S satisfies 2L0 ≤ S ≤ τ2 − L1 .

(11.297)

We claim that x  > }) ≤ . mes({t ∈ [S, S + L1 ] : y(t) −

(11.298)

If S ≥ τ˜1 , then (11.298) follows from (11.260), (11.270), (11.294), and the choice of τ˜1 , τ˜2 . Consider the case with S < τ˜1 ≤ T − L + L0

(11.299)

(see (11.268)). By (11.262), (11.295), and (11.299), S + 2L0 ≤ T − L + 3L0 < T − L0 ≤ τ˜2 = τ2 .

(11.300)

0 Consider the program (x(t), y(t))S+2L t=S−2L0 . By (11.264), (11.297), (11.298), and Lemma 11.21,

+

S+2L0

w(by(t))dt ≥ U (x(S − 2L0 ), x(S + 2L0 ), 0, 4L0 ) − γ .

S−2L0

By (11.292)–(11.294), (11.296), and (11.300), x(S + 2L0 ) − x  ≤ . Combined with (11.255) this implies that

(11.301)

11.12 Proof of Theorem 11.29

423

ax(S + 2L0 ) < Γ d −1 .

(11.302)

Equations (11.267), (11.301), (11.302), and (P2) imply that (11.298) holds. Thus for every number S satisfying (11.297), relation (11.298) is valid. Assume that (11.284) is valid, S ∈ [0, 2L0 ]

(11.303)

0 and consider the program (x(t), y(t))4L t=0 . By (11.262), (11.284), (11.292), (11.295), and (11.296),

x(0) − x ≤ γ ,

(11.304)

x(4L0 ) − x  ≤ .

(11.305)

In view of (11.255) and (11.305), ax(4L0 ) < Γ d −1 .

(11.306)

It follows from (11.262), (11.264), and Lemma 11.21 that +

4L0

w(by(t))dt ≥ U (x(0), x(4L0 ), 0, 4L0 ) − γ .

(11.307)

0

In view of (11.303)–(11.307) and (P2), inequality (11.298) is valid. This completes the proof of Theorem 11.26.

11.12 Proof of Theorem 11.29 We suppose that the sum over the empty set is zero. In view of Proposition 11.1, there exists a positive number M0 > M such that the following property holds: (P6) for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ Me, the inequality x(t) ≤ M0 e is true for all t ∈ [0, T ], and if S ∈ [0, T ], τ > 0 and (u(t), v(t))S+τ t=S is a program with u(S) = x(S), then u(t) ≤ M0 e for all t ∈ [S, S + τ ]. We may assume without loss of generality that 

a x < Γ d −1 ,

(11.308) 

n sup ay : y ∈ R+ and y − x  ≤  < Γ d −1 .

(11.309)

424

11 The Continuous-Time Robinson–Solow–Srinivasan Model

Lemma 11.19 implies that there exists a positive number m0 such that the following property holds: (P7) for every positive number T and every program (x(t), y(t))Tt=0 satisfying x(0) ≤ M0 e, we have +

T

[w(by(t)) − w(b x )]dt ≤ m0 .

0

Lemma 11.17 implies that there exists a number m1 > m0 such that for every n and every positive number T , we have z ∈ R+ U (z, T ) ≥ T w(b x ) − m1 .

(11.310)

Lemma 11.18 implies that there exists m2 > m1 such that the following property holds: n , every z ∈ R n satisfying az ≤ Γ d −1 , and every (P8) for every z0 ∈ R+ 1 1 + number T > m2 , we have x ) − m2 . U (z0 , z1 , 0, T ) ≥ T w(b Theorem 11.28 and (11.309) imply that there exist M∗ > M0 , L0 > 0, δ1 ∈ (0, ) such that the following property holds: (P9) for each T1 ≥ 0, each T2 > T1 + 2L0 , each L × B-measurable function n → R 1 which is bounded on bounded subsets of [T , T ] × R n W : [T1 , T2 ] × R+ 1 2 + and such that for almost every t ∈ [T1 , T2 ], W (t, ·) − w(b(·))M∗ ≤ δ1 2 and each program (x(t), y(t))Tt=T which satisfies 1

x(T1 ) ≤ M0 e, +

x(Ti ) − x  ≤ δ1 , i = 1, 2, T2

W (t, y(t))dt ≥ U (x(T1 ), x(T2 ), T1 , T2 , W ) − δ1

T1

we have x(t) − x  ≤  for all t ∈ [T1 , T2 ] and for each number S satisfying T1 ≤ S ≤ T2 − L1 , x  > }) ≤ . mes({t ∈ [S, S + L1 ] : y(t) −

