Economic dynamics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Edition: | 3., completely rev. and enl. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 610 S. graph. Darst. |
| ISBN: | 3540609881 |
Staff View
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| 245 | 1 | 0 | |a Economic dynamics |c Giancarlo Gandolfo |
| 250 | |a 3., completely rev. and enl. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1996 | |
| 300 | |a XXIII, 610 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | V
Giancarlo Gandolfo
Economic Dynamics
Third, Completely Revised and Enlarged Edition
With 65 Figures
and 6 Tables
Springer
Contents
PREFACE VII
Introduction 1
1 1 Definition 1
1 2 Functional equations 2
1 3 References 3
LINEAR DIFFERENCE EQUATIONS 5
Difference Equations: General Principles 7
2 1 Definitions 7
2 2 Linear difference equations with constant coefficients 9
221 The homogeneous equation 10
222 The non-homogeneous equation 12
2 3 Determination of the arbitrary constants 13
2 4 References 14
First-order Difference Equations 17
3 1 Solution of the homogeneous equation 17
3 2 Particular solution of the non-homogeneous equation 21
321 g(t) is a constant 21
322 g(t) is an exponential function 22
323 g{t) is a polynomial function of degree m 23
324 g(t) is a trigonometric function of the sine-cosine type 23
325 g(t) is a combination of the previous functions 24
326 The case when g(t) is a generic function of time Back-
ward and forward solutions 24
3 3 General solution of the non-homogeneous equation 27
34A digression on distributed lags and partial adjustment equa-
tions 28
3 5 Exercises 31
351 Example 31
352 Other exercises 32
3 6 References 33
Contents
First-order Difference Equations in Economic Models 35
4 1 The cobweb theorem 35
411 The cobweb model and expectations 38
4111 The normal price 39
4112 Adaptive expectations 40
4 2 The dynamics of multipliers 43
421 The basic case 43
422 Other multipliers 45
4221A foreign trade multiplier 46
4222 Taxation 47
4 3 Exercises 48
4 4 References 51
Second-order Difference Equations 53
5 1 Solution of the homogeneous equation 53
511 Positive discriminant (A 0) 54
512 Null discriminant (A = 0) 55
513 Negative Discriminant (A 0) 56
514 Stability conditions 58
5 2 Solution of the non-homogeneous equation 59
521 The operational method 61
5 3 Determination of the arbitrary constants 63
5 4 Exercises 65
541 Example 65
542 Other exercises 68
5 5 References 69
Second-order Difference Equations in Economic Models 71
6 1 Multiplier-accelerator interaction: the prototype model 71
611 Graphical location of the roots 73
6 2 Market adjustments and rational expectations 75
6 3 Exercises 76
6 4 References 80
Higher-order Difference Equations 83
7 1 Solution of the homogeneous equation 83
7 2 Particular solution of the non-homogeneous equation 84
7:2 1 The operational method 85
7 3 Determination of the arbitrary constants 87
7 4 Stability conditions 87
741 Necessary and sufficient stability conditions
(Samuelson s form) 88
742 Necessary and sufficient stability conditions
(Cohn-Schur form) 89
Contents » XI
7 5 Exercises 91
751 Example 91
752 Other exercises 92
7 6 References 92
8 Higher-order Difference Equations in Economic Models 95
8 1 Inventory cycles 95
8 2 Distributed lags and interaction between the multiplier and
the accelerator 98
8 3 Exercises 100
8 4 References 102
9 Simultaneous Systems of Difference Equations 103
9 1 First-order 2x2 systems in normal form 103
911 General solution of the homogeneous system: first
method 103
912 General solution of the homogeneous system:
second (or direct) method 106
9121 Unequal real roots 106
9122 Equal real roots 108
9123 Complex roots 109
913 Particular solution Determination of the arbitrary
constants 110
9 2 First order nxn systems in normal form I l l
921 Direct matrix solution 114
922 Stability conditions 115
