Journal of Economic LiteratureVol. XXXIX (June 2001) pp. 390–431

Petrongolo and Pissarides: A Survey of the Matching FunctionJournal of Economic Literature, Vol. XXXIX (June 2001)

Looking into the Black Box: A Surveyof the Matching Function

BARBARA PETRONGOLO and CHRISTOPHER A. PISSARIDES1

1. Introduction

FRICTIONS HAVE MADE important in-roads in modern macroeconomics.In the labor market they are used to ex-plain the existence of unemployment and(sometimes) wage inequality. In businesscycle models they are used to explain theamplification of the response of employ-ment to aggregate shocks. In coordination-failures models they are used to justifythe dependence of the strategy of oneagent on that of another. In monetarymodels they are used to explain the exis-tence of money.2 In the majority ofcases, the modeling tool used to capturethe influence of frictions on equilibriumoutcomes is the aggregate matchingfunction. This paper surveys recent work

on the existence and stability of the ag-gregate matching function, with empha-sis on microfoundations and empiricalfindings.

The attraction of the matching func-tion is that it enables the modeling offrictions in otherwise conventionalmodels, with a minimum of added com-plexity. Frictions derive from informa-tion imperfections about potential trad-ing partners, heterogeneities, theabsence of perfect insurance markets,slow mobility, congestion from largenumbers, and other similar factors.Modeling each one of these explicitlywould introduce intractable complexi-ties in macroeconomic models. Thematching function captures their effectson equilibrium outcomes in terms of asmall number of variables, usually with-out explicit reference to the source ofthe friction.

Frictions also introduce monopolyrents in competitive markets, which in-fluence behavior. The matching functionhas been used to study their implicationsfor wage and price determination.3 Theappendix traces the history of frictionsin economic modeling, leading up tothe recent generation of equilibriummodels with matching frictions. It arguesthat although the matching function

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1 Petrongolo: University Carlos III, Madrid,Centre for Economic Performance, and CEPR.Pissarides: Centre for Economic Performance,London School of Economics, and CEPR. We aregrateful to Alan Manning, Tony Venables, ÉtienneWasmer, seminar participants at ESSLE 2000, andan anonymous referee for useful comments. Fund-ing for this project was provided by the Centre forEconomic Performance, the Economic and SocialResearch Council, and the Spanish Ministry ofEducation (Grant No PB97–0091).

2 Representative references include Pissarides(2000) and Dale Mortensen and Pissarides (1999b)on the labor market; James Montgomery (1991)and Daron Acemoglu and Robert Shimer (2000)on wage inequality; Monika Merz (1985) andDavid Andolfatto (1986) on business cycles;Russell Cooper and Andrew John (1988) and PeterDiamond (1982a) on coordination failures; andDiamond (1984) and Nobuhiro Kiyotaki andRandall Wright (1989) on monetary exchange.

3 See, for example, Diamond (1982b) and thelabor market references in the preceding footnote.

was not “discovered” by the recent vin-tage of models, in the sense that theidea (and sometimes functional form)were present in earlier models, it wasnot until the late 1970s that it explicitlyappeared in equilibrium models andwas given a new and far more importantrole in the characterization of equilib-rium than had previously been the case.Influential in this respect were equilib-rium models of wage and employmentdetermination (Peter Diamond 1982a,b;Dale Mortensen 1982a,b; and Pissarides1984, 1985).

Virtually all the work that we surveyfocuses on the labor market. This ispartly explained by the fact that fric-tions are likely to be more important inthe labor market than in other markets.But it also has to do with the fact that inlabor markets there are data sets thatcan be used to estimate and test thematching function. A lot of the recentinterest in the matching function stemsfrom the realization that modern labormarkets are characterized by large well-documented flows of jobs and workersbetween activity and inactivity.4 Thematching of workers to new jobs is one-half of the explanation for these flows.Its outcome, in conjunction with theoutcome of the process that separatesworkers from jobs, is often showngraphically in vacancy-unemploymentspace by the “Beveridge curve” (Pis-sarides 2000, ch. 1; Olivier Blanchardand Diamond 1989). Estimated Beveridgecurves can shed light on the nature ofthe aggregate matching function, andwe discuss some below. Most of the evi-dence that we discuss, however, is in

studies that estimate a matching func-tion directly, either at the aggregate orthe sectoral level.

