PARTIAL ADJUSTMENT WITHOUT APOLOGY∗ BY Boston University, Federal Reserve Bank of ...

31 Pages·2006·0.33 MB·English

INTERNATIONALECONOMICREVIEW
Vol.47,No.3,August2006
∗
PARTIALADJUSTMENTWITHOUTAPOLOGY
BY ROBERTG.KINGANDJULIAK.THOMAS1
BostonUniversity,FederalReserveBankofRichmond,andNBER,U.S.A.;
UniversityofMinnesota,FederalReserveBankofMinneapolis,
andNBER,U.S.A.
Econometricpartialadjustmentmodelsperformrelativelywellattheaggre-
gatelevel;however,manykindsofmicroeconomicbehaviorinvolvediscreteand
occasionalchoices.Analyzingtheclassicemploymentadjustmentproblem,we
showhowageneralizedpartialadjustmentmodeltractablyaccommodatesboth
observationsbyaggregatingtheactionsofheterogeneousproducersfacingfixed
adjustmentcosts.Aggregatedisturbancescausechangesinestablishment-level
targetemploymentandinthemeasureofestablishmentsactivelyadjustingto
theirtarget,whereasaggregateresponsesexhibitpartialadjustment.Ourframe-
workalsocanbeappliedingeneralequilibriumsettingswithpersistentidiosyn-
craticshockswithoutforfeitingtheconvenientuseoflinearsolutionmethods.
1. INTRODUCTION
Inmanycontexts,actualfactordemandsclearlyinvolvecomplicateddynamic
elements absent in static demand theory. For example, empirical studies of the
marketdemandforlabortypicallyfindthatlags,eitherofdemandorofthede-
terminantsofdemand,contributesubstantiallytotheexplanationofemployment
determination. The most frequent rationalization of such lags is that individual
plantsfacemarginalcoststhatareincreasingintheextentofadjustment,leading
themtochoosepartialadjustment towardthelevelssuggestedbystaticdemand
theory.Manyempiricalstudiesalsoindicate,however,thatthepartialadjustment
modelisinconsistentwiththebehaviorofindividualplantsorfirms.Forexample,
Hamermesh (1989) shows that individual plants undertake discrete and occa-
sionalworkforceadjustmentsratherthanthesmoothchangesimpliedbypartial
adjustment.Nonetheless,themodelcontinuestobeavehicleforappliedwork,es-
sentiallybecauseitisatractablewayofcapturingsomeimportantdynamicaspects
ofmarketdemand.Itisfrequentlythusemployedinanapologeticmanner,with
theresearchersuggestingthatitisadescriptionofmarket,ratherthanindividual,
factordemand.2
We present a generalized partial adjustment model in which individual pro-
duction units adjust in a discrete and occasional manner, yet there is smooth
∗ManuscriptreceivedJune2004;revisedFebruary2005.
1WethankRandallWrightandtwoanonymousrefereesfortheircomments.Anyremainingerrors
areourown.Theviewsexpressedinthisarticlearethoseoftheauthorsanddonotreflectthoseofthe
FederalReserveBankofRichmondorMinneapolis,ortheFederalReserveSystem.Pleaseaddress
correspondenceto:JuliaK.Thomas,DepartmentofEconomics,UniversityofMinnesota,1035Heller
Hall,27119thAvenueSouth,Minneapolis,MN55455.Phone:[phone].E-mail:[email protected].
2See,forexample,Kollintzas(1985).
779
780 KINGANDTHOMAS
adjustmentattheaggregatelevel.Specifically,individualunitsfacedifferingfixed
costs of adjustment, so the timing of their adjustments is infrequent and asyn-
chronizedwhileaggregationacrossplantsleadstoasmoothpatternofaggregate
factordemandwellapproximatedbythestandardpartialadjustmentmodel.Our
exposition of this model’s relation to the traditional model commonly used in
empiricalworkisuniquetothisarticle.
Our basic framework is sufficiently tractable that it has already been applied
toexamineseveraltopics,amongthempriceadjustmentandcapitalinvestment.3
Here, we apply it to employment, which, relative to the investment application,
requires a different timing to trace the resulting distribution of production. We
providethefirstcomprehensivepresentationoftheframeworksothatresearchers
mayconvenientlyadaptittostudyotherproblems.Tofacilitateitsbroadapplica-
tion,wethenextendthemethodtoallowforpersistentidiosyncraticshocks.
