OLG Model with Strategic Job-Seeking and Credit Rationing Davide Fiaschi - Raimondello Orsini y 29May 1998 Abstract
Inthehumancapitalliterature,itisusuallyassumedthathuman
cap-italis paid according to its marginalproductivity. Nevertheless, inthe
real world labor compensation is linkedto a xedhierarchy dueto the
divisionand organization oflabor. Accesstoprivilegedpositionsinthe
hierarchydependsonschoolingcredentials, whichinturnare afunction
of individual learning abilities and of individual spending ineducation.
Peoplecompeteineducation inorderto achieve the best jobpositions:
positionalcompetitionislikearent-seekingactivity,basedonthe
di er-entlevelsof credentials. Inthis paper, asimpleOLGeconomywithtwo
agentsand twokindsof jobs ismodeled, andthe strategic solutionsare
analyzed. Themodelshowsdi erentoutcomesdependingonthe
hypothe-sesregardingthetypeofstrategicinteraction(sequentialorsimultaneous)
andthe characteristics ofthe capitalmarket. Inthe sequential
equilib-rium,thepresenceofcreditmarketimperfectionsandrisk-aversionmakes
theasympthoticwealthdistributiondependentoninitialconditions(non
ergodicity). Inthe simultaneousequilibrium,a nonmonotonic
relation-shipbetweenincomeinequalityandlongrungrowthisshown;inthelong
run,joballocationis mainlydeterminedbythe innatelearningabilities
anditisunrelatedtotheinitialwealthdistribution(ergodicity).
UniversitadiPisa,e-maild aschi@ec.unipi.it
y
Recentliteratureongrowththeoryhasrenewedeconomists'interestinthe
deci-sionsofinvestmentineducationandaccumulationofhumancapital,underlining
therelationshipsbetweendistribution of income,humancapitaland economic
growth. 1
From the point of view of a single agent, investing in education is
mainly a way to signal his individual ability; that is each subject invests in
educationinordertogetabetterjob. 2
Ifindividualhumancapitalisnot
trans-ferable and the capital market is imperfect, the amount of inherited wealth
is crucial in establishingthe investment in education. Since education
repre-sentsthemainwaytobetterone'seconomiccondition 3
,acomplexrelationship
emergesbetweeneducationalchoicesand thedynamicsofincomedistribution.
Inthispaper,weaimtoinvestigatethisrelationshipbymeansofamodelof
po-sitionalcompetitiononthelabourmarket 4
. AccordingtoHirsch(1976),agents
try to acquire credentialsby investing in education, to distinguish themselves
from othersandtogetaprivilegedrole intheorganizationofproduction(and
thereforeahigherincomeormoresocialprestige 5
). Iftheorganizationofroles
is xedand, withthe developmentoftheeconomicsystem, thenumberofthe
privileged roles does not increase at the same pace as the number of agents
investing in education, then, astime passes, anincreasing levelof investment
in education is required to get thesamesocio-economic roles (the quantity of
"intermediate"goodincreaseswithoutanychangeinthe naloutput: an
ineÆ-cientoutcome). Asweknow,intherealworldthenumberofhighpaying(high
prestige)positionsdoesnotincreasewiththenumberofindividualswithahigh
degreeof education 6
. Thisyields toacompetition amongindividuals in order
togetsuchpositions,competitionwhichtakesplace throughtheinvestmentin
education. Suchaninvestment,independentlyfromtheincreaseofproductivity
that itmightinvolve,actslikeasignal.
Althougheachagentcompetes againstaninde nitenumberofsubjects(all
the potential aspirants for the same role), it is possible to assume that in a
labour market characterized by a xed number of high-paying positions, the
behavior of agents should be strategic. It would be useless for a pooragent
to invest money and time (which also involves an opportunity cost equal to
thetimespentwithoutworkingasanunskilledworker)tocompetewith other
agentswhoselevelof educationisfar greater. Inthis respect,it isrationalfor
thepooragentstoimmediatelyundertakeanunskilledactivity,choosingnotto
invest 7
.
Weassume that the economy is characterised by aparticular institutional
setting,wheretheavailableworkingpositionsare xedovertime(thetypesand
number of available jobs and relativewagesare exogenous 8 ). In this institu-1 See[11 ],[10 ],[8],[7],[5]. 2 See[20 ],[4],[23]. 3 See[18 ]. 4
Onpositionalcompetition,see[14]. Orsini(1997)presentsasimulationofamodelwith
positionalcompetitionbyeducationamongheterogeneousagents.
5
See[9].
6
Davis-Haltiwanger(1991)providesomeinterestingempiricalevidenceaboutthistrend
intheUnitedStatesafter1975.
7
Inthemodel,wealwaysassumethatagentsworkonlyoneperiod.Therefore,the
opportu-nitycostofinvestingineducationisrepresentedonlybytheamountofforgoneconsumption.
8
education (credentials), likewise in rent-seeking models. 9
Anotherpeculiarity
ofthemodelisthat thesignalroleof schoolachievementsdoesnotcomefrom
the equilibrium of a game with incompleteinformation, asin signalling
mod-els, but it is taken for granted, asa startingassumption. There are di erent
jobs withdi erentpay,andtheirallocationis such thatmoreeducated agents
have higher chances of getting better jobs. 10
We assume that the process of
job assignmentis uncertain: anagentcaninvest in education and, depending
on hislearningabilities, can reach aparticularlevelof credentials. Thelatter
in uence the probability of getting ahigh paying job, but it is never
conclu-sive. Credentialsareobtainedatschool,severalyearsbeforetheapplicationfor
theposition: therefore thefuture structure ofthelabourmarket (the demand
side) isunknown. Investing in education increasesthechances of winning the
positionalcompetitioninthefuture,buttheprocessisneverdeterministic: the
agentwithbettercredentialscannotbesureto getthebestposition. 11
Ourmain interestregardstheconsequenceof theinstitutional setting(the
schooling system and the labour and credit markets) on the distributive
dy-namics. Ifweconsideraneconomywithrisk-neutralagentsandperfectcapital
markets,independently from individual wealth,the individual innate abilities
determinetheprobabilitiesofgettingthehigh-payingpositions: inthelongrun
the distribution of wealth among the di erent dynasties will be ergodic, that
is, everydynasty willalwayshaveapositiveprobabilityof gettingthehighest
pay. However,if agentsare riskaverseand credit is rationed (due to possible
imperfections in thecapitalmarket), theasympthotic wealth distributionmay
dependontheinitialwealth,whichmeansthatitmaynolongerbeergodic.
Theeducational system canbeseen asan institution whose purpose is to
providepeopleasignalforthejobmarketthatindicatestheirpotential
produc-tivity. Intheactualeconomytherearemanydi erencesamongtheeducational
systemsof di erent countries (more or lessprivate, moreorless expensive 12
);
the resultsof themodel canhelp explain, at least partially, these di erences.
Forexample,intheAmericaneducationalsystemthereisasmallsetofelitarian
scholasticinstitutions,whichareveryexpensiveanda ordableonlyby
individ-ualswithhighwealth,orwithanotablegrant(Harvard,Yale,Princeton,etc..).
These institutions help e ectively in the competition for a good job. 13
Rich
[13]. Astimepasses,thetypeofcredentialsnecessarytogetthetoppositionschange. This
factleadstothe rstperioduncertaintyaboutthefutureassignmentprocess.
9
See[21 ].