9221A digression on not-wholly-unstable systems 118
9222 Proof of the stability conditions 120
923 Particular solution 121
9231 The operational method 122
924 Determination of the arbitrary constants 125
9 3 General systems 126
931 First-order systems not in normal form 126
932 Higher-order systems 127
9321 An example 127
9322 The general case 129
9323 Transformation of a higher-order system into
a first-order system in normal form 130
9324 Stability conditions for higher-order systems 132
9 4 Exercises 132
941 Example 132
942 Other exercises : 133
9 5 References 134
XII Contents
10 Simultaneous Difference Systems in Economic Models 137
10 1 Cournot oligopoly 137
10 2 Multiplier effects in an open economy 140
10 3 Exercises 143
10 4 References 143
11 LINEAR DIFFERENTIAL EQUATIONS 145
11 Differential Equations: General Principles 147
11 1 Definitions 147
11 2 Linear differential equations with constant coefficients 148
11 2 1 The homogeneous equation 149
11 2 2 The non-homogeneous equation 150
11 3 Determination of the arbitrary constants 152
11 4 References 154
12 First-order Differential Equations 155
12 1 Solution of the homogeneous equation 155
12 2 Particular solution of the non-homogeneous equation 158
12 2 1 g(t) is a constant 158
12 2 2 g(t) is an exponential function 159
12 2 3 g{t) is a polynomial function of degree m 159
12 2 4 g(t) is a trigonometric function of the sine-cosine type 160
12 2 5 g(t) is a combination of the previous functions 160
12 2 6 g(t) is a generic function of time The method of vari-
ation of parameters 161
12 3 General solution of the non-homogeneous equation 162
12 4 Continuously distributed lags and partial adjustment equations 163
12 5 Exercises 165
12 5 1 Example 165
12 5 2 Other exercises 167
12 6 References 167
13 First-order Differential Equations in Economic Models 169
13 1 Stability of supply and demand equilibrium 169
13 2 Theneoclassical growth model 175
13 2 1 Existence of a growth equilibrium 176
13 2 2 Stability of growth equilibrium 178
13 2 3 Refinements 181
13 231 Depreciation and technical progress 181
13 232 Golden rule 183
13 2 4 Further developments 184
13 241 Adjustment time or, how long is the long run? 184
Contents XIII
V •
13 242 /3-convergence, ^-convergence, and all that 187
13 243 Endogenous growth 189
13 3 Exercises 189
13 4 References 191
14 Second-order Differential Equations 193
14 1 Solution of the homogeneous equation 193
14 1 1 Positive discriminant (A 0) 194
14 1 2 Null discriminant (A = 0) 195
14 1 3 Negative discriminant (A 0) 196
14 1 4 Stability conditions 198
14 2 Particular solution of the non-homogeneous equation 199
14 2 1 Variation of parameters 200
14 3 General solution of the non-homogeneous equation 202
14 4 Determination of the arbitrary constants 203
14 5 Exercises 203
14 5 1 Examples 203
14 5 2 Other exercises 205
14 6 References 205
15 Second-order Differential Equations in Economic Models 207
15 1 The second-order accelerator 207
15 2 Exercises 210
15 3 References 212
16 Higher-order Differential Equations 213
16 1 Solution of the homogeneous equation 213
16 2 Solution of the non-homogeneous equation 215
16 2 1 Variation of parameters 215
16 3 Determination of the arbitrary constants 218
16 4 Stability conditions 219
16 4 1 Necessary and sufficient stability conditions
(Routh-Hurwitz) 221
16 4 2 Necessary and sufficient stability conditions
(Lienard-Chipart) 223
16 5 Exercises 223
16 5 1 Example 223
16 5 2 Other exercises 225
16 6 References f 225
17 Higher-order Differential Equations in Economic Models 227
17 1 Feedback control and stabilisation policies 227
17 1 1 Introduction 227
17 1 2 Three types of stabilisation policy 228
XIV Contents
17 121 Proportional stabilisation policy 231
17 122 Mixed proportional-derivative stabilisation pol-
icy 232
17 123 Integral stabilisation policy 233
17 2 Exercises 234
17 3 References 235
