Labor Turnover,Wage Structures,
and Moral Hazard: The Inefficiency
of CompetitiveMarkets
RichardJ.Arnott,
Queen'sUniversity
JosephE. Stiglitz,PrincetonUniversity
A multiperiod,generalequilibriummodel of the labor marketis
developed in which risk-averseworkersare faced with job-related
and labor turnoveris costly.If a workeris unluckyand
uncertainty
suffersa bad job match,he quits and joins anotherfirm,hoping
more.Because the qualityof
thathe will like its work environment
a job matchis unobservable,workerscannotinsureagainstthe risk
of a bad match.The firmprovidesimplicitinsuranceagainstjob
dissatisfaction,
typicallyby paying workersmore than their net
marginal products in their early years with the firm and less
events
subsequently.Since the probabilitiesof the insured-against
(the quit rates over time) are affectedby the amount of such
insuranceprovided,thisimplicitinsuranceis characterized
by moral
hazard. Individuals quit when in the absence of insurancethey
in risk
would not. The equilibriumcontractbalancesout efficiency
bearingwith efficiencyin turnoverincentives.We show that the
and indicatewhy.
equilibriumcontractis not (constrained)efficient
and extendsArnott(1982)and Stiglitz(1975).Stiglitz
Thispapersynthesizes
We
assistance.
of Laborforfinancial
wouldliketo thanktheU.S. Department
andtheSSHRCC forjointfinancial
boththanktheNationalScienceFoundation
comments
Jim
The paperbenefited
fromhelpful
fromBarryNalebuff,
support.
Mirrlees,and seminarparticipantsat Chicago, Princeton,Queen's University
and Tel-Aviv.Remainingerrorsare our responsibility.
[JournalofLabor Economics,1985,vol. 3, no. 4]
(? 1985 by The Universityof Chicago. All rightsreserved.
0734-306X/85/0304-0005$01.50
434
LaborTurnover
andMoralHazard
435
This paper is concernedwith the relationshipbetweenwage structure
and labor turnover.We are concerned,in particular,with the question
who effectively
pays forspecifictrainingand hiringcosts.Becker(1962)
provided the classic solution for this problem when individualsand
firmsare risk neutral.'Individualspay the costs when theyare hired,
but the wages (which equal the posttrainingvalue of theirmarginal
high for
product)that theyare subsequentlypaid must be sufficiently
the individualsto recoup theseexpenditures
(withinterest).Under these
circumstances,
individualshave the correctincentivesformoving;when
they move, theirspecifichuman capital is, in effect,destroyed;hence
or the
theyshould only move when the increasein theirproductivity,
incrementin the utilitytheyreceivefromthe nonpecuniaryaspects of
the job, at least compensatesfor this; when individualshave paid the
trainingcosts themselves,and when their currentwages fullyreflect
theirtotal productivity(includingthe incrementin theirproductivity
resultingfromtheirspecifictraining),thentheywill move only when it
is efficient
to do so.
On the other hand, when individuals are risk averse, this wage
structureimposes a heavy,undesirablerisk on workers.It means, in
particular,thatif,for one reason or another,individualshave to move
or findit desirableto move, theymust bear a large loss. Assume,for
instance,that thereare importantnonpecuniaryaspects of a job that
individualsonly findout about afterpaying for theirspecifictraining
costs.If the individualfindsout he is badly matched,efficiency
requires
that he move; but the entirerisk of whetherthe individualis well or
badly matchedis borne by the worker.Since workersare more risk
aversethanfirms,thisdistribution
of riskbearingis clearlyinefficient.
There is thus a trade-off:
the firmcan bear some partof the training
costs, lowering the wage of the trainedworker below his marginal
productivity
(to pay for the training);but to the extentthe firmdoes
this,individualswill not have the correctincentiveto move. Workers'
productivityexceeds their wages; they move wheneverthe expected
incrementin nonpecuniarybenefitsis enough to offsetany wage
but thismeansthatsome of thetimetheywill actuallymove
differences;
to a job forwhich the incrementin nonpecuniarybenefitsis not enough
in productivity.
to compensateforthe differences
If therewereperfectinformation
and it werecostlessto writecomplete
contracts,the contractwould specifythe conditionsunderwhich individualscould leave or thecompensationthattheywould have to provide
the firmif they leave under each specificset of circumstances;such
' The seminalpaperon the topicis Becker(1962). Subsequentpapersthathave
extendedand formalizedBecker'sanalysis(e.g., Parsons 1972; Hashimoto 1978)
have assumedthatworkersare riskneutral.
436
Arnott/Stiglitz
contracts(thoughtheymightbe interpreted
as formsof indentured
servitude)would ensure economic efficiency.But perfectinformation
does not exist, and it would be hopeless to attemptto write such
complete contracts.The firmthus must design its wage contractsto
in labor turnoverversusthe
balance thesetwo considerations:efficiency
of riskbearing.
efficient
distribution
We have just describedan example of a wide class of problemsthat
have recentlybeen discussed-called "moral hazard" problems.They
arise wheneverthe provisionof insuranceaffectsindividuals'incentives
(see Pauly 1974; Marshall 1976; Arnottand Stiglitz1982; Stiglitz1983).
It has long been believed that the privatebalancing of the trade-off
betweenriskbearingand incentiveswould lead to an efficient
contract:
thoughobviouslyworkersmay be worse offand profitsmay be lower
than they would be with first-best
insurance,still given the inherent
informationproblems,theremay be no governmentintervention
that
would effecta Paretoimprovement.
This paper has, then,two objectivesto characterizewithina simple
generalequilibriummodel: the equilibriumcontract,and to show that
the marketequilibriumis not in generalPareto efficient;
theredo, in
general,existgovernment
interventions
thatwould improvewelfare.
The intuitionbehindour resultson theinefficiency
ofthedecentralized
marketequilibriumcontractscan be seen as follows.Considertwo firms,
the second of which hires the workersthat quit the first.Both offer
long-termcontractswithimplicitinsurance.The secondfirm,in deciding
what contractto offer,will ignore the effectits contracthas on the
proportionof workerswho quit the firstfirm.This would not matterin
a first-best
economy.But here workerswho quit are being subsidized.
Thus the second firmfailsto take into accountthatits actionsaffectthe
profitabilityof the firstfirm. The phenomenon entails a form of
externalitythat causes the competitiveequilibriumto differfromthe
constrainedoptimum.
It turnsout thatthisline of reasoningis correct,as faras it goes, but
incomplete.Consider firstthe case where utilityis intertemporally
separable.There externalities
operateonly via workers'savings.To see
this, perturbthe second firm'scontractin such a way that it offers
slightlymoreinsurancewhile stayingon its budgetconstraint,
and hold
workers'first-period
savingsfixed.A workerwill quit the firstfirmif
his expectedutilityfromquittingexceeds his utilityfromstaying.The
perturbation
does not affecthis utilityfromstaying;nor,sincethesecond
firm'scontracthas been chosen to maximizeprofits,does it affecthis
expectedutilityfromquitting(i.e., the envelopetheoremapplies).Thus,
when workers'first-period
savingsare fixed,the perturbation
does not
its profitability.
But in fact,
affectthe firstfirm'squit rateor, therefore,
will affectworkers'first-period
the perturbation
savings;savingsare a
and will be influencedby the amountof insurance
formof self-insurance
LaborTurnover
andMoralHazard
437
firm.
The changein a worker's
offered
bythesecond-period
first-period
savingswill, in turn,alter his utilityfromstayingand expectedutility
withsavingsvariablewill influence
fromquitting.Thus, theperturbation
the firstfirm'squit rate and profitability.