10
Spence(1973)isthe rstmodelinwhicheducationisonlyasignal. Educationworksas
asignalrelatedtothe unobservableinnate abilityofworkers; inthisrespect,moreyearsof
educationcanmeanahigherperseverance(whichmeansfewerabsenteeismsandasmallerrisk
ofabandoningthejob)andothercharacteristicsgenerallycorrelatedwiththescholastic
cur-riculum;Weiss(1995)writes: "better-educatedworkersarenotarandomsampleofworkers:
theyhavelowerpropensitiestoquitortobeabsent,arelesslikelytosmoke,drinkoruseillicit
drugs,andaregenerallyhealthier. [...] wewouldexpectemployerstofavorbetter-educated
workersasameansofreducingtheircostsofsicknessandjobturnover". Employershireon
thebasisofthecredentials,butthelatterareonlyscreeningdevices: thestudiescarriedout
signalthattheworkerhasahigherprobabilityofbelongingtothesetofagentswithgreater
productivity.
11
Wethinkthatthisapproachisquiterealistic:educationincreasesthechancesofsuccess,
butitcannotassureabetterjobwithcertainty.
12
See[3]and[12].
13
TalentedindividualswhostudyatHarvardareboundto ndamuchbetterjobin
competition forhigh wagepositions. Sincethis willingnesstopayexists, there
aretheconditionsfortheriseofaninstitutioncapableof givingaparticularly
strongsignal,thatcanexcludethepoorbyimposinghighfees 14
. Ofcourse,this
kind of analysis of the factors that determine the degree of social mobility is
onlypartial,becauseitneglectsmanyotherinstitutionalaspects. Nevertheless,
therelationshipbetweenthelevelofinitialwealth,theeducationalchoices,and
theincomedynamics,seemsto becrucial. 15
Wewillshowthatinastrategicenvironment,alowwealthagentmightnot
beabletocompetewith highwealthagentsforthehigh-payingpositions,due
to capital market imperfections andrisk-aversion. Moreover,wewill highlight
howtheeducationalinstitutions cansomehowfavorthecompetitorswithhigh
wealth,andhowthiscouldleadtoamisallocationofresources 16
;asaresultthe
societycanbepartitionedindi erentincomeclasses,withaverylowdegreeof
socialmobility.
Thepaperis organizedas follows. Section 2 developsthe structure of the
model,payingattentiontothespeci cassumptionsabouttherulesofthe
posi-tional competition inthelabourmarket. Insection 3acasewithriskaversion
and credit market imperfections is analyzed. Section 4 analyzesthe
distribu-tional dynamics. Section 5concludes with somecomments. In theAppendix,
theresultsin thesimplecasewithriskneutralagentsarebrie yillustrated.
2 The model
The economy is composed of N agentswho live for twoperiods. In the rst
periodagentjhastodecidehowtoallocatehisowninheritanceb
j
(t)amongthe
investmentineducationa
j
(t),the nancialinvestments
j ( t) andconsumption c j (t),thatis 17 : b j ( t)=a j ( t)+s j (t)+c j (t): (1)
Inthesecondperiodtheagentwillhavetodecidehowtoallocatehisincome
derivingfrompreviousinvestmentsbetweenconsumptionandinheritance. The
Harvard. Inthe modelwepropose,learningabilitiesareheterogeneous, thereforeanyagent
canchoose anyamount ofinvestmentineducation,butthe credentialseventually achieved
dependuponindividualabilities.
14
These institutions,havinggreaterresources,areableto attractthebestprofessorstoo.
Rarelyintherealworldtheselectionofcandidatesisbasedonlyontheirwillingnesstopay,
sincetheaccessbymeansofgrantsofthemosttalentedindividuals,indipendentlyfromtheir
wealth,allowsforaraiseinthevalueofentering intotheinstitutionitself. Fromthispoint
ofview,attractingthemosttalented studentsraisesthevalueofthe signalawardedbythe
institution; full exclusion of the (most talented) poors bymeansof prohibitivefees would
thereforenotbecoherentwiththemaximizationofpro tbytheistitution(see[1]).
15
See[5]forempiricalevidenceaboutItalyandtheUnitedStates. Ahighqualityprivate
schoolingsystemcanbeseenasameansofperpetuatinginequality,whiletheobserveddegree
ofsocialmobilitycanbeseenasanunavoidable concessionwhichisnecessarytolegitimate
the existing inequality. Lasch (1995) about thispoint says: \A high level of mobilityis
notcontradictorywitha systemofstrati cationwhichconcentratespowerand privilegesin
a dominantelite. Onthe contrary, the circulation ofthe elite strengthens the hierarchical
principle, because it always provides new talented elite and legitimates their ascent as a
functionofmeritandnotofbirth". Seealso[22].
16
See[17 ].
17
Notice that a simpleway tomodelcredit rationingisto assumethat s
j
(t) cannot be
vidual labour compensation) depends on the investments of theother agents,
thatisapositionalcompetitioninthejobmarketexists. LetA
j
bethevector
ofinvestmentsin educationof alltheagentswith theexclusionofagentj and
V = (v
1 ;:::;v
N
) be the vector of individual innate learningabilities, so that
the return of the investment in education for agent j can be represented as
w(a
j (t);A
j
(t);V(t)) . Thereforein thesecondperiodthebudget constraint
is: y j (t+1)=R (t+1)s j (t)+w(a j (t);A j (t);V (t))=b j (t+1)+c j (t+1); (2) wherey j
(t+1)istheagentj'sincomeinperiodt+1;b
j
( t+1) isthebequest
forhis o spring, andR (t+1) istherateof returnofthe nancialinvestment.
ThevectorV(t)iscommonknowledge. 18
The simplest way to model the positional competition in education is to
supposethatthereturnoftheinvestmentineducationisdeterminedbyalottery,
whoseprobabilitiesarefunctionsoftheindividualinvestmentsineducationand
theinnateabilities,sothatagentj 0 sproblembecomes: max cj(t);cj(t+1);bj(t+1);aj(t) E[U j ]=E[U(c j (t);c j (t+1);b j ( t+1))]; subjectto: b j (t)=a j (t)+s j ( t)+c j (t) R (t+1)s j (t)+w(a j (t);A j ( t);V ( t))=b j ( t+1)+c j (t+1) c j (t)0 c j (t+1)0 a j (t)0 b j (t+1)0: Aftersubstitutingc j (t)andc j ( t+1), wehave: max b j (t+1);a j (t) E[U j ]=E[U(b j ( t) a j (t) s j (t);R (t+1)s j (t)+w(a j (t);A j (t);V (t) )+ b j ( t+1);b j (t+1))]; subjectto: b j ( t) a j (t) s j (t)0 R (t+1)s j ( t)+w(a j (t);A j (t);V (t) ) b j (t+1)0 a j (t)0 b j (t+1)0: 18
Assuminginstead thatV(t)isunknown,wouldhavedi erentimplicationsaccordingto
the hypotheses about agents' attitude torwards risk. With risk neutral agents, and with
homogeneouspriorbeliefsaboutthevectorV(t);nothingwouldchange. Withriskaversion,
agents'behaviourwouldbedi erent,becausetheinvestmentineducationwouldassumethe
roleofaformofinsuranceagainstpossibleunfavourablerealizationsofthevectoroflearning
j U 0 bj(t+1) U 0 cj(t+1) =1;
where the index representsthevariable with respectto which U hasbeen
de-rived. Noticethat whenthere isno moreuncertaintyabouttheincome ofthe
second period, thechoice about theamount of bequest is deterministic (even
if stillconditionedto the realization ofthe returnof theinvestmentin
educa-tion). UndergeneralhypothesesontheformofU (i.e. omotheticpreferences),
inheritance will be equal to a constant fraction of second period income 19 : c j (t+1)=(1 )y j ( t+1). Thereforewehave: b j ( t+1)=max[ [R (t+1)s j ( t)+w(a j (t);A j (t);V (t) )];0]: (3)
Replacing(3)intheexpectedutilityfunction,wehave:
max s j(t) ; a j (t) E[U j ]=E h ^ U(b j ( t) a j (t) s j (t);R (t+1)s j (t)+w(a j (t);A j (t));V (t) ) i ; where ^
U istheindirectutilityfunctionmodi edin ordertoconsider(3).