In section 2 we discuss the mainideas behind the matching function andwe give some pertinent evidence. Wethen take a look at the theoretical foun-dations of the matching function anddiscuss some of the more importantvariables that are likely to be influentialin empirical work (section 3). Section 4discusses empirical results in the con-text of the methods most frequentlyadopted in the estimation of the match-ing function. Section 5 deals with theconceptual and measurement issues dueto search on the job and to workers’transitions from out of the labor forceto employment. Aggregation problemsacross time and space are discussed insection 6. The main conclusions arebrought together in section 7. The ap-pendix gives a brief historical overviewof the literature on the role of labormarket frictions, leading to the birth ofthe matching function.

2. The Key Idea and Some Evidence

The matching function summarizes atrading technology between agents whoplace advertisements, read newspapers andmagazines, go to employment agencies,and mobilize local networks that even-tually bring them together into produc-tive matches. The key idea is that thiscomplicated exchange process is sum-marized by a well-behaved function thatgives the number of jobs formed at anymoment in time in terms of the numberof workers looking for jobs, the numberof firms looking for workers, and asmall number of other variables.

The matching function is a modelingdevice that occupies the same place inthe macroeconomist’s tool kit as otheraggregate functions, such as the pro-duction function and the demand for

4 See Steven Davis, John Haltiwanger, and ScottSchuh (1996) on the importance of job flows andBlanchard and Diamond (1990a) on the impor-tance of worker flows in the United States; andMichael Burda and Charles Wyplosz (1994) andBruno Contini, Lia Pacelli, Michelangelo Filippi,Graciela Lioni, and Riccardo Revelli (1995) on theimportance of flows in Europe.

Petrongolo and Pissarides: A Survey of the Matching Function 391

money function. Like the other aggre-gate functions its usefulness depends onits empirical viability and on how suc-cessful it is in capturing the key impli-cations of the heterogeneities and fric-tions in macro models. In this surveywe will focus on the microfoundationsunderlying the matching function andon its empirical success but we will notdiscuss its modeling effectiveness.

The simplest form of the matchingfunction is

M = m(U,V), (1)

where M is the number of jobs formedduring a given time interval, U is thenumber of unemployed workers lookingfor work and V the number of vacantjobs. The matching function is assumedincreasing in both its arguments andconcave and usually homogeneous of de-gree 1. Testing for homogeneity, or con-stant returns to scale, has been one ofthe preoccupations of the empirical lit-erature. Other restrictions usually im-posed are m(0,V) = m(U,0) = 0, and indiscrete-time models where M is theflow of matches during an elementaryperiod and U and V are the stocks at thebeginning of the period, m(U,V) ≤min(U,V). In continuous time models, Mis the instantaneous rate of job matchingand U and V the instantaneous stocks ofunemployment and vacancies. In theabsence of frictions, M = min(U,V) indiscrete-time formulations and M → ∞ incontinuous-time models. Under constantreturns to scale, M, U, and V are usuallynormalized by the labor force size, anddenoted by lower-case letters.

On average, an unemployed workerfinds a job during a period of unitlength with probability m(U,V)/U. Simi-larly, a vacant job is filled with prob-ability m(U,V)/V. In a stationary envi-ronment, the inverse of each probabilityis the mean duration of unemploymentand vacancies respectively. Of course, if

workers and jobs are heterogeneous,the transition probabilities (or hazardrates) will differ across the labor mar-ket, as will the mean durations. The ag-gregate matching function is a usefuldevice for introducing heterogeneitiesacross workers, by making the prob-ability m(U,V)/U depend on individualcharacteristics. This has been a themeof the empirical literature that esti-mates hazard functions for individualworkers.

