A Simple Model of Inequality, Occupational Choice, and Development

www.elsevier.com/locate/econbase

Asimplemodelofinequality,occupationalchoice,

anddevelopment

MaitreeshGhataka,*,NevilleNien-HueiJiangbabDepartmentofEconomics,UniversityofChicago,Chicago,IL60637,USADepartmentofEconomics,VanderbiltUniversity,Nashville,TN37235,USA

Received1February2000;accepted1September2001

Abstract

Weanalyzeasimpleandtractablemodelofoccupationalchoiceinthepresenceofcreditmarketimperfections.Weexaminetheeffectofparametersgoverningtechnologyandtransactioncosts,andhistory,intermsoftheinitialwealthdistribution,indeterminingthelong-termwealthdistributionandthelevelofpercapitaincomeofaneconomy.D2002ElsevierScienceB.V.Allrightsreserved.

JELclassification:D31;D82;O10

Keywords:Wealthinequality;Occupationalchoice;Povertytraps

1.Introduction

Awell-knownimplicationofneoclassicalgrowththeoryisthateconomiesthathavesimilarpreferencesandtechnologiesconvergetothesamesteadystatepercapitaincome.1Incontrast,indevelopmenteconomics,wefrequentlyencountertheideaofpovertytraps:poorindividualsandeconomiestendtoremainpoorbecausetheystartpoor.Onespecificmechanismleadingtothepersistenceofpovertythathasrecentlyreceivedalotofattentionoperatesthroughborrowingconstraints.2Becausethreatsofpunishmentworklesswellagainstthepoor,theyfacegreaterborrowingconstraints.Thisinturnprevents

*Correspondingauthor.

E-mailaddress:m-ghatak@uchicago.edu(M.Ghatak).1SeeBarroandSala-i-Martin(1995).2SeeGalorandZeira(1993),BanerjeeandNewman(1993,1994),AghionandBolton(1997),Piketty(1997),andMookherjeeandRay(2000).

0304-3878/02/$-seefrontmatterD2002ElsevierScienceB.V.Allrightsreserved.PII:S0304-3878(02)00059-7

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themfromadoptingefficienttechnologiesorchoosingprofitableoccupations,andhencetheyremainpoor.Attheaggregatelevel,thisimpliesthatunlikeinneoclassicalgrowthmodels,twoeconomiesthatareidenticalintermsofallparametersmayendupwithdifferentlevelsofpercapitaincomesinthesteadystateifinitiallytheyhavedifferentdistributionsofwealthandhencedifferentsizesoftheclassofcreditrationed.Thisargumentisofteninvokedtoexplaintheevidencefromcross-countryanalysissuggestingthatvariousmeasuresofinitialinequalityarenegativelycorrelatedwithgrowth.3However,itturnsoutthatthedynamicbehaviorofaneconomyinthepresenceofcreditmarketimperfectionsisfairlycomplicated,andevenunderstrongsimplifyingassumptionsregardingtechnology,preferencesandmarketstructure,itisdifficulttogiveclear-cutanswerstoquestionssuchaswhendoinitialconditionsmatter,andiftheydo,whatistherelationshipbetweeninitialinequalityandthesteady-statelevelofpercapitaincomeofaneconomy.Inthispaperwetrytoanswerthesequestionsbyanalyzingasimpleandtractabledynamicmodelofoccupationalchoiceinthepresenceofcreditmarketimperfections.

OurpaperiscloselyrelatedtotheimportantcontributionsofGalorandZeira(1993)andBanerjeeandNewman(1993).Theyprovidethefollowinginsight:inthepresenceofcreditmarketimperfections,thecurrentdistributionofwealthwilldeterminetheproportionofcredit-constrainedindividualsintheeconomy,whichinturnmayaffectequilibriumreturnstovariousoccupationsinawaythataffectsthefuturewealthdistributionthroughintergenerationaltransfers.Asaresult,thetransitionofthewealthdistributionfortheeconomyasawholeisnonlinearandhencethewealthdistributiondynamicsisquitecomplex.Inparticular,itisdifficulttosaymuchexceptformultiplestationarywealthdistributionsmayexist,andthattheinitialdistributionofwealthmaydeterminewhichsteady-stateequilibriumtheeconomyconvergesto.BanerjeeandNewman(1993)offersomesimpleexamplestoshowinstancesofhysteresis.However,evenintheseexamples,itisnotalwaysthecasethatthegreateristhesizeofthepoorrelativetothatoftherichintheinitialdistribution,thelowerwillbethesteady-statelevelofincome.

WeconsiderasimplifiedversionofthemodelofBanerjeeandNewman(1993).Inparticular,wehaveasimpleroccupationalstructure.Itturnsout,asaresultofthisoneneedsnomoreinformationaboutthewealthdistributionthantheproportionofpeoplewhosewealthisbelowthelevelneededtostartanenterprise.EventhoughgeneralresultsinthisclassofnonlineardynamicmodelsofwealthdistributionarehardtoobtainasdemonstratedbytheBanerjee–Newmanmodel,thissimplificationallowsustocharacterizepreciselyallthesteady-stateequilibriacorrespondingtovariousconfigurationsofparametersgoverningtechnology,preferencesandtransactionscosts.Italsoallowsustocalculatetheeffectofchangesinparametersofinterestandtheinitialdistributionofwealthonsteadystatepercapitaincome.However,asaresultofthissimplification,welosesomeoftherichnessoftheBanerjee–Newmanmodel,whichallowsforalternativeinstitutionalformsassociatedwiththemoderntechnologythatdifferintermsofagencycosts.

SeeBenabou(1996)foradiscussionoftheempiricalliteratureaswellasothertheoreticalargumentsconsistentwiththeobservednegativerelationshipbetweeninequalityandgrowthsuchasthosebasedonpoliticaleconomyconsiderations.

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Someofourfindingsareasfollows:first,whetherhysteresisoccursdependsonthesizeofthethresholdlevelofwealthneededtostartanenterpriserelativetotheproductivityofthemodernandthesubsistencetechnologies.Inparticular,thelargeristheproductivitydifferencebetweenthemodernandsubsistencetechnologies,thegreateristhelikelihoodofmultiplesteadystates.Second,forparametervaluesunderwhichinitialconditionsmatter,thegreateristhefractionofthepopulationwhoareinitiallypoor,theloweristhesteady-stateincome.Third,whilesomeformsoftechnologicalprogresscaneliminatepovertytraps,allkindsoftechnologicalimprovementsdonotnecessarilyincreasesteady-stateincome.Forexample,anincreaseintheproductivityofthesmallscaleorsubsistencesectorthatpushesupwagescanactasadragonthegrowthofthemodernsector.

Theplanofthepaperisasfollows.InSection2weanalyzethebasicmodel.InSection3weextendthebasicmodel,whichisnonstochastic,byallowingthesavingratetobesubjecttorandomshocks.InSection4wemakesomeconcludingremarksandAppendixAcontainssometechnicalproofs.