2.1 Imperfect capital market
The capital market can provide agent j with additional resources, so that it
becomespossibletoinvestanamountofmoneyineducationthatisgreaterthan
thebequest b
j
(t). However,ifthere aremarket imperfections, theinvestment
in education a
j
(t) , besidestheinferior limit equalto 0,also showsasuperior
limitrelatedtothelevelofinheritanceb
j
(t)(initialwealth). 20
A simplehypothesis aboutthe form of credit market imperfections, is not
toallowagentstoborrowatallinthecapitalmarket,thatiss
j
(t)0. Insuch
away,investmentineducationisconstrainedinthefollowingmanner:
0a
j (t)b
j
(t): (4)
Thisseemslikearestrictivehypothesis,becausein thesecondperiodagent
j getsanincomewhichisatleastequaltotheminimumwage,sothatthe
max-imumborrowingthresholdmightbehigher;howeverthisdoesnotqualitatively
a ect theresults.
2.2 Positional competition
Thereturnoftheinvestmentineducationdoesnotfollowthestandardmarginal
productivityrule. Weassumethattheprocessofjobassignmentisuncertain: an
19
However, thispartition rule of incomeis very favorable to the fraction of income set
asideforbequest; a morerealisticformulationwould considerthat,belowa certainincome
threshold,alltheresourcesaredevotedtoconsumption(aminimumconsumptionexists): in
such acasethe ratiobetween inheritanceand consumptionwould beequalto zeroforlow
levelsofincomeandincreasingwithrespecttoit(i.e. almost-linearpreferences).
20
Forexample,theseimperfectionscouldbecausedbytheimpossibilityofmakingthechild
paypossibledebtsofthe parents(indeedfor hypothesis b
j
( t+1)0), sothat the lender
resultsasbeingsubjecttothe riskofdefault(secondperiodconsumptioncannotclearlybe
a particular credential. The probabilityof getting ahigh paying job depends
ontheindividualrelativelevelofcredentials,butthislatterisneverconclusive.
Investingineducationincreasesthechancesofwinningthepositional
competi-tion in the future,but the process isnot deterministic: theagent withbetter
credentialscanneverbesuretogetthebest position.
Moreover,wesupposethatthereisalimitednumberofjobpositionsrelative
toadi erentlevelofwages. Thereforethetwocrucialpointsare:
1. thespeci cationofthewagescale
2. the determination of the probability of each agent of getting a certain
position,givenhiscredentialsandthose ofhiscompetitors.
Wesuppose that thewagescaleis composed of K di erentpossiblewages
w
k
(wherek=1;:::K),relativeton
k
positionsforeachlevelofwage.
Letp k (a j (t);A j (t);V(t);n k
)betheprobabilitythatagentjgetsareturn
equal to w
k
, givenhis investment a
j
(t) , the vector of the investments of the
otheragentsA
j
( t);andthevectorofindividualinnatelearningabilitiesV (t).
Therefore,agentj'sproblem canbeexpressedas:
max a j (t);s j (t) E[U j ]= K X k =1 p k ( a j ( t);A j (t);V(t);n k ) ^ U(b j ( t) a j (t) s j (t);R (t+1)s j (t)+w k ); subjectto: b j (t) a j ( t) s j ( t)0 R (t+1)s j ( t)+w k 0 8k a j ( t)0 s j ( t)0:
Theform ofthefunctionp
k
isdeterminedbythelotteryused toassignthe
available positions to the di erent agents. The assignmentprocess, given the
investmentsineducation,takesontheformofasamplingwithoutreplacement,
in which thepositionsrelativeto highwagesare drawnout rst 21
. Sinceafter
eachsamplingitis necessarytoremovetheinvestmentof thedrawnoutagent
fromthetotalcalculation,andthereforetorecalculatethesingleprobability,the
calculus ofthe relativeprobabilitiesis quite complicated. Some simpli cation
isdesirableatthispoint.
3 Job seeking with risk aversion: a simple case
with two agents
We choose to limit our attention to an economy with two agents, (agent i
and agentj) and with twolevels of wages (w
H (t+1) and w L ( t+1), where w H ( t+1)>w L ( t+1)). 22 21 See[2]. 22
Asthenumberofagents(andofpossiblejobpositions)increases,thepositional
compe-titionbecomesmorecomplex. Westudyonlythesimplestcase(22). Forthesimulationof
to theinvestmentin educationandto theindividualinnateabilities 23
. In
par-ticular, we assumethat, ifone agent invests a(t) in education, and he hasan
abilitytolearnequaltov,hewillobtainaschoolingcredentialequaltova(t).
Normalizingagenti'sinnateability(v
i
=1),theparameterv
j
becomesanindex
ofagentj'srelativelearningability. Agentj'sprobabilityofachievingthehigh
wagew H (t+1)isgivenby: p j (t)= v j a j (t) v j a j (t)+v i a i (t) = v j a j (t) v j a j (t)+a i (t) (5)
The probability of getting the high-paying job is directly proportionalto the
investment in education 24
. Theanalogywith rent-seeking models isquite
ap-parent. Agentj'sproblemthereforebecomes:
max a j (t);s j (t) E[U j ]= v j a j (t) v j a j ( t)+a i (t) ^ U(b j ( t) a j (t) s j (t);R (t+1)s j (t)+w H ( t+1))+ + a i ( t) v j a j (t)+a i ( t) ^ U(b j ( t) a j (t) s j (t);R (t+1)s j (t)+w L (t+1)) subjectto 25 : b j (t) a j ( t) s j ( t)0 (6) R (t+1)s j (t)+w k ( t+1)0 k=H;L (7) a j ( t)0 (8) s j ( t)0: (9)
Nowwehaveto specify which kindofequilibrium weare lookingfor. Two
typesofstrategicequilibriumarepossible: theCournot-Nashequilibrium(with
simultaneousmoves),andtheStackelbergequilibrium(withsequentialmoves).
Inourmodel, thehypothesisofsequentialmovesmeans thattherichestagent
can actas aleader, investing in education and so setting the standardof the
expenseswhicharenecessarytoacquireusefulschoolingcredentials. Therefore,
when moves are sequential, we assume that the leader is always the wealthy
agent 26
. Weexpect theoutcomewithoneof thecompetitorsthatchoosesnot
23
By"innate abilities"wemeanthe learning abilities thatagents haveafter compulsory
(and free)education. They representthe eÆciency ofthe processwhich transformshigher
educationintocredentials.
24
Amoregeneralformulationisthefollowing:
p j (t)= (v j a j (t) ) ( v j a j ( t)) +( a i ( t) )
where representsthedegreeofmeritocracyoftheprocessofpositions'assignment. Indeed,
as increases,agentsinvestingmoreresourcesineducationwillbemorefavouredinthe
com-petition. Asanextremecase,when =0,thelevelofinvestmentdoesnota ectprobabilities
atall.
