Analytical methods in economics 0745010369, 9780745010366, 0472081357

Table of contents :
Cover......Page 1
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Preface......Page 7
Contents......Page 11
Introduction......Page 17
Part 1 Preliminaries......Page 21
1 Some Basic Mathematics......Page 23
1.1.1 Sets......Page 24
1.1.2 Logical Statements......Page 28
1.1.3 Functions......Page 30
1.2 Euclidian Space......Page 33
1.3.1 Convergence......Page 45
1.3.2 Continuity......Page 48
1.3.3 Some Topological Concepts and Results......Page 52
1.4 Quadratic Forms......Page 56
1.5.1 Differentiation......Page 60
1.5.2 Some Important Results......Page 65
1.5.3 Higher Order Derivatives and Hessians......Page 69
1.5.4 Concave and Convex Functions......Page 72
1.5.5 Quasi-Concave Functions......Page 80
Exercises......Page 85
References......Page 90
Part 2 Nonlinear Programming and Microeconomics......Page 93
2.1 Introduction......Page 95
2.2 Characterization of Optimum by First-Order Condition......Page 105
2.3 Saddle-Point Characterization of Optimum......Page 123
2.4.1 Classical Optimization......Page 131
2.4.2 Comparison to Nonlinear Programming......Page 137
Exercises......Page 140
References......Page 142
3.1 Comparative Statics: Conventional Procedure......Page 145
3.2 The Envelope Theorem......Page 151
3.3 Elements of Microeconomic Theory......Page 159
3.4 Elasticity of Factor Substitution and Its Estimation......Page 171
3.5 On the Concept of Returns to Scale......Page 177
3.6 The Lc Chatclier-Samuelson Principle......Page 183
Exercises......Page 191
References......Page 201
4.1.1 Inferior Inputs......Page 207
4.1.2 Factor Prices and Shifts of the Average Cost Curve......Page 213
4.2 Marginal Cost Pricing......Page 216
4.3 Factor Prices and the Shape of Average Cost Curves......Page 221
4.4.1 The Case without Noivwagc Income......Page 226
4.4.2 The Case with Nonwagc Income......Page 230
4.5 Behavior of the Firm under Regulatory Constraint......Page 232
4.6.1 Introduction......Page 239
4.6.2 The Case of a Welfare-Maximizing Monopoly......Page 241
4.6.3 The Case of a Profit-Maximizing Monopoly......Page 250
4.7.1 Introduction......Page 255
4.7.2 Externality and the Pigouvian Scheme......Page 257
4.7.3 The Coase Theorem......Page 262
References......Page 267
Part 3 Economics of Uncertainty......Page 275
5.1 The Expected Utility Hypothesis......Page 277
5.2.1 Behavior toward Risk......Page 291
5.2.2 Arrow-Pratt Measures of Risk Aversion......Page 293
5.2.3 Examples......Page 298
5.2.4 Indifference Curves......Page 300
5.3.1 Insurance......Page 301
5.3.2 The Theory of the Firm......Page 306
5.3.3 Portfolio Choice......Page 310
5.3.4 Consumption and Savings Decision......Page 316
5.4.1 Asymmetry of Information, the Lemon Principle, and Signaling......Page 325
5.4.2 Preventive Activity and Asymmetry of Information in Insurance......Page 327
5.4.3 Adverse Selection......Page 332
5.5 Concluding Remarks......Page 335
References......Page 336
Part 4 Differential Equations and Economic Analysis......Page 343
6.1 Basic Concepts and Existence of a Solution......Page 345
6.2 Stability......Page 353
6.3.1 The Stability of Competitive Equilibrium: the Walrasian Process......Page 365
6.3.2 The Stability of Macrocconomic Equilibrium......Page 370
6.3.3 Neoclassical Growth Model......Page 375
6.3.4 On the Phillips Curve......Page 378
6.3.5 Tobin's "Walras-Keynes-Phillips" Model: An Application of the Routh-Hurwitz Theorem......Page 385
6.4 Competitive Equilibrium for the Three-Commodity Case......Page 390
References......Page 396
7.1 Linear Systems......Page 403
7.2 Homogeneous Linear Systems on the Plane......Page 405
7.3 Dynamic Behavior of the Solution on the Plane......Page 415
7.4.1 Local Behavior of the Trajectories on the Plane......Page 427
7.4.2 Stability of the Nonlinear System: Liapunov's Direct Method......Page 431
7.4.3 Local Asymptotic Stability......Page 434
Exercises......Page 435
References......Page 436
8.1 Static Macroequilibrium and Its Stability......Page 439
8.2.1 Model......Page 443
8.2.2 Steady State......Page 447
8.2.3 Stability and Instability......Page 450
8.2.4 The Source of Instability......Page 454
8.3.1 Introduction......Page 456
8.3.2 Model and Momentary Equilibrium......Page 458
8.3.3 Long-Run Analysis......Page 460
8.3.4 The Source of Stability......Page 464
8.3.5 Appendix to Section 8.3......Page 465
References......Page 467
Part 5 Optimal Control Theory and Applications......Page 469
9.1 Pontryagin's Maximum Principle......Page 471
9.2 Various Cases......Page 481
9.3.1 Calculus of Variations......Page 488
9.3.2 An Illustrative Problem by Pontryagin et al......Page 492
9.4 Optimal Growth Problem......Page 495
References......Page 508
10.1 A General Theorem......Page 513
10.2 Optimal Monetary Policy......Page 518
10.3 Savings: Permanent Income Hypothesis......Page 524
10.4 Investment......Page 534
10.4.1 The "Neoclassical" Theory......Page 535
10.4.2 The Adjustment Cost Approach......Page 542
10.4.3 Tobin's $q$, the Value of the Firm, and Keynes's Rule of Marginal Efficiency of Capital......Page 545
References......Page 555
11.1 The Main Theorem......Page 561
11.2 Consumer's Lifetime Allocation Process: Finite Horizon Case......Page 567
11.3 Isoperimetric Problem......Page 573
11.4 Spatial Pricing Problem......Page 577
Appendix to Section 11.4: Spatial Pricing Problem and Nonlinear Programming......Page 584
References......Page 588
Appendices......Page 591
A.1 Matrices and Basic Operations......Page 593
A.2 Transposes, Vectors, Simultaneous Equations, and Nonsingular Matrices......Page 596
A.3 Linear Independence, Rank, and Linear Subspaces......Page 601
Linear Subspaces and Ranks......Page 602
A.4 Matrices as Linear Transformations......Page 605
A.5.1 Determinants, Laplace Expansion, and Cramer's Rule......Page 610
A.5.2 Partitioned Matrices and Determinants......Page 614
A.6 Eigenvalues, Eigenvectors, and Symmetric Matrices......Page 615
A.7 Negative or Positive Definite Matrices......Page 620
References......Page 623
B.1 Seven Kinds of Concavity......Page 625
B.2 The Role of Nonvanishing Gradients......Page 631
B.3 Quasi-Concave Programming......Page 635
References......Page 639
C.1 Introduction......Page 641
C.2 Some Basics......Page 646
C.3 Computing the Hicksian Measures......Page 647
C.4 The Correct Welfare Measure......Page 650
C.5 Hicksian Measures under Constancy of Marginal Utility of Income......Page 660
Concluding Remarks......Page 663
References......Page 664
Indices......Page 669
Name......Page 671
Subject......Page 677
Back cover......Page 693