Differential Equations, Discrete Systems and Control Economic Models

by

Aristide Halanay Faculty ofMathematics, University ofBucharest, Bucharest, Romania

and

Judita Samuel Centre of Mathematical Statistics, Romanian Academy, Bucharest, Romania

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

Contents

Preface ix

About the Notations xi

Introduction xiii

1 Linear and Affine Differential Equations.

Equations with Separated Variables 1

1.1 Differential Equations Modelling Growth Processes 1 1.2 Linear Differential Equations 5 1.3 Linear Affine Differential Equations 6 1.4 Simplest Models of Price Evolution in a Market Economy 3 1.5 Discrete - Time Models for Price Evolution 12 1.6 Simplest Models for Economic Growth 14 1.7 Discrete - Time Models for Economic Growth 15 1.8 Production Functions 16 1.9 Equations with Separated Variables 20 1.10 Notes and References 22

2 Linear Differential Equations with Constant

Coefficients 23

2.1 Second Order Differential Equations with Constant Coefficients 23

2.2 Discrete - Time Second Order Linear Equations 28 2.3 Price Evolution in the Presence of Inventories 31 2.4 Economic Growth Models 35 2.5 Second Order Linear Affine Equations 41 2.6 The Phillips Model with Several Types of Autonomous

Investment 48 2.7 Higher Order Linear Differential Equations with Constant

Coefficients 57 2.8 Discrete - Time Linear Affine Equations 63 2.9 The Samuelson - Hicks Model for Economic Growth 68 2.10 Notes and References 73

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3 L i n e a r S y s t e m s w i t h C o n s t a n t C o e f f i c i e n t s 74

3.1 General Form of Solutions 74 3.2 Matrix Exponential 80 3.3 Linear Affine Systems 84 3.4 Economic Models 88 3.5 Leontieff - type Models 92 3.6 Phase - Portrait for Second Order Linear Systems with

Constant Coefficients 96 3.7 Notes and References 102

4 General Theory of Nonlinear Systems. Stability 103

4.1 Existence and Uniqueness Theorem for the Initial Value Problem 103

4.2 Equilibria. Stability. Continuous Time 114 4.3 Stability. Discrete Time 120 4.4 Discrete-Time Logistic Equation 125 4.5 Stahle Polynomials 129 4.6 Some Properties of Matrices that occur

in Economic Models 133 4.7 Notes and References 143

5 Numerical Solution of Differential Equations 144

5.1 Euler Method 144 5.2 Richardson Extrapolation 147 5.3 Predictor - Corrector Methods 151 5.4 Numerical Quadrature 154 5.5 Adams Type Methods 156 5.6 Stiff Systems 158 5.7 Some Applications of Differential Equations in Numerical

Analysis and Optimization 160 5.7.1 Implicit Functions 160 5.7.2 Nonlinear Equations 162 5.7.3 Free Optimization 164 5.7.4 Linear Programming 167

5.8 Notes and References 170

6 Control Systems. Stabil ization of Linear Systems 171

6.1 Stabilization Problem. Stabilization by Linear State Feed-Back 171

6.2 Stabilization of Linear Systems by Using a Controller 180

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6.3 Stabilization in an Economic Growth Model 181 6.4 A Monetary Policy Model 184 6.5 Stabilization of Discrete-Time Systems 190 6.6 A Discrete-Time Monetary Policy Model 193 6.7 Notes and References 197

7 Optimal Stabilization 198 7.1 Linear-Quadratic Optimization on Infinite Horizon.

Continuous Time 198 7.2 Application to a Price Model 209 7.3 Optimal Stabilization in Discrete Time 211 7.4 Optimal Stabilization in a Discrete-Time Model of Price

Evolution 224 7.5 Notes and References 227

8 Linear-Quadrat icOpt imizat ion on Finite Horizon 228

8.1 Continuous Time 228 8.2 Applications 237 8.3 Discrete Time 247 8.4 Applications in Discrete Time 250 8.5 ATracking Problem 254 8.6 A Simple Differential Game 263 8.7 Notes and References 269

9 Some Unconstrained Dynamic

Optimization Problems 270

9.1 The Simplest Problem of the Calculus of Variations 270 9.2 A Macroeconomic Growth Model 278 9.3 A Discrete - Time Variational Problem 284 9.4 An Application 286 9.5 Unrestricted Optimal Control Problem in Discrete Time 288 9.6 An Application 291 9.7 Optimization with Linear Dynamics and Linear Cost.

Continuous Time 294 9.8 Some Microeconomic Models 297

9.8.1 Optimization of the Maintenance of an Equipment 297

9.8.2 A Financial Policy Model at Enterprise Level 298 9.9 Optimization with Linear Dynamics and Linear Cost.

Discrete Time 301

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9.10 Applications 303 9.11 Notes and References 305

10 General Problem of Optimal Control 306 10.1 Problem Statement. General Theorems 30G

10.1.1 The Optimal Control Problem 306 10.1.2 Necessary optimality condit.ions 309

10.2 Optimum Capital Accumulation under the Minimum Time Objective 315

10.3 Reduction of Problems with Free Initial and Final Time to Problems on Fixed Horizon 320

10.4 An Abstract Multiplier Rule 327 10.5 Proofof Theorem 10.1 339 10.6 Notes and References 350

References 351

Index 355