25
Noticethatthesecondconstraintisredundanttakingthefourthconstraintons
j (t)into
account. However,thesecondconstraintisnoteliminatedsinceitwillbecomeusefulwhen,
analyzing theresultswithriskneutral agentsinthe Appendix,wecomparethe two cases,
withandwithoutcreditrationing.
26
The rich dinasties alwaysact as leaders insetting the standardsof consumptionand
investment. Ourhypothesisofarichleaderisthenjusti edonthebasisoftheobservationof
Inthecaseofsimultaneousmoves,onthecontrary,itisnotlikelythatoneagent
will notinvestin educationatall. Under uncertainty,agents'attitudetowards
risk is crucial. We focus onthe behaviourof risk averse agents,con ning the
riskneutralcasein theAppendix.
Riskaversioncanbemodeled by meansof anyconcaveutility function;we
consider thissimpleform:
^ U()=log(b j (t) a j ( t) s j ( t))+ ^ log(R (t+1)s j (t)+w k (t+1)); k=H;L where ^ = 1
and<1istheintertemporalpreference factor 27
.
Tosimplify theanalysis,weconsider onlytheextremecasewithmaximum
borrowing constraints: agents are not allowed to borrow and agents always
preferconsumptionto theinvestmentincapitalmarket,which meansthat the
variable s
j
(t)canbeeliminated from themenu ofchoice (asif capitalmarket
wereabsent).
Therefore,agentj hasto solvethefollowingproblem:
max aj(t) E[ U j ]=log(b j (t) a j ( t))+ (10) + ^ v j a j (t) v j a j (t)+a i ( t) log[w H (t+1)]+ a i (t) v j a j (t)+a i ( t) log[w L (t+1)] subjectto: b j (t) a j ( t)0 (11) a j ( t)0 (12)
Weanalyze rsttheCournot, andthentheStackelbergequilibrium.
3.1 Cournot equilibrium
In this sectionwederive theCournot equilibrium, correspondingto the
inter-section of the reaction curves of the two agents. The reactioncurves canbe
derivedfromtheFOC.Foragentj,theFOCis:
@E[U j ] @a j = 1 b j a j + ^ 2 4 v j a i log w H w L (v j a j +a i ) 2 3 5 =0: (13)
With the logarithmic utility function, the FOC automatically implies the
constrainton rstperiod consumption. If therewereaminimumconsumption
threshold,b
j
wouldhavetobereplacedwith ~ b j =max[b j ~ c;0]. Ananalogous
conditionwithinverseindex holdsforagenti.
27
Withalogarithmicfunction,inordertohaveb
j
(t+1)= y
j
(t+1),theutilityfunction
mustbe: U=logc j ( t)+ logc j (t+1)+ 1 logb j (t+1) +D;
whereDisaconstantwhichdependsonthevaluesofand ,sothat ^
=
1 .
@E[U j ] @a j =0, ( v j a j +a i ) 2 v j a i ( b j a j ) = ^ log w H w L : (14) Onesolutionisa j =0anda i =^a i =v j ^ log w H wL b j
; thismeansthat
if agent i invests more than a^
i
, the optimal choice of agent j is to abandon
thecompetition(the optimalvalueofa
j is 0whenevera i ^a i ) 28 . Noticethat ^ a i
is a positive function of agent j's learning ability, which means that the
investmentineducationismadetocounterbalancetheopponent'sability. Each
agentsvalueshisowninvestmentineducationaccordingtothelevelofhis own
learningability, which isthedegreeof eÆciencyof his "credentialsproduction
function".
IntheCournotequilibrium,givenbytheintersectionofthereactioncurves
ofthetwoagents 29 ,wehave: a j = b j b i a i ; (15)
which,substitutedinto (14)yields:
a j =b j 0 @ v j ^ log w H wL b i b j v j ^ log wH wL b i b j +(v j b j +b i ) 2 1 A : (16)
Equations(16)and(15)highlightthat,ifthecapitalmarketisabsent,each
agentinvestsaquantityofresourcesineducationthat isapositivefunction of
hiswealth. 30
Noticethattheoptimalchoiceofagenti(likewiseforj)belongsto
theinterval[0;^a
i
),aswecan easilydrawfromthecalculationofthemaximum
valueofa i ,thatis lim bi!1 a i =v j ^ log w H w L b j
which assures that (14) represents the part of the reaction curve where the
solutionlies.
ThefollowingPropositiondescribestheCournotequilibriumwithrisk
aver-sion.
Without capitalmarket,the Cournotequilibrium ofan economywith risk
averseagentsisgivenby:
a j =b j vj ^ log w H w L bibj v j ^ log w H w L b i b j +(v j b j +b i ) 2 anda i =b i vj ^ log w H w L bibj v j ^ log w H w L b i b j +(v j b j +b i ) 2 : 28
Analytically,thisresultisdueto thenegativesignof the rstderivativeof agentj for
a
j
=0andtothenegativesignofthesecondderivative @ 2 E[U j ] @a 2 j intheinterval[0;b j ]. 29
Agenti'sreactioncurveis:
@E[U i ] @a i =0, (vjaj+ai) 2 v j a j (b i a i ) = ^ log w H w L : 30
If there were aminimum consumptionc;~ that is @c q (t) = +18c q (t) ~c, b q
wouldhavetobereplacedwith ~ b q =max[b q ~ c;0];forq=i;j.
Therefore,theborrowingconstraintsa ectthequantityofinvestmentin
ed-ucationbut,ifthereisnominimumconsumption,bothagentswilltakepartin
thecompetition. Intuitively,iftherewereaminimumconsumption,itwouldbe
possiblefortherationedagenttonothaveenoughresourcestoconsumeand
in-vestsothat,evenwithapositiveinheritance,hewouldnotmakeanyinvestment
ineducation. Weobservethatalowerinheritancecanbecounterbalancedbyan
higherlearningability: from(16)we ndthat p
j = vjbj v j b j +b i > 1 2 ,v j b j >b i .
Being a ected by both investment and learning abilities, the job assignment
processissuchthatthemosttalentedagenthasthegreatestprobabilityof
get-ting thebest job ifand only ifhis initial wealth is nottoolowerthan that of
his opponent.
3.2 Stackelberg equilibrium
Inthepresentsectionwewillshowthat,ifagentsareriskaverse,theStackelberg
equilibriumdi ersfromtheCournotequilibrium. Inmodelsofrent-seeking,the
peculiaraspectoftheStackelbergequilibriumisthepossibilitythatthefollower
does notinvest at all. Linster(1993)showsthat ifthe prizeis equallyvalued
by the two players, the Stackelberg equilibrium coincides with the Cournot
equilibrium,butiftheleadermakesahigherevaluationoftheprize,thefollower
could refrain from taking partin the competition. Analogously, in ourmodel
this meansthat withriskaversion,thepooragentcanabandonthepositional
competitionsinceintuitivelyhe"evaluates"theadditionalincomehecouldearn
investingineducationlessthantheleader. Inparticular,wewillshowthatthe
leader can be able to exclude the follower from the competition by making
an investment that is greater than acertain threshold, which is afunction of
follower'swealth.Insomecases,learningabilityheterogeneitycanonlymitigate
thise ect and,aswewillshowinsection4,itmightnota ectthedistributive
dynamics.
Inthefollowingwewill alwaysassumethatagentiis theleaderandagent
j thefollower, analyzing thecasewith b
i >b
j .
