Matching, Vacancies, and Mobility Author(s): Harrison C. White Source: Journal of Political Economy, Vol. 78, No. 1 (Jan. - Feb., 1970), pp. 97-105 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1829623 . Accessed: 29/08/2014 18:00 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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Matching, Vacancies, and Mobility

HarrisonC. White Harvard University

Keeping men paired to jobs over time is a basic economic process. It is notexplainedadequatelyby orthodoxtheoriesof supplyand demandin whichwagestructures evolveso as to keepmarketsforlaborcleared.Classical economicsexcludesindivisibility and thus individualityand fixed physicaltermsof trade.1These aspectsof the pairingprocessare emphasized herein a newtypeof mobilitymodel similarin partto input-output modelsofproduction. Considera systemof menand jobs whereineach job is to be filledby a fixednumberof menand each manholdsbutone job. A largebureaucracy witha well-defined table of organizationwould be sucha system,another could be higherstrataof skilledjobs and managerialpositionsin thecompaniesin someindustry, and a thirdcould be all jobs in a givenspecialtysay computerprogrammersacross a whole economy.The departureof menfromthesystemis treatedas an exogenousprocess,partlydependenton theage structure throughdeathand retirement, and partlyon the attractionsofcareersin othersystems.Jobs,likemen,are identifiable individuals whichare constantly entering and leavingthesystem.The entryofjobs is a second exogenousprocess which may reflectthe state of the economy, technological change,plansforgrowth,and so on. Let us call each pairingof man to job a tenure.To simplify description let each job hold but one incumbent, eventhoughthemodelis applicable to the generalcase. The basic dynamicassumptionis asymmetric;an This articlederivesfroma largerprogramof researchreportedin the author's Opportunity Chains,to be publishedby the HarvardUniversity Press in 1970. Financial supportfromthe National Science Foundationunder grantGS-448 is gratefully acknowledged. 1 Koopmans(1957,p. 154) concludes"In regardto theallocationproblems raised by indivisiblecommodities,with or withoutlocational distinctions, theoretical analysishas not yet absorbedand digestedthe simplestfactsestablishedby the mostcasual observation."For one attemptto deal withindivisibilities in thelabor marketsee Holtand David (1966.p. 79).

97

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unoccupiedjob can draw a man fromhis tenurein anotherjob, but an unoccupiedman cannot draw a job fromits tenurewith anotherman. can pull men out of theirjobs but theycannotbe Specificopportunities pushedout by newcomers.Whereincumbents havetherightof indefinite tenuretheassumptionis patentlyvalid, but it also seemsto be a realistic approximationto the situationin many large organizations.Even in centralizedbureaucracieslike armies with policies of reassigningmen afterlimitedtours of duty the assumptionis cogent, because almost always a man is moved only throughassignmentto a definitevacant job. Moves occurin chains,each man fillingthevacantjob leftby themove of his predecessorto stillanothervacantjob. A chain can end eitherbecause a newcomeris called fromoutsidethesystemto fillits last vacancy or becausethisjob is abandonedand leaves thesystem.Joblessmenhave no activerole in such a system,whichcomprisesat any giventimea set oftenuresplussomejobs temporarily vacant. Discretemen are being matchedto discretejobs at a fixedratio. Because each manandjob is unique,identifiable byindividualidiosyncrasies, one asks morethanwhichmenand whichjobs are occupied.Mobilityand turnoverrates,frequencieswithwhich men changejobs, are important measuresof systembehavior.Men may tryto develop themselvesand buildcareersbymovingon to different jobs, and different jobs maytendto call formen of different levelsand kindsof priorjob experience.In any case, thedesireforand effects of anychangesofjobs by menare reflected in moraleand productivity inthesystem.2 Each chainof movescan be seenas thecareerofan identifiable vacancy in thesystem.A vacancynormallymovesfromone job to thenextwithin a matterof days or months.If a vacancyremainslongerthejob may be consideredas abandoned,sincesubsequentreactivation requiresthesame kind of decisionsand searchfora new man as in the case of a newjob. It is reasonableto assume one can classifyjobs in broad strataso that, whilethe chancesof terminating a vancancyby abandonmentof thejob and thechancesofterminating it byrecruitment of an outsiderdiffer from one stratumto another,theyare nearlythesame withina givenstratum. The cognate chances of the vacancy moving instead to anotherjob surelythendependon boththestratumof theoriginjob and thestratum ofthedestination job. Moves bymenbetweenjobs mustinteractwhenthenumberof menin a job is fixed,but movesby different vacanciesmay be independentof one 2 An unusuallyextensive and thoroughsurveyof mobilityof men amongbroad stratain a wholeeconomywillbe foundin Blau and Duncan (1967). Froma studyof somelocal labormarkets, one economisthas cometo conclusionsabouttheroleand importanceof vacanciesconsistentwiththe modelsproposedhere.(See Reynolds