Consider nextthe case where
utilityis not intertemporally
separableand where workers'first-period
It will cause
savingsare again fixed,and performthe same perturbation.
workerswith the second firmto alter their consumptionprograms;
because the utilityfunctionis not intertemporally
separable,this will
affectworkers'propensityto quit the firstfirm.The argumentpresented
in the previousparagraphalso failsto recognizethatthe typesof market
of capital markets.2It
failuresthat arise depend on the characteristics
does, however, correctlyidentifythe central issue whether, in this
handlethetrade-off
between
context,competitivemarketswill efficiently
riskbearingand incentives.
The argumentabove suggeststhatthe equilibriumset of employment
contractsin a 2-period economy may be efficient.The second firm,
whichhiresworkerswho have quit thefirstfirm,employstheseworkers
for only I period. It has no choice but to set the wage equal to net
externalities
betweenthetwo firms
marginalproduct.The uninternalized
identifiedabove are therefore
inoperative.They do, however,arisein an
economyof 3 or more periods.
I. The Model
We employthe same model,withslightmodifications,
throughoutthe
paper. A singlenumerairecommodityis producedaccordingto a single
technology,which exhibitsconstantreturnsto scale to the sole factor,
labor.The risk-averse
workersare equally productiveand have the same
utilityfunctionwithrespectto income and job satisfaction.
They differ,
however,in what they like and dislike about a particularjob; some
enjoyrotatingshifts,othersdo not; some enjoyMusak on thejob, others
do not; and so on. Firms are competitiveand risk neutral.They offer
different
work environments;
some have rotatingshifts,othersdo not;
some have Musak, othersdo not; and so on. At the timea workerjoins
a firm,he does not know what the firm'swork environment
is like,nor
can thefirmjudge ex ante whichworkerswill enjoy the particularwork
environment
it offers.Thus, when joininga firm,a workeris uncertain
how muchhe will enjoy his workenvironment
(or, synonymously,
what
his level of job satisfactionwill be withthe firm,or what the qualityof
the job match will be). This is the only kind of uncertaintyin the
2 Justas firms
provideinsuranceto theirworkers,if capitalmarketsare
firmscan profitby settingup an implicitcapitalmarket,
imperfect,
paying
ofincome
whentheirmarginal
workers
morethantheirmarginal
products
utility
willdependon thenatureof
is highandlesswhenitis low.Thuswageprofiles
in theeconomy.
riskandcapitalmarkets
availableelsewhere
438
Arnott/Stiglitz
economy.While workers'tastesvis-a-visthe work environment
differ,
they are symmetricin the sense that the probabilitydistributionof a
worker'sjob satisfactionwhen the startsemploymentwith any firmis
independentof both the firmand the worker.A workerquits a firmif
doing so increaseshis expectedutility.The replacementof this worker
by anotherentailsa fixedcost to the firm.
The qualitativecharacteristics
of the set of equilibriumemployment
the
contractsdepend on
informationtechnology.We assume that the
firmknows a worker'semploymenthistoryat the time he joins the
firm,3
but cannotascertainhis job satisfaction
while he is withthe firm.
A worker,meanwhile,has fullinformation
on the contractsofferedby
all firms,but does not know, at the timehe joins a firm,how much he
will enjoy his job. Finally, the firm can acquire insurance-relevant
informationconcerningits workersat a substantiallylower cost than
can any agentexternalto the firm.
Beforepresentingthe model formally,we commenton some of its
characteristics
and on the role played by some of the assumptions.We
have chosen the assumptionsto provideas simple as possible a general
equilibriummodel with multiperiodemploymentcontracts,labor mobility,moral hazard,and incompleteinsurancemarkets.The qualitative
resultsto be derivedwill clearlygeneralizeto morecomplexand realistic
economies.
1. The firmis unable to observea worker'sjob satisfaction,
but it can
observe whetheror not he quits. Suppose that full insurancewere
providedagainstjob dissatisfactiono.
When asked about his level of job
the workerwould have an incentiveto lie; specifically,he
satisfaction,
would state that his job satisfactionwas the lowest possible consistent
with his observed behavior. In fact, there is no efficientmethod of
informationrevelation.The firm can do no better than offerquit
insurance;thus,insurancemarketsare incomplete.Since the provision
of quit insuranceaffectsthe probabilityof the insured-against
event,
moral hazard is present.And with moral hazard, if an insured'stotal
insurancepurchasesare observable,an insurancecontractwill specify
both premiumand payout.We assume thatthe firmprovidesinsurance
implicitlyby settingeach period's wage above or below the marginal
product.4
2. An importantcharacteristic
of our economy is that a firmmust
breakeven on the insurancepolicy it offerseach groupof workerswho
3The same qualitativeresultswould obtainifit were assumedinsteadthatthe
firmknowsonly theworker'sage or yearsor employmentat thetimehe applies
fora Job.
are possible(see Arnottand Stiglitz1981). Treating
4 More complexcontracts
themwould complicatethe analysisand not alterthe qualitativeresults.
andMoralHazard
LaborTurnover
439
join the firmwith a particularemploymenthistory.Cross-subsidization
is not possible. Suppose it were otherwiseand a firmcould use the
profitfromthe insuranceprovidedto a lucky group (e.g., 50-year-olds
who had only one previousjob) to subsidizethe insuranceprovidedto
an unluckygroup (e.g., 30-year-oldswho had alreadyhad fiveprevious
jobs). If anotherfirmwere to offera slightlymore attractivepolicy to
the formergroup and a less attractiveone to the latter,it would attract
only the formergroup, therebymaking a profit.This implies an
analyticallyconvenientdefinitionof a firm,whichwe adopt,as a group
of workersin some job who had the same employmenthistorywhen
theystartedthe job.5
3. We assumedthatfirmsare identicaland thatworkersare identical
in all relevantrespects,exceptthoserelatedto thequalityof a job match,
in order to circumventthe adverse selection problems that would
otherwisearise. These problemshave been treatedin Rothschildand
fromadverseselectionnot only simplifiesan
Stiglitz(1976). Abstracting
alreadycomplicatedanalysis,but also allows us to ascribethe market
failureswe identifyto moral hazard.
4. The moralhazard problemfacingthe firmis as follows:ifit offers
more insurance,it bears moreof its workers'job matchrisk,but it also
experiencesa higherquit rate. The formereffectdecreasesthe firm's
costs, since workersare willing to take a reductionin averagepay in
exchangeforsome insurance;the lattereffect,however,increasescosts,
sinceworkerswho quit are subsidized.The firmwill tradeoffthesetwo
effectsso as to maximizeprofits.
The social optimumentailsa similartrade-off.
Increasingthe amount
of insuranceofferedreducesworkerrisk,which is beneficial,but causes
excessiveturnover,
which is harmful.
The centralquestionis, Will a competitiveeconomybe at the optimal
frontier?
trade-off
pointon the workerrisk/quit-rate
II. The Analysis
For reasons that we have suggested,some of the sources of market
failurewe identifydo not arise in a 2-periodeconomy.The resultsfor
the 3-period economy do, however,generalize,and this is the case
we treat.
In our 3-periodeconomy,we allow a workerthreepossibleemployment
histories.A workermay staywiththe same firmforall 3 periodsof his
5If we introduced
as theimperfect
suchrealistic
substitutability
complications
in theirmix of skills,and
becauseof differences
of workersin production
of scale and scope,we wouldobtainfirmsthat
economiesand diseconomies
contained
determinate
numbers
ofworkers
ofdifferent
types.All we requirefor
is thattherebe some
thesourcesof market
failurewe identify
to be operative
interfirm
mobility.
440
Arnott/Stiglitz
working
life.Or he mayquitthefirmhe started
withafter1 periodand
2 periods.Or he maywork
staywiththenextfirmfortheremaining
firmforeach of the 3 periodsof his workinglife.We
fora different
excludeby assumption
thepossibility
thata workerwill staywithhis
firstfirmfor2 periodsand thenjoin anotherfirmforthethirdperiod;
thissimplifies
notationand does notaffect
results.
