Economic dynamics: phase diagrams and their economic application

724 Pages·2003·4.628 MB·English

by  Ronald Shone

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Economic Dynamics
PhaseDiagramsandTheirEconomicApplication
SecondEdition
ThisisthesubstantiallyrevisedandrestructuredsecondeditionofRonShone’s
successfulundergraduateandgraduatetextbookEconomicDynamics.
The book provides detailed coverage of dynamics and phase diagrams in-
cluding:quantitativeandqualitativedynamicsystems,continuousanddiscrete
dynamics,linearandnonlinearsystemsandsingleequationandsystemsofequa-
tions.ItillustratesdynamicsystemsusingMathematica,Mapleandspreadsheets.
Itprovidesathoroughintroductiontophasediagramsandtheireconomicappli-
cationandexplainsthenatureofsaddlepathsolutions.
Thesecondeditioncontainsanewchapteronoligopolyandanextendedtreat-
mentofstabilityofdiscretedynamicsystemsandthesolvingoffirst-orderdiffer-
enceequations.DetailedroutinesontheuseofMathematicaandMaplearenow
containedinthebodyofthetext,whichnowalsoincludesadviceontheuseof
Excelandadditionalexamplesandexercisesthroughout.Thesupportingwebsite
containsasolutionsmanualandlearningtools.
ronaldshoneisSeniorLecturerinEconomicsattheUniversityofStirling.He
istheauthorofeightbooksoneconomicscoveringtheareasofmicroeconomics,
macroeconomics and international economics at both undergraduate and post-
graduatelevel.HehaswrittenanumberofarticlespublishedinOxfordEconomic
Papers, the Economic Journal, Journal of Economic Surveys and Journal of
EconomicStudies.

