Foundations of mathematical and computational economics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 542 S. graph. Darst. 260 mm x 193 mm |
| ISBN: | 9783642137471 |
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MARC
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| 100 | 1 | |a Dadkhah, Kamran |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Foundations of mathematical and computational economics |c Kamran Dadkah |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2011 | |
| 300 | |a XVI, 542 S. |b graph. Darst. |c 260 mm x 193 mm | ||
| 336 | |b txt |2 rdacontent | ||
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| _version_ | 1805094152148353024 |
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IMAGE 1
CONTENTS
PART I BASIC CONCEPTS AND METHODS
1 MATHEMATICS, COMPUTATION, AND ECONOMICS . . . . . . . . . . . . 3
1.1 MATHEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 PHILOSOPHIES OF MATHEMATICS . . . . . . . . . . . . . . . . . . 9
1.3 WOMEN IN MATHEMATICS . . . . . . . . . . . . . . . . . . . . . 11
1.4 COMPUTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 MATHEMATICS AND ECONOMICS . . . . . . . . . . . . . . . . . . 14
1.6 COMPUTATION AND ECONOMICS . . . . . . . . . . . . . . . . . . 14
2 BASIC MATHEMATICAL CONCEPTS AND METHODS . . . . . . . . . . . . . 17
2.1 FUNCTIONS OF REAL VARIABLES . . . . . . . . . . . . . . . . . . . 17
2.1.1 VARIETY OF ECONOMIC RELATIONSHIPS . . . . . . . . . . 22
2.1.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2.1 SUMMATION NOTATION * . . . . . . . . . . . . . . . . 24
2.2.2 LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 CONVERGENT AND DIVERGENT SERIES . . . . . . . . . . . 27
2.2.4 ARITHMETIC PROGRESSION . . . . . . . . . . . . . . . . 29
2.2.5 GEOMETRIC PROGRESSION . . . . . . . . . . . . . . . . . 31
2.2.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 PERMUTATIONS, FACTORIAL, COMBINATIONS, AND THE BINOMIAL EXPANSION .
. . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 LOGARITHM AND EXPONENTIAL FUNCTIONS . . . . . . . . . . . . . 38
2.4.1 LOGARITHM . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 BASE OF NATURAL LOGARITHM, E . . . . . . . . . . . . . 40
2.4.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 MATHEMATICAL PROOF . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 DEDUCTION, MATHEMATICAL INDUCTION, AND PROOF BY CONTRADICTION . .
. . . . . . . . . . . . . . . 42
2.5.2 COMPUTER-ASSISTED MATHEMATICAL PROOF . . . . . . . . 44
2.5.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 45
IX
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X CONTENTS
2.6 TRIGONOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6.1 CYCLES AND FREQUENCIES . . . . . . . . . . . . . . . . 50
2.6.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 COMPLEX NUMBERS . . . . . . . . . . . . . . . . . . . . . . . 51
2.7.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 56
3 BASIC CONCEPTS OF COMPUTATION . . . . . . . . . . . . . . . . . . . 57
3.1 ITERATIVE METHODS . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 NAMING CELLS IN EXCEL . . . . . . . . . . . . . . . . . 60
3.2 ABSOLUTE AND RELATIVE COMPUTATION ERRORS . . . . . . . . . . . 61
3.3 EFF ICIENCY OF COMPUTATION . . . . . . . . . . . . . . . . . . . 62
3.4 O AND O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 SOLVING AN EQUATION . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 BASIC CONCEPTS AND METHODS OF PROBABILITY THEORY AND STATISTICS 69 4.1
PROBABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS . . . . . . . . 72
4.3 MARGINAL AND CONDITIONAL DISTRIBUTIONS . . . . . . . . . . . . 74
4.4 THE BAYES THEOREM . . . . . . . . . . . . . . . . . . . . . . . 79
4.5 THE LAW OF ITERATED EXPECTATIONS . . . . . . . . . . . . . . . . 81
4.6 CONTINUOUS RANDOM VARIABLES . . . . . . . . . . . . . . . . . 82
4.7 CORRELATION AND REGRESSION . . . . . . . . . . . . . . . . . . . 85
4.8 MARKOV CHAINS . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
PART II LINEAR ALGEBRA
5 VECTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 95
5.1 VECTORS AND VECTOR SPACE . . . . . . . . . . . . . . . . . . . . 96
5.1.1 VECTOR SPACE . . . . . . . . . . . . . . . . . . . . . . 100
5.1.2 NORM OF A VECTOR . . . . . . . . . . . . . . . . . . . . 102
5.1.3 METRIC . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.4 ANGLE BETWEEN TWO VECTORS AND THE CAUCHY-SCHWARZ THEOREM . . . . .