31
The rst stepis to nd the
follower'sreactioncurve,whichisgivenbytheFOC(14). Consideringalsothat
fora i ^a i =v j ^ log wH wL b j
thecurve(14)lies inthenegativequadrant,
the reactioncurveof agentj for a
i ^a
i is a
j
= 0(when leader's investment
31
exceeds^a
i
,theoptimalchoiceofthefolloweristonotinvest) . Therefore,the
follower'sreactioncurveis:
a j (a i )= ( a i (2+) + p a i (a i +4a i +4v j b j ) 2v j ifa i <v j b j 0 ifa i v j b j (17) where= ^ log wH wL .
Theleadermaximizes utility, taking into accountthat his own decision
af-fectsthechoiceofthefollower:
max a i ;s i E[ U i ]=log(b i a i )+ ^ a i a i +v j a j (a i ) log(w H )+ v j a j (a i ) a i +v j a j ( a i ) log( w L ) :
TheFOCfortheleaderisthen:
@U i @a i = 1 b i a i +v j ^ log w H w L " a j ( a i ) a i da j da i (v j a j (a i )+a i ) 2 # =0:
From(14)wecancalculate a
j (a i )and daj dai
, whichsubstituted intothe
pre-viousequationyield 33 : @U i @a i = 1 b i a i + 4 2 a i b j v j M(a i ;b j ;v j ;)[M(a i ;b j ;v j ;) a i ] 2 ; (18) whereM(a i ;b j ;v j ;)= p a i ( a i +4a i +4v j b j ) . Leta i bethevalueofa i which makes @U i @a i equalto zero: @U i @a i ai= ai =0 Noticethata i
isnevergreaterthan^a
i
,since^a
i
isenoughtoexcludeagentj.
Moreoveritcanbeshownthat @ 2 Ui @a 2 i <0, lim a i !0 @Ui @a i =+1and lim ai!bi @Ui @a i = 1, 32
Anotherwaytoseethispointisbyde ningthe di erencebetween theutilityvaluesin
thetwocases: withandwithoutapositiveinvestmentineducation:
E[U j ]=E[U j (a j ;a i )] U j (0;a i );
whichisequalto:
E[Uj]=log b j a j bj + 2 4 v j a j log w H w L vjaj+ai 3 5 :
BycalculatingE[Uj]withaj=0weget:
E[U j ]0,a i ^a i =v j log w H w L ;
whichmeansthat,ifa
i ^a
i
;thenitisnotconvenientforagentjtoinvest.
33 Inparticular: da j da i = 2+ 2v j + 2 a i +4a i +2v j b j 2v j p a i (a i +4a i +4v j b j ) :
sothatthesignof i @ai in a i =a^ i
allowsus todistinguishifbothagentsinvest
in educationorifonlytheleaderdoes:
@U i @a i ai= ^ai 0,a i =^a i =)a j =0 (19)
InFigure1anexampleofthefollower'sreactioncurve(inlightgrey)andof
theleader'sutilityU
i
(indarkgrey)is shown:
Figure1-Thegraphisdrawnassumingb
j =1,v j =1,b i =4,w H =4,w L =1and ^ =0.6,so that^a i =0.8318and @Ui @ai a i = ^a i
=0.0375.Ontheorizontalaxisthevariablea
i
ismeasured.It
canbenotedthatU
i
reachesitsmaximumwhena
i =^a
i
,asweknowfrom(19).
Replacingthevalueof^a
i
into(18),itispossibletodemonstratethat
@U i @a i ai= ^ai 0,b i 2 1+ ^ log w H w L v j b j ;
thatistosay,inorderto havea
i =^a i anda j =0(see(19)),b i hastobemuch greaterthanb j (ifv j >1). 34
The following Proposition describes the Stackelberg equilibrium with risk
aversion.
Without capital market, the Stackelberg equilibrium of an economy with
risk averseagents (whereagent i is the leader and agent j is thefollower), is
34
Thisresultisquiteintuitive: givenagentj'sinvestmentandlearningability,thehigher
isagenti'swealth,the loweristherisk thatagenti perceivesinvesting ineducation. The
1)ifb i <2 h 1+ ^ log w H w L i v j b j thena i =a i anda j = a i (2+)+ p a i ( a i +4 a i +4v j b j ) 2v j , wherea i solves @Ui @ai a i = a i =0and= ^ log wH wL ; 2)ifb i 2 h 1+ ^ log w H w L i v j b j thena i = ^ log w H w L v j b j anda j =0.
Iftherewereaminimumconsumption~c;that is @E[U q ] @cq(t) =+18c q (t)~c,then b q
wouldhaveto bereplacedwith ~ b q =max[b q ~ c;0];forq=i;j.
From Proposition 3.2 we canconclude that agreat distributiveinequality
involvesanineÆcientlevelofinvestmentineducation;indeed wenoticethatif
thericheragenthasanamountofwealthb
i greaterthan2 h 1+ ^ log wH wL i v j
timesthatofthepoorer(agentj),hewillalwaysinvest ^ log w H w L v j b j . By
increasingagentj'sresources,wewouldget,besidesadecreaseofinequality,an
increaseoftheinvestmentineducation(case1ofProposition3.2). Alsonotice
that ahigherlearningabilityof thefollower(thatisv
j
>1) isnotasuÆcient
conditionfor apositive investmentin educationby bothagentsifthe follower
ismuchpoorerthantheleader.
4 Distributive dynamics
Toanalyzethedynamicsofthewealthdistribution betweenthetwodynasties,
weneedtospecifythetemporalevolutionofthetwoexogenousvariables: wH(t) w L (t) andv j
(t):Weassumethatthewageratioremainsconstantovertime( w H (t) wL(t) = wH w L
8t) and that the relativelearning ability v
j (t) is uniformly distributed on theinterval[v min ;v max ], withv min >0and E[v j
(t)]=1(the twoagentshave
thesamelearningabilityonaverage).
From(3)weknowthattheinheritanceisequaltoaquota ofsecondperiod
income: itisthenequalto w
H
orto w
L .
Thedynamics relativeto thetwotypesof strategicequilibriaisdi erent 35
.
We rstanalyzetheCournotcase.
4.1 Cournot equilibrium
From Proposition 3.1 we calculate the transition probabilities for agent j,
re-portedinthefollowingTable 36 : wH(t+1) wL( t+1) wH(t) vjwH v j w H +w L wL v j w H +w L wL(t) vjwL vjwL+wH wH vjwL+wH 35
Fortheriskneutralcase,seetheAppendix.
36
If therewerea minimumconsumptionthresholdc~and w
L
<~c,then the transition
probabilitiesforagentjwouldbecome:
wH( t+1) wL( t+1)
wH( t) 1 0
wL(t) 0 1
Being w L (t) a constant and v j
(t) > 0 8t; every transition probability is
positiveanddependsonthevaluesof wH
wL andv
j
. Thismeansthatthedynamics
isergodicbecauseeveryagentalwayshasapositiveprobabilityof gettingany
kindofjob.
FromProposition3.1wecancalculatetheaggregateinvestmentineducation
A(t)=a j ( t)+a i
(t)andtheaggregatecredentialsC(t)=v
j (t)a j (t)+a i (t),
which can be interpreted as a proxy of the aggregate human capital. It is
possible to show that @A @vj > 0 , v j < b i bj and that @C @vj > 0 8v j . The rst
result comes from the strategic nature of the model: the less talented agent
investsmoreineducationasv
j
increases,butonlyuptoapoint: ifthedistance
in the learning abilities becomes greater than the wealth ratio, he decidesto
investlessresourcesinpositionalcompetition,leadingtoanaggregatedecrease
of investment. The most talented agent alwaysinvests moreresources ashis
relative learning eÆciency increases, leading to greater aggregate credentials
(but not alwaysto greater aggregateinvestment) asthedistance betweenthe
individualabilitiesincreases.