Economic Dynamics
Phase Diagrams and Their Economic Application
Second Edition
RONALD SHONE
UniversityofStirling
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambrid ge.org/9780521816847
© Ronald Shone 2002
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2002
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Contents
Prefacetothesecondedition pagexi
Prefacetothefirstedition xiii
PARTI Dynamicmodelling
1 Introduction 3
1.1 Whatthisbookisabout 3
1.2 Theriseineconomicdynamics 5
1.3 Stocks,flowsanddimensionality 8
1.4 Nonlinearities,multipleequilibriaandlocalstability 12
1.5 Nonlinearityandchaos 15
1.6 Computersoftwareandeconomicdynamics 17
1.7 MathematicaandMaple 20
1.8 Structureandfeatures 24
Additionalreading 25
2 Continuousdynamicsystems 26
2.1 Somedefinitions 26
2.2 Solutionstofirst-orderlineardifferentialequations 37
2.3 Compoundinterest 39
2.4 First-orderequationsandisoclines 41
2.5 Separablefunctions 45
2.6 Diffusionmodels 53
2.7 Phaseportraitofasinglevariable 54
2.8 Second-orderlinearhomogeneousequations 59
2.9 Second-orderlinearnonhomogeneousequations 64
2.10 Linearapproximationstononlineardifferentialequations 66
2.11 SolvingdifferentialequationswithMathematica 70
2.12 SolvingdifferentialequationswithMaple 73
Appendix2.1Plottingdirectionfieldsforasingleequation
withMathematica 77
Appendix2.2Plottingdirectionfieldsforasingleequation
withMaple 79
Exercises 80
Additionalreading 84
vi Contents
3 Discretedynamicsystems 85
3.1 Classifyingdiscretedynamicsystems 85
3.2 Theinitialvalueproblem 86
3.3 Thecobwebmodel:anintroduction 87
3.4 Equilibriumandstabilityofdiscretedynamicsystems 88
3.5 Solvingfirst-orderdifferenceequations 99
3.6 Compoundinterest 105
3.7 Discounting,presentvalueandinternalratesofreturn 108
3.8 Solvingsecond-orderdifferenceequations 110
3.9 Thelogisticequation:discreteversion 118
3.10 Themultiplier–acceleratormodel 123
3.11 Linearapproximationtodiscretenonlineardifference
equations 127
3.12 Solowgrowthmodelindiscretetime 130
3.13 SolvingrecursiveequationswithMathematica
andMaple 131
Appendix3.1Two-cyclelogisticequationusingMathematica 135
Appendix3.2Two-cyclelogisticequationusingMaple 137
Exercises 138
Additionalreading 141
4 Systemsoffirst-orderdifferentialequations 142
4.1 Definitionsandautonomoussystems 142
4.2 Thephaseplane,fixedpointsandstability 145
4.3 Vectorsofforcesinthephaseplane 149
4.4 Matrixspecificationofautonomoussystems 156
4.5 Solutionstothehomogeneousdifferentialequation
system:realdistinctroots 160
4.6 Solutionswithrepeatingroots 162
4.7 Solutionswithcomplexroots 164
4.8 Nodes,spiralsandsaddles 166
4.9 Stability/instabilityanditsmatrixspecification 178
4.10 Limitcycles 179
4.11 Euler’sapproximationanddifferentialequations
onaspreadsheet 183
4.12 Solvingsystemsofdifferentialequationswith
MathematicaandMaple 186
Appendix4.1Parametricplotsinthephaseplane:
continuousvariables 194
Exercises 196
Additionalreading 200
5 Discretesystemsofequations 201
5.1 Introduction 201
5.2 BasicmatriceswithMathematicaandMaple 204
5.3 Eigenvaluesandeigenvectors 208
Contents vii
5.4 MathematicaandMapleforsolvingdiscretesystems 214
5.5 Graphingtrajectoriesofdiscretesystems 220
5.6 Thestabilityofdiscretesystems 223
5.7 Thephaseplaneanalysisofdiscretesystems 235
5.8 Internalandexternalbalance 239
5.9 Nonlineardiscretesystems 245
Exercises 247
Additionalreading 250
6 Optimalcontroltheory 251
6.1 Theoptimalcontrolproblem 251
6.2 ThePontryaginmaximumprinciple:continuousmodel 252
6.3 ThePontryaginmaximumprinciple:discretemodel 259
6.4 Optimalcontrolwithdiscounting 265
6.5 Thephasediagramapproachtocontinuoustime
controlmodels 270
Exercises 283
Additionalreading 285
7 Chaostheory 286
7.1 Introduction 286
7.2 Bifurcations:single-variablecase 287
7.3 Thelogisticequation,periodic-doublingbifurcations
andchaos 293
7.4 Feigenbaum’suniversalconstant 301
7.5 Sarkovskiitheorem 302
7.6 VanderPolequationandHopfbifurcations 304
7.7 Strangeattractors 307
7.8 Rationalchoiceanderraticbehaviour 312
7.9 Inventorydynamicsunderrationalexpectations 315
Exercises 319
Additionalreading 321
PARTII Appliedeconomicdynamics
8 Demandandsupplymodels 325
8.1 Introduction 325
8.2 Asimpledemandandsupplymodelincontinuoustime 326
8.3 Thecobwebmodel 332
8.4 CobwebswithMathematicaandMaple 338
8.5 Cobwebsinthephaseplane 339
8.6 Cobwebsintwointerrelatedmarkets 346
8.7 Demandandsupplywithstocks 349
8.8 Stabilityofthecompetitiveequilibrium 353
8.9 Thehousingmarketanddemographicchanges 358
8.10 Chaoticdemandandsupply 363
viii Contents
Appendix8.1ObtainingcobwebsusingMathematica
andMaple 367
Exercises 371
Additionalreading 374
9 Dynamictheoryofoligopoly 375
9.1 Staticmodelofduopoly 375
9.2 Discreteoligopolymodelswithoutputadjusting
completelyandinstantaneously 377
9.3 Discreteoligopolymodelswithoutputnotadjusting
completelyandinstantaneously 389
9.4 Continuousmodellingofoligopoly 398
9.5 Anonlinearmodelofduopolisticcompetition(R&D) 405
9.6 Schumpeteriandynamics 414
Exercises 419
Additionalreading 423
10 Closedeconomydynamics 424
10.1 Goodsmarketdynamics 425
10.2 Goodsandmoneymarketdynamics 429
10.3 IS-LMcontinuousmodel:version1 431
10.4 TrajectorieswithMathematica,MapleandExcel 437
10.5 Someimportantpropositions 442
10.6 IS-LMcontinuousmodel:version2 447
10.7 NonlinearIS-LMmodel 453
10.8 Tobin–Blanchardmodel 455
10.9 Conclusion 465
Exercises 467
Additionalreading 469
11 Thedynamicsofinflationandunemployment 470
11.1 ThePhillipscurve 470
11.2 Twosimplemodelsofinflation 472
11.3 Deflationary‘deathspirals’ 484
11.4 ALucasmodelwithrationalexpectations 490
11.5 Policyrules 493
11.6 Money,growthandinflation 494
11.7 Caganmodelofhyperinflation 500
11.8 Unemploymentandjobturnover 506
11.9 Wagedeterminationmodelsandtheprofitfunction 509
11.10 Labourmarketdynamics 513
Exercises 516
Additionalreading 518
12 Openeconomydynamics:stickypricemodels 519
12.1 Thedynamicsofasimpleexpendituremodel 519
12.2 Thebalanceofpaymentsandthemoneysupply 524