Ourmodelprovidesamicroeconomicfoundationforthevarietyofplant-level
adjustmentexaminedintheempiricalworkofCaballeroandEngel(1992,1993)
andCaballeroetal.(1997).There,individualproductionunitsareassumedtoad-
justemploymentprobabilistically,withadjustmentprobabilitiesbeingafunction
ofthedifferencebetweenatargetlevelofemploymentandactualemployment.
Aggregating from such adjustment hazard functions, which are their basic unit
of analysis, they examine the implications of the resulting state-dependent ad-
justment behavior for aggregate employmentdemand dynamics. In the absence
ofamicroeconomicfoundationforsuchprobabilisticadjustment,Caballeroand
Engel(1993,p.360,paragraph2)explainthatthey“tradesomedeepparameters
forempiricalrichness.”Incontrast,weexplicitlymodeltheplant’sadjustmentde-
cisionasageneralized(S,s)problemandderivetheadjustmenthazardfunctions
thatarethestartingpointofpreviousresearch.4
Onekeystylizedfactuncoveredintheempiricalliteratureisthatanimportant
routethroughwhichaggregateshocksaffectaggregateemploymentisbychang-
ingthefractionofplantsthatchoosetoadjust.Accordingly,wedevelopamodel
wheretheaggregateadjustmentrateisanendogenousfunctionofthestateofthe
economy. Although our generalized model is not observationally equivalent to
thetraditionalpartialadjustmentmodelwithtime-invariantaggregateadjustment
rates,impulseresponsesestablishthatitretainsthebasicfeaturesofgradualpar-
tialadjustment.Anotherdistinguishingfeatureofourtheoreticalapproachisthat
itisconvenienttoundertakegeneralized(S,s)analysisinageneralequilibrium
environment,sotheinfluenceofaggregateshocksonequilibriumadjustmentpat-
ternsmaybesystematicallystudied.Finally,ourapproachissufficientlytractable
soastoaccommodateadditionalsourcesofheterogeneity.Thus,beyondachieving
consistencywiththestylizedfactshighlightedhere,itnaturallyextendstoallow
forthericherheterogeneityofactionsthatisessentialinmatchingotheraspectsof
themicroeconomicdataonfactoradjustment.Moreover,thisgeneralizationtoin-
cludediscreteindividualadjustmentsalongsidepersistentidiosyncraticelements
3Dotseyetal.(1999)usetheframeworktoanalyzepriceadjustment;Thomas(2002)usesitto
examineinvestment.
4Generalized(S,s)modelswerefirststudiedbyCaballeroandEngel(1999)toexplaintheobserved
lumpinessofplant-levelinvestmentdemand.
PARTIALADJUSTMENTWITHOUTAPOLOGY 781
makes our framework amenable to a broader set of applications beyond those
consideredthusfar.
Thearticleisorganizedasfollows.Section2brieflyreviewstheessentialproper-
tiesofthestandardpartialadjustmentmodel,andSection3describestheevidence
onmicroeconomicadjustmentpatternsthatthestandardmodelfailstoexplain.
Next,Section4developsamodelthatisconsistentwiththeobservationthatin-
dividualestablishmentshirevaryingamountsoflaborindiscreteandoccasional
episodes,anditillustratesaresultinghedgingeffectonthedemandforlabor.Our
modelmakesthetimingofdiscreteindividualemploymentchangesendogenous
by assuming that plants face fixed costs of adjustment that are random across
both time and plants. Establishments respond to this by adopting generalized
(S,s)decisionruleswithrespecttolabor.Atthesametime,theframeworkisread-
ilyembeddedwithinafullyspecifiedgeneralequilibriummacroeconomicmodel,
whichallowsustoexaminetheinfluenceofdeepparametersontheadjustment
process.Moreover,withalargenumberofplants,themodelissimilartothetra-
ditionalpartialadjustmentmodelinthatityieldsasmoothmarketlabordemand.