11.12 Proof of Theorem 11.29

425

In view of Lemma 11.23, there exists an integer L2 ≥ 1 such that the following property holds: 2 (P10) for every program (x(t), y(t))L t=0 which satisfies + x(0) ≤ M0 e,

L2

w(by(t))dt ≥ L2 w(b x ) − M1 − m 0 − m 2 − 1

0

there exists t ∈ [0, L2 ] such that x(t) − x  ≤ δ1 . Choose l > 4L1 + 4L2 + 4m2 + 4L0 ,

(11.311)

0 < δ < 8−1 δ1 , 4δm2 + 4δL2 < 1,

(11.312)

Q > 4 + 4δ1−1 M1 .

(11.313)

T1 ≥ 0, T2 > T1 + Ql,

(11.314)

n z 0 ∈ R+ , z0 ≤ Me,

(11.315)

a number δ such that

and a natural number

Assume that

n → R 1 is an L × B-measurable function which is bounded on W : [T1 , T2 ] × R+ n and such that for almost every t ∈ [T , T ], we bounded subsets of [T1 , T2 ] × R+ 1 2 have

W (t, ·) − w(b(·))M∗ ≤ δ

(11.316)

n satisfy 2 and z1 ∈ R+ and that a program (x(t), y(t))Tt=T 1

+

T2

x(T1 ) = z0 ,

(11.317)

az1 ≤ Γ d −1 , x(T2 ) ≥ z1 ,

(11.318)

W (t, y(t))dt ≥ U (z0 , z1 , T1 , T2 , W ) − M1 .

T1

Property (P6), (11.315), and (11.317) imply that

(11.319)

426

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x(t) ≤ M0 e for all t ∈ [0, T ].

(11.320)

We show that the following property holds: (P11) Let a number S satisfy T1 ≤ S ≤ T2 − 2L2 − m2 , and let there exists n such that x(S ˜ + 2L2 + m2 ) ∈ R+

+

x(S ˜ + 2L2 + m2 ) ≤ x(S + 2L2 + m2 ),

(11.321)

a x(S ˜ + 2L2 + m2 ) ≤ Γ d −1 ,

(11.322)

S+2L2 +m2

W (t, y(t))dt ≥ U (x(S), x(S ˜ + 2L2 + m2 ), S, S

S

+2L2 + m2 , W ) − M1 .

(11.323)

Then there exists τ ∈ [S, S + L2 ] such that x(τ ) − x  ≤ δ1 . Let a number S satisfy T1 ≤ S ≤ T2 − 2L2 − m2 , and let (11.321)–(11.323) be n . Property (P6), (11.312), (11.316), and (11.320)– valid with x(S ˜ + 2L2 + m2 ) ∈ R+ (11.323) imply that +

S+2L2 +m2

w(by(t))dt S

+ ≥

S+2L2 +m2

W (t, y(t))dt − δ(2L2 + m2 )

S

≥ U (x(S), x(S ˜ + 2L2 + m2 ), S, S + 2L2 + m2 , W ) − M1 − δ(2L2 + m2 ) ≥ U (x(S), x(S ˜ + 2L2 + m2 ), 0, 2L2 + m2 ) − M1 − 2δ(2L2 + m2 ) ≥ U (x(S), x(S ˜ + 2L2 + m2 ), 0, 2L2 + m2 ) − M1 − 1.

(11.324)

Combined with (11.310), (11.321)–(11.323), and property (P8), this implies that +

S+2L2 +m2

w(by(t))dt ≥ (2L2 + m2 )w(b x ) − m2 − M1 − 1.

(11.325)

S

In view of (11.325) and property (P7), we have + S

S+L2

+ w(by(t))dt =

S+2L2 +m2

+ w(by(t))dt −

S

S+2L2 +m2

w(by(t))dt S+L2

x ) − m 2 − M1 − 1 ≥ (2L2 + m2 )w(b − (L2 + m2 )w(b x ) − m0

11.12 Proof of Theorem 11.29

427

= L2 w(b x ) − m2 − M1 − m0 − 1.

