Mathematical economics

Title Page......Page 1
Copyright and Dedication......Page 2
Preface......Page 3
Contents\0......Page 7
Some Frequently Used Notations\0......Page 12
A. Scope of the Book\0......Page 15
B. Outline of the Book\0......Page 18
a. Some Basic Concepts and Notations\0......Page 25
b. Rn and Linear Space\0......Page 29
c. Basis and Linear Functions\0......Page 34
d. Convex Sets\0......Page 40
e. A Little Topology\0......Page 43
B. Separation Theorems\0......Page 59
C. Activity Analysis and the General Production Set\0......Page 69
A. Introduction\0......Page 79
B. Concave Programming-Saddle-Point Characterization\0......Page 86
a. Differentiation\0......Page 99
b. Unconstrained Maximum\0......Page 106
D. The Quasi-Saddle-Point Characterization\0......Page 110
Appendix to Section D: A Further Note on the Arrow-Hurwicz-Uzawa Theorem\0......Page 126
E. Some Extensions\0......Page 132
a. Quasi-Concave Programming\0......Page 133
b. The Vector Maximum Problem\0......Page 136
c. Quadratic Forms, Hessians, and Second-Order Conditions\"\0......Page 141
F. Some Applications\0......Page 153
a. Linear Programming\0......Page 154
b. Consumption Theory\0......Page 157
c. Production Theory\0......Page 160
d. Activity Analysis\0......Page 164
e. Ricardo\'s Theory of Comparative Advantage and Mill\'s Problem\0......Page 166
a. The Classical Theory of Optimization\0......Page 175
b. Comparative Statics\0......Page 178
c. The Second-Order Conditions and Comparative Statics\0......Page 179
d. An Example: Hicks-Slutsky Equation\0......Page 180
e. The Envelope Theorem\0......Page 184
A. Introduction\0......Page 193
a. Consumption Set\0......Page 199
b. Quasi-Ordering and Preference Ordering\0......Page 200
c. Utility Function\0......Page 203
d. The Convexity of Preference Ordering\0......Page 205
C. The Two Classical Propositions of Welfare Economics\0......Page 209
a. Introduction\0......Page 228
b. Some Basic Concepts\0......Page 231
c. Theorems of Debreu and Scarf\0......Page 237
d. Some Illustrations\0......Page 242
e. Some Remarks\0......Page 248
D. Demand Theory\0......Page 258
a. Various Concepts of Semi continuity\0......Page 273
b. The Maximum Theorem\0......Page 277
a. Historical Background\0......Page 279
b. McKenzie\'s Proof\0......Page 289
Appendix to Section E: On the Uniqueness of Competitive Equilibrium\0......Page 304
F. Programming, Pareto Optimum, and the Existence of  Competitive Equilibria\"\0......Page 309
A. Introduction\0......Page 319
B. Elements of the Theory of Differential Equations\0......Page 326
C. The Stability of Competitive Equilibrium-The Historical Background\0......Page 337
D. A Proof of Global Stability for the Three-Commodity Case (with Gross Substitutability)-An Illustration of the Phase Diagram Technique\0......Page 345
E. A Proof of Global Stability with Gross Substitutability-The n-commodity Case\0......Page 349
a. An Example of Gross Substitutability\0......Page 355
b. Scarfs Counterexample\0......Page 357
c. Consistency of Various Assumptions\0......Page 359
d. Nonnegative Prices\0......Page 360
G. The Tatonnement and the Non-Tatonnement Processes\0......Page 363
a. The Behavioral Background and the Tiltonnement Process\0......Page 364
b. The Tiltonnement and the Non-Tatonnement Processes\0......Page 365
H. Liapunov\'s Second Method\0......Page 371
A. Introduction\0......Page 383
B. Frobenius Theorems\0......Page 391
C. Dominant Diagonal Matrices\0......Page 404
a. Summary of Results\0......Page 415
b. Input-Output Analysis\0......Page 418
c. The Expenditure Lag Input-Output Analysis\0......Page 420
d. Multicountry Income Flows\0......Page 421
e. A Simple Dynamic Leontief Model\0......Page 422
f. Stability of Competitive Equilibrium\0......Page 423
g. Comparative Statics\0......Page 427
a. Statement of the Problem\0......Page 434
b. Euler\'s Equation\0......Page 437
c. Solutions of Illustrative Problems\0......Page 439
a. Introduction\0......Page 443
b. Spaces of Functions and Optimization\0......Page 445
c. Euler\'s Condition and a Sufficiency Theorem\0......Page 450
C. A Digression: The Neo-Classical Aggregate Growth Model\0......Page 456
a. Introduction\0......Page 468
b. The Case of a Constant Capital: Output Ratio\0......Page 474
c. Nonlinear Production Function with Infinite Time Horizon\0......Page 483
a. Introduction\0......Page 492
b. Model\0......Page 494
c. The Optimal Attainable Paths\0......Page 498
d. Sensitivity Analysis: Brock\'s Theorem\0......Page 504
a. Introduction\0......Page 510
b. Major Theorems\0......Page 515
c. Two Remarks\0......Page 521
a. Introduction\0......Page 527
b. The Output System\0......Page 531
c. The Price System\0......Page 541
d. Inequalities and Optimization Model (Solow)\0......Page 546
e. Morishima\'s Model of the Dynamic Leontief System\0......Page 551
Appendix to Section B: Some Problems in the Dynamic Leontief Model-The One-Industry Illustration\0......Page 565
a. Introduction\0......Page 583
b. The Basic Model and Optimality\0......Page 80
c. Free Disposability and the Condition for Optimality\0......Page 587
d. The Radner Turnpike Theorem\0......Page 591
a. Introduction\0......Page 599
b. The Model\0......Page 601
c. Finite Horizon: Optimality and Competitiveness\0......Page 604
d. Optimal Stationary Program\0......Page 607
e. O.S.P. and Eligibility\0......Page 611
f. Optimal Program for an Infinite Horizon Problem\0......Page 618
a. Optimal Control: A Simple Problem and the Maximum Principle\0......Page 624
b. The Proof of a Simple Case\0......Page 630
c. Various Cases\0......Page 633
d. An Illustrative Problem: The Optimal Growth Problem\0......Page 641
a. Regional Allocation of Investment\0......Page 651
b. Optimal Growth with a Linear Objective Function\0......Page 662
a. Constraint: g[x(t), u(t), t] ;:;; 0\"\0......Page 670
b. Hestenes\' Theorem\0......Page 675
c. A Sufficiency Theorem\0......Page 684
a. Optimal Growth Once Again\0......Page 691
b. Two Peak-Load Problems\0......Page 695
a. Introduction\0......Page 709
b. The Case of No Adjustment Costs\0......Page 712
c. The Case with Adjustment Costs\0......Page 721
d. Some Remarks\0......Page 727
Name Index......Page 745
Subject Index......Page 751