2.Themodel

2.1.Demographicsandpreferences

Consideraneconomyinhabitedbyinfinitelyliveddynastiesrepresentedbysuccessivegenerationsofagentswholiveforoneperiod.Thepopulationislargeanditssizeisnormalizedto1.Thereisnopopulationgrowth.Therearetwogoodsintheeconomy,labor,andsomefinaloutputwhichcanservebothasaconsumptiongoodandacapitalgood.Inperiodtadynastyiisendowedwith1unitoflaborandaninitialwealthai,t.Itearnsincomebysupplyinglaborandcapitalandtheresultingincomeyi,tisdividedattheendoftheperiodbetweenconsumptionci,tandsavings,orbequesttothenextgeneration,bi,t.Therefore,

ai;tþ1¼bi;t:

Followingtheliterature,weassumethatindividualshaveidenticalCobb–Douglas

Àssutilityfunctionsoverconsumptionandbequests,withUi(ci,t,bi,t)=ci,1tbi,t,wheresa(0,1)andthebudgetconstraintisyi,t=ci,t+bi,t.Thismeansthatthecurrentgenerationsavesaconstantfractionsofitsincomeandleavesitasbequest:

ai;tþ1¼syi;t:

Wealsoassumethatallagentsarerisk-neutral.

Inperiodt,wealthisdistributedaccordingtotheprobabilitymeasurekt(Á),andforconvenience,wedefine

GtðaÞuktððÀl;aÞÞ:

ThefunctionGtisverysimilartothedistributionfunctionexceptthatitdoesnotincludethemeasureatpointa.

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2.2.Productiontechnologies

Therearetwoproductiontechnologiesbothofwhicharedeterministic.Oneusesnocapitalandoneunitoflabortoproducewunitsofoutput.Thiswillbedescribedasasubsistence(oragricultural)technology.TheotherusesI>0unitsofcapitalandtwounitsoflabor(oneunitofsupervisorylaborandoneunitofordinarylabor)toproducequnitsofoutput.Onesupervisor(orentrepreneur)canperfectlymonitoroneworkerspendingherentirelaborendowment.Thiswillbedescribedasanentrepreneurial(orindustrial)technology.4Assumption1.Weassumethatthistechnologyissuperiorinthesensethatthenetoutputofusingthistechnologyisgreaterthanweretwounitsoflaborusingthesubsistencetechnology.Thatis,

qÀrI>2wwherer(z1)istheexogenouslygivengrossinterestrate.52.3.Occupations

Therearethreepossibleoccupationsopentoanindividualwhohasinheritedwealthai,t:(a)Subsistence:Theagentearnssomeincomebyusingherlaborendowmenttoproducewwiththesubsistencetechnology.Sheputsherinheritedwealthinthebank,whichyieldsrai,t.Therefore,herincomeis

ySi;t¼wþrai;t:

(b)Worker:Theagentworksforanentrepreneurforwageincomewt(whichisdeterminedendogenously).Sheputsherinheritedwealthinthebank,whichyieldsrai,t.Therefore,herincomeis

yWi;t¼wtþrai;t:

(c)Entrepreneur:TheagentinvestsanamountItostartafirmandhiresoneworkertoproduceanoutputqwithcertainty.Herjobistomonitortheworker.Theagent’sincomeasanentrepreneuristheoutputoftheprojectlesswageandcapitalcosts:

yEi;t¼qÀwtþrðai;tÀIÞ:

IncontrastintheBanerjeeandNewman(1993)model,apartfromthesetwotypesoftechnologies,thereis

athirdonewhichinvolvessomecapitalandoneunitoflabor(‘‘self-employment’’).

5Wecanthinkofthecreditmarketasaninternationalmarketwherethegiveneconomyis‘small’.

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2.4.Creditandlabormarkets

Thecreditmarketissubjecttotransactionscostsonthelendingsideduetoimperfectenforcementofloancontracts.6Thisresultsincreditrationingofthefollowingform:ifanindividual’swealthisbelowacertainminimumlevel,shewouldnotgetaloannomatterhowhightheinterestratesheoffers.FollowingBanerjeeandNewman(1993),asimplewaytogeneratethisformofcreditrationingisasfollows:aborrowermaydefaultonherloan(namely,r(IÀa)),butthecostofthisactionisthatshegetscaughtwithsomeprobabilitypandthenhastopayafixednonmonetarycostofFduetoimprisonmentorsocialsanctions.Thus,onlythoseindividualsgetloanswhosewealthsatisfiestheincentivecompatibilityconstraint(ICC)7:

ðqÀwtÞÀrðIÀai;tÞzqÀwtÀpFor;ai;tzIÀ

pF:r

ð1Þ

Thelowerisanindividual’swealth,thegreaterisherincentivetodefaultbecauseshehastoborrowagreateramounttostartanenterprise,andthelevelofsanctionsagainstdefaultisthesameforallborrowers.Hence,onlythosewhohaveacertainminimumamountofwealth(namely,IÀpF/r)canborrow.8Withoutlossofgenerality,wesetp=0sothatonlythosewhohaveenoughwealthtofullyfinancetheirownenterprisesareabletobecomeentrepreneurs.

Thewagerateatwhichentrepreneursareindifferentbetweenworkingaswagelaborersandhiringworkersisgivenby:

¯þrðai;tÀIÞ¼w¯þrai;tqÀw¯¼or;w

qÀrI

:2

ByAssumption1,wWeareassumingtherearenoimperfectionsonthedepositsideofthecreditmarket:thereisaconstantrateofreturnofrirrespectiveoftheamountdeposited.

7Itisbeingassumedthatevenifaborrowergetscaughttryingtoavoidrepayingherdebt,shegetstoconsumeherprofits.

8Animplicationofthisformofcreditrationingisthatthethresholdwealthleveldoesnotdependonthewagerate.Otherwise,thethresholdwealthlevelwillchangewiththewagerate.Thistendstocomplicatethedynamicssomewhat,butthebasicresultsarenotaffected.

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refertothoseindividualswhosewealthislessthanIascapital-constrained,orsimply,poor,andtherestasunconstrained,orrich.

TheICCtellsuswhatfractionofthepopulationiscapital-constrained,namely,Gt(I).Noticethatthisfollowsfromourassumptionthatallentrepreneursareself-financedandthecreditmarketdoesnotoperateasp=0.Otherwise,therelevantfractionofthepopulationthatiscapital-constrainedwouldbeGt(IÀpF/r).

Forwt0ifwtw

Conversely,toderivethedemandcurveforlabor,wenoticethatforwt>w¯thereisno

demandforlabor;aswtfallstow¯,thedemandforlaborjumpstoanyvaluebetween0and1ÀGt(I).Whenwt¯0ifwt>w

¯½0;1ÀGtðIÞ󰀈ifwt¼w¯:1ÀGtðIÞifwtFromthelabordemandandsupplyscheduleswecaneasilyfindtheequilibriumwage

rateinperiodt:

1

¯ifGtðIÞ21

wt*¼½w;w¯󰀈ifGtðIÞ¼

21

wifGtðIÞ>:

2Sinceeachentrepreneurhiresexactlyoneworker,iftherearemorepeoplewhoarecapital-constrained(unconstrained),thenthecompetitionforentrepreneurs(workers)

M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226211

amongthemwilldrivetheequilibriumwageratedown(up)toitslower(upper)bound.WhenGt(I)=1/2,theequilibriumwagerateisindeterminate,andthroughoutthispaper,wearegoingtoassumethatthewagerateisequaltow¯inthiscase.