(11.326)

In view of (11.320), (11.326), and (P10) there exists a number τ ∈ [S, S + L2 ] such that x(τ ) − x  ≤ δ1 . Hence property (P11) holds. By property (P11), Lemma 11.21, (11.309), (11.318), and (11.319), the following property holds: (P12) Let a number S satisfy T1 ≤ S ≤ T2 − 2L2 − m2 and x  < . x(S + 2L2 + m2 ) − x  ≤ δ1 . Then there exists a number τ ∈ [S, S + L2 ] for which x(τ ) − Property (P11), (11.318), and (11.319) imply that there exists a number S1 such that x  ≤ δ1 . S1 ∈ [T2 − 2L2 − m2 , T2 − L2 − m2 ], x(S1 ) −

(11.327)

By induction applying (P12), we construct a finite strictly decreasing sequence of real numbers S1 , . . . , Sp where p is a natural number such that x  ≤ δ1 , i = 1, . . . , p, x(Si ) −

(11.328)

Sp ∈ [0, 2L2 + m2 ), Equation (11.327) is valid, and for all integers i satisfying 1 ≤ i < p, we have Si − Si+1 ∈ [L2 , 2L2 + m2 ].

(11.329)

By induction we construct a finite strictly increasing sequence of real numbers τ1 , . . . , τq ∈ {Si : i = 1, . . . , p}

(11.330)

such that τ1 = Sp , for each integer i ∈ {1, . . . , q} satisfying i < q we have: if +

S1

τi

W (t, y(t))dt ≥ U (x(τi ), x(S1 ), τi , S1 , W ) − δ1 ,

(11.331)

then q = i + 1, τq = S1 ; otherwise

(11.332)

428

11 The Continuous-Time Robinson–Solow–Srinivasan Model

+

τi+1 = min{τ ∈ {S1 , . . . , Sp } : τ > τi and τ

τi

W (t, y(t))dt < U (x(τi ), x(τ ), τi , τ, W ) − δ1 }.

(11.333)

It follows from (11.318), (11.319), (11.330)–(11.333), and Lemma 11.21 that +

T2

M1 ≥ U (x(T1 ), x(T2 ), T1 , T2 , W ) − ≥



W (t, y(t))dt +

{U (x(τi ), x(τi+1 ), τi , τi+1 , W ) −

T1 τi+1

W (t, y(t))dt :

τi

i ∈ {1, . . . , q} and i < q − 1} ≥ (q − 2)δ1 , q ≤ 2 + δ1−1 M1 .

(11.334)

Define A = {i ∈ {1, . . . , q} : i < q, τi+1 − τi ≥ l}.

(11.335)

Let i ∈ A be given. In view of (11.311), (11.329)–(11.333), and (11.335), there exists a natural number j ∈ {1, . . . , p} such that +

τi+1 − 2L2 − m2 ≤ Sj (i) ≤ τi+1 , Sj (i)

τi

W (t, y(t))dt ≥ U (x(τi ), x(Sj (i) ), τi , Sj (i) , W ) − δ1 .

(11.336) (11.337)

It follows from (11.312), (11.316), (11.328), (11.330), (11.336), (11.337), and (P9) that x(t) − x  ≤ , t ∈ [τi , Sj (i) ] and that for every number S satisfying τi ≤ S ≤ Sj (i) − L1 we have x  ≥ }) < . mes({t ∈ [S, S + L1 ] : y(t) − In order to complete the proof, it is sufficient to note that [T1 , T2 ] \ ∪{[τi , Sj (i) ] : i ∈ A}

11.13 Optimal Programs over Infinite Horizon

429

is a finite number of intervals, their maximal length does not exceed l, and in view of (11.313) and (11.334), their number does not exceed 2(q + 2) ≤ 4 + 4δ1−1 M1 < Q. This completes the proof of Theorem 11.29.

11.13 Optimal Programs over Infinite Horizon A program (x(t), y(t))∞ t=0 is called weakly maximal if for every positive number T , +

T

w(by(t))dt = U (x(0), x(T ), 0, T ).

0

We prove the following result. Theorem 11.31 A program (x(t), y(t))∞ t=0 is overtaking optimal if and only if it is good and weakly maximal. Proof Assume that a program (x(t), y(t))∞ t=0 is overtaking optimal. Let us show that it is good. Assume the contrary. Then in view of Proposition 11.2, it is bad and +

T

(w(by(t)) − w(b y ))dt = −∞.

lim

T →∞ 0

(11.338)

∞ which Proposition 11.3 implies that there exists a good program (x(t), ¯ y(t)) ¯ t=0 1 satisfies x(0) ¯ = x(0). Then there exists a number M ∈ R such that

+

T

(w(by(t)) ¯ − w(b y ))dt ≥ M for all positive numbers T .