1951, pp. 212, 227, 239, 240, 242, 244-46.)

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anotherto a firstapproximation.It also is reasonable to regardeach vacancyas of shortmemory,its nextmove influenced onlyby its current location.The stratumof a vacancy'scurrent job maywell be an adequate descriptionof its currentstate,withoutany specification of the personal attributes ofa prioror subsequentincumbent in thatjob. The monographfromwhich this paper is derived develops in full detail a stochasticmodel for this mobilityprocess.Vacancies followa Markov renewalprocess(Pyke 1961), but since vacancies move quickly and exact timingis not important,it provesto be sufficient to treatthe imbeddedMarkov chain,one withan absorbingstateand an exogenous input. Parametersare estimatedfrom personnelassignmentdata for two largesystems.Large samplesof vacancychainsare tracedin thedata. The model is then validatedby comparingpredictedand observeddistributions forindividualchainsofmovesbyvacancies. A simpledeterministic versionof the model is developedhere forcohortsofvacancies,whichis closelyanalogousto theinput-output modelsof Leontief.An exampleis givenin termsofthismodel. AnInput-Output ModelforVacancies Let the numberof vacancieswhicharrivein year t, stratumi of jobs be designatedFi(t). There are s strata. Total arrivalsare specifiedby a vectorF(t), withone part contributedby arrivalsof new jobs, FjOb(t), and theotherpart,Fman(t),resultingfromdeathsand otherdeparturesof menfromthesystem F(t)

= Fjob(t)

+

Fman(t)*

(1)

Let the fractionof vacanciesin jobs in stratumi whichmove nextto jobs in stratum j be qij, and arraythemin a square matrixQ in the usual way.The fractionwhichleave thesystemon theirnextmoveis designated in a columnvectorp. Bydefinition qij; arraythesefractions Q1 + p = 1,

(2)

where1 is a columnvector,each ofwhoses componentsis unity. Assume the cohort of vacancies which arrivesin year t have all left the systembeforeyear t + 1, so that each cohort may be treatedas a separatepopulation.The cohortdisappearswhentherehave been enough exposuresof vacancies to moves so that all F(t) vacancies have leftthe system.Let M(t) countthetotalcumulativenumberofarrivalsofvacancies to jobs by stratum,includingthe F(t) creationsof vacancies.Also, M(t) countsthetotalnumberof departures;thusM(t)Q countsthe numberof arrivals,by stratum,of vacancieswhichmovedfromotherstrata.Hence, M(t) = M(t)Q + F(t).

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(3)

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The fractionsof movesp whichare terminalwill each be positive;so the determinant of Q is lessthanunity.Therefore,

M(t) = F(t)(I- Q)1.

(4)

The inversematrix(I - Q) -1 is a multiplieranalogous to that in the Leontiefmodel,exceptthatit specifiestotalsin termsofinputsratherthan exogenousoutputs,or as a scheduleof finishedgoods to be produced.3 The analogue to the amount of labor in Leontief'smodel is the total numberof vacancymovesto the outside,Mp, whichcan be construedas the total "supplies" demanded fromthe environmentto "produce" the vacancycreations.Justas prices are fixedin the Leontiefmodel in termsof the price of a numerairegood, say labor, so one can pointout that exactlyone vacancy terminationis requiredfor the "production" of one vacancyarrival.The fixedpricesin the Leontiefmodel would be numericallyunityalso if the technologicalcoefficients qij were fractions summingto unity (Dorfman, Samuelson, and Solow 1958, p. 239). The analogyis to theunreducedformof theLeontiefmodelin which,for example,the use of iron in producingiron is representedby nonzero in diagonalcoefficients qii(Dorfmanet al., p. 205, n. 2). These coefficients thevacancymodelcorrespondto movesfromone job to anotherwithina stratum,and theymustbe includedto obtainvalid countsof totalmoves. to The termination fractionsp are the sum of a portioncorresponding arrivalsof men abolition of vacant jobs, Pjob, and a portionreflecting Total moves by men intojobs duringthe fromoutsidethe system,Pman. year t, Min(t),are thus smaller than total moves of vacancies in that cohortout ofjobs: in stratumi Min(t)

