Mathematical Economics

Preface to the second edition (iii)......Page 5
Preface to the first edition (vi)......Page 7
Contents (viii)......Page 9
A. Scope of the Book (xiii)......Page 14
B. Outline of the Book (xvi)......Page 17
Some Frequently Used Notations (3)......Page 24
a. Some basic concepts and notations (5)......Page 26
b. ��ⁿ and linear space (9)......Page 30
c. Basis and linear functions (14)......Page 35
d. Convex sets (20)......Page 41
e. A little topology (23)......Page 44
Footnotes (36)......Page 57
References (38)......Page 59
B. Separation Theorems (39)......Page 60
References (48)......Page 69
C. Activity Analysis and the General Production Set (49)......Page 70
Footnotes (57)......Page 78
References (58)......Page 79
A. Introduction (59)......Page 80
Footnotes (65)......Page 86
B. Concave Programming—Saddle-Point Characterization (66)......Page 87
Footnotes (76)......Page 97
References (78)......Page 99
a. Differentiation (79)......Page 100
b. Unconstrained Maximum (86)......Page 107
Footnotes (89)......Page 110
D. The Quasi-Saddle-Point Characterization (90)......Page 111
Footnotes (104)......Page 125
References (105)......Page 126
Appendix to Section D: A Further Note on the Arrow-Hurwicz-Uzawa Theorem (106)......Page 127
Footnotes (111)......Page 132
E. Some Extensions (112)......Page 133
a. Quasi-concave programming (113)......Page 134
b. The vector maximum problem (116)......Page 137
c. Quadratic forms, Hessians, and second-order conditions (121)......Page 142
Footnotes (131)......Page 152
References (132)......Page 153
a. Some applications (133)......Page 154
b. The envelope theorem (137)......Page 158
c. Elements of micro theory (141)......Page 162
d. Elasticities of factor substitution, duality, and translog estimation (144)......Page 165
Footnotes (150)......Page 171
References (153)......Page 174
G. Linear Programming and Classical Optimization (155)......Page 176
a. Linear programming (156)......Page 177
b. The classical theory of optimization (159)......Page 180
c. Comparative statics (161)......Page 182
d. The second-order conditions and comparative statics (162)......Page 183
e. An example: Hicks-Slutsky equation (163)......Page 184
Footnotes (166)......Page 187
References (167)......Page 188
A. Introduction (169)......Page 190
Footnotes (174)......Page 195
a. Consumption set (175)......Page 196
b. Quasi-ordering and preference ordering (176)......Page 197
c. Utility function (179)......Page 200
d. The convexity of preference ordering (181)......Page 202
Footnotes (183)......Page 204
References (184)......Page 205
C. The Two Classical Propositions of Welfare Economics (185)......Page 206
Footnotes (201)......Page 222
References (203)......Page 224
a. Introduction (204)......Page 225
b. Some basic concepts (207)......Page 228
c. Theorems of Debreu and Scarf (213)......Page 234
d. Some illustrations (218)......Page 239
e. Some remarks (224)......Page 245
Footnotes (229)......Page 250
References (232)......Page 253
D. Demand Theory (234)......Page 255
Footnotes (248)......Page 269
a. Various concepts of semicontinuity (249)......Page 270
b. The maximum theorem (253)......Page 274
Footnotes (254)......Page 275
a. Historical background (255)......Page 276
b. McKenzie's proof (265)......Page 286
Footnotes (274)......Page 295
References (278)......Page 299
Appendix to Section E: On the Uniqueness of Competitive Equilibrium (280)......Page 301
Footnotes (283)......Page 304
References (284)......Page 305
F. Programming, Pareto Optimum, and the Existence of Competitive Equilibria (285)......Page 306
Footnotes (291)......Page 312
References (293)......Page 314
A. Introduction (295)......Page 316
Footnotes (301)......Page 322
B. Elements of the Theory of Differential Equations (302)......Page 323
Footnotes (311)......Page 332
C. The Stability of Competitive Equilibrium—the Historical Background (313)......Page 334
Footnotes (319)......Page 340
References (320)......Page 341
D. A Proof of Global Stability for the Three-Commodity Case (with Gross Substitutability)—An Illustration of the Phase Diagram Technique (321)......Page 342
Footnotes (324)......Page 345
E. A Proof of Global Stability with Gross Substitutability—the ��-Commodity Case (325)......Page 346
Footnotes (329)......Page 350
References (330)......Page 351
a. An example of gross substitutability (331)......Page 352
b. Scarf's counterexample (333)......Page 354
c. Consistency of various assumptions (335)......Page 356
