LectureNotesManuelli - Notes on Discrete Time Economic...

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Notes on Discrete Time Economic Models: The Growth Model Rody Manuelli Department of Economics. University of Wisconsin-Madison August 8, 2007
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ii
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Contents Introduction vii 1 Basic Results in Dynamic Optimization 1 1.1 Concavity and Optimization . . . . . . . . . . . . . . . . . . . . . 1 1.2 Optimization Subject to Constraints . . . . . . . . . . . . . . . . 2 1.3 A Two Period Model . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 An Extension of the Kuhn-Tucker Theorem . . . . . . . . . . . . 8 2 The Basic Growth Model: Steady States 15 2.1 The Basic Growth Model . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The Planner’s Problem . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . . . 19 3 The Basic Growth Model: Dynamics 27 3.1 Global Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.1 The E ff ects of Permanent Shocks . . . . . . . . . . . . . . 36 3.1.2 The E ff ects of Temporary Shocks . . . . . . . . . . . . . . 37 3.2 Local Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.1 Constant Saving Rates . . . . . . . . . . . . . . . . . . . . 49 3.3.2 Population Growth and Technical Change . . . . . . . . . 49 4 Competitive Equilibrium 53 4.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Taxes and De fi cits . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 The Structure of Taxation: Steady State Implications . . . . . . 75 4.3.1 Basic Steady State Analysis . . . . . . . . . . . . . . . . . 77 4.3.2 Changing the Tax Rate on Capital Income . . . . . . . . 79 4.3.3 A Digression: The User Cost of Capital . . . . . . . . . . 81 4.3.4 Changing the Tax Rate on Labor Income . . . . . . . . . 82 4.3.5 Digression: Partial Equilibrium Analysis . . . . . . . . . . 83 4.3.6 Individual and Corporate Taxation . . . . . . . . . . . . . 87 4.3.7 A Balanced Budget (Revenue Neutral) Change in Tax Rates 93 4.3.8 Taxation and Human Capital . . . . . . . . . . . . . . . . 97 iii
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iv CONTENTS 4.4 Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4.1 The Distribution of Individual Consumption . . . . . . . 104 4.4.2 The Time Path of Consumption Inequality . . . . . . . . 107
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Preface These notes were originally developed as part of the fi rst year Macroeconomic Theory course at the University of Wisconsin. Several generations of students deserve credit for fi nding numerous errors, and pointing alternative ways of discussing and presenting the material. They deserve recognition. The last iteration has bene fi tted from detailed comments by Nick Mader. v
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vi PREFACE
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Introduction The purpose of these notes is to brie fl y present some results from dynamic optimization as they apply to standard exogenous growth models. In addition, they also describe some basic techniques that can be used to study the dynamics of these models. The main application is the standard growth model. This basic model is used to study a variety of issues, ranging from population growth, the impact of new technologies on income distribution, and the e ff ects of alternative tax and spending policies. The notes are designed so that the exercises are an integral part of them (even though no answers are supplied at this time). It is suggested that the reader should try to work through as many of these exercises as possible. vii
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viii INTRODUCTION
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Chapter 1 Basic Results in Dynamic Optimization This section presents —with no proofs— some elementary results in dynamic opti- mization. The objective is to provide a brief overview of the basic mathematical techniques necessary to analyze dynamic models. Even though these tools will su ce to follow the arguments in these notes, they are not a good substitute for the formal study of dynamic optimization.
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