Noticethatononehand,theequilibriumwageratedependsonthecurrentwealthdistributionbutontheotherhand,italsoinfluencesnextperiod’swealthdistributionthroughthesavingsbehaviorofcurrentlyactiveagents.2.5.Dynamicsofindividualwealth

Considerthefactorsgoverningdynastyi’sbequest.Firstofall,theinitialwealthlevelofanagentdetermineshercapitalincomeandheroccupationalchoice.Secondly,thecurrentwagerateisdeterminedbytheeconomy-widewealthdistribution.Withtheknowledgeofanindividual’soccupationalchoiceandthatthewageratecantakeonlytwovalues(wandw¯),wecanwritedownthedifferenceequationsdescribingtheevolutionofadynastyi’swealthas:

ai;tþ1ðai;tjwt¼wÞ¼s½rai;tþw󰀈

ifai;t¼s½rðai;tÀIÞþqÀw󰀈ifai;tzI

¯Þ¼s½rai;tþw¯󰀈ai;tþ1ðai;tjwt¼w

bai;t:

Fig.1showswhatthesedifferenceequationslooklike.Noticethattherearetworegimes

ofwealthtransitionscorrespondingtothetwowagelevels.Whenthewagerateislow,anagentwhoiscapital-constrainedcanonlychoosebetweenbeingaworkerandengaginginsubsistenceandineithercase,herlaborincomeisw.Afractionsofthesumofherlaborincomeandhercapitalincomerai,tisleftforhernextgeneration.Anagentwhoisnotcredit-constrainedwillstrictlyprefertobeanentrepreneurandhertotalincomewillber(ai,tÀI)+qÀw.Whenthewagerateishigh,nobodywillengageinsubsistenceandallagentswillbeindifferentbetweenbeingentrepreneursandworkers.

Fig.1.Dynastyi’swealthtransitionsunderdifferentwageregimes.

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Assumption2.Weassumethatitisnotpossibleforadynastytogetarbitrarilyrichovertimemerelybysavingaconstantfractionofitsincomeeveryperiodandearninginterestonit:

sr<1

Assumptions1and2willberetainedthroughoutthissection.2.6.Stationarywealthdistributionsandwages

Inthissection,weexaminethelong-runbehaviorofthiseconomy.Ifthedifferenceequationsgoverningthewealthtransitionsarestable,itwouldbeeasytoprovetheexistenceofastationarywealthdistribution.However,thefactthatthesedifferenceequationsdependonthewagelevelsraisesthepossibilitythattheprocessmaynotbestable.Inparticular,theconcernhereisthatthewageratemaychangeinfinitelyoften.Thefollowinglemmarulesoutthispossibility.

Lemma1.Thewageratecanchangeatmostonce.

Proof:Noticethatthedifferenceequationsareorder-preserving.Thatis,ai,t+1>aj,t+1ifandonlyifai,t>aj,t.Therefore,inordertostudythewagedynamics,wecanonlylookatthewealthdynamicsofthedynastywhichhasthemedianwealth.Defineatmumax{a:Gt(a)V1/2}.NotethatatmiswelldefinedbecauseG(Á)iscontinuousfrombelowaccording

1toourdefinition.Thenam¯.Similarly,amtzIZGtðIÞV2whichimplieswt=wt2whichimplieswt=w.Nowifwt=wandwt+1=w¯,thenwemusthaveatis,oncethehigh-wagerateisreached,therewillnotbeanydownwardmobilityandhence

mthehighwagewillprevailforever.Ifwt=w¯andwt+1=w,thenwemusthaveatzIand

m

¯Þwe{w,w¯}.Thatis,therewillnotbeanyupwardmobilityandoncethelow-wagerateisreached,itwillprevailforever.Therefore,wecanconcludethatstartingwithanyinitialdistributionofwealth,thewageratecanchangeatmostonce.5Lemma1showsthatthewagerateisconstantinthelongrunandrulesoutthepossibilityofcyclesorchaoticwagedynamics.Oncethewagerateswitchesfromlowtohigh,therewillbenodownwardmobilityandsothehighwageprevailsforeverandsimilarly,oncethewagerateswitchesfromhightolow,therewillbenoupwardmobilityandthelow-wageprevailsforever.Asaresult,althoughwehavetworegimesofthewealthtransitionprocess,therewillnotbeinfiniteswitchesfromonetotheother.Onlyoneofthemwillprevailinthelongrun.However,forthesameparametervalues,bothwealthtransitionprocessescouldbecandidatesforthelong-runequilibriumandwhichoneisarrivedatcoulddependoninitialconditions.TogetherwithAssumption2,whichimpliesthereexistsastationarypointforeachdifferenceequation,weimmediatelyhave:Proposition1:Givenanyinitialwealthdistribution,thereexistsauniquestationarywealthdistributiontowhichitconverges.

ByLemma1inthelongrunthewagerateisconstantandcorrespondingtothiswagerate,oneofthetwopossiblewealthtransitionprocesseswillprevail.Thedifferenceequationsassociatedwiththeseprocesseshaveuniquestationarypointsandsothewealth

M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226213

distributionoftheeconomywillconvergetoastationarydistribution.Thisstationarywealthdistributionwillhaveallmassconcentratedononepoint(forthehigh-wageequilibrium)ortwopoints(forthelow-wageequilibrium)whichisaconsequenceofthemodelbeingnonstochastic.NoticethattheLemma1andProposition1donotsuggestthatgiventheparametersofthemodelthereisauniquelong-runwagerate,andacorrespondinglong-runstationarywealthdistribution.Indeed,oneofourmaingoalsistocharacterizeparameterconditionsunderwhichmultiplelong-runequilibriacouldexistandtoshowwhichequilibriumtheeconomyconvergestodependsoninitialconditions.Whattheseresultsdoistoruleoutcyclesorchaoticbehavior.Nowweproceedtocharacterizehowthelong-runequilibriumoftheeconomydependsonvariousparametersandtheinitialwealthdistribution.

LetaJ(w)bethestationarypointofthedifferenceequationdescribingthewealthtransitionofadynastyengagedinoccupationJ(whereJ=S,W,Edenotesthethreeoccupations:subsistence,worker,andentrepreneur)whenthewagerateisw.Thenwehave

aSðwÞ¼

swforallw:

1Àsrsw1ÀsrsðqÀrIÀwÞ1ÀsrsðqÀrIÞ

:

2ð1ÀsrÞ

aWðwÞ¼aEðwÞ¼

¯Þ¼aEðw¯Þ¼aWðw

WWByAssumption1,aE(w)>aE(w¯)=a(w¯)>a(w).