With
anyqualitative
our definition
of a firm,thereare as manyfirmtypesas employment
We shalldefinea type1 firmto be one thathiresworkers
histories.
whentheyfirst
jointhelaborforce;a type2 firm,
onethathiresworkers
who havequit a type1 firmafterworkingforit for1 period;and a
one thathiresworkerswho havequit successively
type3 firm,
type1
at thebeginning
and type2 firms.
Thus,a type1 firmhiresworkers
of
the firstperiod,a type2 firmhiresworkersat the beginning
of the
at thebeginning
of thethird.
second,and a type3 firmhiresworkers
The following
is employed:
notation
thewageoffered
by a firmof typej duringtheithperiodof a
worker's
life;
0'
thejob satisfaction
of a workerin a firmoftypej;
thelowestlevelof job satisfaction
forwhicha workerwill stay
O'
witha typej firm;
0,
thejob satisfaction
of a workerin periodi;
of 0', withF [0'] the cumulative
f[0O] the densityfunction
density
to enclose
thepaperwe use squarebrackets
function
(throughout
w'
s,
U[*]
T'
ci
m
the argumentsof a function);
cumulativesavingof a workeremployedby a typej firmat the
end of the ith periodof his workinglife;
utilityfunction;
the costs to a firmof typej of hiringand traininga worker
consumptionof a workerin periodi; and
marginalproductof labor.
We make the followingsimplifying
assumptions:
betweenwage
i) For each employmenthistory,workersare indifferent
streamswith the same presentdiscountedvalue (PDV) of wages. We
termthisthe perfectcapital marketscase.6(Later we shall commenton
how our resultsare modifiedwhen capital marketsare imperfect.)
6 We may say that capital marketsare imperfect
when (and only when)
workersare concernedabout the timingof theirincome.This is not a primitive
definition(we hope to deriveone in subsequentwork) but in the contextof the
paper will suffice.Accordingto thisdefinition,
capital marketsare imperfectif
borrowingor lending rates differ,or if the individualis constrainedin the
amounthe can lend or borrow.Imperfectcapital marketsmay arise because of
regulationsor thenatureof the transactions
technology(e.g.,administrative
costs
LaborTurnover
andMoralHazard
0 for01 E (-oo, oc) and is continuous.
ii) f[0']>
iii)
U[.]
441
=
z$_1(u[cd]
+ Oi), and
u[ ] is strictly concave and twice
differentiable.
Utility is additivelyseparable in consumptionand job
satisfaction,
and overtime.7Furthermore,
job satisfaction
is measuredby
. .
its utility.
iv) The expectedjob satisfaction
froma randomjob matchis zero.
v) In his firstperiodwitha firm,a workermustmakehis consumption
decisionbeforefindingout his job satisfaction.8
vi) The interestrateis zero.
We shall proceedfirstby characterizing
the worker'schoice problem,
thensolvingforthe (constrained)optimumin whichtheplannerhas the
same informationas firms.Finally,we shall solve for the competitive
equilibriumand compareit withthe planningsolution.
A. The Worker'sProblem
At the beginningof the firstperiod of his workinglife,the worker
mustdecide how muchto consume.Then, havingdiscoveredhow much
he likes his job, he mustdecide whetheror not to quit at the end of the
period and join another firm.If he decides not to quit, his future
decisionsare simple;because of the concavityof uz ], he will equalize
second-and third-period
consumption.
If, instead,he decides to quit at the end of the firstperiod,he must
decide how much to consumeduringthe second period.Then, when he
findsout the quality of his job match with the second firm,he must
decide whetherto quit at the end of the second period.In eitherevent,
his third-period
consumptionis determinedas a residual.
Under our assumptions,the workerwill quit at the end of the first
period if his job satisfactionis below some criticallevel 0'. Similarly,a
workerwho joins a second firmwill quit that firmat the end of the
second periodif 0 is below 02.
As usual, the intertemporal
maximizationproblemmay be set up in a
numberof different
ways. It turnsout thatforour purposesit is most
in borrowing
and lending,
theabsenceof financial
assets).Moreinterestingly,
theymay also resultfrominformational
asymmetries;
for example,capital
markets
willtypically
be imperfect
whenthereis a finite
probability
of default
andwhenthisprobability
canbe affected
byactionsoftheborrower
thatcannot
be observedat reasonable
costby thelender;thisis, of course,an exampleof
moralhazard.Thatperfect
capitalmarkets
arenotinconsistent
withourmodel
is arguedin Arnott
andStiglitz
(1981).
7 Theseassumptions
resultin a worker's
choiceofemployment
contract
being
dependent
on his employment
history,
butnotdirectly
on his job satisfaction
history.
8 Relaxing
thisassumption
theanalysisby makingconsumption
complicates
dependent
notonlyon employment
butalsoon realizedJobsatisfaction,
history,
anddoesnotalterthequalitative
results
ofthepaper.
442
Arnott/Stiglitz
convenientto treatthe workeras choosingthe amountsto save and the
criticallevelsof job satisfactionbelow which to quit (i.e., sO,s ', 01, and
In makingthischoice,theworker
02) so as to maximizeexpectedutility.
takes the parametersof the employmentcontractsofferedas given. If
firmsof the same typesare offering
different
contracts,he will choose
the contractthatmaximizeshis expectedutility.
A worker'sexpectedutilityis calculatedas theprobabilitythathe will
staywith the same firmfor3 periodstimesexpectedutilityconditional
on this employmenthistory,plus the probabilitythat he will quit his
firstfirmand stay with his second firmfor 2 periods times expected
utilityconditionalon thisemploymenthistory,plus the probabilitythat
he will work fordifferent
firmsin each of the 3 periodstimesexpected
utilityconditionalon thisemploymenthistory.
The probabilitythat a workerwill stay with the same firmfor all 3
periods is the probabilitythat first-period
job satisfactionexceeds 01,
1-F[01]. Expected utilityis J3=(I(u[ci]+ EO3). With this employment
history,Cl =
C2 = C3=
(w2 + w3 + s1)/2,and EOi = (fi2
I- s,
X O'dF[Ol]) . (1 -F[O']) fori = 1, ..., 3. Since it is only the sum of
w. Thus, expected
w2 and w3 thatmattersto the worker,we set w=
utilitycontingenton thisemploymenthistoryis
uawI
-
sI] + 2u [I
w]
J-
00
+3
O'dF[0l]
F[O']
Proceedingsimilarlyforthe othertwo employmenthistories,we obtain
EU = u[wl - sI] + 2(1 - F[
2
+ 2
O'dF[O']
+ F[O'](U[w2+ sI - s2] + (1-F [02])U[w2+ s2] +
f
02dF[02]
+ F[02]U[w3
(1)
+ S2]).
LettingMudenotethe utilityof a workerin periodi when witha firmof
typej and 'us' the correspondingmarginalutilityof consumption,we
conditionsof the worker'sproblemas:
may writethe first-order
+ F[O1](
SI:-(lu') + (1 - F[O1])(1u')
2u') = 0;
S2:
F[01](-(2U')
+ (1 - F[02])(2u')
+ F[02](3u'))
(2a)
= 0;
(2b)
andMoralHazard
LaborTurnover
01: f'[1]{ -2((1u)
443
+ 01) + (2U) + (1-F[02])(2u)
+
J 02dF[02]
02: F[0I]f[02]{-((2U)
+
+ 02) +
(3U)}
(2c)
F[02](3u)} = 0;
=
0.
(2d)
Equations (2a) and (2b) statethatthe workerwill save so as to equalize
the expectedmarginalutilityof consumption.Equations (2c) and (2d)
statesimplythatthe workerwill quit if doing so increaseshis expected
utility.On the basis of (2a)-(2d) and the budgetconstraints
containedin
(1), we can expressthe worker'ssavingsand quit decisionsas functions
of the wage vector
SI=
2=
SI[WI,
WI,
w2,
W2,
W3]
s2[w, wI, w2, W2, w3]
(3a)
(3b)
01 =OI[WI,
WI,
W2,
W32, W3]
(3c)
02 =
wI
W2,
W32, W3].