= Mi(t)(I

-

[qiO]30b)

(5)

Let N#(t)be the total numberof men in jobs in stratumi at the end of year t - 1 and thus the beginningof year t; and let [i be the mobility rateintostratumi: = Min(t)/N#(t+ 1). (6) By definition the systemdoes not includemen not in jobs, so thecomponentsNi(t) arrangedin a vectorN(t) count all men in the systemby stratumat thebeginningof yeart. At thebeginningof each yearand at its end afterthecohortof vacancieshas departedNi(t) also countsthe numberofjobs. The changein thevectorN(t) overa yearcan be computedby countingchangesinjobs: N(t + 1) = N(t) +

Fjob(t)

-

M(t)PJob,

(7)

wherePjob is simplya diagonal matrixwith(qio)jobin the ith row and ith 3References to theLeontiefmodelare based on theaccountin chaps.9 and 10 of matrixsee p. 254 there. Dorfman,Samuelson,and Solow (1958). For themultiplier

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column.Aftersubstitution fromequation(4) a simpledifference equation results: AN(t) = Fjob(t)F(t)(IQ) - 1PJob. (8) Changein thesizesofstratadependson therelativesizesoffourquantities foreach stratum:rate of creationof newjobs, rateof departureof men, fractionof vacancy moves which representabolition of jobs, and the fraction whichcorrespondto recruitment ofmen. Considerthe simplestcase, whereit provessufficient to treatall jobs as equivalent,lumpedin one stratum.Then Q is a scalar q and p is a scalar,p, such thatq + p = 1. Furthermore p = Piob + Pman- Equation (8) becomes AN(t) = Fjob(t)-

Or in moresymmetric form, AN(t) = Fjob(tPan

p

F(t)Piob p

(9)

_ Fman(t)PJob*

p

(10)

An Example Crucialto thevalue of themodel is stabilityof theparameters, thetransition fractionqij. By hypothesis,theymust be constantwithina given year,buttheymustalso remainthesameacrosslongperiodsifthemodelis to be usefulfor predictions.The exogenousvariables,the total arrivals of vacanciesF(t), mayfluctuate widelyfromyearto year,and thisfluctuation should completelyaccount for changesin mobilitypatternsand in growthof the system.Then, if the transitionfractionsforvacancies are constant,the turnoverfractionsfor men fromone stratumto another, theusual focusin modelsofmobility,4 willvaryovertime. The clergiesof severalnationalchurcheshave been used as testcases, because long time-seriesof detailed assignmentsof clergyto individual jobs areavailablein publishedform.Table 1 reportsestimatesof transition fractionsin widelyspaced decades in the Episcopal church,using three stratadefinedby thesize of theparish.Also givenis thefractionof total vacancy creationswhich occur in each stratum.The parametersin Q and p are remarkably whilethe stable,evenignoringsamplingvariability, distribution of vacancycreationsamong stratachangesdrastically.Table 2 partitionsthetermination fractionsin each decade intothe partdue to ofnewmenand thepartforabolitionofjobs.5 recruitment 4 An excellentrecentsurveyof thesemodelscan be foundin D. Bartholomew (1967, especiallychaps. 2 and 3). In chap. 7, Bartholomewintroducesthe idea of vacanciesin a new model, whichstill bases its dynamicson the probabilitiesof movements bymen.(See sec.9.6 ofWhite1970.) 5 Tables 1 and 2 are adaptedfromWhite(1970,tables4.15 and 5.10 respectively). The principaldata sourcewas theofficial LivingChurchAnnual.Codingand sampling inchap.4. rulesarespecified inchap.3 and thechoiceofstratajustified

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1

TABLE

FRACTIONS FOR VACANCY MOVES CLERGY, VARIOUS DECADES*

TRANSITION EPISCOPAL

ESTIMATED

DESTINATION

Big (1)

ORIGIN STRATUM

IN THE

FOR VACANCIES

STRATUM

Outside (0)

Small (3)

Medium (2) 1912-23

fl/F .