ComparingthevaluesofthesethresholdlevelsofwealthwithI,wecancompletelycharacterizethelong-runoutcome(intermsofthestationarydistributionofwealth,theequilibriumwagerateandthelevelofnetoutput)oftheeconomy.

Proposition2:Theinitialdistributionofwealthmattersindeterminingthestationarydistributionofwealthandthelongrunequilibriumwagerateifandonlyif

sðqÀwÞzI>

sw:1ÀsrOtherwisetheeconomyconvergestoahigh-wageequilibrium(ifIVsw/(1Àsr))orasubsistenceequilibrium(ifI>s(qÀw))irrespectiveofinitialconditions.

Proof:Theproofconsistsofthefollowingtwosteps

Step1.Thefollowingfourcasescharacterizethesteady-stateequilibriumoftheeconomycorrespondingtovariousparametervalues:

Case1.I>sðqÀwÞZI>aEðwÞ.Thisisasituationwherethesteady-statewealthoftheentrepreneurialclasscannotfinancetheoperationoftheindustrialtechnologyeven

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whenwagesareaslowaspossible.TheonlyequilibriuminthiseconomyisthereforeonewhereeveryoneisengagedinsubsistenceproductionirrespectiveoftheinitialwealthdistributionG0.Asaresultthestationarywealthdistributiondisplaysnoinequality.

WWCase2.s(qÀw)zI>sq/(2Àsr)ZaE(w)zI>aE(w¯)=a(w¯)>a(w).TheconditionthataE(w)zIimpliess[r(aÀI)+qÀw]zIbazI.Itsayswhenthewagerateislow,offspringofindividualswhoareabletostartanenterpriseinthecurrentperiodwillalsobeabletodosointhenextperiod,i.e.,thereisnodownwardmobility.Similarly,I>aW(w)impliess(ra+w)1/2),therewillnotbeanymobilityineitherdirection.Thisimpliesthatthewageratewillalwaysbeequaltow;thewealthofthosedynastiesthatareinitiallycapital-constrainedwillconvergetoaW(w);thewealthofthosethatarenotwillconvergetoaE(w);andtherewillbe1ÀG0(I)firmsoperatingineachperiod.Nowsupposetheeconomystartsoutwiththehigh-wagerate(G0(I)V1/2).The

Wcondition,I>aE(w¯)=a(w¯),impliesw¯isnotsustainable.Thereexistsafinitessuchthatws=w¯andws+1=w.ThereafterthestoryisthesameasaboveifwetakeGs+1(Á)astheinitialwealthdistributioninthenewlow-wageregime.Andofcourse,Gs+1dependsonG0.WWCase3.sq/(2Àsr)zI>sw/(1Àsr)ZaE(w)>aE(w¯)=a(w¯)zI>a(w).Again,sinceaE(w)>I>aW(w),thereisnoupwardordownwardmobilitywhenwagerateislow.Therefore,iftheeconomystartsoutatlow-wagerate(G0(I)>1/2),thestoryisthesame

WasinCase2.However,thecondition,aE(w¯)=a(w¯)zI,impliess(ra+((qÀrI)/2))zIbazI.Hence,whenthewagerateishigh,peoplewhoarenotcapital-constrainedwillremainunconstrained,i.e.,thereisnodownwardmobility.Therefore,iftheeconomystartsoutwithG0(I)V1/2,thehighwagew¯willlastforever.Asaresult,everydynasty’swealthwill

Econvergetoa(w¯).

Case4.sw/(1Àsr)zIZaW(w)zI.Thehigh-wageequilibriumwillresultirrespectiveofG0becauseevenwhenwagesarelow,thesteady-statewealthleveloftheworkingclasspermitsthemtostartafirm.Asaresult,theuniquestationarywealthdistributiondisplaysnoinequality.

Step2.Nextweshowthatthesetsofparametervaluesthatcorrespondtothefourcasesanalyzedabovearemutuallyexclusiveandexhaustivewithrespecttothesetofalladmissibleparametervalues(i.e.,thosesatisfyingAssumptions1and2).

Supposesw/(1Àsr)VI.Thisinequalityimplies[(2Àsr)/(1Àsr)]wV2w+Ir.Asaresult,Assumption1,whichguaranteesq>2w+Ir,alsoimpliesq>[(2Àsr)/(1Àsr)]w,i.e.,q/(2Àsr)>w/(1Àsr).Thelastinequalityinturnimplies,uponrearranging,s(qÀw)>sq/(2Àsr)andsq/(2Àsr)>sw/(1Àsr).Thus,wehavethefollowinginequalitywhichisderivedfromAssumptions1and2:

sðqÀwÞ>

sqsw>

2Àsr1Àsrwhichholdssolongassw/(1Àsr)VI.Ifinstead,IM.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226215

Fig.2.Long-runwageratesunderdifferentparameterconfigurations(Non-StochasticModel).

Iftherewerenofrictionsinthecreditmarket,solongasthemoderntechnologyismoreproductivethanthesubsistencetechnology(whichisensuredbyAssumption1),itwillbeusedbytheentireeconomy.Theinitialdistributionofwealth,theproductivityofthesubsistencetechnologyorthepropensitytosavewouldnotberelevantindeterminingtotaloutput.Ifcreditmarketsareimperfect,Proposition2showsthatthelong-runequilibriumoftheeconomycannotbepredictedbyasimplecomparisonoftheproductivityofthetwotechnologies.Ifthesizeofthewealththresholdneededtostartanenterprise(I)isveryhigh,thentheeconomywillcollapsetosubsistencesectorsincethesteady-statewealthlevelofevenarichdynastyinalow-wageequilibriumwillfallshortofit.Conversely,ifIisverylow,thenthesteady-statewealthlevelofevenapoordynastyinalow-wageequilibriumwillexceedit.Inthiscase,inthelongruntheeconomywillconvergetoahigh-wageequilibriumwherethewholepopulationisengagedinthemodernsector.ForintermediatevaluesofI,thelong-runequilibriumoftheeconomycannotbepredictedfromtheparametersgoverningtechnologyandpreferencesonly.Theinitialwealthdistributionalsomatters.Iftheparametersaresuchthatthelow-wageequilibriumistheuniquelong-runequilibrium(i.e.,thisisthecasewherethesteady-statewealthlevelofadynastyunderthehigh-wageequilibriumislessthanI),thenumberoffirmsusingthemoderntechnologyinthelong-runequilibriumisthesameasthoseatt=0andthisishowtheinitialwealthdistributionmatters.Moreinterestingly,iftheparametersaresuchthatboththelowandhigh-wageequilibriumarepossible,thentheinitialdistributionofwealthalsodetermineswhichequilibriumwillbechosen.IfinitiallytherearemanydynastieswhohavewealthhigherthanI,thenthehigh-wageequilibriumwillresult,andthiswillenableotherstoaccumulateenoughwealthsothatinthelongruneveryonecanbecomeanentrepreneur.Ifontheotherhand,ifinitiallythecredit-constraineddynastiesareinamajority,theywillpushthewagedowninthelabormarketwhichwillcontinuetokeepthempoorinsuccessivegenerations.