. . . . . 105
5.1.5 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 ORTHOGONAL VECTORS . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.1 GRAMM-SCHMIDT ALGORITHM . . . . . . . . . . . . . . 109
5.2.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 111
6 MATRICES AND MATRIX ALGEBRA . . . . . . . . . . . . . . . . . . . .
113
6.1 BASIC DEF INITIONS AND OPERATIONS . . . . . . . . . . . . . . . .
113
6.1.1 SYSTEMS OF LINEAR EQUATIONS . . . . . . . . . . . . . 118
6.1.2 COMPUTATION WITH MATRICES . . . . . . . . . . . . . . 121
6.1.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 THE INVERSE OF A MATRIX . . . . . . . . . . . . . . . . . . . . .
123
6.2.1 A NUMBER CALLED THE DETERMINANT . . . . . . . . . . 127
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CONTENTS XI
6.2.2 RANK AND TRACE OF A MATRIX . . . . . . . . . . . . . . 132
6.2.3 ANOTHER WAY TO FIND THE INVERSE OF A MATRIX . . . . . 133
6.2.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 SOLVING SYSTEMS OF LINEAR EQUATIONS USING MATRIX ALGEBRA . . 137
6.3.1 CRAMER'S RULE . . . . . . . . . . . . . . . . . . . . . 139
6.3.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 142
7 ADVANCED TOPICS IN MATRIX ALGEBRA . . . . . . . . . . . . . . . . .
143
7.1 QUADRATIC FORMS AND POSITIVE AND NEGATIVE DEF INITE MATRICES . . . .
. . . . . . . . . . . . . . . . . . . . 143
7.1.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 GENERALIZED INVERSE OF A MATRIX . . . . . . . . . . . . . . . . .
147
7.2.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 ORTHOGONAL MATRICES . . . . . . . . . . . . . . . . . . . . . . 150
7.3.1 ORTHOGONAL PROJECTION . . . . . . . . . . . . . . . . . 150
7.3.2 ORTHOGONAL COMPLEMENT OF A MATRIX . . . . . . . . . 152
7.3.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 EIGENVALUES AND EIGENVECTORS . . . . . . . . . . . . . . . . . . 153
