Mathematical methods in theoretical economics

Instead of the cover......Page 1
Series......Page 2
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
CONTENTS......Page 7
Preface......Page 15
Acknowledgments......Page 17
Notation......Page 19
Part I POINT SET TOPOLOGY......Page 21
1.1 Introduction......Page 23
1.2 Preliminary Notions......Page 24
1.3 Functions......Page 26
1.4 Operations with Sets......Page 28
1.5 Algebra of Sets......Page 31
1.6 Duality Principle......Page 32
Problems......Page 34
Notes......Page 36
2.1 Products of Sets......Page 37
2.2 Sets of Points......Page 39
2.3 Metric Spaces......Page 44
2.4 Binary Relations......Page 45
2.5 Classes of Relations......Page 47
Exercises......Page 51
Problems......Page 53
Notes......Page 54
3.1 Preorderings and Orderings......Page 55
3.2 Preordered and Ordered Subsets......Page 58
3.3 Elements of Preordered and Ordered Sets......Page 59
3.4 Preorderings on Product Sets......Page 61
3.5 Lower and Upper Bounds......Page 62
Exercises......Page 64
Problems......Page 65
Notes......Page 66
4.1 Denumerable and Countable Sets......Page 67
4.2 Closedness......Page 70
4.3 Boundedness......Page 80
4.4 Compactness......Page 83
4.5 Connectedness......Page 90
4.6 Convexity......Page 92
4.7 Sum of Sets in $E^n$......Page 94
Exercises......Page 96
Problems......Page 98
Notes......Page 99
5 Point-to-Point Mappings......Page 100
5.1 Single-Valued Functions......Page 101
5.2 Classes of Mappings......Page 104
5.3 Continuous Mappings......Page 109
5.4 Mappings and Set Properties......Page 112
5.5 Convex and Concave Functions......Page 115
5.6 Linear Single-Valued Function......Page 120
Exercises......Page 121
Problems......Page 122
Notes......Page 123
6 Point-to-Set Mappings......Page 124
6.1 Multivalued Function......Page 125
6.2 Continuity......Page 129
6.3 Some Theorems on Maxima......Page 134
Exercises......Page 138
Problems......Page 139
Notes......Page 140
7.1 Introduction......Page 141
7.2 Simplicial Topology: Elementary Concepts......Page 144
7.3 Point-to-Point Mappings......Page 155
7.4 Point-to-Set Mappings......Page 157
Exercises......Page 160
Notes......Page 162
8 Algebraic Structures......Page 163
8.1 Abstract Systems and Homomorphisms......Page 164
8.2 Groups......Page 165
8.3 Rings, Integral Domains, and Fields......Page 167
Exercises......Page 169
Problems......Page 170
Notes......Page 171
9 General Equilibrium For Economics with a Finite Number of Agents and Commodities......Page 172
9.1 Frame of the Problem and Basic Assumption......Page 173
9.2 Producers and Supply......Page 175
9.3 Consumers and Demand......Page 176
9.4 General Equilibrium......Page 182
Exercises......Page 185
Problems......Page 187
Notes......Page 188
Part II VECTOR SPACES AND VECTOR-SPACE HOMOMORPHISMS......Page 189
10 Vector Spaces and Subspaces......Page 191
10.1 Vector Spaces over a Field......Page 192
10.2 Vectors and Operations with Vectors......Page 193
10.3 Metrics......Page 196
10.4 Subspaces......Page 199
10.5 Linearly Independent Set of Vectors......Page 203
10.6 Basis and Dimension......Page 206
10.7 Vector-Space Homomorphisms......Page 209
Exercises......Page 211
Notes......Page 212
11.1 Linear Transformations......Page 213
11.2 Analysis of Linear Transformations......Page 216
11.3 Nonsingular and Inverse Transformations......Page 218
11.4 Linear Functional and Dual Spaces......Page 220
11.5 Transpose of a Linear Transformation......Page 223
11.6 Linear Algebras......Page 225
Exercises......Page 227
Notes......Page 228
12.1 Operational Description of Linear Mappings......Page 229
12.2 Matrix over a Field......Page 231
12.3 Basic Operations with Matrices......Page 232
12.4 The Transpose of a Matrix......Page 237
12.5 Square Matrices......Page 239
12.6 Miscellaneous Square Matrices......Page 242
Exercises......Page 246
Problems......Page 247
Notes......Page 248
13 Rank, Equivalence, and Similarity......Page 249
13.1 The Rank of a Matrix......Page 250
13.2 Matrix Equivalence......Page 252
13.3 Matrix Similarity......Page 259
Exercises......Page 260
Notes......Page 261
14.1 The Determinant Function......Page 262
14.2 Evaluation of Determinants......Page 271
14.3 The Adjoint of a Square Matrix......Page 274
14.4 Methods of Matrix Inversion......Page 275
Exercises......Page 279
Problems......Page 280
Notes......Page 282
15.1 Introduction......Page 283
15.2 General Case: The Nonhomogeneous System......Page 286
15.3 Special Case: The Homogeneous System......Page 291
15.4 A Fundamental Property of Linear Systems of Equations......Page 293
Exercises......Page 294
Problems......Page 296
Notes......Page 297
16 Characteristic Vectors, Diagonalization, and Triangularization......Page 298
16.1 Characteristic Polynomials, Roots, and Vectors......Page 299
16.2 The Cayley-Hamilton Theorem......Page 305
16.3 Diagonalization and Triangularization......Page 307
16.4 Matrices of Functions and Series of Matrices......Page 313
Exercises......Page 317
Problems......Page 318
Notes......Page 319
17 Real Quadratic Forms......Page 320
17.1 Unconstrained Quadratic Forms......Page 321
17.2 Constrained Quadratic Forms......Page 329
Exercises......Page 335
Problems......Page 336
Notes......Page 337
18 Linear Inequalities, Hyperplanes, and Convex Cones......Page 338
18.1 Linear Inequality Systems......Page 339
18.2 Bounding Hyperplanes......Page 343
18.3 Convex Cones......Page 353
Exercises......Page 361
Problems......Page 362
Notes......Page 364
19 Nowtegative Square Matrices......Page 365
19.1 Indecomposable and Decomposable Matrices......Page 366
19.2 Nonnegative Indecomposable Matrices......Page 370
19.3 Nonnegative Decomposable Matrices......Page 371
19.4 The Matrices $\\mu I - A$ and $(\\mu I - A)^{-1}$......Page 372
Exercises......Page 375
Problems......Page 376
Notes......Page 377
20 Multisectoral Balanced Growth......Page 378
20.1 Framework and Assumptions......Page 379
20.2 Technological Expansion......Page 382
20.4 Von Neumann\'s Theorem......Page 383
20.5 Indecomposability and Duality......Page 385
Exercises......Page 387
Problems......Page 388
Notes......Page 389
References......Page 390
Index......Page 397