Optimization in Economics and Finance

- Some Advances in Non-linear, Dynamic, Multi-criteria and Stochastic Models

by

BRUCE D. CRAVEN
University of Melbourne, VIC, Australia

and
SARDAR M. N. ISLAM
Victoria University, Melbourne, VIC, Australia

 published by Springer 2005

It is VOLUME 7 of the collection of Dynamic Modeling and Econometrics in Economics and Finance.

Table of Contents

Preface ix
Acknowledgements and Sources of Materials xi

Chapter One: Introduction :
Optimal Models for Economics and Finance 1
1.1 Introduction 1
1.2 Welfare economics and social choice: modelling and applications 2
1.3 The objectives of this book 5
1.4 An example of an optimal control model 6
1.5 The structure of the book 7

Chapter Two: Mathematics of Optimal Control 9
2.1 Optimization and optimal control models 9
2.2 Outline of the Pontryagin Theory 12
2.3 When is an optimum reached? 14
2.4 Relaxing the convex assumptions 16
2.5 Can there be several optima? 18
2.6 Jump behaviour with a pseudoconcave objective 20
2.7 Generalized duality 24
2.8 Multiobjective (Pareto) optimization 29
2.9 Multiobjective optimal control 30
2.10 Multiobjective Pontryagin conditions 32

Chapter Three: Computing Optimal Control:
The SCOM package 35
3.1 Formulation and computational approach 35
3.2 Computational requirements 37
3.3 Using the SCOM package 40
3.4 Detailed account of the SCOM package 41
3.4.1 Preamble 41
3.4.2 Format of problem 41
3.4.3 The SCOM codes: The user does not alter them 42
3.5 Functions for the first test problem 46
3.6 The second test problem 47
3.7 The third test problem 49

Chapter Four: Computing Optimal Growth
and Development Models 55
4.1 Introduction 55
4.2 The Kendrick-Taylor growth model 56
4.3 The Kendrick-Taylor model implementation 57
4.4 Mathematical and economic properties of the results 60
4.5 Computation by other computer programs 64
4.6 Conclusions 64

Chapter Five: Modelling Financial Investment
with Growth 66
5.1 Introduction 66
5.2 Some related literature 66
5.3 Some approaches 69
5.4 A proposed model for interaction between investment and physical
capital 70
5.5 A computed model with small stochastic term 72
5.6 Multiple steady states in a dynamic financial model 75
5.7 Sensitivity questions concerning infinite horizons 80
5.8 Some conclusions 81
5.9 The MATLAB codes 82
5.10 The continuity required for stability 83

Chapter Six: Modelling Sustainable Development 84
6.1 Introduction 84
6.2 Welfare measures and models for sustainability 84
6.3 Modelling sustainability 87
6.3.1 Description by objective function with parameters 87
6.3.2 Modified discounting for long-term modelling 89
6.3.3 Infinite horizon model 90
6.4 Approaches that might be computed 92
6.4.1 Computing for a large time horizon 92
6.4.2 The Chichilnisky compared with penalty term model 92
6.4.3 Chichilnisky model compared with penalty model 94
6.4.4 Pareto optimum and intergenerational equality 95
6.4.5 Computing with a modified discount factor 95
6.5 Computation of the Kendrick-Taylor model 96
6.5.1 The Kendrick-Taylor model 96
6.5.2 Extending the Kendrick-Taylor model to include a long time
horizon 97
6.5.3 Chichilnisky variant of Kendrick-Taylor model 98
6.5.4 Transformation of the Kendrick-Taylor model 98
6.6 Computer packages and results of computation of models 99
6.6.1 Packages used 99
6.6.2 Results: comparison of the basic model solution with results for
modified discount factor 99
6.6.3 Results: effect of increasing the horizon T 101
6.6.4 Results: Effect of omitting the growth term in the dynamic
equation 103
6.6.5 Results: parametric approach 103
6.6.6 Results: the modified Chichilnisky approach 105
6.7 Existence, uniqueness and global optimization 108
6.8 Conclusions 109
6.9 User programs for transformed Kendrick-Taylor model for
sustainable growth 110

Chapter Seven : Modelling and Computing a Stochastic
Growth Model 111
7.1 Introduction 112
7.2 Modelling stochastic growth 112
7.3 Calculating mean and variance 113
7.4 Computed results for stochastic growth 114
7.5 Requirements for RIOTS 95 as M-files 116

Chapter Eight: Optimization in Welfare Economics 123
8.1 Static and dynamic optimization 123
8.2 Some static welfare models 123
8.3 Perturbations and stability 125
8.4 Some multiobjective optimal control models 126
8.5 Computing multiobjective optima 128
8.6 Some conditions for invexity 129
8.7 Discussion 130

Chapter 9: Transversality Conditions for Infinite
Horizon Models 131
9.1 Introduction 131
9.2 Critical literature survey and extensions 131
9.3 Standard optimal control model 135
9.4. Gradient conditions for transversality 136
9.5 The model with infinite horizon 139
9.6 Normalizing a growth model with infinite horizon models 139
9.7 Shadow prices 141
9.8 Sufficiency conditions 142
9.9 Computational approaches for infinite horizon 143
9.10 Optimal control models in finance: special considerations 146
9.11 Conclusions 146

Chapter 10: Conclusions 147

Bibliography 149

个人认为可作为Dixit一书的进阶.