7.4.1 COMPLEX EIGENVALUES . . . . . . . . . . . . . . . . . 159
7.4.2 REPEATED EIGENVALUES . . . . . . . . . . . . . . . . . 160
7.4.3 EIGENVALUES AND THE DETERMINANT AND TRACE OF A MATRIX . . . . . .
. . . . . . . . . . . . . . . . . . 164
7.4.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 166
7.5 FACTORIZATION OF SYMMETRIC MATRICES . . . . . . . . . . . . . . 167
7.5.1 SOME INTERESTING PROPERTIES OF SYMMETRIC MATRICES . 167 7.5.2
FACTORIZATION OF MATRIX WITH REAL DISTINCT ROOTS . . . 170 7.5.3
FACTORIZATION OF A POSITIVE DEF INITE MATRIX . . . . . . 172
7.5.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 176
7.6 LU FACTORIZATION OF A SQUARE MATRIX . . . . . . . . . . . . . . 176
7.6.1 CHOLESKY FACTORIZATION . . . . . . . . . . . . . . . . 181
7.6.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 182
7.7 KRONECKER PRODUCT AND VEC OPERATOR . . . . . . . . . . . . . . 183
7.7.1 VECTORIZATION OF A MATRIX . . . . . . . . . . . . . . . 185
7.7.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 185
PART III CALCULUS
8 DIFFERENTIATION: FUNCTIONS OF ONE VARIABLE . . . . . . . . . . . . .
189
8.1 MARGINAL ANALYSIS IN ECONOMICS . . . . . . . . . . . . . . . . 189
8.1.1 MARGINAL CONCEPTS AND DERIVATIVES . . . . . . . . . . 190
8.1.2 COMPARATIVE STATIC ANALYSIS . . . . . . . . . . . . . . 192
8.1.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 LIMIT AND CONTINUITY . . . . . . . . . . . . . . . . . . . . . . 194
8.2.1 LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.2.2 CONTINUITY . . . . . . . . . . . . . . . . . . . . . . . 196