(3d)
02[WI,
B. The Planner'sProblem
We assumethattheplannerhas thesameinformation
as thecollectivity
of firmsand can do no betterthancontrolwages.9Since thereare perfect
capital markets,all that mattersto a workeris the sum of wages for
each of thethreepossiblewage histories.We could therefore
characterize
theplanner'sproblemas thechoiceof thesethreewage sumsto maximize
expectedutility,subjectto an overallresourceconstraint.But doing so
would make it difficultto compare the planning solution with the
competitiveequilibrium.We have chosen insteadto treatthe planneras
selectingthe wage vector {w'}. Since thereare only threewage sums
but fivewages, we should be able to specifytwo additionalconstraints
9If theplannerwereableto do so, andwe haveimplicitly
assumedthathe is
not,he coulddo evenbetterby intervening
in notonlythelabormarketbut
also thecapitalmarket.
(Thispointis madein Diamondand Mirrlees
[1978]in
thecontextof theeffect
of socialsecurity
on theretirement
decision.)To see
If thegovernment
this,consider
thecaseof perfect
capitalmarkets.
wereto tax
or subsidize
savings(raiseor lowertheinterest
rate),thiswouldaltertheamount
a worker
saved.Thisinturnwould,fora givensetofwages,causesomeworkers
to changetheirquitdecisionsandhencewouldalterthequitrate.We arguedin
theintroduction
thatas a resultof themoralhazardarisingfromtheprovision
of quit insurancetherewould be "excessive"(morethan in the first-best
The government's
theinterest
ratecan reducethe
optimum)
quitting.
adjusting
amountofexcessive
quitting.
444
Arnott/Stiglitz
to the planningproblemwithoutshrinkingthe solutionset. For reasons
thatwill become apparent,we choose the followingtwo constraints:
w
=
(2
-
F[02])m
(12-F
_
-
2
[02])W
(4a)
and
w=
m - T3.
(4b)
LEMMA 1. Adding constraints
(4a) and (4b) to the planner'smaximization problemdoes not shrinkthe solutionset. See AppendixA forthe
proof.
The economy-wideresourceconstraintis
w} + T1 + 2(1 - F[Ol])wl + F[01](T2 + w2
+ (1
F[02])w 2 + F[02](T3 + w3)) -3m
-
=
0.
(5)
Subtracting(4a) and (4b) from(5) gives
wI + T1 + 2(1
-
F[0l])wl
-
(3
-
2F[01])m
=
0,
(6)
which we writein compactformas
A[wl, 01,W2; m, T1] = 0.
(6')
To further
simplifynotation,we write(1) as
w2w},wI
EU = EU[sl, s2, 01 02
, w?
w3],
(1')
and (4a) as
B(w2, 02
w32;
m, T2) = 0,
(4a')
and we substituteout forw3 using (4b). Then the planner'sproblemis
to maximize(1') with respectto (wi, wI, w2, w 2), subjectto equations
(4a') and (6'). Let X be the Lagrange multiplieron (4a') and 4 the
multiplieron (6'). Then the Lagrangianof the planner'sproblemis
L = EU+ XB + (A.
(7)
The correspondingfirst-order
conditionsare
dB
dA
X dB=?d=Q
+j
w:wijdEU
dwl
w2:
dw 1
dwl
(8a)
dwl
+ X dB 1 + 0~ dA1
dw
dw
0
(8b)
LaborTurnover
andMoralHazard
2dEU
+
2dw2
W2.
X
445
dB
dA
dB
dA
2
d 3
+ d) = 0,
dw2
dw2
(8c)
and
dEU
2
W3:
d2
d3
+
dW2
d3
(8d)
wheredx/dydenotesthe derivativeof x withrespectto y, holdingother
wage ratesfixedbut treatingthe dependenceof sI, s2, 01, and 02 on y.
Equations (8a)-(8d) statesimplythatthe optimalwI, w2, w2, w23is at a
pointin (w 1, wb w2, w2) space at whichan indifference
curveis tangent
to the resourceconstraint's
The planningoptimumis thereforecharacterizedby (4a'), (4b), (6'),
(8a)-(8d), and (2a)-(2d).
The variable X is the social benefitin utiles from the planner's
receiving$1.00 per workeremployedin type 2 firmsfromoutside the
economy;0 is thecorresponding
social benefitwhentheplannerreceives
$1.00 per workeremployedin type I firms.It followsthat
X=
F[01]0.
(8e)
Equivalently,the economy'sresourceconstraintmay be writtenas A +
F[01]B = 0, a resultwe shall use below.
C. Characterizationof the CompetitiveEquilibrium
A type 3 firmhas no choice. It mustpay workersthe value of their
marginalproduct,w3 = m - T3.
A type2 firmchooses w2 and w2 so as to maximizeits profitssubject
to providingworkerswith the competitivelydeterminedutilitylevel.
This utilitylevel will adjustuntilin equilibriumfirmsmake zero profits.
Thus, we may view the firminsteadas maximizinga worker'sutility,
subjectto its zero profitconstraint,which is given by (4a) or (4a'). In
thismaximization,it takesw' and wl as fixedsince it treats
performing
the equilibriumcontractofferedby type I firmsas exogenous. We
assume that a type 2 firmgets a random group of workerswho have
quit type I firms.As a result,a singletype2 firmby itselfcan influence
neitherthe first-period
savingsof its enteringworkersnor the quit rate
of the firmsfromwhich theycome, and will therefore
treats} and 01 as
fixed.It will, however,recognizethatits choice of w2 and w2 affectsits
workers'second-periodsavingsand quit decisions.
10
As pointedout in Arnott(1982), theremay be multiplelocal optimato the
planner'sand firm'smaximization
problems.Dependingon thepathof adjustment
to equilibrium,a competitiveeconomycould finditselfat an inferioroptimum.
This is anotherpotentialsourceof marketfailure.
446
Arnott/Stiglitz
Thus, a type 2 firm'sproblem is to maximize (1') with respectto
s', 0', wl, w as fixed.WhereX is
{w23,w} and subjectto (4a'), treating
the Lagrangianmultiplieron (4a'), the Lagrangianof the type 2 firm's
problemis
L = EU + 1B,
(9)
conditionsare
and the correspondingfirst-order
W2
28EU +
W3:
28EU
2
8B
8 2(ia
=? 0
6
(1Oa)
and
81?
W3
83
=0
(lOb)
wherethe 8x/8ydenotesthe derivativesof x with respectto y, holding
otherwage ratesand s} and 01 fixed,but treatingthe dependenceof s2
and 02 on y.
A type 1 firmacts as if it chose wI and w to maximizea worker's
expectedutility,subjectto the zero profitconstraint,
which is givenby
thismaximization,it takesw2 and w2 as fixed,
(6) or (6'). In performing
since it treats the equilibriumcontractofferedby type 2 firmsas
exogenous,and O1, 02, s}, and s2 as varyingwith wl and wl according
to (2a)-(2d), since it realizes that its contractchoice will affectits
workers'first-and second-periodsavingsand quit decisions.WhereX is
the Lagrange multiplieron (6'), the Lagrangianof the type 1 firm's
problemis
L = EU+
A,
(11)
and the correspondingfirst-order
conditionsare
1 dEU
XdA
~dw~ 0,
WI.di+
(12a)
and
dEU ~dA
W2:d
dwi
+
d
dw 1
= O.
(12b)
The competitiveequilibriumis thereforecharacterizedby (4a'), (4b),
(6'), (i0a), (lob), (12a), (12b), and (2a)-(2d).
