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Intertemporal Choice and Competitive Equilibrium Intertemporal Choice and Competitive Equilibrium Kirsten I.M. Rohde °c Kirsten I.M. Rohde, 2006 Published by Universitaire Pers Maastricht ISBN-10 90 5278 550 3 ISBN-13 978 90 5278 550 9 Printed in The Netherlands by Datawyse Maastricht Intertemporal Choice and Competitive Equilibrium PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Maastricht, op gezag van de Rector Magni¯cus, prof. mr. G.P.M.F. Mols volgens het besluit van het College van Decanen, in het openbaar te verdedigen op vrijdag 13 oktober 2006 om 14.00 uur door Kirsten Ingeborg Maria Rohde UMP UNIVERSITAIRE PERS MAASTRICHT Promotores: Prof. dr. P.J.J. Herings Prof. dr. P.P. Wakker Beoordelingscommissie: Prof. dr. H.J.M. Peters (voorzitter) Prof. dr. T. Hens (University of ZÄurich, Switzerland) Prof. dr. A.M. Riedl Contents Acknowledgments v 1 Introduction 1 1.1 Intertemporal Choice . 2 1.2 Intertemporal Behavior . 4 1.3 General Equilibrium . 5 I Intertemporal Choice 9 2 Koopmans' Constant Discounting: A Simpli¯cation and an Ex- tension to Incorporate Economic Growth 11 2.1 Introduction . 11 2.2 The Result . 13 2.3 Related Literature . 18 2.4 Conclusion . 21 2.5 Appendix A. Proofs . 21 2.6 Appendix B. Example . 25 3 The Hyperbolic Factor: a Measure of Decreasing Impatience 27 3.1 Introduction . 27 3.2 The Hyperbolic Factor De¯ned . 30 3.3 The Hyperbolic Factor and Discounted Utility . 32 3.3.1 Constant Discounting . 33 3.3.2 Generalized Hyperbolic Discounting . 33 3.3.3 Harvey Discounting . 34 i Contents 3.3.4 Proportional Discounting . 34 3.3.5 Quasi-hyperbolic Discounting . 35 3.4 Conclusion . 35 3.5 Appendix . 36 4 Time-Tradeo® Sequences for Quantifying and Visualising the De- gree of Time Inconsistency, Using only Pencil and Paper 39 4.1 Introduction . 39 4.2 Discounted Utility . 42 4.3 Deriving the Degree of Time Inconsistency from TTO-Sequences . 44 4.4 TTO-Sequences to Measure Proneness to Arbitrage . 46 4.5 Illustrations of TTO-Sequences . 49 4.6 Deriving the Discount Function from TTO-Sequences . 52 4.7 Method of Experiment . 55 4.8 Results . 57 4.9 Discussion . 61 4.10 Appendix . 63 II General Equilibrium 67 5 Time-inconsistent Preferences in a General Equilibrium Model 69 5.1 Introduction . 69 5.2 The Model . 72 5.3 Naijve Societies . 75 5.4 Sophisticated Societies . 79 5.5 E±ciency - the Myopic Case . 83 5.5.1 Myopic Overall Pareto E±ciency . 84 5.5.2 Constrained Myopic Periodical E±ciency . 88 5.6 E±ciency - the Forward-looking Case . 89 5.6.1 Overall Pareto E±ciency . 91 5.6.2 Constrained Periodical E±ciency . 93 5.7 Conclusion . 96 5.8 Appendix . 97 6 On the Completeness of Complete Markets 111 6.1 Introduction . 111 6.2 The Primitives of the Economy . 115 6.3 Arrow-Debreu Markets . 118 ii Contents 6.4 Sequentially Complete Markets . 119 6.4.1 Naijve Behavior . 120 6.4.2 Sophisticated Behavior . 121 6.4.3 Properties of Sequentially Complete Equilibria . 123 6.5 Complete Markets . 126 6.5.1 Naijve Behavior . 127 6.5.2 Sophisticated Behavior . 128 6.5.3 Properties of Complete Equilibria . 130 6.6 Conclusion . 133 6.7 Appendix . 135 7 Arbitrage Opportunities in Frictionless Markets with Sophisti- cated Investors 147 Nederlandse Samenvatting (Summary in Dutch) 153 References 161 Author Index 169 Subject Index 173 Short Curriculum Vitae 179 iii Acknowledgments Economics and mathematics were my favorite subjects at high school. It was a pleasure to ¯nd the Dutch econometrics study program, which combines both ¯elds. Consequently, I began my studies in econometrics at Maastricht University in 1998. The ¯rst one and a half years of the curriculum were dominated by mathematics with very little economics. This caused me to consider switching over completely to economics. Then I became convinced of pursuing the subject of mathematical economics after taking the ¯rst mathematical economics course, which was taught by Jean-Jacques Herings. As a result of this course I began to consider pursuing a PhD degree in mathematical economics. I have always had a great interest in psychology. Thus, in formulating my the- sis topic with Jean-Jacques Herings, I expressed my wish to combine psychology with mathematical economics. We agreed that I would do research on general equilibrium and on decision theory, in particular intertemporal choice, where psy- chology is involved. We engaged Peter Wakker, an expert in decision theory to be my second promoter. I would like to express my profound gratitude to Jean-Jacques Herings and Peter Wakker for the very pleasant cooperation. I appreciate the due diligence with which they attended to my work. I would also like to thank Riccardo Welters for his support and the useful perspectives about PhD research that he shared with me during our walks to and from the railway station. I am grateful to Hans Peters for his support and career advise. Special thanks go to Arthur Attema and Han