It is worth makinga morein depth analysis of dynamics, focusing on the
possiblee ectofthewagescaleonthegrowthrate. Ifwesupposethataggregate
credentials C are a proxy of the level of human capital and that this latter
is the only productive factor, then the output of period t can be expressed
as Y (t) = C(t 1) where > 0; moreover letting > 1
2
be the quota
of output attributed to the high wage position, we get w
H
= Y (t) and
w
L
= (1 )Y( t). To analyze the long run dynamics we set v
j
equal to
its expected value (v
j
= 1) and, without being lessgeneral, b
j = w L and b i = w H . Thenweobtain: Y (t+1)= C(t)= Y(t) 2 4 ^ (1 ) log 1 ^ (1 ) log 1 +1 3 5
fromwhichthegrowthrateisobtained:
g Y = Y(t+1) Y(t) 1= 2 4 ^ (1 ) log 1 ^ (1 ) log 1 +1 3 5 1 Wenoticethat g Y ispositivelya ected by , ^
and andin anonlinear
wayby . Derivingg
Y
withrespectto weobtain
dg Y d = ^ 2 6 4 1 (2 1)log 1 h 1+ (1 ) ^ log 1 i 2 3 7 5 that is, dg Y d 0, ^ (20)
where ^ 0:823959 solves log 1 = 1 2 1 37
. Therefore, for = ^ the
economy performs the maximum growth rate. In other words, to maximize
37
Itiseasytodemonstratethat ^istheonlysolutionintheinterval 1 2 ;1 .
a di erence betweenhigh and lowwagesdiscouragesagentsfrom investing in
education to get the best job and, if education also determines the level of
output,thisunderinvestmentcausesalowergrowthrate.
ThefollowingPropositiondescribesthedistributivedynamicswithrisk
aver-sion.
InaneconomywhereagentsareriskaverseandequilibriumisCournot,the
distributivedynamicsisergodic,exceptinthecasewithaminimum
consump-tionthreshold~cand w
L <~c.
Aggregateinvestmentineducationcanbebothapositiveandanegative
func-tion of learning ability v
j
, while aggregate credentials are always a positive
function ofv
j .
Finally,ifaggregatecredentialsareaproxyofthelevelofhumancapital, then
the growthrate is maximized when the high wagereceivesa quota of output
equalto0:823959.
Therelationshipbetweenwageinequalityand growthisnotmonotonic. It
isnecessarytohaveacleardistinctionbetweenhigh-payingpositions,whichare
mainlyallocatedtoeducatedworkers,andlow-payingpositions. Thisdi erence
givesagentsthe incentivesto invest in education. Ifthe school is nanced by
means ofprogressiveincometaxes, theselatter should notbetooprogressive,
in order to mantain theright degreeof(net) wagedi erentiation. If thewage
di erenceistoohigh,theaggregateinvestmentineducationcandecrease,due
tothefact thatthepoor dynastieshavenotenoughresourcestocompete 38
.
4.2 Stackelberg equilibrium
IntheStackelbergequilibrium,assumingthatagentj isthefollowerandagent
i is theleader (b
i b
j
), we ndthat the followerwill notinvest in education
ifb i 2 h 1+ ^ log w H wL i v j (t)b j
(seeProposition 3.2),whereb
i = w H and b j = w L
. Tohave a stable dichotomy in the wealth distribution it is
necessarythat: w H 2 1+ ^ log w H w L v j (t) w L ; that is, w H w L 2v j (t) 1+ ^ log w H w L : De ne wH w L suchthat wH w L =2v j (t) h 1+ ^ log wH w L i ,so thatfor w H wL w H wL
the followerneverinvests in education. As onecouldexpect,
thewagescale,togetherwiththelearningabilities,iscrucialinestablishingthe
investmentineducation 39
. Wenotethatif
38
ComparingItalyandtheU.S.A.,weseethattheitalianwagedi erentialsaremuchlower,
butthehumancapitalgrowthrateishigher(see[5]).Thisevidencecouldberationalisedon
thebasisofavalueof whichistoohighintheU.S.A.
39
Notethatthevalueof w H w L
v max < wL 2 h 1+ ^ log wH wL i
thenthedistributivedynamicsisnonergodic, 40
thatis
wH(t+1) wL( t+1)
wH(t) 1 0
wL(t) 0 1
ThefollowingPropositiondescribesthedistributivedynamicswithrisk
aver-sion.
Inaneconomywhere agentsareriskaverseandequilibrium isStackelberg,
thedistributivedynamicscanbenonergodic,inparticular:
1)if wH w L < wH w L
thedistributivedynamicsisergodic;
2)if w H wL w H wL
thedistributivedynamicsisnotergodicbecausethepoorer
agentneverinvests,where wH w L solves wH w L =2v max h 1+ ^ log wH w L i .
Likewise in the Cournot equilibrium, when there is a minimum consumption
threshold~c,thedistributivedynamicsisnonergodicif w
L <c.~
Thisresultcanhaveinterestingeconomicimplications. Thepossibilitythat
thedistributivedynamics isnon ergodicinvolvesboth astaticand adynamic
ineÆciency. In particular, if the richest agent moves rst (competition is
se-quential),thegrowthratecanbesub-optimal. By nancingschoolswithpublic
funds,thepooragentcanbecomeabletoinvestineducation,andthedicothomy
can be overcome. However, the amountof public nancing is constrained by
theconditionontheoptimalvalueof ;whichmustnotbelessthan .^
5 Conclusions
Inthis paperweanalyzed thee ects ofcredit constraintsand riskaversionin
a model of positional competition in the labour market. The investment in
education has been modeled as a rent-seeking activity, which determines the
probabilitiesofgettingahigh-payingjob.
Withriskaverseagents,theborrowingconstraintscauseastaticineÆciency
in boththe Cournot equilibrium andthe Stackelbergequilibrium, but onlyin
this latter a non ergodic distributive dynamics can emerge. 41
Indeed, if the
wage di erence is rather high, it can be optimum for the leader to exclude
thefollowerfromthecompetition, makinganinvestmentineducationwhich is
directlyproportionaltothefollower'sabilityandwealth. 42
Thisresultdoesnot
depend ontheabsolutelevelsofthe twopossiblewages,but only ontheratio
between them; therefore it remains valid if we assume that the wages evolve
40
Withthe parameterschoosenintheexampleofFigure1,theconditionforergodicityis
v
max
1;092:ItissuÆcientthatthepoordynastygivebirthtoadescendantwithalearning
abilitywhichis10%greaterthanthatofthericherdynasty.
41
IntheAppendixweshowthatwithriskneutralagents,thedistributivedynamicsforboth
equilibriaisalwaysergodic(exceptforthecasewithaminimumconsumptionthreshold).