WeillustrateseveralpropertiesofourgeneralizedpartialadjustmentmodelinSec-
tion 5 through a series of numerical examples.5 Beyond its consistency with the
evidenceonemploymentadjustmentinSection3,themodelisalsodistinguished
by its potential to reproduce the sharp changes in market employment demand
foundinthedataduringepisodesinvolvinglargechangesinproductivity.6Moving
fromourmarketdemandexamples,weprovidecounterpartresultsthatillustrate
theroleofequilibriuminshapingtheaggregateresponsetoshocks.Finally,Sec-
tion6demonstrateshowourframeworkisextendedtoaccommodatepersistent
differencesinproductivityacrossestablishments,andSection7concludes.
2. STANDARDPARTIALADJUSTMENT
Thestandardpartialadjustmentmodelrelatescurrentemployment,N ,totarget
t
ordesiredemployment,N∗t,throughNt−Nt−1=κ [N∗t −Nt−1],whereκ ∈(0,1)
isthefractionofthegapclosedintheperiod.Thisspecificationimpliestheinflu-
enceofpastactualordesiredemploymentoncurrentemployment.
(cid:2)∞
(1) Nt =κN∗t +(1−κ)Nt−1 =κ (1−κ)jNt∗−j
j=0
5Ourmodelisdistinguishedfromearliergeneralizedcostofadjustmentmodels,assummarized,
extended,andcritiquedinMortensen(1973),inthatitsuggestsverydifferentdynamicsatthees-
tablishmentlevel.Nonetheless,becauseourmodelisessentiallyonewithmanydynamicallyrelated
factordemands,itiscapableofgeneratingsomeoftheaggregatedynamicsthatmotivatedresearchers
inthisearlierarea.Forexample,underunrestrictedparameters,interrelatedfactordemandmodels
werefoundtobeconsistentwithoscillatoryapproachestothelong-runposition.Ourmodelcanalso
generatesuchrichdynamics,althoughitdoesnotdosoundertheparametersselectedhere.
6Thisisbecausetheeconomy-widerateofadjustmentimpliedbyourmodelvarieswithaggregate
conditions.Thetraditionalmodelunderpredictsemploymentchangesduringsuchepisodesprecisely
becausetheadjustmentratethereisconstant.
782 KINGANDTHOMAS
As shown by Sargent (1978), this empirical partial adjustment model may be
derived as the solution to a firm’s dynamic profit maximization problem under
the assumption that there are quadratic costs of adjusting the workforce. In the
absence of costly adjustment, assume that the firm’s workforce declines at the
rated∈[0,1)duetoquitsormismatches.Ife employeesarehiredattimet,then
t
Nt = (1 − d)Nt−1 + et, and the cost of the workforce adjustment is (cid:3)(et)=
Be2, where B > 0.7 Let z reflect current productivity, and w be the real wage,
2 t t t
(bothseriallycorrelatedrandomvariablesknownatdatet),andletproductionbe
mf(aNxitm,zitz)e.Ditissceoxupnetcitnegdfuptruerseenetardniisncgosubnyteβd∈va(0lu,e1,),Eth[(cid:3)efi∞rmβcth(ofo(Nses,z{N)−t,e(cid:3)t}(∞te=0)t−o
t=0 t t t
wtNt)|z0,w0], subject to Nt = (1 − d)Nt−1 + et and given initial employment
N−1.Iftheproductionfunctionisquadraticinemployment,itisstraightforward
toshowthat
(cid:2)∞
(2) Nt∗ = Et (β/κ)j(χaat+j −χwwt+j)
j=0
demonstratingthatthepresenceoflagsinemploymentimpliesleadsunderratio-
nalexpectations,asstressedbySargent(1978).8
Thekeyimplicationsofthemodelare:(i)currentemployment,N ,isdirectly
t
relatedtolaggedemployment,Nt−1,becauseadjustmentsarecostly,and(ii)expec-
tationsoffuturewagesandproductivityinfluencecurrentemploymentthroughthe
∗
target,N ,because,givenadjustmentcosts,itschoicewillinpartdeterminefuture
t
employment.Takentogether,thesefeaturesimplythatadjustmentcostsdampen
theresponsetochangesincurrentwageandproductivityandyieldsmooth,grad-
ualchangesinemploymentovertime.