447
LaborTurnover
andMoralHazard
D. Comparisonof the CompetitiveEquilibrium
and the PlanningOptimum
To provide a thoroughtreatmentof the comparisonwould greatly
lengthenthe paper and generatelittleadditionalinsight.For thisreason,
we concentrateon what we term the normalcase, for which, at the
social optimum,
3m - T1 > w1 + 2w' > wI
+W2+W2>
wI
+W2+W3
> 3m -T
- T2-
T3
m> wl > m - T1, and m> W2 > m - T2. In the normalcase, workers
who stay with the same firmthroughouttheirworkinglives receivea
sum of wages thatis higherthanif theywere to quit once or twice,but
lower than theirnet marginalproduct.Workerswho quit twice during
theirworkinglives receivea sum of wages that is lower than if they
were to quit only once or not at all, but higherthantheirnet marginal
both type 1 and type 2 firmsprovide partial
product. Furthermore,
insuranceagainstquitting,in the sense thatthesefirmscover only part
of the worker'shiringcosts.
The normal case will certainlyobtain when the TP's are large and
workersare only slightlyriskaverse.Because the Ti's are large,3m - T1
> 3m - T1 - T2 > 3m - T1 - T2 - T3 (where > indicatesmuch
largerthan);and because workersare only slightlyriskaverse,m > wI
m - T2 (where > indicatesonly slightly
> m - T1 and m >
2
largerthan),which togetherimply(13).
LEMMA 2. In the normal case at the social optimum,(?u') < (Iu')
<
(2u') and
<
(2U')
<
W3 .
Thus W32
(2U')
(33U).
For the normal case, the sum of wages when a workerquits once,
3 exceeds the sum of wages when the workerquits twice,
wI + w + w2,
W I+
W2 +
>
W3.
Now
c2 = s2 + w2
and C3
=
S2 +
W3.
Hence
C3, whichimplies(3Ut) > (2Ut'). From (2b), theworkerchooses
s2 to equalize the second-periodand expected third-periodmarginal
utilitiesof consumption,implying(3Ut) > (2U ) > (32U').
The proofof the firstinequalityis similar.For the normalcase, the
sum of wages when a workerneverquits,wI + 2w exceeds thatwhen
+
+ w2. Thus, 2w > w2 + w2. Now 2c'
he quits once, wlwI
=2w' + s' and c +c3=w+
sw++ . Hence, 2c >c +c2. From
above, c2 > C2. It follows that cI > c2, and therefore(I U') < (2U'). From
and expected
(2a), the worker chooses s' to equalize the first-period
second-period marginal utilities of consumption, implying (lu') <
(1u') < (2U').We can now establish
C2 >
Arnott/Stiglitz
448
PROPOSITION 1. In thenormalcase, at thesocial optimum,competitive
type I firmswill (locally) want to offeradditionalinsurance.
Proof. Hold w' and w3 at theirsociallyoptimallevels,and plot in
contour
(w25,W) space: (i) the social optimum,P; (ii) the indifference
going throughthe optimum,EU; (iii) the true resourceconstraint,R;
A. Since type 1
and (iv) the budgetconstraintperceivedby type 1 firms,
contour(recallthattheycorrectly
perceivetheindifference
firmscorrectly
determinehow workerswill adjust theirsavingsand quit decisionsin
responseto changesin wl and w'), then to prove the proposition,we
need to demonstrate(in termsof fig.1) thatimmediatelyabove P, A lies
above R.
and
Let superscript
p denoteevaluationalong the resourceconstraint,
let c denote evaluationalong the budgetconstraintperceivedby competitivefirms.
we have
RecallingthatA = 0 is the type 1 firmbudgetconstraint,
(dw}C__
(d@, )=
dA
dw'2
_ tdA2)
(i)
dwl/
The social calculation,however,takes account of the factthatchanges
in w} and w affectthe costs incurredon type2 contractsas well. We
demonstrated
earlierthatthe plannerfacesthe budgetconstraintF[01]B
+ A = 0. Thus
WI
R
w2
FIG.
I.-The
competitivetype I contractprovides"too much" insurance
Labor Turnoverand Moral Hazard
449
F
w2 p
F[01]
dw(
dB
dA
dB + dA
dwid
i
It mustbe the case that(dEU)/(dw}) > 0 and (dEU)/(dw ) > 0. From
(8a) and (8b), it follows that both the numeratorand denominatorof
the term in bracketson the right-handside of (ii) are negative.And
using the resultsfromlemma 2 and those presentedin Appendix B, it
can be shown that(dB/dw})< 0 and (dB/dw) > 0-raising w} increases
the costs associated with type 2 contractsbut raisingw lowers these
costs.(The economicsof theseand otherresultswill be explainedin the
next subsection.)Hence, (dA/dw1)p tic0, while (dA/dw1)p< 0. If (dA/
dwDp < O, then
dA
dwjl
d-A
(FO]dB
dw1 dw1
]dB
dA
d I
+
dw I
<
PP
dA(iii)
dw I
whileif (dA/dw})p> 0, then (dw /dw')p > 0 and (dw}/dw2)p< 0. In
eithercase, at the social optimum,competitivefirmswould want to
increasewI and decreasew2, therebyprovidingmoreinsurance.Q.E.D.
We have drawn in the indifference
curvesand the budgetconstraints
in figure1 as havingstandardcurvatureproperties.However,theremay
be multiplelocal optima.For thisreason,the statementthattype 1 firm
contractsprovidetoo much insuranceis local.
The result in proposition 1 occurs because, for type 1 firms,the
privatecosts of providinginsuranceare less than the social costs. We
shall explain this resultin the next subsection.Proposition1 has an
immediate
COROLLARY. In the normal case, the competitiveequilibriumset of
employmentcontractsis Paretoinefficient.
PROPOSITION 2. In thenormalcase,at thesocial optimum,competitive
type2 firmsmay want to offereithermore or less insurance.
Proof. The proofis analogous to thatforproposition1. Competitive
contour.Proving
type 2 firmsagain correctlyperceivethe indifference
the propositionthereforeagain entails comparingthe slopes of the
resourceconstraintand the budgetconstraintperceivedby type2 firms
at the social optimum,but this time in (w2, w2) space. From (4a'), the
slope of the budgetconstraintperceivedby type2 firmsis
450
Arnott/Stiglitz
(
dw~
/B2
8w3
8B
\C__
t~w3}p
(i)
,
2w2 p
while the slope of the resourcecontraintis
kd
W0
dB
dA
dB
dA
(ii)
Aftera great deal of tedious manipulation,it can be shown that
p maybe eitherlargeror smallerthan-(dw2/dw2)c. Q.E.D.
These are thebasic analyticalresults.More detailis providedin Arnott
and Stiglitz(1981).
-(dw2 dw2)
E. The Causes of MarketFailures:The Normal Case
Let us suppose thatw2 and w2 are at theirsociallyoptimalvalues and
investigatewhy a type 1 firm would choose a socially inefficient
employmentcontract(w', wD).
Consider the firm'scalculationswith respectto raisingw1, holding
w2 constant.The firmwill take w2 and w2 as fixed.When it raiseswl
it causes the workerto alterhis savingsbehavior,changingboth s' and
2
s2. It can be shown that
49S22
awl
=(
2([01])2
[OI]f [02](22U1)(lut
2
A
ul>
O.
where A is the determinantof the Hessian correspondingto (2a)-(2d)
and is positive. As one would intuitivelyexpect, in response to an
increasein w 1,theworkerwill increasehis consumptionin everyperiod,
whatever his employmenthistory,and c2 and C3 are increased by
S22
Now, this increasein s2 makes workersmore willingto
increasings2.
quit type2 firms.In particular,from(2d), with w3 and W3 fixed,(d02/
ds2) = (3u') - (3U') > 0 (using lemma 2). In responseto theirincreased
quit rates,type2 firmswill be forcedto offera less attractive
employment
contractin order to restorebudget balance. In takingw2 and W3 as
fixed,type 1 firmsignorethis and hence underestimate
the social cost
of raisingwI.