Proposition2alsosuggeststhattheeffectofchangesinparametervaluesregardingtechnologyandpreferencesmaydependontheinitialwealthdistribution,andinparticular,canpushtheeconomyfromonetypeofsteady-stateequilibriumtoanother.Letusconsidertheeffectsofchangesinvariousparametersofthemodel.

Anincreaseintheproductivityofthemoderntechnologyq(asaresultoftechnologicalchangeoreconomicpolicies,suchasliberalizingtheeconomy)willincreasetheincomegeneratedbyexistingenterprisesusingthemoderntechnology.Theeffectofthisonpercapitaincomewilldependontheinitialwealthdistributionunderthelow-wageequilibriumasthatdeterminesthenumberoffirmsusingthemoderntechnology,butnotinthehigh-wageequilibrium.Moreover,ifasaresultofanincreaseinqthesteady-statewealthlevelofsomeindividualsarepushedaboveI,thenumberofenterprisesusingthemoderntechnologyinasteady-stateequilibriummayincrease.Thiswillbethecaseif

216M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226

initially[s/(2Àsr)]qAnincreaseintheproductivityofthesubsistencetechnologyw(asaresultoftechnologicalchange,orgovernmentpoliciessuchasminimumwagelawsorsubsidytosmall-scaleindustry)willincreasepercapitaincomebyraisingtheincomesofthoseengagedinthesubsistencesector.9However,whileanincreaseinwincreasesthesteady-statewealthlevelofworkersandthoseengagedinsubsistence,itreducesthesteady-statewealthlevelofentrepreneursinalow-wageequilibrium.Asaresult,theeffectofitonsteadystateincomeisambiguous.Forexample,startingwithasituationwheres(qÀw)zI(sothatthelow-wageequilibriumexists),anincreaseinwcanleadtos(qÀw)Theeffectofanincreaseinsisstraightforward.Itdoesnotraisesteady-stateincomedirectlyinthismodel,butraisesthesteady-statewealthlevelofeverydynasty.Ifanincreaseinspushesthesteady-statewealthlevelofsomeindividualsaboveI,thenumberofenterprisesusingthemoderntechnologyinasteady-stateequilibriumwillincrease.Previously,wehaveassumedthatthechanceofbeingcaughtfromdefaultiszero(p=0).Therefore,thereisactuallynocreditmarketinthiseconomy—oneneedstoownthewholeamountofcapitalrequired(I)tostartupamodernfirm.Nowsupposep>0,sothatonly(IÀ(pF/r))isneededtobecomeanentrepreneur.Otherthingsbeingequal,sinceitiseasiertoreachthisthreshold,theeconomyismorelikelytoendupwithahigh-wageequilibrium.Forthesamereason,anincreaseinthepunishment(F)wouldhavethesameeffect.Changesintheinterestrate(r),however,havetwooppositeeffects.Adecreaseinrwouldreducethewealththresholdforborrowingononehand,butontheotherhand,itbecomeshardertoaccumulateone’swealth.Thissuggeststhatimprovingtheenforcementtechnology(i.e.,increasesinpandF)hasanunambiguouslypositiveroleineliminatingpovertytraps,whereastheeffectoflowercapitalscarcityintheinternationalcreditmarket(i.e.,adecreaseinr)hasanambiguouseffect.

Letusdefinethetotalincomeoftheeconomy,thesumofwageandprofitincome,as:

Y¼GðIÞwþf1ÀGðIÞgðqÀwÀIrÞ

Thefollowingresultcomparestheequilibriaintermsoftotalincome.

Itwillalsoincreasethewagesofworkersengagedinthemodernsector.Butthiswillbematchedbyadecreaseintheprofitsofentrepreneursandtherewillbenoeffectonpercapitaincome.

9M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226217

Proposition3:Forparametervaluesforwhichinitialconditionsmatter,thegreateristhefractionofthepopulationwhoareinitiallypoor,thelowerissteady-stateincome.Proof:Underasubsistenceequilibrium,totalincomeisY=w.Inalow-wageequilibrium,totalincomeisY=(qÀIr){1ÀG(I)}À{1À2G(I)}w.Finally,inahigh-wageequilibrium,totalincomeisY=(qÀIr)/2.SinceqÀIr>wbyAssumption2,andunderalow-wageequilibriumG(I)z1/2

qÀIr

zðqÀIrÞf1ÀGðIÞgÀf1À2GðIÞgw>w:2

Hence,thetotalincomeoftheeconomyunderahigh-wageequilibriumexceedsthatunderalow-wageequilibrium,whichinturnexceedsthatunderasubsistenceequilibrium.Proposition2showsthatfortheparametervaluess(qÀw)zIzsw/(1Àsr)(correspondingtoCases2and3),ifG0(I)>1/2,thentheeconomyconvergestoalow-wageequilibriumwhereonly1ÀG0(I)firmsoperate.Hence,thepropositionfollows.5Whatthisresultshowsisthatevenifthelow-wageequilibriumistheuniqueequilibrium,thegainfromhavingonelesscredit-constrainedpersonisonemorefirmthatusesthemoderntechnologyandgeneratesgreaterincome.Whenmultipleequilibriaexist,thelongrungainsfromhavingasmallernumberofpeoplewhoarecredit-constrainedaremuchgreaterthaninthepreviouscase,sincethismightunleashmarketforcesthatpushtheeconomytoahigh-wageequilibriumwherethewholepopulationisengagedinthemodernsector.

Theaboveresultalsoshowsthattotheextentgreaterequalityofthedistributionofwealthreducesthefractionofthepopulationwhoarecapital-constrained,bothgreaterequityandgreaterefficiency(intermsoftotalincome)areachieved.Asaresult,one-shotredistributivepoliciescanraisethetotalincomeoftheeconomypermanentlyforparametervaluesforwhichtheinitialwealthdistributionmattersforthelong-termperformanceoftheeconomy,asBanerjeeandNewman(1993)pointout.Toseethisassumethatthepolicyisimplementedaftertheeconomyhassettleddowninasteady-stateequilibrium.Supposethegovernmenttaxesbequestsofrichdynastiesandredistributestherevenue(sothatthegovernmentbudgetisbalanced)topoorerdynastieswhosewealthislessthanIwiththegoalofmakingasmanyindividualstobeabletostarttheirownenterprisesaspossible.Naturally,thispolicywillhavenoeffectwhentheeconomyisinahigh-wageorsubsistenceequilibriumbecauseeveryonehasequalwealthtostartwith.Forthecaseoflow-wageequilibrium,itcanhaveaneffect.ConsiderCase3.Thepolicymoveseveryone’swealthclosertothemean,whereaswhetherthewealthofthemedianpersonisgreaterthanorlessthanIdetermineswhetherthereisahigh-oralow-wageequilibrium.Startingfromalow-wageequilibrium,ifthemeanisgreaterthanI,thensucharedistributivepolicywillpushtheeconomytowardsahigh-wageequilibrium.EvenifthemeanislessthanIinwhichcasethehigh-wageequilibriumcannotbeachieved,thepolicywillincreasethenumberofenterprisesthatareoperatedandhenceraisetotalincome.Similarly,inCase2suchapolicywillincreasethenumberofenterprisesthatareoperatedandhence,raisetotalincome.