d. Nonnegative prices (336)......Page 357
References (338)......Page 359
G. The Tǎtonnement and the Non-Tǎtonnement Process (339)......Page 360
a. The Behavioral background and the tǎtonnement process (340)......Page 361
b. The tǎtonnement and the non-tǎtonnement processes (341)......Page 362
Footnotes (345)......Page 366
References (346)......Page 367
H. Liapunov's Second Method (347)......Page 368
Footnotes (356)......Page 377
References (357)......Page 378
A. Introduction (359)......Page 380
References (366)......Page 387
B. Frobenius Theorems (367)......Page 388
Footnotes (378)......Page 399
References (379)......Page 400
C. Dominant Diagonal Matrices (380)......Page 401
References (390)......Page 411
a. Summary of Results (391)......Page 412
b. Input-output analysis (394)......Page 415
c. The expenditure lag input-output analysis (396)......Page 417
d. Multicountry income flows (397)......Page 418
e. A simple dynamic Leontief model (398)......Page 419
f. Stability of competitive equilibrium (399)......Page 420
g. Comparative statics (403)......Page 424
Footnotes (406)......Page 427
References (408)......Page 429
a. Statement of the problem (410)......Page 431
b. Euler's equation (413)......Page 434
c. Solutions of illustrative problems (415)......Page 436
References (418)......Page 439
a. Introduction (419)......Page 440
b. Spaces of functions and optimization (421)......Page 442
c. Euler's condition and a sufficiencompany theorem (426)......Page 447
Footnotes (430)......Page 451
References (431)......Page 452
C. A Digression: the Neo-Classical Aggregate Growth Model (432)......Page 453
Footnotes (442)......Page 463
References (443)......Page 464
a. Introduction (444)......Page 465
b. The case of a constant capital:output ratio (450)......Page 471
c. Nonlinear production function with infinite time horizon (459)......Page 480
Footnotes (464)......Page 485
References (466)......Page 487
a. Introduction (468)......Page 489
b. Model (470)......Page 491
c. The optimal attainable paths (474)......Page 495
d. Sensitivity analysis: Brock's theorem (480)......Page 501
Footnotes (484)......Page 505
References (485)......Page 506
a. Introduction (486)......Page 507
b. Major theorems (491)......Page 512
c. Two remarks (497)......Page 518
References (501)......Page 522
a. Introduction (503)......Page 524
b. The output system (507)......Page 528
c. The price system (517)......Page 538
d. Inequalities and optimization model (Solow) (522)......Page 543
e. Morishima's model of the dynamic Leontief system (527)......Page 548
Footnotes (535)......Page 556
References (539)......Page 560
Appendix to Section B: Some Problems in the Dynamic Leontief Model—The One-Industry Illustration (541)......Page 562
Footnotes (554)......Page 575
References (556)......Page 577
a. Introduction (559)......Page 580
b. The basic model and optimality (561)......Page 582
c. Free disposability and the conditions for optimality (563)......Page 584
d. The Radner turnpike theorem (567)......Page 588
Footnotes (572)......Page 593
References (573)......Page 594
a. Introduction (575)......Page 596
b. The model (577)......Page 598
c. Finite horizon: optimality and competitiveness (580)......Page 601
d. Optimal stationary program (583)......Page 604
e. O.S.P. and eligibility (587)......Page 608
f. Optimal program for an infinite horizon problem (594)......Page 615
References (598)......Page 619
a. Optimal control: a simple problem and the maximum principle (600)......Page 621
b. The proof of a simple case (606)......Page 627
c. Various cases (609)......Page 630
d. An illustrative problem: the optimal growth problem (617)......Page 638
Footnotes (625)......Page 646
a. Regional allocation of investment (627)......Page 648
b. Optimal growth with a linear objective function (638)......Page 659
Footnotes (644)......Page 665
References (645)......Page 666
a. Constraint: g[x(t), u(t), t] ≥ 0 (646)......Page 667
b. Hestenes' Theorem (651)......Page 672
c. A sufficiency theorem (660)......Page 681
Footnotes (665)......Page 686
a. Optimal growth once again (667)......Page 688
b. Two peak-load problems (671)......Page 692
Footnotes (683)......Page 704
References (684)......Page 705
a. Introduction (685)......Page 706
b. The case of no adjustment costs (688)......Page 709
c. The case with adjustment costs (697)......Page 718
d. Some remarks (703)......Page 724
Footnotes (712)......Page 733
References (715)......Page 736
Name index (721)......Page 741
Subject index (725)......Page 745