Now considerthe firm'scalculationswith respectto loweringw2l
holdingwl constant.It can be shown that
LaborTurnover
andMoralHazard
2F [0']2f [02]f
Os2
aw2
[0](2u)
{-(I
451
-F[1])(")
A
+ f [O1] [(2u')
-(IU)](I
U)
< 0.
Suppose that the workerrespondsto the fall in w2 by reducingc' by
the fullamountof the fall and holdingall other {c,}'s at theiroriginal
level.This would increase(lu'), and to restoreequalitybetweenthe firstperiod and expected second-periodmarginalutilitiesof consumption,
theworkerwould haveto decreasefirst-period
consumptionand increase
first-periodsavings. It can in fact be shown that (Oas'/aw')
<
0; thus, in
responseto the fall in w', workersjoiningtype2 firmswill have more
savings.As a resultC2, C3, and c' will rise,whichrequiresthats2 rise.As
we have seen, this increase in s2 forces type 2 firmsto offera less
attractivecontract.Hence, type 1 firmsoverestimatethe social saving
fromloweringw'. Puttingthesetwo resultstogether,we have thattype
I firms,by ignoring the effectsof their contract changes on the
profitability
of type 2 firms,will set w} higherand w' lower than is
sociallyoptimal(in a local sense).This entailsoffering
excessiveinsurance.
We termthisuninternalized
theforward-directed
externality,
externality.
We argued earlier that when the utilityfunctionis intertemporally
separable,which we have assumed,this externality
operatesexclusively
via workers'savings.To check this,let us retracethe argumentof this
subsectionholding workers'savings fixed.Raise w}, holding w2 w2,
and w3 fixed.The increasein wI altersneitherthe budgetconstraints
of
type2 firmsor thecharacteristics
(savings,tastes)of itsenteringworkers.
In consequence,workerswill respondin the same way as beforeto type
2 contracts,and type2 firmswill have no incentiveto alterthecontracts
theyoffer.The same argumentapplies to thefallin w . When theutility
functionis not intertemporally
separable,the past, present,and future
are linked not only via savings but also directly,throughthe utility
function.
We now turn to the type 2 firm'scalculations.We shall not go
throughthe algebrabut insteadshall just outlinethe sourcesof market
failure.Thereare two of them.First,each type2 firm,becauseit receives
a random selectionof workers,treatssl as fixed.However, all type 2
firmstogetheraltersl throughthe contractstheyoffer.And s in turn
affects
02. Thus, type2 firms
collectivelyignoretheeffectof thecontracts
theyofferon theirown quit rates,whichoperatesvia first-period
savings.
We termthis the sidewaysexternality,
because each type2 firmignores
the effectsof the contractit offerson s and hence on the profitability
of all othertype2 firms.Second, type2 firmsfail to considerthat the
contracttheyofferaffectsthe profitability
of type 1 firmsand hence the
attractiveness
of the contractstheycan offer.This marketfailure,too,
452
Arnott/Stiglitz
operatesvia workers'
externality,
which we termthe backward-directed
savings.
The forward-and backward-directedexternalitiesare examples of
whatwe havetermedelsewhere(Arnottand Stiglitz1982) the"seemingly
unrelatedeventsmarketfailure,"which stemsfrommoral hazard. We
shall now give an example of this marketfailureand then show how
the problemat hand is analogousto the example.Suppose thatthereare
two statistically
independentsets of statesof nature.One set influences
the probabilitythatone's housingwill burndown, the otherthatof an
information
(i.e., theinsurer
automobileaccident.Because of asymmetric
can observethe outcome or event,but neitherthe underlyingstate of
effort)
insuranceis provided
naturenor theinsured'saccident-prevention
statesof nature,
againstthetwo eventsratherthanagainsttheunderlying
and both formsof insuranceare, as a result,characterizedby moral
that an individualpurchasesinsurance
hazard. Suppose, furthermore,
againstone accidentfromone agentand againstthe otheraccidentfrom
another.The provisionof insuranceagainsthis housing'sburningdown
may cause the individualto drive less carefully,therebyincreasingthe
probabilityof an automobileaccident.In this case, the providerof fire
insuranceignoresin his calculationsthatby providingmorefireinsurance
he affectsthe profitability
of the contractofferedby the providerof
the private cost he
automobile accident insurance.Put alternatively,
perceivesin providingan extraunit of insurancewill in generaldiffer
fromthe social cost; as a result,he providesthe wrong amountof fire
insurance.The two agentsfailto take into accountthisinterdependence
betweenthe insurancecontractstheyoffer.
In the problemat hand,the two statistically
independentrisksare the
qualitiesof two job matchesforthe same worker,one correspondingto
a job earlierin his career,the otherto a laterjob. The interdependence
comes about throughsaving.The firmthatemploysthe workerearlyin
his career fails to take into account that the contractit offers,by
influencingthe amount the worker saves, will in general affectthe
profitability
of the firmthat employs the workerlater in his career.
Similarly,the latterfirmfails to take into account that the contractit
offerswill in generalaffectthe profitability
of the formerfirm.
As we suggested earlier, the corresponding2-period economy is
In such an economy,thereare type 1 and type2 firms.Since
efficient.
type2 firmshireworkersforonly 1 period,theyhave no choice but to
pay workerstheirnet (of T) marginalproducts.Since type2 firmshave
of type 1 firms.Nor is
no discretion,theycannotaffectthe profitability
theirown profitability
affectedby the type 1 contract.
F. ImperfectCapital Markets
It is difficultto make generalstatementsabout how our resultsare
because thereare so many
alteredwhen capital marketsare imperfect,
453
LaborTurnover
andMoralHazard
and so manyformsof imperfectcapital
possible causes of imperfection
market.One can, however,identifyadditionalsourcesof marketfailure
in capitalmarketsmaygiverise.First,borrowing
to whichimperfections
and lending(whetherthis is done explicitlyby banks or implicitlyby
firms)11
resultsin another marketfailureof the seeminglyunrelated
events variety:The borroweror lender will ignore the effectof his
of
actions,throughtheireffectson workersavings,on the profitability
otherfirms.Second,loans maybe characterizedby moralhazard;ifthey
points
agentsat different
are,and if a workercan borrowfromdifferent
in time,or frommore than one agent at a point in time,therewill be
externalitiesanalogous to
forward-,backward-,and sideways-directed
thosediscussedin the previoussubsection.
marketfailuremay arise, which we
Third, a qualitativelydifferent
term"the cross-subsidization
marketfailure."Recall that with perfect
capital markets,workerswere concernedwith only the sum of wages
associated with each of the threeemploymenthistories,and that this
in the planningproblem.The
allowed us to set two wages arbitrarily
one foreach firm
competitiveequilibriumhad threebudgetconstraints,
type,while the planningproblemhad only one resourceconstraint.But
these two extra constraintsin the marketproblem ''did not matter,"
sincetheplanningproblemhad thetwo degreesof freedomnotedabove.
With imperfectcapital markets,however,the workeris concernednot
only with the sum of wages for a particularemploymenthistory,but
with the timingof wages as well. In this case, the planningproblem
in the market
may have no degreesof freedomand the extraconstraints
because a
potential
market
failure
arises
problem may "matter."The
can
subsidize
firm
but
the
hence
cannot;
across
types,
market
planner
the term cross-subsidizationmarket failure. The cross-subsidization
marketfailureis particularly
easy to see in the 2-periodanalogue to the
3-periodmodel treatedin the paper. Suppose thatexpectedutilitytakes
on the special form
EU = min(cl,c2,c2) + E01 + E02,
(i)
and thatthereis no borrowing,lending,or saving.Under competition,
c2 =m
- T2+
(ii)
where i9 is the size of the subsidy to type 2 firmsper worker they
"This would entailpayingworkersmorein periodswhentheirmarginal
of consumption
is highand less in periodswhenit is low. If younger
utility
loantoyounger
workers
areliquidity
a firm
willprovidean implicit
constrained,
workersthatis repaidby olderworkerswho staywiththe firm.The older
workers
whoquitthefirmin effect
on theirimplicit
default
loans.