The dependence of the mean transi-tion rates on the number of workersand firms engaged in search is an exter-nality that has played an important rolein the analysis of the efficiency ofsearch equilibrium. The average timethat it takes a firm to find a worker de-pends on what searching workers do be-fore they meet the firm. Similarly, theprobability that an unemployed workerfinds a job depends on what hiringfirms do, for example on whether theyadvertise or not and where they adver-tise. Generally, search equilibrium is in-efficient because when firms and work-ers meet, the costs of their search,which influence the transition prob-abilities, are sunk. Estimated matchingfunctions can give a measure of the ex-tent of the externalities. If the elasticitywith respect to unemployment in thematching function is ηU and the elastic-ity with respect to vacancies ηV (notnecessarily constants), ηU − 1 measuresthe negative externality (congestion)caused by the unemployed on other un-employed workers, and ηV measures thepositive externality (thick-market ef-fect) caused by firms on searchingworkers. Similarly, ηU measures thepositive externality from workers tofirms, and ηV − 1 measures the negativeexternality by firms on each other.Higher elasticity estimates indicate lesscongestion and more positive externalities.

The returns to scale in the matching

392 Journal of Economic Literature, Vol. XXXIX (June 2001)

function play an important role in modelswith endogenous search effort. If thereare increasing returns to matching (inthe notation above, if ηU + ηV > 1), as theauthors of some early models assumed(Diamond 1982a; Peter Howitt andPreston McAfee 1987), there could bemore than one equilibrium, because ofthe strong positive externalities: in oneequilibrium firms and workers put moreresources into search, pushing up thereturns from search available to theother side, which justify the bigger in-puts; in another they put less effort intosearch with lower returns from search,lower matching rate, and higher unem-ployment. Increasing returns to scalecan support the high and low activityequilibria even when there are increas-ing marginal costs to search effort,whereas constant returns cannot (al-though the complementarity betweenthe actions of firms and workers is stillpresent).

Evidence on the key matching-functionidea comes from four sources. The firstone uses aggregate data on stocks of un-employment and vacancies and esti-mates an equilibrium relation, theBeveridge (or UV) curve. The seconduses aggregate data on employment andunemployment flows and estimates theaggregate matching function, either forthe whole economy or for a particularsector (usually manufacturing). Thethird uses data on local labor markets,which can be either a time-series or apanel, and estimates the matching func-tion for each. The fourth uses data onindividual transitions and estimates haz-ard functions for unemployed workers.We discuss each approach in some de-tail in subsequent sections. Here wesummarize the main implications of theempirical research for the simplematching function in (1).

The Beveridge curve is an equilib-rium relation that equates flows in with

flows out of unemployment. In vacancy-unemployment space it slopes down-ward if the outflow from unemploymentis given by the matching function in (1).Estimated Beveridge curves slopedownward but shift over time, espe-cially in cases where there have beensecular increases in unemployment, asin most European countries since themid-1970s. So the matching functionin (1) is not contradicted by theBeveridge-curve evidence, but this evi-dence is indirect; it is consistent withother mechanisms, and points to othervariables that influence job matching too.