However,theimplicationofthisexerciseisnottosupportanyegalitarianredistributivepolicytoincreasetotalincome,ratheronlythosethatincreasethenumberofenterprises

218M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226

operatingintheeconomy.Forexample,inCase3,ifthemeanwealthlevelislessthanI,thenacompleteredistributionwillpushtheeconomytosubsistence.

3.Extension:stochasticmodelwithmobility

AnimportantfeatureofthemodelinSection2isthattheincomesofallagents,andthebequestsoftheirprogenyarealldeterministic.Thisisunsatisfactoryasthelong-runwealthdistributionhasallprobabilitymassconcentratedontwopoints(foralow-wageequilibrium)oronepoint(thehigh-wageequilibriumorthesubsistenceequilibrium).Asaresult,thereisnomobilityacrossclasses.Inthissection,weexaminetheimplicationsofallowingupwardanddownwardmobilitythroughrandomshocks.

Inparticular,weassumethateveryindividual’ssavingrateissubjecttoanidiosyncratici.i.d.shock.Ineveryperiod,eachindividual’ssavingratecouldbehigh(s¯)withprobabilityp

10orlow(s)withprobability1Àp.Ifs¯(s)ishigh(low)enough,wewillhaveupward(downward)mobilitywhichisabsentinthestationarydistributionsdiscussedinSection2.11Wemakethefollowingassumptionsabouttheparameterssands¯:Assumption3

¯>s

I

ands¼0

wþrIThefirstpartoftheequationofAssumption3ensuresthereisupwardmobilityinthiseconomy.NoticethatI/(w+rI)(\"#)

nÀ1X

¯ÞiwzI:¯ðrsm¼minnaN:s

i¼0

Thatis,ittakesatmostmconsecutiveperiodsofgoodluckforadynasty—evenifit

startedwithnoinitialwealthandevenifwageratesremainedlow—tobecomerich.Thesecondpartoftheassumption,ofcourse,ensuresthereisdownwardmobilityinthiseconomy,butmoreimportantly,itgreatlysimplifiestheanalysis.Bysettings=0,anindividualdynasty’swealthdynamicsdependsonthehistoryonlyuptothelasttimeitreceivedabadshock.Togetherwiththefirstpart,Assumption3impliesthefractionofthepoor,andthereforethecurrentwageratedependsonthewagedynamicsonlyuptothepreviousmperiods.Togetherwiththefactthatthewageratecanonlybehighorlow,thisimpliesthattheresultingwagedynamicsmustbeeitheroneofthefollowingthreetypes:alwayshighwage,alwayslowwage,oracycle.

Theseshockscouldbetasteshocksorshocksrelatedtothetechnologyofsaving(e.g.,anegativeshockcouldbeinterpretedasanindividual’ssavingsbeingstolenorexpropriated).

11Analternativewaytointroducerandomshocksinthemodelwouldbetoletproductionbestochastic(asinBanerjeeandNewman,1993).However,givenourassumptionsabouttheproductiontechnologyandpreferences,thecontractualformofpaymenttoworkerswillbeindeterminate(forexample,wagecontractsorprofit/outputsharingcontractswillbeequivalent).Thisisunsatisfactorysincethespecificcontractualformwillbecrucialindrivingtheextentofupwardanddownwardmobilityinthemodel.

10M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226219

Tobemorespecific,givendatet,wecandefinefunctionat(Á):{0,1,...,t}!R+asatð0Þ¼0;

\"

#

ifnaf1;...;tg:

nÀ1X¯¯ÞiwtÀiÀ1atðnÞ¼sðrs

i¼0

Therefore,at(n)representstheinitialwealthlevelatdatetofadynastywhichreceived

exactlynconsecutiveperiodsofgoodluckandwasawage-earnerduringthesenperiods.Thedistributionthatat(n)describesdiffersfromtherealwealthdistributionatdatetsincewealthypeoplecouldbeearningentrepreneurialprofitsinsteadofwages.However,fordynastiesthatwerewageearners,at(n)correctlyrepresentstheirinitialwealthlevelsatdatet.Therefore,atdatet(zm),therewillbeaprobabilitymasspn(1Àp)atat(n),bn=0,1,...,l(t),wherel(t)=min{n:at(n)zI}.Sincethereisnopoordynastywhosewealthlevelisdifferentfromat(n),bn=0,1,...,l(t)À1,

GtðIÞ¼

lXðtÞÀ1n¼0

À1Becausel(t)Vm,bt,Gt(I)dependsonatmost{wtÀiÀ1}im=0.Withoutlossofgenerality,wecanuseafunctionf:Wm!W,whereW={w,w¯},todescribetherelationshipbetweencurrentwagerateandthewageratesinthepreviousmperiods.Tworesultsfollowimmediately:

pnð1ÀpÞ:

Lemma2.Thewagedynamicscanbeeitherstationary(withhighorlowwages),ordisplaycycles.

Proof:SeeAppendixA.

Lemma3.fisweaklyincreasingineachofitselements.

Proof:SeeAppendixA.Wedivideourdiscussionfortherestofthissectionintotwocases,constant-wagedynamicsandcycles.3.1.Constant-wagedynamics

Inthefollowingproposition,weshowthatifs¯isnottoolarge,thestationarywagedynamicscanonlybeoftwotypes:alwayshighwageoralwayslowwage.

Proposition4:InadditiontoAssumption3,ifs¯V1/r,thenthestationarywagedynamicscanonlybeeitheralwayshighwageoralwayslowwage.Proof:SeeAppendixA.

Whichcasewillemergedependsonhowfastthepoorbecomerichwhenwagesarehighorlow,andinsomecases,theinitialwealthdistribution.ThischaracterizationisprovidedbyProposition5.Ifrs¯V1,intheexpressionforthewealthlevelofacurrentlypoordynasty,wageratesintherecentpastreceivegreaterweightthanwageratesinthedistantpast.FromtheproofofProposition4,ifwt=w¯,thewealthdistributionofthepoor

220M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226

isgoingtoremainthesameorshiftstotheright(first-orderstochasticdominance),andineithercase,wt+1=w¯.Ifwt=w,theoppositehappens.Thewealthdistributionatdatet+1isthesameasthewealthdistributionatdatet,orisfirst-orderstochasticallydominatedbyitandhence,wt+1=w.