454
Arnott/Stiglitz
employ,whichis collectedfromtype1 firms.Suppose,furthermore,
thatin theabsenceofa subsidymin(c', c', c') = c'. From (i),
dEU
(iii)
dEO2
0,
Thus, in this extremeexample,a $1.00 subsidypayable to type2 firms
for each worker they employ increases every worker's utilityfrom
consumptionby $1.00.
In sum, while precise characterizationwill have to await further
research,we assertwith confidencethatwith imperfectcapital markets,
too, multiperiodemploymentcontractsare inefficient.
III. Concluding Comments
A. Discussion
The model can be generalizedto N periodsor to continuoustimeand
our simplifying
assumptionsof a zero interestrate and of an intertemporallyseparableutilityfunctionwithunchangingtastescan be relaxed,
and the same qualitativeresultsobtained.
the policy implicationsof the marketfailureswe have
Unfortunately,
identifiedare far from clear. One cannot say a priori whetherfirms
providetoo littleor too muchinsurancein theircontracts,12northerefore
a prioriwhetherthe government
should tax or subsidizelabor turnover.
In principle,one could computethe optimal set of turnovertaxes and
subsidiesfor the economy treatedin the paper, but the informational
requirements
to do thisare unrealistically
large.Thus,themarketfailures
we have identifiedshould be viewed as potentialmarketfailures;they
warrantgovernmentintervention
are
only if the costs of intervention
less than the benefits.Furthermore,
the
results
one would expect
to be
substantiallyalteredwhen the realisticcomplicationsarisingwith heterogeneousworkers,notablyadverseselectionproblems,are considered.
It is by now well known thatin economieswithincompletemarkets,
institutional
structure"matters"and is endogenous.One should derive,
ratherthan assume,who will provideinsurancein what form,what the
characteristics
ofcapitalmarketswill be,whatformofmarketorganization
will arise, and so on. We have shown in a specificcontextthat the
We wereable to obtainunambiguous
resultsfortype1 firms.
But type1
firmsare special,sincetheyhireworkerswho havemadeno prioreconomic
decisions.
12
LaborTurnover
andMoralHazard
455
institutional
structure
generatedby the marketis (potentially)inefficient.
Optimal correctiveaction may entail not only the conventionaltaxes
To treat
and subsidiesbut also the regulationof institutional
structure.
such considerations,
a new welfareeconomicswill have to be developed
that is more sophisticatedand complex, but at the same time more
interesting
and realistic,than the old.
What mightbe a "Chicago economist's"defenseof the marketagainst
in a competitiveeconomymay be inefficient?
our claim thatinstitutions
He mightpoint out that the externalitieswe have identifiedcould be
internalizedby a superfirmthat sets all wages and, in the case of
imperfect
capitalmarkets,subsidizesacrossfirms,and arguefurthermore
that the profitmotivewould lead to the creationof such a superfirm.
This is an intellectually
respectabledefense,since thereis nothingin our
model to preclude the creationof such a superfirm.It is, however,a
ratherstrangedefenseof the marketthat requiresthat there be one
which all workersmustjoin forlife.One wondershow the
superfirm,
forcesof competitionwould operate-potentialentryby othersuperfirms
would reallybe
stretchesone's credulity-andwhethersuch a superfirm
if we were to enrich
very different
froma government.Furthermore,
our model to include manyproducts,and economiesand diseconomies
of scale and scope (the diseconomiesreflecting
the costs of coordination
and administration),
we would findthatthe optimumentaileda group
of superfirms.
Competitiveequilibriumwould be characterizedby all
the marketfailureswe have identifiedas long as there is any intersuperfirmmobilityof labor. The Chicago economistwould probably
also stressthat we have only identifiedpotentialmarketfailures;we
have acknowledgedthispoint.
B. MacroeconomicImplications
In this paper we have focusedon only one aspect of the contractual
betweenworkersand employers,albeitan importantone.
arrangements
The contractualrelationshiphas recentlybeen the centerof discussions
to explainwage rigiditiesand unemployment.
Elsewhere,we
attempting
have argued that at least the conventionalformsof implicitcontract
theorydo not succeed in providingverypersuasivemicro foundations
forthe theoryof unemployment
(Stiglitz1984).Though in the presence
of perfectinformation
implicitcontracttheoryexplainswage rigidities,
it does not explainunemployment:
workerswould stillalwayswork up
to the pointwherethevalue of theirmarginalproductivity
was equal to
their marginalrate of substitutionbetween goods and leisure. More
recentdevelopments,
information
focusingon imperfect
(see, e.g.,Grossman and Hart 1981), are equally unconvincing:the contractstypically
than is reasonablyavailable
analyzedentailtherebeing less information
(the termsof the contractare not made contingenton variables,such as
456
Arnott/Stiglitz
rateorimports
ofcars,etc.,which,iftheinformational
theunemployment
problemson which theyfocus were central,theyshould be) and more
information
than is reasonablyavailable (workersmust know the total
numberof hourshiredby thefirmand,moreover,mustmonitortransfers
of capitalgoods fromthefirmto otherfirms).Underplausibleconditions,
ratherthan underemployment.
the contractslead to overemployment
to be enforceable,they
Moreover,the contractsare not self-enforcing;
mustbe explicitratherthan implicit,and, in fact,we see few contracts
of the indicatedform.The theoriesdo not explaineitherthe patternsof
unemploymentrates
unemployment(i.e., why there are differential
different
or
forms
of
(i.e., why there
unemployment
groups)
among
should be lay-offsratherthan work sharing).When thereare lay-offs,
the theoriespredict(again underplausible assumptions)thatthose laid
offare betteroffthanthosewho are retained(providedthefirmprovides
optimalseverancepay).Finally,thoughthetheoriesprovidean explanation
which
of layoffs,
theydo not providean explanationof unemployment,
explainlayoffsand hiring."3
mustsimultaneously
The fact that these conventionaldevelopmentsof implicitcontract
does not,of
theoryhave not provideda good theoryof unemployment
between
course,implythatthe structure
of thecontractualarrangements
of unemworkersand theiremployersis not an importantdeterminant
ployment.Severalextensionsof our model can, we believe,provideat
least partof the microfoundations
of a theoryof unemployment.
It is easy, for instance,to constructa simple model of frictional
by havinga multiperiodmodel in which thereare lags
unemployment,
betweenquittingand beinghiredand in whichthe expecteddurationof
depends on search intensity.Such a model should give
unemployment
rise to a ratherinterestingset of additional marketfailures.When a
13 Thedifference
between
ourimplicit
contract
theory
andthestandard
implicit
withasymmetric
contract
theories
information
shouldnowbe clear;boththeories
are,in a sense,basedon incomplete
information.
(As we have argued,with
serveonlyto smoothdemandanddo not
perfect
information,
implicit
contracts
give rise to any unemployment.)
The information
problemon whichthe
asymmetric
information
modelshavefocusedconcerns
theproductivity
of the
firm.Risk-averse
firms,
wishingto reducetheirwagesin bad states,needto
persuadetheworkers
thatthestateis in factbad; to do so, theyagreethatif
theysay the stateis bad, theywill reducetheiremployment
belowwhatit
otherwise
wouldbe. Becausethecostofreducing
in good
employment
is greater
statesthanin bad states,thisrestriction
on employment
inducesfirmsto say
thatit is a bad stateonly whenit is in factbad. We are concernedwith
information
problemsthatarisewheneverlabor turnover
is possible.These
includeproblems
associated
withascertaining
theindividual's
assessment
of the
nonpecuniary
benefits
ofthefirm
(inthispaper),andascertaining
theindividual's
searchintensity
(in Arnott,
Hosios,andStiglitz1982).