Bleichrodt for their cooperation in a few of the chapters of this dissertation. I am also grateful to Benjamin Arthur for correcting part of this thesis for the English language. I thank my o±ce mates Alexander Sebald and Jeroen van den Berg for the pleasant atmosphere they fostered in the o±ce. I am grateful to all my other v Acknowledgments colleagues from Maastricht University. Finally, special thanks go to my parents, Paul Rohde and Gea Rohde-Kotter, and my sister Ingrid Rohde for their constant support and endurance. vi 1 Introduction The majority of decisions that individuals make throughout their lives directly or indirectly concern tradeo®s over time. In our childhood going to school prevents us from immediately earning money by having a job, in return for an investment in our skills that will enable us to earn a higher income in the future. Many people save money or invest their money in stocks, which gives them fewer possibilities to spend money today in return for more possibilities to spend money in the future. Environmental issues mostly concern a tradeo® between bene¯cial consumption today and environmental damage in the future. With the current ageing of the population, increasing attention is paid to our pensionsystem. The costs to society caused by the ageing of the population are huge (Feldstein, 2005). For governments it is an increasingly important issue how to design an optimal pension and social security system. Thus, it is important to understand the intertemporal tradeo®s a society and its members are willing to make. This thesis studies such intertemporal tradeo®s. In economic and psychological models it is commonly assumed that individuals weigh immediate rewards and costs more than future ones. Thus, individuals dis- count future rewards and costs. Since Samuelson's (1937) introduction of constant discounted utility, this model has been a common assumption in economic analy- ses. Constant discounting, also known as exponential discounting, implies, under the additional assumption that preferences over costs and rewards do not change, that preferences over intertemporal tradeo®s do not change over time either. Un- der constant discounting, a decision made today will not have to be reconsidered in the future as long as there are no unforeseen events. Thus, constant discounting implies time-consistent preferences. Strotz (1956) was the ¯rst to discuss an economic model with time-inconsistent preferences. Since the publication of his paper, many psychological and economic studies have found that individuals do not discount constantly (Frederick, Loewen- stein, and O'Donoghue, 2002). Instead, hyperbolic discounting seems to be more 1 Introduction accurate in describing the intertemporal tradeo®s that individuals make. When discounting is not constant, but, for instance, hyperbolic, preferences typically do change over time, i.e. preferences are time-inconsistent. Thus, the extensive ¯ndings of non-constant discounting require the dismissal of the assumption of time-consistent preferences in economic models if we want these models to cap- ture reality. Today, much research is being done on this topic. The introduction of hyperbolic discounting in economic models has enabled economists to explain phenomena that could not be explained before, such as, for instance, a sharp drop in consumption at retirement and consumers reporting to save too little. The introduction of time-inconsistent preferences in economic models not only implies that we have to redo the analyses done before. It also requires a reconsi- deration of the concepts that constitute economic models. It is, for instance, not obvious how traditional welfare concepts should be extended in the case of time- inconsistent preferences, as will be discussed later. In order to design and evaluate governmental policies and economic systems, welfare criteria are essential. Until now, only very little research has been done on de¯ning appropriate welfare crite- ria when preferences change over time. In the second part of this thesis, welfare criteria are introduced and economic systems are analyzed under the assumption of time-inconsistent preferences. The development of more welfare criteria and further analysis of economic systems under the assumption of time-inconsistent preferences, should eventually enable governments and policy makers to, for in- stance, design appropriate pension systems that can deal with the ageing of the population. The ¯rst part of this thesis deals with intertemporal preferences at an indi- vidual level. The second part deals with dynamic general equilibrium models and associated welfare criteria. The next three sections give an introduction to the ¯rst part, the link between the two parts, and an introduction to the second part. 1.1 Intertemporal Choice The ¯rst part of this thesis considers intertemporal preferences of individuals.
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