42
With heterogeneous abilities, the possibilitythat the poor agentchooses not toinvest
arisesinthecasewithriskneutralitytoo,butitisacaseoflittleinterest,becauseit stems
in time, preserving the value of the ratio. The temporal evolution of the
dynasties'wealth also depends upon the(exogenous) realization of theinnate
abilities of thedescendants; in thesequentialgame (Stackelbergequilibrium),
thedistributivedynamicscanbecomeergodiciftherangeofpossiblevaluesfor
theinnatelearningabilityiswideenough. Finally,weshowthatintheCournot
equilibrium, in orderto maximizethegrowthrate, acertainwagedi erence is
necessaryto giveagentsincentivestoinvestin education.
An interestingextension would come from the introduction of atime
con-straint, allowingfora rst-period unskilledjob. Inthiscase, the
opportunity-cost of investing in education would be increased by the amountof resources
forgonebytheagentifhewasnotworkingin the rstperiod. Thewagescale
wouldthenbeenriched,introducingatleastthreewagelevels: fortheunskilled,
for theskilled andfor thetoppositions. A singleagentwould then face three
di erentperspectives: a)nottoinvest,andworkfortwoperiodsasunskilled 44
;
b) to invest, losing the positional competition; c) to invest, winning the
po-sitionalcompetition. Theincome strati cation and the distributive dynamics
wouldbedi erent,andcertainlymorecomplex.
43
Notethataconstantwageratioiscompatiblewithanincreasingwagedi erence.
44
Evenafourthlevelofwagecouldbeintroduced,assumingthatthe rstandthesecond
periodwages,fortheunskilledworker,mightdi er(i.e.,forlearningbydoingconsiderations).
Thesimplestform ofriskneutralpreferencesisthelinearone: ^ U j ()=b j (t) a j ( t) s j (t)+[R (t+1)s j ( t)+w k (t+1)]; k=H;L (21)
where,intermsoftheprecedingnotation,underthehypothesisofalinearutility
function,= ^
( 1 ):
Therefore,withriskneutralagentstheproblembecomes:
max aj(t);sj(t) E[U j ]=b j (t) a j (t) s j ( t)+ (22) + R (t+1)s j (t)+ v j a j (t)w H (t+1)+a i (t)w L (t+1) v j a j (t)+a i (t)
A.1 Cournot equilibrium
Inorderto ndtheCournotequilibrium,itisnecessarytocalculatetheagents'
reaction curves by means of the two rst order conditions (FOC) of problem
(22),which arethefollowing:
@E[U j ] @s j = 1+R (t+1)=0 (23) @E[U j ] @a j = 1+ " v j a i ( w H w L ) (v j a j +a i ) 2 # =0 (24)
where the timeindex is omittedwhenever thisdoesnotgive riseto confusion
in thenotation.
Ontheotherhand,noticethat, ifborrowingwerepossibleandb
j <a j (a i ), where a j (a i
) indicates the optimum value of a
j
, then agent j would borrow
s j = a j ( a i ) b j
( rst period consumption has to be non negative, see (6)).
Moreover,iftherewereaminimumlevelofconsumption~c,thentheloanwould
beequaltos j =a j ( a i ) b j
+~c. (ignoring theconstraintonthe consumption
ofthesecondperiod).
Condition(24)statesthattheoptimumvalueofa
j
hastosatisfythe
follow-ing: v j a i (w H w L ) (v j a j +a i ) 2 = 1 : (25)
Thisonlyrepresentstheimplicitreactioncurveofagentj,ignoringpossible
borrowingconstraints. Since @ 2 E[U j ] @a 2 j
is negative, equation (25) admits only onesolution, that we
indicate witha j (a i ) . Ifa j (a i )>b j
, fortheconstraints(6)and (9)(non
nega-tivityof consumption in the rst period and of s
j
), theadmissible choicewill
beconstrainedandequaltob
j
. Moreover,iftherewereaminimumlevelof
con-sumption~c(so thatifinheritanceisinferiorto thisthreshold,alltheresources
mustbedevotedtoconsumption,whichinturnmeansthatthemarginalutility,
when consumptionisunder thethresholdc;~ approaches+1),theinvestment
in educationwouldbeconstrainedtomax[b
j ~ c;0].
j i
strained, so that the intersection of the two reactioncurves 45
determines the
optimalamountofinvestmentineducation,whichisthesameforbothagents:
a j =a i = v j [w H w L ] ( 1+v j ) 2 : (26)
Ifagentjisconstrained,that isa
j =b
j
,agentimodi es hisownchoice;in
particular: a i = q v j [w H w L ]b j v j b j :
ThefollowingPropositiondescribestheCournotequilibriumforaneconomy
withriskneutralagents:
IntheCournotequilibriumwithriskneutralagents,investmentineducation
dependsonthecharacteristicsofthecapitalmarket:
1) ifthecapitalmarket isimperfect (thereare borrowingconstraints: s
j 0), then a j =min h b j ; p v j [w H w L ]b j v j b j ; v j [w H w L ] (1+v j ) 2 i and a i =min h b i ; p v j [w H w L ]b j v j b j ; vj[wH wL] (1+vj) 2 i ;
2)ifthecapitalmarket isperfect, thena j =a i = vj[wH wL] (1+v j ) 2 .
If there were aminimumconsumption ~c;that is, @E[U q ] @c q (t) =+18c q (t)c,~ b q
wouldhavetobereplacedwith ~ b q =max[b q ~ c;0];forq=i;j.
Wenoticethattheborrowingconstraintscouldinduceagentstounderinvest
in education: itispossiblethat intheCournotequilibrium oneagentdoesnot
investatall,havinginsuÆcientresourcesinthe rstperiod(forinstanceb
j (t)=
0),whiletheotherinvestsaquantity">0;thatisassmallashewishes,getting
thehighwagew
H
. Finally,ifthereisaminimumlevelofconsumptionforeach
period,itispossiblethattherationedagentmaynotset asideanyresourcesto
invest,sothat,eventhoughhereceivedapositiveinheritance,hedoesnotmake
anyinvestmentin education. Ceterisparibus, heterogeneouslearningabilities
favour the most talented individuals in the positional competition. Without
borrowingconstraints,from (5) and (26)weget that p
j = v j 1+v j , which means that p j > 1 2 ,v j >1
A.2 Stackelberg equilibrium
The Stackelberg equilibrium is found assuming that one agent (the follower)
considers the choice of the other agent (the leader) as given and maximizes
accordingly.
Assumethat agentiis theleaderwhile agentj isthefollower. Inthe rst
stepwe ndtheoptimumchoiceofthefollowerdependingonthechoiceofthe
leader;thisisgivenbytheFOC(25)ifagentj isnotrationed,andbyb
j inthe othercase 46 . 45
Agenti'simplicitreactioncurveisspecularto(25):
v j a j (w H w L ) (v j a j +a i ) 2 = 1 : 46
Iftherewereaminimumconsumption,itwouldbegivenbymax[b
j ~ c;0].
decisionofthefollowerintoaccount: max ai;si U i =b i s i a i + a i wH +v j a j (a i )wL v j a j (a i )+a i +R (t+1)s i :
TheFOCwith regardtos
i
is thesameasitis in theCournotcase(see(23)),
whiletheFOCwithregardto a
i is: @U i @a i = 1+ 2 4 v j ( w H w L ) a j (a i ) a i d aj da i ( v j a j (a i )+a i ) 2 3 5 =0:
From(25)wecancalculate a
j (a i )and d aj da i
, whichsubstituted intothe
pre-viousequationyield:
@U i @a i =0,a i = ( w H w L ) 4v j (27) Replacinga i
in (25),we ndthe optimalinvestmentin education ofagent
j: a j = (2v j 1)(w H w L ) 4v j (28) Noticethat,if v j < 1 2
; thenthe followerdoesnotinvest atall: if thepoor
agentalso haslesslearningability,he abandonsthecompetition forthe
high-payingjob. Moreover,in theStackelberg equilibrium without borrowing
con-straints,weknowfrom (27)and(28)that,ifv
j
wereequalto1,thetwoagents
wouldinvest thesameamountof resourcesin education. Ifthe followeris
ra-tioned,a j ( a i )=b j < (2v j 1)(w H w L ) 4v j and da j da i =0,sothat @U i @a i =0,a i = q v j (w H w L )b j v j b j :
Finally, if there is a minimum consumption c,~ then b
q has to be replaced with ~ b q =max[ b q ~ c;0];forq=i;j.
