Contents
Preface to the Classics Edition xv
Preface xvii
PART ONE
INTRODUCTION
1
1 Economizing and the Economy 2
1.1 The Economizing Problem 2
1.2 Institutions of the Economy 3
1.3
PART TWO
STATIC
OPTIMIZATION
7
2 The Mathematical
Programming Problem 8
2.1 Formal Statement of the Problem 8
2.2 Types of Maxima, the Weierstrass Theorem, and the
Local-Global Theorem 12
2.3 Geometry of the Problem 16
ix
Economies 4
CONTENTS
3 Classical Programming 20
3.1 The Unconstrained Case 22
3.2 The Method of Lagrange
Multipliers 28
3.3 The Interpretation of the Lagrange
Multipliers 36
Problems 38
4 Nonlinear Programming 44
4.1 The Case of No Inequality
Constraints 46
4.2 The Kuhn-Tuclcer Conditions 49 4.3 The Kuhn-Tucker Theorem 56
4.4 The Interpretation of the
Lagrange Multipliers 60
4.5 Solution Algorithms 62
Problems 64
5 Linear Programming 72
5.1 The Dual Problems of Linear
Programming 77
5.2 The Lagrangian Approach;
Existence, Duality and Complementary
Slackness Theorems 79
5.3 The Interpretation of the Dual 86
5.4 The Simplex Algorithm 89
Problems 96
6 Game Theory 106
6.1 Classification and Description of
Games 107
6.2 Two-person, Zero-sum Games 110
6.3 Two-person Nonzero-sum Games 120
6.4 Cooperative Games 123
6.5 Games With Infinitely Many
Players 130
Problems 131
PART THREE
7 Theory of the Household 142
7.1 Commodity Space 142
7.2 The Preference Relation 143
7.3 The Neoclassical Problem of the
Household 148
APPLICATIONS OF
STATIC OPTIMIZATION
141
X
CONTENTS
7.4 Comparative Statics
of the Household 154
7.5 Revealed Preference 163
7.6 von Neumann-Morgenstern Utility 166
Problems 169
8 Theory of the Firm 178
8.1 The Production Function 178
8.2 The Neoclassical Theory of the
Firm 189
8.3 Comparative Statics of the Firm 196
8.4 Imperfect Competition.- Monopoly
and Monopsony 201
8.5 Competition Among the Few:
Oligopoly and Oligopsony 205
Problems 213
9 General Equilibrium 220
9.1 The Classical Approach: Counting
Equations and Unknowns 221
9.2 The Input-Output Linear
Programming Approach 227
9.3 The Neoclassical Excess Demand
Approach 238
9.4 Stability of Equilibrium 241
9.5 The von Neumann Model of
an Expanding Economy 246
Problems 249
10 Welfare Economics 258
10.1 The Geometry of the Problem
in the 2 X 2 X 2 Case 259
10.2 Competitive Equilibrium and
Pareto Optimality 269
10.3 Market Failure 278
10.4 Optimality Over Time 279
Problems 282
PART FOUR
DYNAMIC
OPTIMIZATION
291
11 The Control Problem 292
11.1 Formal Statement of the
Problem 293
xi
CONTENTS
11.2 Some Special Cases 298
11.3 Types of Control 299
11.4 The Control Problem as One of Programming in an Infinite
Dimensional Space; the Generalized
Weierstrass Theorem 302
12 Calculus of Variations 306
12.1 Euler Equation 308
12.2 Necessary Conditions 312
12.3 Transversality Condition 315
12.4 Constraints 317
Problems 320
13 Dynamic Programming 326
13.1 The Principle of Optimality and
Bellman's Equation 327
13.2 Dynamic Programming and the
Calculus of Variations 330
13.3 Dynamic Programming Solution
of Multistage Optimization Problems 333
Problems 338
14 Maximum Principle 344
14.1 Costafe Variables, the
Hamiltonian, and the Maximum Principle 345
14.2 The Interpretation of the
Costate Variables 351
14.3 The Max/mum Principle and the
Calculus of Variations 353
14.4 The Maximum Principle and
Dynamic Programming 355
145.
Problems 362
15 Differential Games 370
15.1 Two-Person Deterministic
Continuous Differential Games 371
15.2 Two-Person Zero-Sum Differential
Games 373
15.3 Pursuit Games 377
15.4 Coordination Differential Games 383
15.5 Noncooperative Differential Games 387
Problems 388
xii
Examples 357
CONTENTS XIII
PART FIVE
APPLICATIONS OF DYNAMIC OPTIMIZATION
397
16 Optimal Economic Growth 398
16.1 The Neoclassical Growth Model 399
16.2 Neoclassical Optimal Economic Growth 405
16.3 The Two Sector Growth Model 416
16.4 Heterogeneous Capital Goods 430
Problems 435
APPENDICES
449
Appendix A Analysis 450
A.I Sets 450
A.2 Relations and Functions 452
A.3 Metric Spaces 454
A.4 Vector Spaces 457
A.5 Convex Sets and Functions 460 A.6 Differential Calculus 465
A.7 Differential Equations 467
Appendix B Matrices 476
B.I Basic Definitions and Examples 476
B.2 Some Special Matrices 478
B.3 Matrix Relations and Operations 479
B.4 Scalar Valued Functions Defined on Matrices 484
B.5 Inverse Matrix 487
B.6 Linear Equations and linear
Inequalities 488
B.7 Linear Transformations;
Characteristic Roots and Vectors 493
B.8 Quadratic Forms 495
B.9 Matrix Derivatives 497
Index 501
