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Mathematical Economics and Control Theory: Studies in Policy Optimisation

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Mathematical Economics and Control Theory: Studies in Policy Optimisation University of London School of Oriental and African Studies Department of Economics Mathematical Economics and Control Theory: Studies in Policy Optimisation Masoud Derakhshan-Nou Thesis Submitted for the Degree of Doctor of Philosophy November 1996 ProQuest Number: 10731732 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10731732 Published by ProQuest LLC(2017). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 Abstract Chapter 1 deals with the origin and limitations of mathematical economics and its implications for economic applications of optimal control theory. Using an historical approach, we have proposed a hypothesis on the origin and limitations of classical and modern mathematical economics. Similar hypotheses proposed by Cournot, Walras, von Neumann-Morgenstern and Debreu are shown not to be convincing. Conditions are established under which applications of mathemati­ cal methods, in general, and optimal control theory, in particular, may produce economic results of value. Chapter 2 concerns the formation and development of optimal control ap­ plications to economic policy optimisation. It is shown that the application of mathematical control theory (as compared with engineering control) may signifi­ cantly contribute to mathematical economics (as compared to econometrics ). The development of optimal growth theory has been examined as an example. Within the context of economic policy optimisation, a critical examination of the recent developments in macroeconomic modelling, the relationship between theory and observation, rational expectations, the Lucas critique and the problem of time- inconsistency is presented. Chapter 3 provides the first illustration of the main theme of the earlier chap­ ters. Using the generalised Hamiltonian in Pontryagin’s maximum principle, as well as using Bellman’s dynamic programming, we have obtained a number of new results on the mathematical properties of optimal consumption under liquid­ ity constraints. For example, we have demonstrated how the response of optimal consumption to liquidity constraints is conditioned by the consumer’s intertem­ poral elasticity of substitution. Considered as a mathematical structure, this is shown to capture the effects of the following variables on the optimal consumption path: pure preference parameters, the interest rates variations and the structural parameters prevailing in the credit markets. In chapter 4, the dynamic Leontief model, which according to the conditions established in chapter 1, is one of the most successful applications of mathematical methods to economic policy analysis, is first considered as a control problem. We have then obtained the optimal consumption path for deterministic and stochastic dynamic Leontief models with substitute activities which are in turn formulated in deterministic and stochastic environments. Our solution uses Pontryagin’s max­ imum principle, Bellman’s method and Astrom’s Lemma on stochastic dynamic programming. Acknowledgements I wish to express my sincere gratitude to my supervisors Professor Laurence Harris and Dr. Massoud Karshenas. Their support and encouragement through­ out the progression of this work are highly appreciated. I have also benefited greatly from the Weekly Seminars by Economic Research Students. The advice and critical remarks of Professor Ben Fine, who supervised these seminars, to­ gether with the comments of the participants, have been a continuous source of inspiration. Most of all, I am grateful to my wife Parvin and to our chil­ dren, Nazanin and Jamshid, for their patience, understanding and help during the writing of this dissertation. To Professor Laurence Harris my supervisor to whom I shall remain indebted Contents Chapter 1 The Origin and Limitations of Mathematical Economics and its Implications for Economic Applications of Optimal Control: An Historical Approach 1.1 Introduction 9 1.2 Mathematisation of economics: hypotheses on the origin and significance of mathematical economics 12 1.2.1 Debreu’s incidentality hypothesis of early developments in mathematical economics 14 1.2.2 Cournot’s hypothesis: erroneous presentations and the poor mathematical knowledge 17 1.2.3 Walras’s hypothesis: the narrowness of ideas 20 1.2.4 von Neumann and Morgenstem’s hypothesis: the unfavourable circumstances 22 1.2.5 The hypothesis of one-dimensionalisation of economic analysis: the reduction of economic life to mechanical economic science 25 1. Mathematical economics and the formation of mechanical economic science 27 i) The nature of classical mathematical economics 28 ii) Mathematical economics: a remedy to multi-dimensional political economy 30 2. Mathematical economics and Marxian economics 35 1.3 The origin and formation of modern mathematical economics 39 1.3.1 Refutation of Debreu’s hypothesis on von Neumann and Morgenstem’s epoch-making contribution 41 1.3.2 The hypothesis of co-ordinated research programmes 44 1.4 Classical vs modern mathematical economics: attitudes and limitations 1.4.1 Attitudes 49 1.4.2 Limitations 53 -Can mathematical methods discover economic truths? 