....

. 0

.0880

.2503

.6616

...

1 2 3

.3751 .0456 .0243

.3873 .3229 .1139

.0741 .2098 .1826

.1634 .4217 .6792

1954-55 Fl/F .

....

. 0 .

1 2 3

.

.

.

.3231

.5157

.1612

.

.

.

.

.

.

.

.

.

.

.3383 .2918 .1789

.3897 .0540 .0665

.

.

.

.

.

... .

.

.

.0947 .1985 .1484

.1774 .4556 .6063

* Above the dotted line in each panel is the distribution of vacancy creations among strata. In the "Outside" column are the termination probabilities, qO ; the remaining 3 x 3 matrix is the Q of equation 2. Parishes with 300 or more communications are big jobs, those with less than 100 are small jobs.

TABLE 2 FRACTIONS OF VACANCY TO NEW RECRUITS

Stratum

MOVES CORRESPONDING AND TO JOB ENDS*

(qiO)job

(qiO)man

1912-23 1 . . . . . . . 2 . . . . . . . 3 . . . . . . .

.0421 .0967 .2632

.1213 .3250 .4160 1954-55

1 . . . . . . . 2 . . . . . . . 3 . . . . . . .

.1683 .4384 .3765

.0091 .0172 .2298

* The sum of the two entries in a row is entered in that row in the last column of table 1.

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TABLE THE

Origin Stratum

1 .... 2 .... 3 ....

. . . . . .

3

MATRIX (I - Q)1 PANEL IN TABLE 1

MULTIPLIER FIRST

FOR THE

Destination Stratum

. . .

1

2

3

1.692 0.135 0.069

1.039 1.627 0.258

0.420 0.430 1.296

Table 3 reportsthemultiplier matrix(I - Q)-1 forthefirstdecade. In theithrowandjth columnappearsthenumberof timesa vacancycreated in stratumi will appear in stratumj. The sum of the ithrow is the total numberof movesset offbycreationof a vacancyin stratumi. This multiplierfactorvariesfromover3 in thetop stratumto about I1 in thestratum of smalljobs. The largerthemultiplier, thelongerthechainsof movesset offby the initialflowof vacancies into the system.The vacancymodel in stochasticformwas validatedby the closenessof fitbetweenthe observed and predicteddistributionsof chain lengthsstartedin a given stratum. Discussion Thereis no labor market,in the usual sense,in a systemof men and jobs whichconformsto thevacancymodel.Net flowof menintothesystemis not matchedwithnetdemandforadditionallabor througha pricemechanism. Demand forlabor is replacedby the numberof vacantjobs, each job requiring exactlyone man.The systemadaptsto thesumoftwostreams of vacanciesimposed fromthe environment. One is the gross (not net) flowof menout of thesystem,and theotheris thegrossflowofjobs into the system.This externalflowsets offtrainsof moves.Fixed fractionsof the vacancies whichoccur duringthis process are neverfilled,the jobs beingabandonedor ejectedfromthe system;otherfixedfractionscall in newmenfromoutsidethesystem. Justas the numberof newjobs createdis arbitrary, so a limitlesspool of menis assumedavailable to be called into thesystemto fillvacancies. If salariesdecreaserelativeto othersystems,fewermen withhighqualificationsmay acceptcalls into the system;but, accordingto the model, the same fractionof vacancies will be filledwith whateverquality of recruitis available. Such apparentlyhas been thecourseof recenthistory in theEpiscopalclergy(Puseyand Taylor1967).