3. DISCONCERTINGEVIDENCE
Althoughthetraditionalpartialadjustmentmodeloffersatractableframework
with which to study gradual aggregate labor adjustment, there is considerable
empiricalevidencetosuggestthatthemodelisnot consistentwiththebehavior
ofindividualproductionunits.Thisevidencesuggestsanumberofstylizedfacts
aboutindividualandaggregateadjustmentthatwesummarizehere.
Stylized fact 1: Adjustment at the plant level is discrete, occasional, and asyn-
chronous.Hamermesh(1989)examinesmonthlydataonoutputandemployment
between 1983 and 1987 across seven manufacturing plants. For each plant, out-
putfluctuatessubstantiallyoverthesample.Employmentexhibitslongperiodsof
constancybrokenbyinfrequentandlargejumpsattimesroughlycoincidingwith
thelargestoutputfluctuations.Hence,theplantdataarenotconsistentwiththe
smoothemploymentadjustmentthatwouldarisefromconvexadjustmentcosts.
7Thiscapturestheideathatthefirm’smarginaladjustmentcostisrisingintheextentofemployment
adjustment;thesameideaisincorporatedinalternativeadjustmentcostfunctionsusedinappliedwork.
8In(1)and(2),theparametersκ,χa,andχw dependontheadjustmentcostparameterB,the
discountfactorβ,andtheparametersoftheproductionfunction.
PARTIALADJUSTMENTWITHOUTAPOLOGY 783
Stylizedfact2:Aggregatesexhibitsmoothandpartialadjustment.Hamermesh
(1989) also examines the behavior of the aggregate of his seven manufacturing
plants.Hefindsthatfluctuationsinaggregateemploymentresemblethedynamics
of aggregate output and appear consistent with smooth adjustment behavior of
aggregates.Morespecifically,hearguesthatthestandardpartialadjustmentmodel
works quite well at the aggregate level, even though it does not describe the
behaviorofindividualproductionunits.9
Stylized fact 3: Adjustment hazards depend on aggregate conditions. Follow-
ingtheeconometricliteratureondiscretechoices,anadjustmenthazardtypically
referstotheprobabilitythataproductionunitwillundertakeadiscretechange,
withthisprobabilitydependingonthepositionoftheunit’sstatevariablerelative
tosometargetvalue.CaballeroandEngel(1993)constructageneralframework
for studying aggregate employment changes that can incorporate a variety of
assumptionsabouthowadjustmenthazardsarerelatedtoaggregateconditions.
UsingU.S.manufacturingdatafrom1961through1983,theyexaminethedynam-
icsofaggregateemploymentchangesundertwoalternativespecificationsforthe
hazard function: (1) a benchmark case with a time-invariant, flat hazard, which
correspondstothetraditionalpartialadjustmentmodel(asshownbyRotemberg,
1987)and(2)analternativehazardmodelinvolvinghighermomentsofthecross-
sectionaldistributionoffirms’“disequilibrium”levels,reflectingstate-dependent
adjustmentbehavior.Theyfindlargeincreasesinexplanatorypowerforaggregate
employmentchangesinmovingfromtheconstanthazardmodeltoageneralized
hazardstructureandattributethistotheeffectsoflargeaggregateshocksupon
theemploymenthazard.
Stylizedfact4:Adjustmenthazardsdependonmeasuresof“microgaps.”More
directevidenceontheimportanceofstate-dependentadjustmenthazardsispro-
vided by Caballero et al. (1997). Studying the direct relationship between the
adjustmenthazardattheleveloftheindividualproductionunitandtheextentof
that unit’s gap between current employment and a measure of desired employ-
ment,theseauthorsshowthattheadjustmenthazarddependsonthesizeofthis
discrepancy. They suggest that individual units may face differential adjustment
costs,sothatthedistributionofadjustmentcostsgovernstheadjustmenthazard.