Next,weaskunderwhatconditionstheeconomywillconvergetothehigh-wageequilibrium.Intuitively,ifthechanceofreceivingahigh-savingshockishigh(plarge)andifitdoesnottakelongforaverypoordynastytobecomerich(msmall),theeconomyshouldendupwithahigh-wageequilibrium.Ontheotherhand,thelow-wageequilibriumwouldemergeifthechanceofanindividualtobebornwithnowealthishigh(psmall)andifittakesmanyperiodstobecomerich.Betweenthesetwoextremes,thereshouldbecaseswheretheinitialdistributionmatters.WeformallyprovethisinLemma4andProposition5.LetusfirstdefinemV,similartom,asthenumberofperiodsneededforazero-wealthdynastytobecomerichunderhighwages.Inotherwords,

(\"#)

nÀ1X¯¯Þiw¯zI:mV¼minnaN:sðrs

i¼0

Naturally,mVVm.

Lemma4.Ifpmz1/2,thentheeconomyconvergestothehigh-wageequilibrium.If

pmV<1/2,thentheeconomyconvergetothelow-wageequilibrium.Proof:SeeAppendixA.

GivenLemma4,wecanprovethefollowingproposition:

Proposition5:Theinitialdistributionofwealthmattersindeterminingthestationarydistributionofwealthandthelong-runequilibriumwagerateinthestochasticmodelifandonlyif

pmVz

1

>pm:2

Otherwisetheeconomyconvergestoahigh-wageequilibrium(ifpmVzpm>1/2)oralow-wageequilibrium(if1/2>pmVzpm)irrespectiveofinitialconditions.Proof:SeeAppendixA.

Inthecasewheretheinitialwealthdistributioncanmatter,itisdifficulttogiveconditionsoninitialdistributionsunderwhichtheeconomywouldendupwithahigh-orlow-wageequilibrium.Intuitively,iftheeconomystartsoutwithmanyrichpeople,thehigh-wageratewouldbelikelytolastformanyperiods.Asaresult,bythetimemostofthosewhowereoriginallyrichwouldbehitbyalowsavingsshock,someofthosewhowereoriginallypoorwouldaccumulateenoughwealth.Therefore,thestationarydistributionismorelikelytobetheoneassociatedwiththehighwage.Conversely,iftheeconomystartsoutwithmanypoorindividuals,thelow-wageratewouldlastforalongtime.Thenonlyafewluckyindividualswillbeabletoaccumulateenoughwealth

M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226221

Fig.3.Long-runwageratesunderdifferentparameterconfigurations(StochasticModel).

beforebeinghitbyalowsavingsshock.Therefore,thelow-wageequilibriumwouldresult.

Proposition5providesconditionsonparametersunderwhichmultiplestationarydistributionsmayexist(alsoseeFig.3).Otherthingsbeingthesame,thegreateristhedifferencebetweentheproductivityofthemodernandthesubsistencetechnology(namely,w,andthemore¯andw),thegreaterwillbethedifferencebetweenmandmV

likelythiscasewilloccur.Also,thiscaseismorelikelywithintermediatevaluesofI.Thehigher(lower)isIthehigher(lower)willbebothmandmVandforgivenpthemorelikelytheeconomywillendupinalow(high)wageequilibrium.

Sincethenumberoffirmsoperatinginalow-wageequilibrium(pm)islessthanthatunderahigh-wageequilibriumforparametervaluesforwhichinitialconditionsmatter,thegreateristhefractionofthepopulationwhoareinitiallypoor,themorelikelytheeconomywillendupinalow-wageequilibriumwithalowerleveloflongrunpercapitaincome.ThisissimilarinspirittoProposition3inthenonstochasticmodel.However,inthelow-wageequilibriumofthenonstochasticmodel,thelong-runnumberoffirmsdependsontheparametersofthemodelaswellastheinitialfractionofpoorindividuals,whereasinthestochasticmodelitdependsonlyontheparameters.3.2.Cycles

mVmFromProposition4andLemma3,ifs¯islargeenough(s¯>1/r)andifpz1/2>p,theeconomydoesnotnecessarilyconvergetoaconstant-wageequilibrium;thewagedynamicsmightdisplaycycles.12Itisdifficulttoprovidegeneralresultsforthiscase

Itturnsoutthatwedonotneeds¯V1/rtoproveLemma4andProposition5.Inotherwords,theyaretrueevenwhens¯>1/r.

12222M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226

andwerestrictourdiscussionaroundasimpleexampleofacycle,thesimplestonewecanfind,wherethehighwageandthelowwagealternatewitheachother.

Example:Supposes=2,andp2z1/2>p3.Ifs¯[(rs¯)w+w¯]1/r,m=3,mV

thewagedynamicsmightdisplayatwo-periodcyclewherehighwageandlowwagealternatewitheachother.13Startingwithahigh-wageperiodwt=w¯,thewealthdistributionmustdisplayaprobabilitymass

(i)(1Àp)ata=0consistingofthosewhoreceivedabadsavingshocklastperiod;(ii)p(1Àp)ata=s¯wconsistingofthosewhoreceivedabadsavingshockintheperiod

beforelastperiod,butagoodshockinthelastperiod(noticethatthewagerateinthepreviousperiodwaslow);(iii)p2(1Àp)ata=s¯[(rs¯)w¯+w],consistingofthosewhoreceivedagoodsavingshockinthe

lasttwoperiods(noticethatthewagerateintheperiodbeforethepreviousperiodwashigh);

(iv)p3consistingofthosewitha>s¯[(rs¯)w¯+w].Sinces¯[(rs¯)w¯+w]zI,and,byassumptionwt=w¯inthecurrentperiod,thefractionofthepoorcannotbemorethanhalf.Thisisindeedthecaseas1Àp2V1/2<1Àp3.Inthenextperiod,thewealthdistributionhasaprobabilitymass(1Àp)at0,p(1Àp)ats¯w¯,and23p(1Àp)ats¯[(rs¯)w+w¯],andpconsistingofthosewitha>s¯[(rs¯)w+w¯].Again,since

231ÀpV1/2<1Àpands¯[(rs¯)w+w¯]Cyclesinourmodelcanoccurinthespecialcasewhenthepositivesavingsshockisveryhigh—sohighthatadynastythatreceivesonlypositivesavingshockswillbecomeinfinitelyrichjustbysaving,howeversmalltheinitialamountitstartedoffwith.Thencurrentwageswillhavealargeimpactonfutureincome.Whenwagesarehightherichestamongthepoorstaypoorbecausepastwages(whicharelow)playadominantrole.Sincethereisonlydownwardmobilitybutnoupwardmobility,thewagerateswitchesfromhightolow.Butthen,eventhoughthewageislow,therichestdynastiesamongthepoor

2becomerichsincethehighwagetheyexperiencedpreviouslyisweightedby(s¯r).Theparameterconfigurationweassumeensuresthatthereismoreupwardmobilitythandownwardmobilitysothatthewageratebecomeshighagain.Thisprocesswillgoonforeverandtheeconomywilldisplaycycles.Aghionetal.(1999)alsoshowthepossibilityofendogenouscyclesinamodelwithimperfectcreditmarkets,butthemechanismthereisverydifferent.Intheirmodel,highinvestmentgenerateshighfutureprofitsandinvest-ment,butitalsopushesuptheinterestrate,whichreducesfutureprofitsandinvestment.Ifthesecondeffectisstrongenoughrelativetothefirst,outputwilldisplaynegativeserialcorrelation.