0

In view of (11.338) and the inequality above, we have ,+

T

lim

T →∞

0

+ w(by(t))dt ¯ −

T

w(by(t))dt = ∞,

0

a contradiction. Hence the program (x(t), y(t))∞ t=0 is good. Let us show that the program (x(t), y(t))∞ t=0 is weakly maximal. Assume the contrary. Then there exists a positive number T0 such that +

T0

w(by(t))dt < U (x(0), x(T0 ), 0, T0 ). 0

430

11 The Continuous-Time Robinson–Solow–Srinivasan Model

T0 This implies that there exists a program (x(t), ¯ y(t)) ¯ t=0 such that

x(0) ¯ = x(0), x(T ¯ 0 ) ≥ x(T0 ), + T0 + T0 w(by(t))dt ¯ > w(by(t))dt. 0

(11.339)

0

∞ Lemma 11.20 and (11.339) imply that there exists a program (x(t), ˜ y(t)) ˜ t=T0 such that

¯ 0 ), y(t) ˜ = y(t), t ∈ [T0 , ∞). x(T ˜ 0 ) = x(T

(11.340)

Set ¯ t ∈ [0, T0 ], x1 (t) = x(t), ˜ t ∈ (T0 , ∞), x1 (t) = x(t), y1 (t) = y(t), ¯ t ∈ [0, T0 ], y1 (t) = y(t) ˜ = y(t), t ∈ (T0 , ∞).

(11.341)

In view of (11.339)–(11.341), (x1 (t), y1 (t))∞ t=0 is a program. By (11.339)–(11.341), for all numbers T > T0 , we have +

T

+ w(by1 (t))dt −

0

T

w(by(t))dt 0

+

T0

=

+

T0

w(by(t))dt ¯ −

0

w(by(t))dt > 0 0

and ,+

T

lim sup T →∞

+

-

T

w(by1 (t))dt −

w(by(t))dt > 0,

0

0

a contradiction. The contradiction we have reached proves that the program (x(t), y(t))∞ t=0 is weakly maximal. Assume that a program (x(t), y(t))∞ t=0 is good and weakly maximal. We show that it is overtaking optimal. In view of Theorem 11.5, there exists an overtaking ∞ such that optimal program (x(t), ˜ y(t)) ˜ t=0 x(0) ˜ = x(0).

(11.342)

Evidently, ,+

T

lim sup T →∞

0

+ w(by(t))dt −

T

w(by(t))dt ˜ ≤ 0.

0

In order to complete the proof, it is sufficient to show that

11.13 Optimal Programs over Infinite Horizon

,+

T

lim sup T →∞

431

+

T

w(by(t))dt ˜ −

0

w(by(t))dt ≤ 0.

0

Assume the contrary. Then there exist a positive number  and a strictly increasing sequence of positive numbers {Tk }∞ k=1 such that lim Tk = ∞

k→∞

and that for all integers k ≥ 1, we have +

Tk

+

Tk

w(by(t))dt ˜ −

w(by(t))dt > .

(11.343)

w(b( x + Δe)) − w(b( x − Δe)) < /2.

(11.344)

0

0

Fix Δ > 0 for which

In view of Lemma 11.22, there exists δ ∈ (0, Δ) such that the following property holds: n which satisfy (P) for every pair of points z, z ∈ R+ x ≤ δ z − x , z − and every τ ∈ [2−1 , 2], there exists a program (u(t), v(t))τt=0 such that u(0) = z, u(τ ) ≥ z , u(t) − x , v(t) − x  ≤ Δ, t ∈ [0, τ ], u (t) ≤ Δ, t ∈ [0, τ ]. ∞ are good, it follows from ˜ y(t)) ˜ Since the programs (x(t), y(t))∞ t=0 and (x(t), t=0 Theorem 11.4 that

lim x(t) = x , lim x(t) ˜ = x,

t→∞

t→∞

and there exists a positive number S such that x(t) ˜ − x  ≤ δ, x(t) − x  ≤ δ for all t ≥ S.