Stylized fact 5: Aggregate shocks are much more important in accounting for
aggregateresponsesthanareshiftsincostdistributions.Theempiricalanalysisof
9Hamermeshcompareslog-likelihoodvaluesfromtheestimationofapartialadjustmentmodel
based on quadratic costs to those from a lumpy adjustment fixed-cost alternative. For plant-level
data,thelatterachievesmuchlargerlikelihoodvalues.Furthermore,hisswitchingmodelestimatesof
thepercentage“disequilibrium”requiredtoinduceadjustmentarelarge,suggestingthatplantsvary
employmentwithanonmarginaladjustmentonlyinthepresenceofsubstantialshockstoexpected
output.However,differencesattheaggregatelevelaretoosmalltodiscriminatebetweenmodels,
asisthecasewhentheyarecomparedusing4-digitSICdata.Thus,lumpyadjustmentbehaviorat
themicroeconomiclevelisobscuredbyaggregation.Fromthisandsimilarevidence,Hamermeshand
Pfann(1996,p.1274)concludethat“observingsmoothadjustmentbasedondatadescribingindustries
orhigheraggregatesovertimeisuninformativeaboutfirms’structuresofadjustmentcostsandinno
waydisprovestheexistenceoflumpycosts.”
784 KINGANDTHOMAS
Caballeroetal.(1997)alsosuggeststhatchangesinthedistributionofadjustment
costs are not central in explaining stylized fact 3. Instead, aggregate shocks in-
ducechangesinhazardsthatareimportantforaggregatesbecausetheyproduce
movementsalongthemicro-distributionofemploymentimbalances.
4. GENERALIZEDPARTIALADJUSTMENT
A number of recent theoretical and empirical studies—notably those of
CaballeroandEngel(1993)andCaballeroetal.(1997)—havearguedforaricher
vision of the adjustment process that can generate the stylized facts discussed
above.Theframeworkwedevelopexemplifiessuchamodel.Inparticular,itdeliv-
erstheimplicationthat,althoughindividualestablishments’employmentadjust-
mentsarediscrete(fact1),theirasynchronoustimingimpliesasmoothaggregate
employment series similar to that implied by the traditional partial adjustment
model (fact 2). Moreover, an individual production unit’s probability of adjust-
ment depends on a measure of the “gap” between its current employment and
anotionofdesiredemployment(fact4),andthemodelcanproducesubstantial
responsesofemploymenttoaggregateshockswithoutrelyingonanyshiftsinthe
distributionofadjustmentcosts(fact5).Atthesametime,ourapproachisreadily
incorporatedintoageneralequilibriummodel,sothattherelationshipbetween
adjustment hazards and macroeconomic conditions can be formally established
(fact3).
Weassumealargeandfixednumberofproductionunits,eachmakingdiscrete
choicesabouttheiremploymentadjustmentovertime.Productionattheplantis
constantreturnsinlaborandafixedinput,whichwenormalizeto1, f(n,1,z).10
t t
Anyunitthatdoesnotadjustitsworkforceseesitdecayatrated:
(3) nt =(1−d)nt−1+et
wheree representsanactiveadjustment.Weendogenizethetimingofsuchad-
t
justmentsbyintroducingfixedcoststhatarestochasticacrossunits,anapproach
adopted by Caballero and Engel (1999) in their study of manufacturing invest-
ment.11 Withineachdate,anyindividualplantfacesarandomcostξ thatitmust
pay in order to adjust its employment prior to current production. This cost is
drawn from a time-invariant distribution over [0, B] that is summarized by the
CDFG(ξ)andassociateddensityg(ξ).
Inthediscussionthatfollows,weintegrateourgeneralizedpartialadjustment
approach into a general equilibrium setting by imposing two restrictions on the
pricesfacedbyestablishmentswithineachdate.First,assetmarketclearingwill
10Thepresenceofthefixedinputallowsdeterminationoftheemploymentchoiceattheproduction
unit.
11Thegeneralizedadjustmentmodeldevelopedherehasbeenusedinseveralgeneralequilibrium
applications.Dotseyetal.(1999)studythedynamicsofpriceadjustment,whereasThomas(2002)and
KhanandThomas(2003)investigateinvestmentdynamicsandKhanandThomas(2004)useasimilar
approachtostudy(S,s)inventoryaccumulation.Inthisstudy,weuselinearapproximationmethodsin
thetraditionofSargent(1978)toexplorethegeneralequilibriumdynamics,asdoDotseyetal.(1999)
andThomas(2002).