8.2.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 198
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XII CONTENTS
8.3 DERIVATIVES . . . . . . . . . . . . . . . . . . . . . . . . . . .
198
8.3.1 GEOMETRIC REPRESENTATION OF DERIVATIVE . . . . . . . . 200
8.3.2 DIFFERENTIABILITY . . . . . . . . . . . . . . . . . . . . 201
8.3.3 RULES OF DIFFERENTIATION . . . . . . . . . . . . . . . . 204
8.3.4 PROPERTIES OF DERIVATIVES . . . . . . . . . . . . . . . . 207
8.3.5 L'HOPITAL'S RULE . . . . . . . . . . . . . . . . . . . . 214
8.3.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 215
8.4 MONOTONIC FUNCTIONS AND THE INVERSE RULE . . . . . . . . . . . 216
8.4.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 219
8.5 SECOND- AND HIGHER-ORDER DERIVATIVES . . . . . . . . . . . . . 220
8.5.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 221
8.6 DIFFERENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . . .
221
8.6.1 SECOND- AND HIGHER-ORDER DIFFERENTIALS . . . . . . . . 223
8.6.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 224
8.7 COMPUTER AND NUMERICAL DIFFERENTIATION . . . . . . . . . . . . 224
8.7.1 COMPUTER DIFFERENTIATION . . . . . . . . . . . . . . . 224
8.7.2 NUMERICAL DIFFERENTIATION . . . . . . . . . . . . . . . 225
8.7.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 226
9 DIFFERENTIATION: FUNCTIONS OF SEVERAL VARIABLES . . . . . . . . . . .
227
9.1 PARTIAL DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . .
227
9.1.1 SECOND-ORDER PARTIAL DERIVATIVES . . . . . . . . . . . 230
9.1.2 DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES USING COMPUTER .
. . . . . . . . . . . . . . 232
9.1.3 THE GRADIENT AND HESSIAN . . . . . . . . . . . . . . . 232
9.1.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 234
9.2 DIFFERENTIAL AND TOTAL DERIVATIVE . . . . . . . . . . . . . . . .
235
9.2.1 DIFFERENTIAL . . . . . . . . . . . . . . . . . . . . . . . 235
9.2.2 TOTAL DERIVATIVE . . . . . . . . . . . . . . . . . . . . 237
9.2.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 240
9.3 HOMOGENEOUS FUNCTIONS AND THE EULER THEOREM . . . . . . . . 240
9.3.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 243
9.4 IMPLICIT FUNCTION THEOREM . . . . . . . . . . . . . . . . . . . 244
9.4.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 248
9.5 DIFFERENTIATING SYSTEMS OF EQUATIONS . . . . . . . . . . . . . . 248
9.5.1 THE JACOBIAN AND INDEPENDENCE OF NONLINEAR FUNCTIONS 248 9.5.2
DIFFERENTIATING SEVERAL FUNCTIONS . . . . . . . . . . . 250
9.5.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 255
10 THE TAYLOR SERIES AND ITS APPLICATIONS . . . . . . . . . . . . . . .
257
10.1 THE TAYLOR EXPANSION . . . . . . . . . . . . . . . . . . . . . .
257
10.1.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 266
10.2 THE REMAINDER AND THE PRECISION OF APPROXIMATION . . . . . . 267
10.2.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 270
10.3 FINDING THE ROOTS OF AN EQUATION . . . . . . . . . . . . . . . .
270
10.3.1 ITERATIVE METHODS . . . . . . . . . . . . . . . . . . . 270
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CONTENTS XIII
10.3.2 THE BISECTION METHOD . . . . . . . . . . . . . . . . . 272
10.3.3 NEWTON'S METHOD . . . . . . . . . . . . . . . . . . . 273
10.3.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 276
10.4 TAYLOR EXPANSION OF FUNCTIONS OF SEVERAL VARIABLES . . . . . . 276
10.4.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 279
11 INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 281
11.1 THE INDEFINITE INTEGRAL . . . . . . . . . . . . . . . . . . . . . .
282
11.1.1 RULES OF INTEGRATION . . . . . . . . . . . . . . . . . . 283
11.1.2 CHANGE OF VARIABLE . . . . . . . . . . . . . . . . . . . 285
11.1.3 INTEGRATION BY PARTS . . . . . . . . . . . . . . . . . . 287
11.1.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 289
11.2 THE DEF INITE INTEGRAL . . . . . . . . . . . . . . . . . . . . . .
289
11.2.1 PROPERTIES OF DEFINITE INTEGRALS . . . . . . . . . . . . 294
11.2.2 RULES OF INTEGRATION FOR THE DEF INITE INTEGRAL . . . . . 298
11.2.3 CHANGE OF VARIABLE . . . . . . . . . . . . . . . . . . . 299
11.2.4 INTEGRATION BY PARTS . . . . . . . . . . . . . . . . . . 300
11.2.5 RIEMANN-STIELTJES INTEGRAL . . . . . . . . . . . . . . . 304
11.2.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 306
11.3 COMPUTER AND NUMERICAL INTEGRATION . . . . . . . . . . . . . . 306
11.3.1 COMPUTER INTEGRATION . . . . . . . . . . . . . . . . . 306
11.4 NUMERICAL INTEGRATION . . . . . . . . . . . . . . . . . . . . . .
307