18 Simultaneous Systems of Differential Equations 237
18 1 First-order 2x2 systems in normal form 237
18 1 1 General solution of the homogeneous system: first
method 238
18 1 2 General solution of the homogeneous system: second
(or direct) method 240
18 121 Unequal real roots 241
18 122 Equal real roots 243
18 123 Complex roots 244
18 1 3 Particular solution Determination of the arbitrary
constants 245
18 2 First order nxn systems in normal form 245
18 2 1 Solution of the homogeneous system 247
18 211 The matrix exponential 249
18 2 2 Stability conditions 251
18 221 D-stability, and stabilisation of matrices 254
18 222 Sensitivity analysis 255
18 223A digression on not-wholly-unstable systems 259
18 224 Proof of the stability conditions 262
18 2 3 Particular solution 264
18 231 Variation of parameters 264
18 2 4 Determination of the arbitrary constants 265
18 3 General systems 266
18 3 1 First-order systems not in normal form 267
18 3 2 Higher-order systems 268
18 321 An example 268
18 322 The general case 270
18 323 Transformation of a higher-order system into
a first-order system in normal form 271
18 324 Stability conditions for higher-order systems 273
18 4 Exercises 273
18 4 1 Example 273
18 4 2 Other exercises 275
18 5 References 277
Contents XV
19 Differential Equation Systems in Economic Models 279
19 1 Stability of Walrasian general equilibrium of exchange 279
19 1 1 Static stability 280
19 1 2 Dynamic stability 283
19 2 Human capital in a growth model 286
19 3 A digression on arrow diagrams 291
19 4 Balanced growth in a multi-sector economy 293
19 5 Exercises : 298
19 6 References 301
III ADVANCED TOPICS 303
20 Comparative Statics and the Correspondence Principle 305
20 1 Introduction 305
20 2 The method of comparative statics 306
20 2 1 Purely qualitatively comparative statics 310
20 2 2 The inverse comparative statics problem 310
20 3 Comparative statics and optimizing behaviour 311
20 4 Comparative statics and dynamic stability of equilibrium: the
correspondence principle 314
20 4 1 Criticism and qualifications 316
20 5 Extrema and dynamic stability 318
20 5 1 An application to the theory of the firm 323
20 6 Elements of comparative dynamics 324
20 7 An illustrative application of the correspondence principle: the
IS-LM model 325
20 8 Exercises 328
20 9 References 329
21 Stability of Equilibrium: A General Treatment 331
21 1 Introduction 331
21 2 Basic concepts and definitions 332
21 2 1 Stability 333
21 2 2 Further definitions 337
21 2 3 Structural stability 338
21 3 Qualitative methods: phase diagrams 341
21 3 1 Single equations 342
21 3 2 Two-equation simultaneous systems 346
21 321 Introduction: phase plane and phase path 346
21 322 Singular points 347
21 323 Graphical construction of the trajectories 350
21 324 Linear systems 356
21 4 Quantitative methods 360
XVI Contents
21 4 1 Linearisation 360
21 5 Elements of the qualitative theory of difference equations 363
21 5 1 Single difference equations 363
21 5 2 Two simultaneous difference equations 368
21 6 Economic applications 368
21 7 Exercises 369
21 8 References 370
22 Saddle Points and Economic Dynamics 373
22 1 Saddle points in optimal control problems 374
22 1 1 Introduction 374
22 1 2 The maximum principle 375
22 2 Optimal economic growth 377
22 2 1 Optimal growth: traditional 378
2212 1 1 The setting of the problem 378
22 212 The optimality conditions in the basic neo-
classical model 381
22 213 Saddle-point transitional dynamics in the ba-
sic neoclassical model 384
22 2 2 Optimal growth: endogenous 386
22 221A model of optimal endogenous growth 386
22 222 The conditions for optimal endogenous growth 388
22 223 Optimal endogenous growth: saddle-point tran-
sitional dynamics 390
22 3 Rational expectations and saddle points 394
22 3 1 Introduction 394
22 3 2 Rational expectations, saddle points, and overshooting 397
22 3 3 Rational expectations and saddle points: the general
