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RATIONAL HERDS
Economic Models of Social Learning
Penguins jumping off a cliff, economic forecasters predicting a recovery in the
business cycle, financial advisors for the stock market speculating against a cur-
rency,andfarmersusingnewseedsinIndiaareallpracticingsociallearning.Such
learningfromthebehaviorofotherscanleadtoherds,crashes,andbooms.These
issueshavebecome,overthelasttenyears,anexcitingfieldofresearchintheoretical
andappliedeconomics,finance,andothersocialsciences.Thisbookprovidesboth
aninformalintroductionandin-depthinsightsintothemostrecentadvances.
Thepropertiesofsociallearningdependonthecontextinwhichlearningand
actionstakeplace.Eachchapterisdevotedtoaseparateissue:Individualslearnfrom
theobservationsofactions,fromtheoutcomesoftheseactions,andfromwhatothers
say. They may delay or make an immediate decision; they may compete against
others or gain from cooperation; they make decisions about capital investment,
cropchoices,andfinancialinvestments.Thebookhighlightsthesimilaritiesandthe
differencesbetweenthevariouscases.Arecurrentthemeisthatsocietymaylearn
moreifindividualsarelessthanperfectlyrationalintheirinterpretationofothers’
behavior.
ChristopheChamleyisProfessorofEconomicsatBostonUniversityandaDirector
ofStudiesattheEcoledesHautesEtudesenSciencesSociales,Paris.Hehasalsoheld
teachingorvisitingpositionsatYaleUniversity,theHooverInstitution,theWorld
Bank,UniversidadCarlosIII(Madrid),theUniversite´ LouisPasteur(Strasbourg),
and MIT. Professor Chamley’s research has appeared in the leading journals in
economics,includingtheAmericanEconomicReview,Econometrica,theJournalof
PoliticalEconomy,theQuarterlyJournalofEconomics,andtheReviewofEconomic
Studies.HewasnamedaFellowoftheEconometricSocietyin1995.Hisresearch
interests continue to focus on the economics of information, theoretical macro-
economics,monetaryeconomics,publiceconomics,andpubliceconomicshistory.
ProfessorChamleyreceivedhisdoctoratefromHarvardUniversity.
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Rational Herds
ECONOMIC MODELS OF SOCIAL LEARNING
CHRISTOPHE P. CHAMLEY
iii
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521824019
© Christophe P. Chamley 2004
This publication 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 2004
isbn-13 978-0-511-16482-8 eBook (EBL)
isbn-10 0-511-16482-3
eBook (EBL)
isbn-13 978-0-521-82401-9 hardback
isbn-10 0-521-82401-x hardback
isbn-13 978-0-521-53092-7 paperback
isbn-10 0-521-53092-x paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
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Tothememoryofmyfather
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Contents
Preface pagexiii
1 Introduction 1
1.1 Overview 3
1.2 ABitofHistory 13
1.3 HowtoUseThisBook 15
PARTONE. Social Learning 17
2 Bayesian Tools 19
2.1 TheBayesianFramework 21
2.2 BinaryandGaussianInformation 23
2.2.1 TheBinaryModel 23
2.2.2 TheGaussianModel 24
2.2.3 ComparisonoftheTwoModels 26
2.2.4 TheRateofConvergenceofLearning 27
2.3 PrivateSignalsandBeliefs 30
2.3.1 EquivalencebetweenPrivateSignalsandPrivateBeliefs 30
2.3.2 ExamplesofDistributionsofBeliefswithTwoStates 31
2.3.3 OtherConstructionsofPrivateInformation 33
2.4 Martingales 33
2.4.1 ConvergenceofBeliefs 35
EXERCISES 38
2.5 Appendix 38
2.5.1 SomeDefinitionsRegardingConvergence 38
2.5.2 TheModelofYoungstersandOldTimers 39
3 Social Learning with a Common Memory 42
3.1 ACanonicalModelofSocialLearning 43
3.1.1 TheModel 43
3.1.2 TheProcessofSocialLearning 45
3.2 EfficientLearning 45
3.2.1 TheGaussian–QuadraticModel 46
3.3 ObservationNoise 48
3.3.1 OneActionperPeriod 49
3.3.2 LargeNumberofAgents 51
3.3.3 Application:AMarketEquilibrium 52
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viii Contents
3.4 Extensions 53
3.4.1 LearningwithaPrivateCostofInformation 53
3.4.2 Policy 54
EXERCISES 55
3.5 Appendix 55
4 Cascades and Herds 58
4.1 TheBasicModelofHerding 62
4.1.1 The2-by-2-by-2Model 63
4.1.2 InformationalCascades 63
4.2 TheStandardModelwithBoundedBeliefs 67
4.2.1 SocialLearning 68
4.2.2 BoundedBeliefs 69
4.3 TheConvergenceofBeliefs 71
4.3.1 UnboundedBeliefs:ConvergencetotheTruth 71
4.4 HerdsandtheSlowConvergenceofBeliefs 74
4.4.1 Herds 74
4.4.2 TheAsymptoticRateofConvergenceIsZero 75
4.4.3 WhyDoHerdsOccur? 76
4.4.4 DiscreteActionsandtheSlowConvergenceof
Beliefs 77
4.5 PricingtheInformationalExternality 77
4.5.1 TheSocialOptimum 78
4.5.2 AMonopoly 81
4.6 CrashesandBooms 84
4.7 BibliographicalNotes 86
EXERCISES 87
4.8 Appendix 90
4.8.1 Proofs 90
4.8.2 AModelofLearningwithTwoTypesofAgents 91
5 Limited Memories 95
5.1 TheSequentialModelwithSampling 97
5.1.1 TheCaseofOneObservation(N =1):Asymptotic
Herding 99
5.1.2 TheCaseofMorethanOneObservation(N ≥2) 100
5.2 TheWelfare-ImprovingPrinciple 103
5.2.1 TheAverageWelfareFunction 103
5.2.2 TheWelfare-ImprovingPrinciple 104
5.2.3 Convergence 105
5.3 SamplinginaVeryLargePopulation 107
5.3.1 TwoExamples 108
5.3.2 Convergence 110
5.4 SocialLearningorSamplinginaLargePopulation? 112
EXERCISES 113
6 Delays 115
6.1 TheSimplestModel 118
6.2 AGeneralModelwithHeterogeneousBeliefs 123
6.2.1 CharacterizationandExistenceofEquilibria 124