11.4.1 THE TRAPEZOID METHOD . . . . . . . . . . . . . . . . . 308
11.4.2 THE LAGRANGE INTERPOLATION FORMULA . . . . . . . . . 310
11.4.3 NEWTON-COTES METHOD . . . . . . . . . . . . . . . . . 312
11.4.4 SIMPSON'S METHOD . . . . . . . . . . . . . . . . . . . 313
11.4.5 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 315
11.5 SPECIAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . .
316
11.5.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 316
11.6 THE DERIVATIVE OF AN INTEGRAL . . . . . . . . . . . . . . . . . .
317
11.6.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 320
PART IV OPTIMIZATION
12 STATIC OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . .
. 323
12.1 MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE . . . . . . 324
12.1.1 INFLECTION POINT . . . . . . . . . . . . . . . . . . . . . 331
12.1.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 333
12.2 UNCONSTRAINED OPTIMA OF FUNCTIONS OF SEVERAL VARIABLES . . . 334
12.2.1 CONVEX AND CONCAVE FUNCTIONS . . . . . . . . . . . . 338
12.2.2 QUASI-CONVEX AND QUASI-CONCAVE FUNCTIONS . . . . . 341 12.2.3
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 342
12.3 NUMERICAL OPTIMIZATION . . . . . . . . . . . . . . . . . . . . .
343
12.3.1 STEEPEST DESCENT . . . . . . . . . . . . . . . . . . . . 343
12.3.2 GOLDEN SECTION METHOD . . . . . . . . . . . . . . . . 344
12.3.3 NEWTON METHOD . . . . . . . . . . . . . . . . . . . . 344
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XIV CONTENTS
12.3.4 MATLAB FUNCTIONS . . . . . . . . . . . . . . . . . . . 345
12.3.5 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 346
13 CONSTRAINED OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . .
347
13.1 OPTIMIZATION WITH EQUALITY CONSTRAINTS . . . . . . . . . . . . 347
13.1.1 THE NATURE OF CONSTRAINED OPTIMA AND THE SIGNIFICANCE OF * . . .
. . . . . . . . . . . . . 354
13.1.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 354
13.2 VALUE FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . .
355
13.2.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 362
13.3 SECOND-ORDER CONDITIONS AND COMPARATIVE STATIC . . . . . . . 362
13.3.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 368
13.4 INEQUALITY CONSTRAINTS AND KARUSH-KUHN-TUCKER CONDITIONS . . 368
13.4.1 DUALITY . . . . . . . . . . . . . . . . . . . . . . . . . 372
13.4.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 375
14 DYNAMIC OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . .
377
14.1 DYNAMIC ANALYSIS IN ECONOMICS . . . . . . . . . . . . . . . . 377
14.2 THE CONTROL PROBLEM . . . . . . . . . . . . . . . . . . . . . . 379
14.2.1 THE FUNCTIONAL AND ITS DERIVATIVE . . . . . . . . . . . 381
14.3 CALCULUS OF VARIATIONS . . . . . . . . . . . . . . . . . . . . . .
384
14.3.1 THE EULER EQUATION . . . . . . . . . . . . . . . . . . 386
14.3.2 SECOND-ORDER CONDITIONS . . . . . . . . . . . . . . . 389
14.3.3 GENERALIZING THE CALCULUS OF VARIATIONS . . . . . . . . 390
14.3.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 391
14.4 DYNAMIC PROGRAMMING . . . . . . . . . . . . . . . . . . . . . 391
14.4.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 396
14.5 THE MAXIMUM PRINCIPLE . . . . . . . . . . . . . . . . . . . . 397
14.5.1 NECESSARY AND SUFF ICIENT CONDITIONS . . . . . . . . . 405
14.5.2 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 405
PART V DIFFERENTIAL AND DIFFERENCE EQUATIONS
15 DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . . .
. 409
15.1 EXAMPLES OF CONTINUOUS TIME DYNAMIC ECONOMIC MODELS . . 409 15.2 AN
OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . 412
15.2.1 INITIAL VALUE PROBLEM . . . . . . . . . . . . . . . . . 414
15.2.2 EXISTENCE AND UNIQUENESS OF SOLUTIONS . . . . . . . . 416
15.2.3 EQUILIBRIUM AND STABILITY . . . . . . . . . . . . . . . 417
15.3 FIRST-ORDER LINEAR DIFFERENTIAL EQUATIONS . . . . . . . . . . . .
418
15.3.1 VARIABLE COEFF ICIENT EQUATIONS . . . . . . . . . . . . 420
15.3.2 PARTICULAR INTEGRAL, THE METHOD OF UNDETERMINED COEFFICIENTS . .
. . . . . . . . . . . 421
15.3.3 SEPARABLE EQUATIONS . . . . . . . . . . . . . . . . . . 424
15.3.4 EXACT DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . 426
15.3.5 INTEGRATING FACTOR . . . . . . . . . . . . . . . . . . . 430
15.3.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 433
IMAGE 7
CONTENTS XV
15.4 PHASE DIAGRAM . . . . . . . . . . . . . . . . . . . . . . . . . 433
15.4.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 436
15.5 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS . . . . . . . . . . 436