56 1.5 The rocky lane to successful co-ordination: the development of the relationship between mathematical economics and econometrics 60 5 1.5.1 The turning point in the rocky lane to co-ordination: the emergence of alternative strategies 64 1.6 Economic applications of optimal control theory as an illustration 66 1.6.1 Limitations of economic applications of optimal control theory 71 1.7 The logic of abstraction: the origin of limitations in mathematisation of economics and its implications for optimal control applications 73 1.8 Summary and concluding remarks 78 Chapter 2 Control Theory and Economic Policy Optimisation: Developments, Challenges and Prospects 2.1 Introduction 88 2.2 The beginning: classical control theory and economic stabilisation 93 2.3 Early applications of modern control theory to optimal economic policies 97 2.3.1 Optimality conditions in models of economic growth 98 2.3.2 Engineering control theory and econometrics 103 1. Contributions of control engineers and control theorists 104 2. Contributions of control engineering institutions 109 2.4 Stochastic and adaptive control applications to optimal economic policy design 2.4.1 Stochastic control applications 110 2.4.2 Adaptive control applications 113 2.5 The relationship between theoiy and observation: a critical analysis of the recent developments in macro-econometric modelling and the role of dynamic optimisation 116 2.5.1 Responses to the Cowles Commission’s traditional strategy 117 2.5.2 Data-instigated vs theory-based econometric models: the origin of the gap, the role of dynamic optimising models and a critical analysis of the attempts to bridge it 124 -A critical examination of Smith and Pesaran on the interplay of theory and observation 128 2.5.3 Speculations on the future course of developments 132 2.6 Rational expectations, the Lucas critique and the policy ineffectiveness debate 137 6 2.7 Time-inconsistency and the optimal control of macro-econometric models with rational expectations 144 2.7.1 Time-inconsistency, reputation and the stochastic environment 151 2.7.2 Rational expectations and econometric modelling in practice 154 2.8 Summary and concluding remarks 157 Chapter 3 Consumption Behaviour Under Liquidity Constraints: An Application of Optimal Control Theory 3.1 Introduction 164 3.2 The importance and implications of liquidity constraints in consumption models 167 3.3 Optimal consumption properties using the maximum principle 173 3.3.1 Consumer’s rate of time preference, interest rates and the optimal path of consumption 174 3.3.2 The Bernoulli case 177 3.3.3 Pontryagin’s maximum principle and Hall’s random walk hypothesis 180 3.4 Optimal consumption properties using the dynamic programming 181 3.4.1 Optimal consumption paths by direct search 184 3.4.2 Properties of the optimal consumption path 190 3.4.3 The Bernoulli case and optimal consumption functions 194 3.5 Properties of the optimal consumption path with liquidity constraints 198 3.6 The generalised Hamiltonian, liquidity constraints and the rejection of Hall’s random walk hypothesis 200 3.7 Time-varying interest rates and the properties of optimal consumption path under liquidity constraints 204 3.7.1 Time-varying interest rates and liquidity constraints 206 3.7.2 Liquidity constraints and the interaction between time-varying interest rate and the utility discount rate 208 3.7.3 Interest rates, intertemporal elasticity of substitution and liquidity constraints 211 3.8 Optimal consumption in a stochastic environment 216 3.8.1 Uncertain lifetimes and the optimal consumption behaviour 217 7 3.8.2 Income uncertainty and the optimal consumption behaviour 219 3.8.3 Implications of income uncertainty on the applications of optimal control theory to dynamic consumption decisions 225 3.9 Summary and concluding remarks 229 Chapter 4 Optimal Control of Dynamic Leontief Models 4.1 Introduction 234 4.2 The Leontief model and mathematical economic
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