LaborTurnover
andMoralHazard
457
on
workerjoins the unemployment
pool, he imposesan externality
otherworkers
by reducing
theprobability
thatan advertised
vacancyis
a job vacancy,imposesan
unfilled.Similarly,
a firm,in advertising
theprobability
of theirfillinga
externality
on otherfirmsby reducing
theirquit rates.
givenjob vacancy,and perhapsalso by increasing
it ignoresthatits
whena firmchoosesitswagestructure,
Furthermore,
thesize of the unemployment
quit rateaffects
pool and thata larger
to recruiton thereturns
unemployment
pool mayhavepositiveeffects
In addition,
theinsurance
thefirmprovidesagainst
mentbyotherfirms.
which
quittinghas a moralhazardeffect
on workersearchintensity,
a workermovesfromone
maybe harmful
to otherfirms.Whenever
is altered.Thus,
firmto another,
theeconomy'sworker-firm
matching
as wellas theexternalities
all oftheseunemployment-related
externalities,
formsof matching
noted previously,
may be regardedas different
in theliterature.
externalities,
whichhavebeendiscussedelsewhere
is theextension
Morecentral
to understanding
cyclicalunemployment
on
of the analysisto examinethe effectof contractual
arrangements
to sectoralshocks.Contracts
thatprovidegoodinsurance
lead
responses
fromlow- to high-productivity
sectors.
workersto move sluggishly
in particular,
Firmsrecognize
thisin designing
contracts;
theyrecognize
thatthetotalcostoftheimplicit
subsidythattheyprovidein bad states
can be reducedifworkers
can be inducedto leave.Workers-exanterecognizethisand thusare willingto sign contracts
thateffectively
committhem(withsome probability)
to searchin the bad state;the
commitment
(theonlyenforceable
commitment
in thiscontext)takes
the formof a layoff(possiblywithseverancepay). ElsewhereArnott,
Hosios,andStiglitz
(1982)haveanalyzedthestructure
ofthesecontracts
in detail;thatanalysisprovidesbettermicrofoundations
forthetheory
of unemployment
thanexistingmodels:theequilibrium
contractmay
entaillayoffs
withwellas wellas worksharing;
thetheory
is consistent
defined
of who getslaid off(whicharein accordwithwhatis
patterns
and thosewho do getlaidoffaremorelikelyto be worseoff
observed);
thanthosewho areretained
withtheparadoxical
(in comparison
results
ofthestandard
models).
modelmaygo someway toward
ThoughtheArnott-Hosios-Stiglitz
a versionofimplicit
contract
thatcan providepartof
developing
theory
a stillbetter
thebasisofa theory
ofunemployment,
wouldentail
theory
of implicitcontract
theintegration
theorywithefficiency
wagetheory.
at present.
Workon thisis underway
C. Conclusion
We have obtaineda ratherstriking
resultfor a particular
model.
that when marketsare
Arrow(1965) and othershave conjectured
458
Arnott/Stiglitz
incomplete,institutions
ariseto filltheholes leftby thoseabsentmarkets
in a mannerthatis efficient,
giventhe transactionscost technologyand
information
available.In our model,becauseof information
asymmetries,
completeinsurancewas not availableagainstthe underlying
uncertainty,
the qualityof job matches.We arguedthat in these circumstancesthe
accommodatinginstitutionthat would arise is the firm'sproviding
implicit insuranceagainst quittingin the employmentcontract.We
demonstrated
thatin thesecircumstances
each firm'schoice of contract
affects
theprofitability
of all otherfirmsthroughworkersavings.Because
firmsignorethis interdependence,
thereare uninternalizedexternalities
that cause potentialmarketfailure.This resultis a counterexampleto
Arrow'sconjecture.If the sourceof marketfailurewe have identifiedis
general,the disturbingimplicationthatfollowsis thatthe wide rangeof
public, charitable,and privatequasi-insuranceinstitutionsin existence
are collectivelyinefficient.
How generalin factare the sources of marketfailurewe identified?
They are generic,of a class we termedseeminglyunrelatedevents,and
theyarisewheneverthe provisionof insuranceis characterizedby moral
hazard and individualsobtain insurancefrommore than one source.
This is truewhetherthe insuranceis providedby insurancecompanies
or the governmentor social institutions-fraternal
and charitableorganizations,family,friends,and so forth.Each providerof insurancewill
generallyignorethatthe insuranceor quasi insurancehe provideswill
affecttheprobabilityof accidentsagainstwhichothersprovideinsurance.
The seeminglyunrelatedeventsmarketfailuresidentified
in thispaper
were of a special typethatmeritattentionin theirown right.Suppose,
in a temporaleconomy,thatthe probabilityof an accidentat a point in
time depends, not only on the contemporaneousamount (flow) of
insuranceprovided,but also on thevalue of some state(or stock)variable
thatis affectedby how much insurancehas been providedin the past
and how muchwill be providedin the future.If different
agentsprovide
insuranceover different
periods,marketfailurewill occur. Each agent
will ignorethe effectthe insurancehe provideshas on the stockvariable
for periods duringwhich he does not provide the insurance.In the
model treatedin the paper,thisstockvariablewas savings.It could also
be stateof health,level of education,or work experience.This typeof
is internalizedifa singleinsurercoversan individualthroughexternality
out his life.To achievethis,eitherlifetimeinsurancewould have to be
providedby a third-party
insurerexternalto firms(which could be the
or
else
individual
would have to be employedby the
each
government),
same superfirmthroughouthis workinglife that would provide such
insurance.The formersolutionsuffers
fromtheproblemthatinformation
costs
are
insurerthan for
acquisition
typicallyhigherfor a third-party
LaborTurnover
andMoralHazard
459
an individual'semployer.The latter solution raises the issue of the
compatibilityof superfirmsand competitivebehavior.Both solutions
informedconcerning
reston theassumptionthatindividualsare perfectly
are best
the menu of lifetimecontractsoffered.How theseexternalities
on the partof
treatedwhen accountis takenof incompleteinformation
individualsis a topic thatwill have to await futureresearch.
Appendix A
Proof of Lemma 1
We may index employmenthistoriesaccordingto the firmtype a
workeris employedwith when he retires.LettingWk denote the sum
of wages associatedwithemploymenthistoryk, we obtain
W1 = wl + 2wL
(Ala)
A Ib)
~~~~~(A
+ W32,
W2 = WI 1+ W2 2
3
and
W3 = w + w+W.
(Ac)
We need to provethatany triple(W1, W2, W3) in the planner'ssolution
set is also in the solution set when (4a) and (4b) are imposed as
constraints.
First,note that02 may be viewed as a functionof (W1, W2,
W3).Next set
wV2=W2-W3+m-T3
WI = W3-
(3 - F1[02])m+ T2+ T3 + (1
(AId)
- F[02])w2,
(Ale)
and
IW _(WI
-WI)I
(Alf)
We assumethatthe utilityfunctionsand the exogenousparametersare
suchthattheoptimal{ W4,}entailsnonnegative
{w,}. It is a straightforward
matterto check that (4a), (4b), and (Ald)-(Alf) satisfy(Ala)-(Alc).
Furthermore,
from(4a), (4b), and (Ald)-(Alf), one can express each
wage as a functionof (W1, W2, W3) and exogenousparameters.Thus,
for any (W1, W2, W3), there is a unique wage vector (wI, wI
(4a), (4b), and (Ala)-(Alc). Q.E.D.
w2, w3, w3) satisfying
Appendix B
Evaluation of Derivatives
of (2a)-(2d) gives
Total differentiation
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Arnott/Stiglitz
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