15.5.1 TWO DISTINCT REAL ROOTS . . . . . . . . . . . . . . . . 437
15.5.2 REPEATED ROOT . . . . . . . . . . . . . . . . . . . . . 439
15.5.3 COMPLEX ROOTS . . . . . . . . . . . . . . . . . . . . . 442
15.5.4 PARTICULAR INTEGRAL . . . . . . . . . . . . . . . . . . . 443
15.5.5 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 446
15.6 COMPUTER SOLUTION OF DIFFERENTIAL EQUATIONS . . . . . . . . . . 447
15.7 NUMERICAL ANALYSIS OF DIFFERENTIAL EQUATIONS . . . . . . . . . 448
15.7.1 THE EULER METHOD . . . . . . . . . . . . . . . . . . . 448
15.7.2 RUNGE-KUTTA METHODS . . . . . . . . . . . . . . . . . 450
15.7.3 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 452
16 DIFFERENCE EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . .
. 453
16.1 AN OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . 454
16.2 EXAMPLES OF DISCRETE DYNAMIC ECONOMIC MODELS . . . . . . . 457
16.2.1 ADAPTIVE EXPECTATIONS . . . . . . . . . . . . . . . . . 458
16.2.2 PARTIAL ADJUSTMENT . . . . . . . . . . . . . . . . . . . 459
16.2.3 HALL'S CONSUMPTION FUNCTION . . . . . . . . . . . . . 460
16.2.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 461
16.3 FIRST-ORDER LINEAR DIFFERENCE EQUATIONS . . . . . . . . . . . . 462
16.3.1 SOLUTION OF FIRST-ORDER LINEAR HOMOGENEOUS DIFFERENCE EQUATIONS .
. . . . . . . . . . . . . . . . . 462
16.3.2 SOLUTION OF FIRST-ORDER NONHOMOGENEOUS EQUATIONS . 465 16.3.3
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 472
16.4 SECOND-ORDER LINEAR DIFFERENCE EQUATIONS . . . . . . . . . . . 472
16.4.1 SOLUTION OF SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENCE EQUATIONS
. . . . . . . . . 473
16.4.2 BEHAVIOR OF THE SOLUTION OF SECOND-ORDER EQUATION . 475 16.4.3
COMPUTER SOLUTION OF DIFFERENCE EQUATIONS . . . . . . 481 16.4.4 THE LAG
OPERATOR . . . . . . . . . . . . . . . . . . . 482
16.4.5 SOLUTION OF SECOND-ORDER NONHOMOGENEOUS DIFFERENCE EQUATIONS . .
. . . . . . . . . . . . . . . . 485
16.4.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 490
16.5 N -TH-ORDER DIFFERENCE EQUATIONS . . . . . . . . . . . . . . . .
491
16.5.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 493
17 DYNAMIC SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . .
495
17.1 SYSTEMS OF DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . 495
17.1.1 EQUIVALENCE OF A SECOND-ORDER LINEAR DIFFERENTIAL EQUATION AND A
SYSTEM OF TWO FIRST-ORDER LINEAR EQUATIONS . . . . . . . . . . . . . 497
17.1.2 LINEARIZING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS 500 17.2
THE JORDAN CANONICAL FORM . . . . . . . . . . . . . . . . . . . 502
17.2.1 DIAGONALIZATION OF A MATRIX WITH DISTINCT REAL EIGENVALUES . . .
. . . . . . . . . . . . . . . . . 502
IMAGE 8
XVI CONTENTS
17.2.2 BLOCK DIAGONAL FORM OF A MATRIX WITH COMPLEX EIGENVALUES . . . .
. . . . . . . . . . 504
17.2.3 AN ALTERNATIVE FORM FOR A MATRIX WITH COMPLEX ROOTS . . . . . . .
. . . . . . . . . . . . . . 505
17.2.4 DECOMPOSITION OF A MATRIX WITH REPEATED ROOTS . . . 506 17.2.5
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 508
17.3 EXPONENTIAL OF A MATRIX . . . . . . . . . . . . . . . . . . . . .
509
17.3.1 REAL DISTINCT ROOTS . . . . . . . . . . . . . . . . . . 510
17.3.2 COMPLEX ROOTS . . . . . . . . . . . . . . . . . . . . . 511
17.3.3 REPEATED ROOTS . . . . . . . . . . . . . . . . . . . . 513
17.3.4 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 515
17.4 SOLUTION OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS . . . . . 515
17.4.1 DECOUPLED SYSTEMS . . . . . . . . . . . . . . . . . . 516
17.4.2 SYSTEMS WITH REAL AND DISTINCT ROOTS . . . . . . . . . 518
17.4.3 SYSTEMS WITH COMPLEX ROOTS . . . . . . . . . . . . . 519