13Thisisnottheonlypossibleoutcome.Dependingontheinitialdistributionwecouldalsogetastationaryequilibriumwithloworhighwages.

M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226223

4.Conclusion

Inthispaper,weanalyzedasimpledynamicmodelofoccupationalchoiceinthepresenceofcreditmarketimperfectionswherewealthinequalityandreturnstovariousoccupationsareendogenous.Weexaminedconditionsunderwhichmultiplesteady-stateequilibriaexistandcharacterizedhowinitialconditionsaffectwhichequilibriumtheeconomyconvergesto.Weconcludewithtwoobservationsbothofwhichsuggestdirectionsforfutureresearch.First,therearemanyinterestingquestionsregardingtherelationshipbetweencreditmarketimperfectionsandeconomicdevelopmentthatthecurrentmodelormodelssimilartoit(suchasBanerjeeandNewman,1993,onwhichitisbased,andalsoGalorandZeira,1993;Piketty,1997)cannotaddress.Asexamples,onecanmentionrecentresearchstudyingconsequencesofdynasticutilitymaximizationinasimilarframeworkandarichersetofpossibleoccupations(seeMookherjeeandRay,2000),allowingentrepreneurstohaveheterogeneoustalent(BernhardtandLloyd-Ellis,2000),andtheinteractionbetweencreditmarketimperfectionsandincentivesandcontractinginthelabormarket(seeGhataketal.,2001).Second,whilethereissomecross-countryevidenceonthenegativeeffectoninequalityandmeasuresofcreditmarketimperfectionsongrowth(Benabou,1996)thatisconsistentwiththepredictionofthismodel,moremicro-levelevidenceontheeffectofborrowingconstraintsoneconomicmobilityisclearlyneeded.14Acknowledgements

Wethanktheanonymousrefereesforhelpfulcomments,andChristianAhlin,AbhijitV.Banerjee,CharlesHunter,JosephKabowski,AlexanderKaraivanov,andAndreasLehnertforusefuldiscussions.Weareresponsibleforallremainingerrors.

AppendixA

ProofofLemma2:First,foranyw=(w1,w2,:::,wm)aWm,wecandefineafunctionM:Wm!Wmas

MðwÞ¼ðw2;w3;:::;wm;fðwÞÞ:

Second,wecomparewwithM(w):ifw=M(w),westop.IfwpM(w),thenwecalculateM(w)uM(M(w))andcheckifitisequaltoeitherworM(w).Ifyes,westop;ifno,thenwecompareM3(w)withw,M(w),andM2(w),andsoon.Sincewageratewonlytakestwo

2TherehasbeensomeworkonthisareausingpaneldatasetsfromtheUSandtheUK(see,forexampleEvansandLeighton,19;BlanchflowerandOswald,1998).Butverylittleisknownaboutdevelopingcountrieswhereborrowingconstraintsarepresumablymuchmoresevere.

14224M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226

values(wandw¯)andmisfinite,thisprocesscannotgoonforever.TheremustexistsomekandkVwith0Vk\"#\"#nÀ1nÀ1XXi

¯ÞwmÀiVs¯Þiwm¯¯ðrsðrsVÀisforn¼1;2;...;m

i¼0

i¼0

ZaðnÞVaVðnÞforn¼1;2;...m

ZlzlV

Z

lÀ1Xn¼0

pð1ÀpÞz

n

lVÀ1Xn¼0

pnð1ÀpÞ

ZGðIÞzGVðIÞ

ZfðwÞVfðwVÞ

5

ProofofProposition4:Foranytzm,thereisaprobabilitymasspn(1Àp)atat(n),

bn=0,1,...,l(t).Compareat+1(n)withat(n),

\"#\"#nÀ1nÀ1XX¯¯ÞiwtÀiÀs¯ÞiwtÀiÀ1¯atþ1ðnÞÀatðnÞ¼sðrsðrs

i¼0

i¼0

\"#)

nÀ2X

¯wtÀð1Àrs¯Þ¯ÞiwtÀiÀ1Àðrs¯ÞnÀ1wtÀn¼sðrs

i¼0

(

¯ðwtÀw¯Þ;s¯ðwtÀwÞ󰀈:a½s

wt=wThisimpliesl(t+1)Vl(t)andsinceGtðIÞ¼¯,at+1(n)Àat(n)z0.PPlIfðtÞÀ1ilðtþ1ÞÀ1i

pð1ÀpÞV1=2.Thatis,wt+1=wt=w¯.Ifi¼0pð1ÀpÞV1=2;Gtþ1ðIÞ¼i¼0

M.Ghatak,N.Jiang/JournalofDevelopmentEconomics69(2002)205–226225

wt=w,at+1(n)Àat(n)V0.Thisimpliesl(t+1)zl(t)andsinceGtðIÞ¼Plðtþ1ÞÀ1i1=2;Gtþ1ðIÞ¼i¼0pð1ÀpÞ>1=2.Thatis,wt+1=wt=w.

PlðtÞÀ1

i¼0

pið1ÀpÞ>

5

PÀ1i

ProofofLemma4:Ifpmz1/2,thenmi¼0pð1ÀpÞV1=2,whichinturnimpliesthatf(w,mmVw,...,w)=w¯.Therefore,fromLemma3,f(Á)=w¯foranyelementinW.Ifp<1/2thenPmVÀ1i

¯,w¯,...,w¯)=w.Therefore,fromi¼0pð1ÀpÞ>1=2whichinturnimpliesthatf(wmLemma3,f(Á)=wforanyelementinW.5ProofofProposition5:The‘‘onlyif’’partisimpliedbythepreviousLemma.Toprovethe‘‘if’’part,itsufficestofindtwowealthdistributionssuchthatoneisconsistentwiththehigh-wageequilibrium,whereastheotherisconsistentwiththelow-wageequilibrium.Theobviouschoiceforsuchadistributionisoneinthesteadystate.First,forthehigh-wageequilibrium,considerawealthdistributionatdatetthathasaprobabilitymass

ð1ÀpÞpnð1ÀpÞ

atat

#

nÀ1X¯¯Þw¯sðrs

i¼0

0

\"

bn¼1;2;...

FromthedefinitionofmV,weknow

\"¯s

mVÀ1Xi¼0

#

mVX

¯Þw¯i¼0

#\"

andtherefore

GtðIÞ¼

mVÀ1X

1

pnð1ÀpÞ¼1ÀpmVV:

2n¼0

Thisimplieswt=w¯whichinturnimpliesthenext-periodwealthdistributionremains

exactlythesame.BychangingmV,w¯tom,w,onecanshowthatthelow-wageequilibriummayalsoemergeinasimilarway.Theonlydifferenceisthatthewealthdistributionsimilarlyconstructedisnotthesteady-statedistributionsincetherichcanearnentrepre-neurialprofitsinsteadofwages.Butsinceitisthewealthtransitionofthepoorthatmattersindeterminingthewagerate,theargumentisstillvalid.5

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