Therefore,wecanconcludewiththefollowing:
The Stackelberg equilibrium with risk neutral agents is equivalent to the
Cournot equilibrium only in thecase of homogeneous learningabilities, orin
the caseof binding borrowingconstraints. With an imperfect capital market,
theStackelbergoptimalchoicesdi erfrom theCournotcase:
a j =min h b j (t); p [w H w L ]b i b i ; (2vj 1)(wH wL) 4v j i and a i =min h b i ( t); p [w H w L ]v j b j v j b j ; (w H w L ) 4vj i :
Iftherewereaminimumconsumption~c,thenb
q
wouldhavetobereplacedwith
~ b q =max[b q ~ c;0];forq=i;j. Ifv j 1 2 ,thena j
In the Stackelberg case, we assume that v
min =
1
2
, allowing us to ignore the
case in which the follower has much lessability in addition to less wealth in
comparisonwiththeleader. 47
Ifagentsare risk neutral,then the Cournot andStackelberg equilibria are
qualitativelysimilarasfarasthedistributivedynamicsisconcerned(see
Propo-sitionsA.1andA.2),sothatweanalyzeonlytheCournotcase.
From(3)weknowthattheinheritanceisequaltoaquota ofsecondperiod
income: itisthenequalto w
H orto w L . De ning H = w H vj(wH wL) (1+vj) 2 and L = w L vj(wH wL) (1+vj) 2
;wehavethreepossiblecases:
1.
H
<0,wherebothagentsareconstrained;
2.
L
>0,where noagentisconstrained;
3.
H
>0and
L
<0,whereonlyoneagentisconstrained,whiletheother
isnot.
Inthe rst case the twoagentsalwaysinvest all their inheritance (b
j and
b
i
respectively). The transition probabilities for agent j are reported in the
followingTable 48 : wH(t+1) wL( t+1) wH(t) vjwH v j w H +w L wL v j w H +w L wL(t) v j w L vjwL+wH w H vjwL+wH
whilein thesecondcasebothinvest
vj(wH wL)
(1+vj) 2
,so thatthethetransition
probabilitiesforagentj are:
wH(t+1) wL( t+1) wH(t) vj 1+v j 1 1+v j wL(t) vj 1+vj 1 1+vj .
The third case is the most interesting one, because we have one agent
who is rationed in every period; if agent j gets w
L
, then his o spring will
receive b
j
(t+1) = w
L
, and as a consequence he will invest a j (t+1) = w L , while a i (t+1) = p v j w L (w H w L ) v j w L , so that p j (wH(t+1)=wL(t)) = v j w L p v j w L (w H w L )
. Therefore the transition
proba-bilitiesforagentjwill be
wH(t+1) wL( t+1) wH(t) v j p v j w L (w H w L ) v j w L v j p v j w L (w H w L )+ w L (1 v 2 j ) w L v j p v j w L (w H w L )+ w L (1 v 2 j ) wL(t) vj wL p vjwL(wH wL) 1 vj wL p vjwL(wH wL) . 47
>FromthelastProposition,weknowthatwithv
j
1
2
thefollowerdoesnotinvest. We
canexcludethiscase,whichisoflittleinterest.
48
If therewerea minimumconsumptionthresholdc~and w
L
<~c,then the transition
probabilitiesforagentjwouldbecome:
wH( t+1) wL( t+1)
wH( t) 1 0
wL(t) 0 1
Since L <0;wehave w L < (1+v j ) 2
; whichmeansthat
p
j
( wH( t+1)=wL(t))< 1
2
Therichestagent hasa greaterprobabilityof getting thehigh-payingjob,
but thedistributivedynamicsisergodicbecausep
j
>08t.
We can observe that if there were not informational problems and credit
markets were competitive, R (t+1) would be a function of the wage scale.
Consider the third case; in the rst period inheritance is equal to w
L , so
thatthelevelofthenecessaryloanisequalto s j = vj(wH wL) (1+v j ) 2 w L :The
probabilityofgettingthehighwageisequaltop
j = v j 1+v j
,sothattheexpected
incomeis equalto 1 1+v j ( v j w H +w L
):This latter,in turn, mustbeenough
torepaythedebt(theamountofloanplustheinterests),thatisR (t+1) s j .
Therefore the loan is granted only if the condition
vj(1+vj)(wH+wL) v j (w H w L ) (1+v j ) 2 w L
R (t+1) is satis ed, from which the competition among banks would involve
that R (t+1)= v j (1+v j )(w H +w L ) vj(wH wL) (1+vj) 2 wL . 49
Likewisein theriskaversecase,itisworthanalysingthedynamicsfocusing
onthepossiblee ectof thewagescaleonthegrowthrate. Ifwesupposethat
aggregate credentials C are a proxy for the level of human capital and that
this latter is the only productive factor, then the output of period t can be
expressed as Y (t) = C( t 1) where > 0: Moreover let > 1
2
be the
quota of output attributed to the high wage position,so that w
H
= Y (t)
and w
L
= ( 1 )Y (t). To analyzethe long run dynamics we set v
j equal
to itsexpectedvalue(v
j
=1) and,withoutbeinglessgeneral,b
j = w L and b i = w H
. Thethirdcase,(distributivedynamicswhichisalwaysergodic,while
anagentisalwaysrationed),whichoccurs when >max h 2( 2 ) ; 4 + 2(2 +) i ,
seemsto be themostinterestingone. Bysimplealgebra,wecancalculate the
growthrateofoutput
g Y =[ ( 2 1)(1 ) ] 2 1 fromwhich @g Y @ 0, 3 4 and therefore = ^= 3 4
is thedivision of output that maximizesthe growth
rate. Thisresultisanalogoustowhatwefoundfortheriskaversecase.
ThefollowingPropositiondescribesthedistributivedynamicswithrisk
neu-trality:
In an economy with risk neutral agents, the distributive dynamics is not
a ectedbythetypeofequilibrium weconsider.
IntheCournotequilibriumthreecasesarepossible:
1) if w H < vj(wH wL) (1+v j ) 2
; then all agents are rationed and the distributive
dynamicsisergodic; 2) if w L > vj(wH wL) (1+v j ) 2
; then nobody is rationed and the distributive
dy-namicsisergodic;
49
Noticethatbankscanknowv
j givens
j .
3) if w H > (1+v j ) 2 but w L < (1+v j ) 2
;then only oneagent is
rationed. Therationedagenthasalowerprobabilityofgettingthehigh-paying
position,but thedistributivedynamicsisstill ergodic.
In thiscase, considering theaggregatecredentialsas aproxyof theaggregate
levelof human capital, the growth rate is maximizedwhen thehigh wage
re-ceivesaquotaofoutputequalto 3
4 .
Ifthere wereaminimumconsumption ~c;thenthedistributivedynamics could
benonergodicif w
L <~c.
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