case 401
22 4 Exercises 403
22 5 References 405
23 Liapunov s Second Me thod 407
23 1 General concepts 407
23 2 The fundamental theorems 408
23 3 Some economic applications 413
23 3 1 Global stability of Walrasian general equilibrium 413
23 3 2 Rules of thumb in business management 420
23 3 3 Price adjustment and oligopoly under product differ-
entiation 421
23 4 Exercises 425
23 5 References 426
Contents XVII
24 Introduction to Nonlinear Dynamics 429
24 1 Preliminary remarks 429
24 2 Some integrable differential equations 431
24 2 1 First-order and first-degree exact equations 431
24 2 2 Linear equations of the first order with variable coeffi-
cients 434
24 2 3 The Bernoulli equation 436
24 3 Limit cycles and relaxation oscillations 437
24 3 1 Limit cycles: the general theory 437
24 3 2 Limit cycles: relaxation oscillations 439
24 3 3 Kaldor s non-linear cyclical model 441
24 331 The model 441
24 332 Kaldor via relaxation oscillations 445
24 333 Kaldor via Poincare s limit cycle 447
24 4 The Lotka-Volterra equations 449
24 4 1 Construction of the integral curves 454
24 4 2 Conservative and dissipative systems, and
irreversibility 456
24 4 3 Goodwin s growth cycle 458
24 431 The model 458
24 432 The phase diagram of the model 461
24 5 Exercises 464
24 6 References 466
25 Bifurcation Theory 469
25 1 Introduction 469
25 2 Bifurcations in continuous time systems 469
25 2 1 Codimension-one bifurcations 471
25 2 2 The Hopf bifurcation 475
25 2 3 Sensitivity analysis and bifurcations: a reminder 479
25 2 4 Kaldor s non-linear cyclical model again 480
25 2 5 Oscillations in optimal growth models 481
25 251 The model 481
25 252 The optimality conditions 483
25 253 Emergence of a Hopf bifurcation 484
25 2 6 Cycles in an IS-LM model with pure money financing 486
25 3 Bifurcations in discrete time systems 488
25 3 1 Codimension-one bifurcations 489
25 3 2 The Hopf bifurcation in discrete time 491
25 3 3 Kaldor s cyclical model in discrete time 492
25 3 4 Liquidity costs in the firm 494
25 341 The model 494
25 342 The dynamics 497
25 4 Exercises 499
XVIII » Contents
25 5 References 501
26 Complex Dynamics 503
26 1 Introduction 503
26 2 Discrete time systems and chaos 505
26 2 1 The logistic map 505
26 2 2 Intermittency 512
26 2 3 The basic theorems 512
26 2 4 Discrete time chaos in economics 515
26 241 Chaos in growth theory 515
26 242 Exchange rate dynamics and chaos 516
26 3 Continuous time systems and chaos 518
26 3 1 The Lorenz equations, strange attractors, and chaos 518
26 3 2 Other routes to continuous time chaos 521
26 321 The Rossler attractor 521
26 322 The Shil nikov scenario 521
26 323 The forced oscillator 522
26 324 The coupled oscillator 522
26 3 3 International trade as the source of chaos 526
26 34A chaotic growth cycle 527
26 4 Significance and detection of chaos: Stochastic dynamics or
chaos? 528
26 5 Other approaches 532
26 5 1 Introduction 532
26 5 2 Fast and slow, and synergetics 533
26 5 3 Catastrophe theory 536
26 6 Exercises 538
26 7 References 540
27 Mixed Differential-Difference Equations 545
27 1 General concepts 545
27 2 Continuous vs discrete time in economic models 545
27 3 Linear mixed equations 550
27 4 The method of solution 550
27 5 Stability conditions 555
27 6 Delay differential equations and chaos 556
27 7 Some economic applications 556
27 7 1 Kalecki s business cycle model 557
27 7 1- 1 The model 557
27 712 The dynamics 559
27 72A formalization of the classical price-specie-flow mech-
anism of balance of payments adjustment 563
27 721 The model 563
27 722 Stability 565
Contents XDC
V
27 8 Exercises 567
27 9 References 568
Bibliography 571 ,
Name Index - 593
Subject Index 599
|
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| discipline | Wirtschaftswissenschaften |
| edition | 3., completely rev. and enl. ed. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV010716527 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:57:43Z |