17.4.4 SYSTEMS WITH REPEATED ROOTS . . . . . . . . . . . . . 520
17.4.5 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 521
17.5 NUMERICAL ANALYSIS OF SYSTEMS OF DIFFERENTIAL EQUATIONS . . . 521
17.5.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 523
17.6 THE PHASE PORTRAIT . . . . . . . . . . . . . . . . . . . . . . . .
523
17.6.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . 527
SOLUTIONS TO SELECTED PROBLEMS . . . . . . . . . . . . . . . . . . . . .
. . 529
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 539 |
| any_adam_object | 1 |
| author | Dadkhah, Kamran |
| author_facet | Dadkhah, Kamran |
| author_role | aut |
| author_sort | Dadkhah, Kamran |
| author_variant | k d kd |
| building | Verbundindex |
| bvnumber | BV036493078 |
| classification_rvk | QH 100 QH 110 SK 980 |
| ctrlnum | (OCoLC)701477090 (DE-599)DNB1002540798 |
| dewey-full | 330.0151 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 330 - Economics |
| dewey-raw | 330.0151 |
| dewey-search | 330.0151 |
| dewey-sort | 3330.0151 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV036493078 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T10:38:51Z |
| institution | BVB |
| isbn | 9783642137471 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020415661 |
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| physical | XVI, 542 S. graph. Darst. 260 mm x 193 mm |
| publishDate | 2011 |
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| publisher | Springer |
| record_format | marc |
| spelling | Dadkhah, Kamran Verfasser aut Foundations of mathematical and computational economics Kamran Dadkah 2. ed. Berlin [u.a.] Springer 2011 XVI, 542 S. graph. Darst. 260 mm x 193 mm txt rdacontent n rdamedia nc rdacarrier Wirtschaftsmathematik (DE-588)4066472-7 gnd rswk-swf Ökonometrie (DE-588)4132280-0 gnd rswk-swf Wirtschaftsmodell (DE-588)4079348-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Wirtschaftsmodell (DE-588)4079348-5 s Wirtschaftsmathematik (DE-588)4066472-7 s Ökonometrie (DE-588)4132280-0 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-13748-8 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3476839&prov=M&dok_var=1&dok_ext=htm Inhaltstext SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020415661&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dadkhah, Kamran Foundations of mathematical and computational economics Wirtschaftsmathematik (DE-588)4066472-7 gnd Ökonometrie (DE-588)4132280-0 gnd Wirtschaftsmodell (DE-588)4079348-5 gnd |
| subject_GND | (DE-588)4066472-7 (DE-588)4132280-0 (DE-588)4079348-5 (DE-588)4123623-3 |
| title | Foundations of mathematical and computational economics |
| title_auth | Foundations of mathematical and computational economics |
| title_exact_search | Foundations of mathematical and computational economics |
| title_full | Foundations of mathematical and computational economics Kamran Dadkah |
| title_fullStr | Foundations of mathematical and computational economics Kamran Dadkah |
| title_full_unstemmed | Foundations of mathematical and computational economics Kamran Dadkah |
| title_short | Foundations of mathematical and computational economics |
| title_sort | foundations of mathematical and computational economics |
| topic | Wirtschaftsmathematik (DE-588)4066472-7 gnd Ökonometrie (DE-588)4132280-0 gnd Wirtschaftsmodell (DE-588)4079348-5 gnd |
| topic_facet | Wirtschaftsmathematik Ökonometrie Wirtschaftsmodell Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3476839&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020415661&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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