PARTIALADJUSTMENTWITHOUTAPOLOGY 785
require that all establishments discount their future profit flows by households’
marginalrateofsubstitutionbetweencurrentandfutureconsumption,denoted
herebyβpt+1.Equivalently,establishmentsvaluetheircurrentoutputbyp,the
pt t
current marginal utility of consumption, and discount their future values by the
householdsubjectivediscountfactorβ.Next,theequilibriumwage,w,willequal
t
households’marginalrateofsubstitutionbetweencurrentleisureandconsump-
tion, D2u(c,1−N). Provided that these restrictions are satisfied, the role of house-
D1u(c,1−N)
holds in the economy is effectively subsumed, and equilibrium allocations are
retrievedastheaggregateofestablishments’decisions.12
4.1. AdjustmentProbabilities. Atthestartofanydatet,eachproductionunit
may be identified as a member of a particular time-since-adjustment group, j,
where j is the number of periods that have elapsed since the unit’s last active
employmentadjustment.Letn representthestart-of-periodlaborstockassoci-
jt
atedwithamemberoftime-since-adjustmentgroupj,andletα beashorthand
jt
representinganysuchestablishment’sprobabilityofacurrentadjustment,asper-
ceived after the observation of the aggregate state but prior to the realization
of adjustment costs.13 We limit the number of time-since-adjustment groups by
restricting the parameters of our model to ensure that there is some maximum
nonadjustmenthorizon,J,thenumberofperiodswithinwhichallproductionunits
willadjusttheiremploymentwithprobability1:α =1.Thisfinitememoryfeature
J
is useful in limiting the size of the aggregate state vector, and will be discussed
furtherbelow.
Let S denote the aggregate state of the economy determining prices and ex-
t
pectations.WeusethenotationV(n ,S)torepresenttheproduction-timevalue
j jt t
ofaplantthatlastadjustedemploymentjperiodsagoandisenteringcurrentpro-
duction with no change to its start-of-period workforce n . Next, we use V (S)
jt 0 t
todenotetheproduction-timevalueofaplantthathaspaiditscurrentfixedcost
inordertoadjustitsemploymentpriortoproduction.Basedonacomparisonof
thesevalues,itisstraightforwardtocharacterizethediscreteadjustmentdecisions
made by individual production units. Given its start of period employment, n ,
jt
and the aggregate state, S, an establishment will actively adjust its workforce if
t
itscurrentfixedcost,ξ,doesnotexceedthevalueofundertakingtheadjustment,
thatis,ifV (S)−V(n ,S)≥ξ.14
0 t j jt t
Becausethereisalargenumberofproductionunitswithineachdifferenttime-
since-adjustmentgroup,eachgroupischaracterizedbyamarginalplantthatfinds
itjustworthwhiletoadjust.Thismarginalplantisassociatedwithathresholdcost
ξ¯ satisfyingthefollowingcondition.
jt
12SeeKhanandThomas(2003)forfurtherexplanation.
13Exceptwherenecessaryforclarity,wesuppresscommasinsubscriptsthroughoutthistext.
14Aswillbemadeexplicitbelow,St includestwoendogenousvectorsthattogetheridentifythe
start-of-perioddistributionofplantsoveremployments,alongsideexogenousaggregateproductivity,
zt.WeassumeztfollowsaMarkovprocessthatistakenasgivenbyallagents,asistheevolutionofthe
endogenousaggregatestate,At,accordingtoamappingAt+1=(cid:8)(At,zt)that,inequilibrium,results
fromtheaggregationofindividualactions.
786 KINGANDTHOMAS
(4) ξ¯ = V(S)−V(n ,S)
jt 0 t j jt t
Allproductionunitsinthejthtime-since-adjustmentgroupwithadjustmentcosts
atorbelowthethresholdin(4)willchoosetoadjust.Asaresult,thefractionof
plantsadjustingoutofanyparticulargroupj,j=1,...,J−1,isgivenby(5).
