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continuation on paga 231
Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
352
Onno van Hilten
Optimal Firm Behaviour in the
Context of Technological
Progress and a Business Cycle
Springer-Verlag Berlin Heidelberg GmbH
Editorial Board
H.Albach M. Beckmann (Managing Editor)
P. Ohrymes G. Fandel G. Feichtinger W. Hildenbrand W. Krelle (Managing Editor)
H. P. Kunzi K. Ritter U. Schittko P. Schi:infeld R. Selten
Managing Editors
Prof. Or. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Or. W. Krelle
Institut fUr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Author
Or. Onno van Hilten
Netherlands Energy Research Foundation ECN
P.O. Box 1, NL-1755 ZG Petten, The Netherlands
ISBN 978-3-540-53563-8 ISBN 978-3-662-02718-9 (eBook)
DOI 10.1007/978-3-662-02718-9
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© Springer-Verlag Berlin Heidelberg 1991
Originally published by Springer-Verlag Berlin Heidelberg New York in 1991
2142/3140-543210 - Printed on acid-free paper
ACKNOWLEDGEMENTS
Writing this thesis has been a great pleasure. Many people contributed to
this pleasure. The first one to be mentioned in this respect is Paul van
Loon, who lead me into the world of economics and dynamic optimisation. His
creativity with regard to generating ideas and solving the many 'small'
problems that occurred, has continued to amaze me in the last four years.
Our cooperation was in many ways optimal.
With many colleagues at the University of Limburg I have had interesting
discussions on all kinds of subjects. Especially with my 'collega proximus'
Kees Cools, who is not a specialist of optimal control theory, but, to me,
a specialist of life.
I would also like to thank: Raymond Gradus, Peter Kort, Piet Verheyen (all
University of Tilburg) and Jan de Jong (University of Eindhoven) for many
discussions on almost every part of the thesis; and Richard Hartl
(Technical University of Vienna), who helped me writing Appendix 4 and who
patiently and accurately answered the many letters I wrote him.
Appendix 4 has been written during a three weeks stay at the Technical
University of Vienna. This stay has been financially supported by the
Netherlands Organisation for Scientific Research (NWO).
Many errors have been removed by Peter Kort, who scrupulously read the
manuscript with expert eyes, and Mieke Lettink, who improved my use of the
English language. All remaining errors are mine.
Finally I want to thank two persons who are almost completely unknowing of
the contents of this book, but whose contribution is nevertheless
invaluable: my mother, from whom I not only inhereted the need for (too)
much sleep, but also a talent for mathematics and a (apparently sufficient)
degree of perseverance, and Ankie, who taught me the irrelevance of a long
distance and who (mostly unaware of it) took my mind off this book at the
right times.
CONTENTS
Notation xi
CHAPTER ONE 1
INTRODUCTION
CHAPTER TWO 5
A SELECTIVE LITERATURE SURVEY
2.1 Introduction 5
2.2 The predecessors of the models in this thesis 6
2.3 Optimal behaviour of a firm facing a business cycle 9
2.3.1 An explicit business cycle 9
2.3.2 An implicit business cycle 12
2.4 Optimal behaviour of a firm facing technological progress 13
2.5 Summary 17
CHAPTER THREE 19
ON DYNAMIC OPTIMISATION MODELS OF THE FIRM AS
A BRANCH OF 'PURE THEORY' AND ON THE USE OF
MATHEMATICS
3.1 Introduction 19
3.2 Theoretical dynamic optimisation models of the firm:
a branch of 'pure theory' 19
3.2.1 Pure theory 19
3.2.2 An illustration 20
3.3 Some guidelines for the use of mathematics 23
3.3.1 Introduction 23
3.3.2 On the role of mathematics in economics 24
3.3.3 The status of assumptions 25
3.3.4 On the mathematical translation of economic
assumptions and the interpretation of
mathematical tools 27
3.4 Summary and conclusions 30
CHAPTER FOUR 33
THE BASIC MODEL
4.1 Introduction 33
VI
4.2 The model and its assumptions 34
4.3 Necessary and sufficient conditions for optimality 38
4.4 A feedback decision rule 43
4.5 ruustrations of the decision rule 47
4.6 Limitations of the decision rule 51
4. 7 Book value and market value 53
4.8 Summary and conclusions 55
CHAPTER FIVE 57
A MODEL WITH A BUSINESS CYCLE
5.1 Introduction 57
5.2 The model and the optimality conditions 59
5.2.1 The model 59
5.2.2 The conditions for optimality 62
5.3 The optimal solution in case i < (1-/)r 63
5.3.1 Introduction 63
*
5.3.2 A 'light' recession (g <msg +a) 64
*
5.3.3 A 'moderate' recession (g +a < m s m ) 65
* **
5.3.4 A 'severe' recession (m <msm ) 71
5.3.5 A fatal recession 76
5.4 The optimal solution in case i>(1-/)r 78
5 .4.1 Introduction 78
*
5.4.2 A 'light' recession (g<msg +a) 78
*
5.4.3 A 'moderate' recession (g+a <msm ) 78
* **
5.4.4 A 'severe' recession (m <m:Sm ) 80
5.4.5 A fatal recession 82
5.5 Summary and conclusions 83
CHAPTER SIX 85
SHADOW PRICES IN A MODEL WITH PURE STATE CONSTRAINTS
6.1 Introduction 85