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In general,thesizesofstratain numbersoffilledjobs willvaryerratically overtheyears,even thoughthe parametersforinternaldynamicsremain constant.Similarly,the compositionof men in a stratumby remarkably Careersforthebulkofthemenin thesystem pastjob historywillfluctuate. will be unpredictableby-productsof thesevacancydynamicsin the eyes of an outside observer,even though endless gossip will still circulate (Thompabout howto getahead. In thelanguageoforganizationtheorists subject as a natural system is best described the system 1), son 1967,chap. closed not a rational system. external contingencies, to Whenjobs are stableentitieswithfixednumbersofoccupants,individual of one another,or, in the language of men cannot move independently cannotbe linearas requiredin the the equations dynamic aggregateflows, however,are much less of Vacancies, models mobility. stochastic usual A dual modelis possible, move independently. nearly and may numerous wheremen move onlyby bumpingothersas in a unionizedplant during throughthe system, layoffs;thenthe "bumps" may move independently to a firstapproximation. Vacancy models in some formmay also be applicable to marketsfor automobilesand houses. Smith(1941, pp. 29-30) comes close to defining a vacancymodel forcars (using a singlelumpedstratumas in equation [9]) "for every100 new cars sold, 90 used cars are acceptedin trade as part payment,... the sale of these 90 involvestakingin approximately 60 otherused cars in partialpaymentand ... in the sale of these60, 30 otherused cars are acceptedwithpossiblya numberof trade-insinvolved in thesale of these30." The totalrateof creationof "vacant" cars would of be new car sales to currentownersadded to deaths and retirements of strata cars, presumably of vacancies Moves among owners. current definedin termsof price ranges,mightwell be describableby a stable matrixof transferprobabilities.One stable fractionof vacant cars in each stratummightbe scrappedand anotherfractionfilledby teenagers buyers.As to housing,a teamat theSurveyResearch and otherfirst-time of Michiganhas "followedeach of 1144vacancies of the Center University fromone housingunitto thenextuntil construction residential createdby it disappeared."(Lansing, Clifton,and Morgan 1969). In analyzingthe to developa vacancymodelapplicableto the results,Cliftonis attempting offamiliesand housesthroughthehousing flows and simultaneous opposite market. 6 The mainempiricalquestionforbothhousingand autos is whether big enough setsof menable to seekvacanciesand vacanciesable to seekmenare awareofeach to requirethe notion of a marriagenegotiationmarketas othersimultaneously "call" ofa manto a vacancy.(Reynolds'sworkcitedearlier opposedto theunilateral how fewavailablejobs are knownto a man changingjobs, and-at least illustrates passiveas jobs are greatlyhandicappedand effectively in higherstrata-menwithout candidatesforvacancies.)Littleworkseemsto havebeendoneon modelsofmarriage forhighratesofdivorceandremarriage. appropriate

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References Bartholomew,D. Stochastic Models of Social Processes. New York: Wiley, 1967. Blau, P. M., and Duncan, 0. D. The American Occupational Structure. New York: Wiley,1967. Dorfman,R.; Samuelson, P. A.; and Solow, R. M. Linear Programmingand EconomicAnalysis.New York: McGraw-Hill, 1958. Holt, C., and David, M. "The Concept of Job Vacancies in a Dynamic Theory of the Labor Market." In The Measurementand Interpretation of Job Vacancies.New York: Nat. Bur. Econ. Res., 1966. Koopmans, T. C. ThreeEssays on theState of EconomicScience. New York: McGraw-Hill, 1957. Lansing, J. B.; Clifton, C. W.; and Morgan, J. N. New Homes and Poor People. Ann Arbor,Mich.: Inst. Soc. Res., 1969. Pusey, N. M., and Taylor, C. L. Ministryfor Tomorrow:Reportof theSpecial Commissionon TheologicalEducation in the Episcopal Church.New York: Seabury,1967. Pyke, R. "Markov Renewal Process." Ann. Math. Statis. 32 (1961): 1231-59. Reynolds,Lloyd. The Structureof Labor Markets: Wages and Labor Mobility in Theoryand Practice. New Haven, Conn.: Yale Univ. Press, 1951. Smith, T. H. The Marketing of Used Automobiles.Columbus.: Bur. Bus. Res., Ohio State Univ., 1941. Thompson, J. D. Organizationsin Action.New York: McGraw-Hill, 1967. White, Harrison C. Opportunityin Chains. Harvard Univ. Press, 1970.

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