| institution | BVB |
| isbn | 3540609881 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007156060 |
| oclc_num | 231662913 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-473 DE-BY-UBG DE-703 DE-19 DE-BY-UBM DE-12 DE-20 DE-706 DE-521 DE-634 DE-188 DE-83 DE-2070s |
| owner_facet | DE-355 DE-BY-UBR DE-473 DE-BY-UBG DE-703 DE-19 DE-BY-UBM DE-12 DE-20 DE-706 DE-521 DE-634 DE-188 DE-83 DE-2070s |
| physical | XXIII, 610 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| spelling | Gandolfo, Giancarlo Verfasser (DE-588)170260941 aut Economic dynamics Giancarlo Gandolfo 3., completely rev. and enl. ed. Berlin [u.a.] Springer 1996 XXIII, 610 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier economia politica - modello matematico tessin-TR Mathematisches Modell Wirtschaft Economics, Mathematical Economics -- Mathematical models Statics and dynamics (Social sciences) Dynamik (DE-588)4013384-9 gnd rswk-swf Wirtschaftsmathematik (DE-588)4066472-7 gnd rswk-swf Dynamische Makroökonomie (DE-588)4200428-7 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Wirtschaftstheorie (DE-588)4079351-5 gnd rswk-swf Ökonometrie (DE-588)4132280-0 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Dynamisches Modell (DE-588)4150932-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Differentialgleichung (DE-588)4012249-9 s Ökonometrie (DE-588)4132280-0 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Wirtschaftsmathematik (DE-588)4066472-7 s Dynamisches Modell (DE-588)4150932-8 s Dynamische Makroökonomie (DE-588)4200428-7 s Dynamik (DE-588)4013384-9 s Wirtschaftstheorie (DE-588)4079351-5 s 1\p DE-604 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007156060&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gandolfo, Giancarlo Economic dynamics economia politica - modello matematico tessin-TR Mathematisches Modell Wirtschaft Economics, Mathematical Economics -- Mathematical models Statics and dynamics (Social sciences) Dynamik (DE-588)4013384-9 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd Dynamische Makroökonomie (DE-588)4200428-7 gnd Differentialgleichung (DE-588)4012249-9 gnd Wirtschaftstheorie (DE-588)4079351-5 gnd Ökonometrie (DE-588)4132280-0 gnd Mathematisches Modell (DE-588)4114528-8 gnd Dynamisches Modell (DE-588)4150932-8 gnd |
| subject_GND | (DE-588)4013384-9 (DE-588)4066472-7 (DE-588)4200428-7 (DE-588)4012249-9 (DE-588)4079351-5 (DE-588)4132280-0 (DE-588)4114528-8 (DE-588)4150932-8 (DE-588)4123623-3 |
| title | Economic dynamics |
| title_auth | Economic dynamics |
| title_exact_search | Economic dynamics |
| title_full | Economic dynamics Giancarlo Gandolfo |
| title_fullStr | Economic dynamics Giancarlo Gandolfo |
| title_full_unstemmed | Economic dynamics Giancarlo Gandolfo |
| title_short | Economic dynamics |
| title_sort | economic dynamics |
| topic | economia politica - modello matematico tessin-TR Mathematisches Modell Wirtschaft Economics, Mathematical Economics -- Mathematical models Statics and dynamics (Social sciences) Dynamik (DE-588)4013384-9 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd Dynamische Makroökonomie (DE-588)4200428-7 gnd Differentialgleichung (DE-588)4012249-9 gnd Wirtschaftstheorie (DE-588)4079351-5 gnd Ökonometrie (DE-588)4132280-0 gnd Mathematisches Modell (DE-588)4114528-8 gnd Dynamisches Modell (DE-588)4150932-8 gnd |
| topic_facet | economia politica - modello matematico Mathematisches Modell Wirtschaft Economics, Mathematical Economics -- Mathematical models Statics and dynamics (Social sciences) Dynamik Wirtschaftsmathematik Dynamische Makroökonomie Differentialgleichung Wirtschaftstheorie Ökonometrie Dynamisches Modell Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007156060&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gandolfogiancarlo economicdynamics |