6.2 The model 86
6.3 A shadow price interpretation for v 89
6.4 Interpretation of the costate variables 90
6.5 Why and when does a costate jump? 92
6.6 Some general guidelines for the interpretation of costates 94
6. 7 Summary and conclusions 95
VII
CHAPTER SEVEN 97
TECHNOLOGICAL PROGRESS IN VINTAGE MODELS OF THE
FIRM: SCRAPPING CONDITION AND STEADY STATE
7.1 Introduction 97
7.2 The model and the optimality conditions 99
7 .2.1 The vintage structure and the role of taxes 99
7 .2.2 The model 101
7.2.3 Some properties of N(t) 102
7 .2.4 The optimisation problem 103
7.2.5 The optimality conditions 104
7.3 The scrapping condition 106
7 .3.1 A general scrapping condition 106
7.3.2 Comparison with other scrapping rules 109
7.3.3 Interpretation of the scrapping condition
in the general model 110
7.3.4 Another way to derive the scrapping condition? 112
7.4 A steady state solution 113
7.4.1 Existence of a steady state solution 113
7 .4.2 Is the steady state identical to the final path? 117
7.5 Summary and conclusions 120
CHAPTER EIGHT 121
OPTIMAL POLICIES IN MODELS WITH TECHNOLOGICAL
PROGRESS, WITH AND WITHOUT A BUSINESS CYCLE
8.1 Introduction 121
8.2 The optimal solution for the simplified model of Chapter 7 121
8.2.1 Limitations of the coupling procedure 121
8.2.2 The optimal policy 124
8.2.3 A decision rule? 127
8.3 The optimal solution for the general model 129
8.4 Technological progress and a business cycle in one model 133
8.4.1 Introduction 133
8.4.2 When does the steady state solution violate I~ 0? 133
8.4.3 The business cycle in the simplified model 136
8.5 Summary and conclusions 140
CHAPTER NINE 141
SUMMARY AND CONCLUSIONS
VIII
APPENDIX ONE 147
OPTIMALITY CONDITIONS FOR THE BASIC MODEL
OF CHAPTER 4
Al.1 Necessary and sufficient conditions 147
A1.2 The coupling procedure 152
A1.2.1 The paths 152
A1.2.2 Derivation of the final paths 155
A1.2.3 The coupling procedure 156
APPENDIX TWO 159
THE MATHEMATICAL DETAILS OF CHAPTER 5
A2.1 General remarks 159
A2.2 The details of section 5.3.3 161
A2.3 The details of section 5.3.4 163
A2.4 The details of section 5.3.5 168
A2.5 Uniqueness of the solution 169
A2.6 Numerical illustrations 170
A2.7 The details of section 5.4 172
APPENDIX THREE 179
ON THE SHADOW PRICE INTERPRETATION OF THE
MULTIPLIERS OF PURE STATE CONSTRAINTS IN
OPTIMAL CONTROL PROBLEMS
A3.1 Introduction 179
A3.2 The class of models to be considered 179
A3.3 An outline of the proof 180
A3.4 A general sensitivity result 180
A3.5 Problem II written as a problem P 181
a
A3.6 The Kuhn-Tucker conditions and Theorem 1 for problem II 182
A3. 7 The shadow price interpretation of v 185
APPENDIX FOUR 187
NECESSARY AND SUFFICIENT CONDITIONS FOR AN
OPTIMAL CONTROL PROBLEM WITH AN ENDOGENEOUSL Y
DETERMINED 'LAG-STRUCTURE'
A4.1 Introduction 187
A4.2 The model 187
IX
A4.3 The tric 188
A4.4 Derivation of the necessary conditions for optimality
for a special case 190
A4.5 Sufficient conditions for the general model 196
APPENDIX FIVE 205
VARIOUS DERIVATIONS
AS .1 The details of section 7.4 205
AS .1.1 Existence of a steady state solution
in section 7 .4.1 205
A5.1.2 Derivation of equation (7 .57) 206
A5.1.3 Convergence of the upper and lower bounds
on MandT 207
A5.2 The optimal policy of section 8.2.2 210
AS.3 The pattern of investments in section 8.4.2 215
A5.4 Discussion of 'zero investment'-periods 219
REFERENCES 223
NOTATION
General remarks
In this thesis small Latin letters are used to denote exogeneous parameters
and functions. Latin capitals are used to denote endogeneous variables.
Greek letters are used to denote auxilliary variables (for instance, shadow
prices). Below is a list of symbols, which is used throughout the thesis.
If a symbol is not explained in the text, it should be in this list.
The figures in this thesis are only rough sketches, mainly to indicate
whether a particular function is increasing, decreasing or constant.
A dot above a letter (as in K) denotes the total time derivative. In
general total derivatives are denoted by d, partial derivatives by a.
If a reference is made to a formula, table or figure in a different
chapter, the number of the chapter is added. For instance, Figure 5.3
refers to Figure 3 of Chapter 5; equation (26) refers to equation (26) of
the current chapter; equation (4.30) refers to equation (30) of Chapter 4.
The formulas in the appendices are denoted as follows: (A3.14) refers to
formula (14) in Appendix 3.
List of symbols
Endogeneous variables
D(t) dividends (dollars/time)
l(t) investments (dollars/time)
K(t) number of capital goods
L(t) number of units of labour
X(t) equity (dollars)
Y(t) debt (dollars)
Q(t) output (numbers/time)
S(t) revenue (dollars/time)
R(K) marginal return on investment
R (K,X) marginal return on equity
e
N(t) birth date of oldest capital goods still in use at time t
V(t) scrapping date of capital installed at time t
M(t) lifetime of capital installed at time t ( = V(t)-t)