(5) α =G(ξ¯ )
jt jt
From(4),notethattheseadjustmentfractionsarefunctionsoftheplant-levelstate
vector,(n ,S).Weassumethatthestochasticprocessforproductivity,alongside
jt t
theparametersassociatedwithproduction,fixedcostdraws,andhouseholdutility,
aresuchthatB<V (S)−V (n ,S)forallrelevantvaluesofthevector(n ,S).
0 t J Jt t Jt t
ThisassumptionfollowsnaturallyfromB<∞andensuresthatα =1.
J
4.2. Production-TimeValues. Havingdescribedthedeterminationofendoge-
nousadjustmentprobabilitiesasfunctionsoftheproduction-timevaluesassoci-
atedwithadjustingandnonadjustingplants,wenowstatethefunctionalequations
that determine these values. We have been explicit above in noting the depen-
dence of prices and adjustment probabilities on the economy’s aggregate state.
Here,however,wesuppressthesedependenciestoreduceequationlength.
Thevalueofaplantthatiscurrentlyadjustingitslaboris
(cid:4) (cid:5)
(6) V0(St)=max f(n0t,zt)−wtn0t +βE pt+1[α1,t+1V0(St+1)−ξ1,t+1
p
t (cid:6)(cid:7)
+(1−α1,t+1)V1((1−d)n0t,St+1)]|St
wheren0tisfreelychosen,andα1,t+1isgivenbyEquations(4)and(5)above.15The
right-handsideofthisBellmanequationinvolvesthreeexpressions.First,thereis
theflowofcurrentprofit.Second,thereistheexpecteddiscountedvalueofbeinga
unitthatadjustsnextperiod,whichoccurswithstate-dependentprobabilityα1,t+1
and imp(cid:8)lies an expected fixed cost payment, ξ1,t+1, conditional on adjustment:
ξ1,t+1 = 0G−1(α1,t+1)xg(dx).Finally,thereisthevalueofbeingaunitthatdoesnot
adjustnextperiod,anoutcomethatoccurswithprobability(1−α1,t+1).
For units choosing not to adjust their workforce, there are no further current
decisions in this simple model, although there would be in more elaborate set-
tings allowing adjustments on other margins, such as in hours-per-worker. The
production-time value of a plant that last adjusted j = 1,...,J − 2 periods ago,
andisnotcurrentlyadjusting,is
15Asn0tisourmodel’scounterparttotargetemploymentinthetraditionalmodel,weoccasionally
refertoitasn∗whenmakingcomparisonsbelow.
t
PARTIALADJUSTMENTWITHOUTAPOLOGY 787
(7) V(n ,S)= f(n ,z)−w n
j jt t jt t t jt
(cid:5)
+βE pt+1[αj+1,t+1V0(St+1)−ξj+1,t+1
p
t (cid:6)
+(1−αj+1,t+1)Vj+1((1−d)njt,St+1)]|St
(cid:8)
whereαj+1,t+1isgivenbyEquations(4)through(5),andξj+1,t+1 = 0G−1(αj+1,t+1)×
xg(dx).Anysuchplantproduceswithitsstart-of-periodlaborn ,thenmovesto
jt
startnextperiodasamemberoftime-since-adjustmentgroupj+1withworkforce
(1−d)n .Inotherrespects,itsBellmanequationisanalogoustothatofcurrent
jt
adjustors described above. Finally, the production-time value of a nonadjusting
plantthatlastadjustedJ−1periodsagoisjustasin(7),butreflectsthecertainty
ofemploymentadjustmentinthenextperiod.
(cid:5) (cid:6)
(8) VJ−1(nJ−1,t,St)= f(nJ−1,t,zt)−wtnJ−1,t +βE pt+1[V0(St+1)−ξJ,t+1]|St
p
t
Beforeproceedingfurther,weshouldoffersomecommentonthetime-since-
adjustmentsubscriptsattachedtotheplantvaluefunctionsabove.Solongasad-
justmentprobabilitiesareoptimallychosen,itisclearthattheplant-levelstateis
fullycapturedby(n ,S),andthesesubscriptsareunnecessary.Wehavechosento
jt t
includethemheretoallowouranalysistoaccommodateaspecialcasethatliesbe-
tweenourgeneralizedmodelwithstate-dependentadjustmentprobabilitiesand
thetraditionalpartialadjustmentmodelcharacterizedbyatime-invariant,flatad-
justmenthazard.Thisintermediatetime-dependentadjustmentmodelreplacesthe
endogenousdeterminationofadjustmentprobabilitiesaccordingto(4)–(5)with
afixedvectorofadjustmentprobabilitiesobtainedfromthedeterministicsteady
stateofourstate-dependentadjustmentmodel.Thus,itresemblesthetraditional
modelinthatitsadjustmenthazardremainsfixedovertime,butdiffersinthatthe
hazardisnotflat,butinsteaddependsuponaunit’stime-since-last-adjustment.16
When our model is specialized to this case of time-dependent adjustment, j be-
comesaseparateindividualstatevariabledeterminingtheprobabilitythataplant
willbeallowedtoadjustitsemployment,andthesubscriptsonourvaluefunctions
areusefulinaccountingforthis.