(11.345)

Fix an integer k ≥ 1 such that Tk > S. In view of (11.345) and (11.346), we have

(11.346)

432

11 The Continuous-Time Robinson–Solow–Srinivasan Model

x(Tk ) − x  ≤ δ, x(Tk + 1) − x  ≤ δ, x(T ˜ k) − x  ≤ δ, x(T ˜ k + 1) − x  ≤ δ.

(11.347)

It follows from (11.347) and property (P) that there exists a program Tk +1 (x0 (t), y0 (t))t=0

such that ˜ y0 (t) = y(t), ˜ t ∈ [0, Tk ], x0 (t) = x(t), x0 (Tk + 1) ≥ x(Tk + 1), x0 (t) − x , y0 (t) − x , x0 (t) ≤ Δ, t ∈ [Tk , Tk + 1].

(11.348) (11.349) (11.350)

In view of (11.342) and (11.348), we have x0 (0) = x(0).

(11.351)

It follows from (11.343) and (11.348) that +

Tk +1 0

+

w(by(t))dt 0

+

Tk

=

+

+

Tk

w(by(t))dt ˜ −

0

+

Tk +1

w(by0 (t))dt −

w(by(t))dt 0

Tk +1

Tk

>+

+

w(by0 (t))dt −

+

Tk +1

w(by(t))dt Tk

Tk +1

Tk

w(by0 (t))dt −

+

Tk +1

w(by(t))dt.

(11.352)

Tk

By (11.345) and (11.346), for for almost every t ∈ [Tk , Tk+1 ], we have y(t) ≤ x(t) ≤ x + δe.

(11.353)

In view of (11.350), for almost every t ∈ [Tk , Tk+1 ], x − Δe(σ ). y0 (t) ≥

(11.354)

It follows from (11.344), (11.353), and (11.354) that for almost every t ∈ [Tk , Tk+1 ], x − Δe(σ ))) − w(b( x + δe)) w(by0 (t)) − w(by(t)) ≥ w(b( ≥ w(b( x − Δe)) − w(b( x + Δe)) > −/2.

11.13 Optimal Programs over Infinite Horizon

433

Combined with (11.352) the relation above implies that + 0

Tk +1

+ w(by0 (t))dt −

Tk +1

w(by(t))dt ≥  − /2.

0

This contradicts (11.349) and (11.351). The contradiction we have reached proves that the program (x(t), y(t))∞ t=0 is overtaking optimal. Theorem 11.31 is proved.

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Index

A Agreeable program, 17, 21, 160 Approximate solution, 6, 16 Asymptotic turnpike property, 17, 19, 79 Autonomous discrete-time control system, 16, 342 Average turnpike property, 25, 32

B Bad program, 22, 41 Baire category approach, 285

C Cardinality of a set, 6, 79 Cauchy sequence, 144 Compact metric space, 16 Compact set, 16 Complete metric space, 337 Concave function, 21 Constrained problem, 1 Continuous-time problem, 383 Convex dynamic unconstrained optimization problems, 2 Convex set, 17

D Differentiable function, 3, 21 Discrete-time dynamic optimization, 1 Discrete-time problem, 16, 225

E Euclidean space, 2, 17, 39

F Finitely optimal, 268

G Golden-rule stock, 21, 28 Good program, 17, 19, 41 Gross investment sequence, 26

H Hausdorff distance, 336

I Increasing function, 1, 21 Infinite horizon, 8, 39 Infinite horizon optimal control problem, 39 Inner product, 2 Interior point, 18

M Maximal program, 277 Metric space, 16 Minimization problem, 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. J. Zaslavski, Turnpike Theory for the Robinson–Solow–Srinivasan Model, Springer Optimization and Its Applications 166, https://doi.org/10.1007/978-3-030-60307-6

441

442 N Neumann path, 2 Norm, 2 O Objective function, 17 Optimality criterion, 1, 14 Overtaking optimal program, 20, 22, 42 Overtaking optimal sequence, 14 P Program, 18, 21 S Strictly concave function, 17, 22 Strictly convex function, 2, 3 Strictly increasing function, 21 T Taylor’s theorem, 71 Turnpike, 8, 17

Index Turnpike phenomenon, 16, 17 Turnpike property, 1, 17 Turnpike result, 18, 19

U Unconstrained problem, 1 Uniform concavity, 229 Uniform equicontinuity, 299 Uniformity, 335 Upper semicontinuous function, 16 Utility function, 125

V von Neumann facet, 37

W Weakly agreeable program, 160 Weakly maximal program, 130 Weakly optimal program, 27, 42