CONTENTS
Preface
Part 1 Introduction
1 The Nature of Dynamic Optimization
1.1 Salient Features of Dynamic Optimization Problems
1.2 Variable Endpoints and Transversality Conditions
1.3 The Objective Functional
1.4 Alternative Approaches to Dynamic Optimization
Part 2 The Calculus of Variations
2 The Fundamental Problem of the Calculus
of Variations
2.1 The Euler Equation
2.2 Some Special Cases
2.3 Two Generalizations of the Euler Equation
2.4 Dynamic Optimization of a Monopolist
2.5 Trading Off Inflation and Unemployment 3 Transversality Conditions
for Variable-Endpoint Problems
3.1 The General Transversality Condition
3.2 Specialized Transversality Conditions
3.3 Three Generalizations
3.4 The Optimal Adjustment of Labor Demand
xi
vii
viH CONTENTS
4 Second-Order Conditions
4.1 Second-Order Conditions
4.2 The Concavity ;convexity Sufficient Condition 81
4.3 The Legendre Necessary Condition 91 4.4 First and Second Variations 95
5 Infinite Planning Horizon 98
5.1 Methodological Issues of Infinite Horizon 98 5.2 The Optimal Investment Path of a Firm 103 5.3 The Optimal Social Saving Behavior 111
5.4 Phase-Diagram Analysis 117
5.5 The Concavity /Convexity Sufficient Condition Again 130
6 Constrained Problems 133
6.1 Four Basic Types of Constraints 134
6.2 Some Economic Applications Reformulated 144
6.3 The Economics of Exhaustible Resources 148
Part 3 Optimal Control Theory 159
7 Optimal Control: The Maximum Principle 161
7.1 The Simplest Problem of Optimal Control 162
7.2 The Maximum Principle 167
7.3 The Rationale of the Maximum Principle 177
7.4 Alternative Terminal Conditions 181
7.5 The Calculus of Variations and Optimal Control
Theory Compared 191
7.6 The Political Business Cycle 193 7.7 Energy Use and Environmental Quality 200
8 More on Optimal Control 205
8.1 An Economic Interpretation of the Maximum Principle 205
8.2 The Current-Value Hamiltonian 210
8.3 Sufficient Conditions 214
8.4 Problems with Several State and Control Variables 221
8.5 Antipollution Policy 234 9 Infinite-Horizon Problems 240
9.1 Transversality Conditions 240
9.2 Some Counterexamples Reexamined 244
9.3 The Neoclassical Theory of Optimal Growth 253 9.4 Exogenous and Endogenous Technological Progress 264
10 Optimal Control with Constraints 275
10.1 Constraints Involving Control Variables 275
10.2 The Dynamics of a Revenue-Maximizing Firm
10.3 State-Space Constraints
10.4 Economic Examples of State-Space Constraints
10.5 Limitations of Dynamic Optimization
Answers to Selected Exercise Problems
Index