4.3. Target Employment. Adjusting production units exit the jth group for
theadjustmentgroupandchooseemploymentsoastomaximizetheright-hand
sideof(6),whichresultsinanefficiencyconditionofthefollowingform.
(cid:5) (cid:6)
D1f(n0t,zt)−wt +βE pt+1(1−α1,t+1)(1−d)D1V1((1−d)n0t,St+1)|St =0
p
t
16Onemayinterpreteachtime-dependentadjustmentprobabilityαjasreflectingtheprobability
thataplantwillbeabletochangeitsemploymentatzerofixedcostversustheprobabilitythatitwill
faceaninfiniteadjustmentcost.
788 KINGANDTHOMAS
A notable feature of this condition is that the optimal employment decision on
thepartoftheadjustingproductionunitisindependentofthelengthoftimesince
itlastadjustedandthesizeofitsworkforceatthestartoftheperiod,sinceneither
j nor n enters into the efficiency condition. This justifies our writing V above
jt 0
in the restricted form that omits these factors. In addition, it implies a common
employment adopted by all units adjusting within the same date, and hence the
common start-of-period workforce across all members of any particular time-
since-adjustmentgroup.
Workingwiththevaluefunctionin(7),wecandeterminethemarginalvalueof
additionalworkersasfollows.
D V(n ,S)= D f(n ,z)−w
1 j jt t 1 jt t t
(cid:5) (cid:6)
+βE pt+1(1−αj+1,t+1)(1−d)D1Vj+1((1−d)njt,St+1)|St
p
t
Thesederivativesmaybeusediterativelytosimplifytheefficiencyconditionand
deriveanalternativeimplicitexpressionfortheoptimalworkforcechosenbyan
adjustingproductionunit.Inparticular,n solves
0t
(9) D f(n ,z)−w
1 0t t t
(cid:5) (cid:6)
+E(cid:2)J−1 pt+j(cid:9)β(1−d)]jϕj,t+j[D1f(cid:10)(1−d)jn0t,zt+j(cid:11)−wt+j(cid:12)(cid:13)(cid:13)St =0
p
j=1 t
whereeachϕj,t+j representstheprobabilitythattheadjustingunitwillmakeno
furtheradjustmentinthenextjperiods.
(cid:14)j (cid:14)j
(10) ϕj,t+j ≡ (1−αk,t+k)= (1−G(ξ¯k,t+k)), for j =1,...,J −1
k=1 k=1
Inpractice,itisconvenienttobreakthelargeforward-lookingconditiondetermin-
ingtargetemploymentintoJfirst-orderstochasticdifferenceequations.Defining
(cid:10)Jt ≡0,anddefining(cid:10)1t,...,(cid:10)J−1,t recursivelyin(12),theconditionin(9)may
bealternativelywrittenasthatinEquation(11).
(cid:5) (cid:6)
(11) D1f(n0t,zt)−wt +β(1−d)E pt+1(cid:10)1,t+1 |St =0
p
t
(cid:4) (cid:5) (cid:6)(cid:7)
(12) (cid:10)jt ≡(1−αjt) D1f(njt,zt)−wt +β(1−d)E pt+1(cid:10)j+1,t+1|St
p
t
Note that the condition above may be explicitly solved for adjusting units’
optimallabordemandinthespecialcaseofaCobb–Douglasproductionfunction,
y=znγ.Inthatcase,