Direct estimates of the matchingfunction give better information aboutthe properties of (1). Table 1 summa-rizes the specifications adopted by ag-gregate studies. Most studies that esti-mate aggregate functions find that alog-linear approximation to (1) withconstant returns to scale fits the datawell, although some estimates withtranslog specifications find increasingreturns. The estimated elasticities withrespect to unemployment and vacanciesvary, depending on whether the depen-dent variable is the outflow from unem-ployment, the flow from unemploymentto employment, or the total number ofhires. When the dependent variable isthe total outflow from unemployment, theestimated elasticity on unemployment isabout 0.7 and the elasticity on vacancies0.3. Precise data on unemployment-to-employment transitions are rarely avail-able, but when an approximation for thematching rate is used the elasticity onunemployment drops, although not bymuch when other flows into employ-ment are ignored. A plausible range forthe empirical elasticity on unemploy-ment is 0.5 to 0.7, showing that perhapsthe congestion effects caused by firmson each other are bigger than the onescaused by workers on each other. Thereare good reasons for the drop in the

Petrongolo and Pissarides: A Survey of the Matching Function 393

TABLE 1AGGREGATE MATCHING FUNCTION STUDIES

AuthorCountry and

coveragePeriod andfrequency

Dependentvariable

Jobseekers

Pissarides 1986 U.K., men 1967–83quarterly

male unempl.outflow rate

unemployedmen

Blanchard andDiamond 1989,1990b

U.S. 1968–81monthly

all new hires unemployed;laid-off;

out of LF;STU and LTU

Layard, Nickell, andJackman 1991

Britain 1968–88quarterly

unempl.outflow rate

unemployed

van Ours 1991 Netherlands 1961–87annual

vacancyoutflow

unemployed

Burgess 1993 U.K., men 1968–85quarterly

male unempl.outflow rate

male unempl.rate

Burda and Wyplosz1994

FranceGermany

SpainU.K.

1971–931968–911977–921985–93

(all monthly)

unemploymentoutflow

unemployed

Warren 1996 U.S.manufacturing

1969–73monthly

all new hires unemployed(from manuf.)

Feve and Langot1996

France 1971–89quarterly

— —

Berman 1997 Israel 1978–90monthly

referrals unemployed

Gross 1997 Germany (West) 1972–94quarterly

all new hires unemployed

Gregg andPetrongolo 1997

Britain 1967–96quarterly

unempl. outflow;vacancy outflow

unemployed

Bell 1997 FranceBritainSpain

1979–941967–851980–95

(all quarterly)

unempl. outflownew hiresnew hires

unemployed

Bleakley andFuhrer 1997

U.S. 1979–93monthly

hires from U unemployed

Coles and Smith1998

Britain 1987–95monthly

unempl. outflowby duration

U stockU inflow

Mumford andSmith 1999

Australia 1980–91quarterly

U outflow rate;outflow rate

from out of LF

unemployed(from manuf.)

Yashiv 2000 Israel 1975–89monthly

all new hires unemployed

TABLE 1 (Cont.)

AuthorJob

vacanciesOther

variables Specification

Pissarides 1986 notified,adjusted

Time trend, mismatch,replacement ratio

linear;log-linear

Blanchard andDiamond 1989,1990b

help-wanted indexadjusted

time trend log-linear;CES

Layard, Nickell, andJackman 1991

notified time trend,search intensity

index

log-linear

van Ours 1991 notified,adjusted

replacement ratio,LTU/U

log-linear

Burgess 1993 — male hires, replacementratio, demographicvariables, LTU/U

log-linear

Burda and Wyplosz1994

notified time trend log-linear

Warren 1996 help-wanted index(in manuf.)

— translog

Feve and Langot1996

notified a general-equilibrium small open economy model isestimated, in which a log-linear matching function

is included

Berman 1997 notified time trend log-linear

Gross 1997 notified real wagesreal energy price

log-linear (withco-integration analysis)

Gregg andPetrongolo 1997

notified time dummies non-linear

Bell 1997 notifiednotifiednotified,adjusted

time trend,benefits, mismatch

demographicvariables, LTU/U

log-linear(with co-integration

analysis)

Bleakley andFuhrer 1997

help-wanted indexadjusted

structural breaks log-linear

Coles and Smith1998

V stockV inflow

time trend log-linear

Mumford andSmith 1999

— new hires,other groups of

job seekers, LTU/U

log-linear

Yashiv 2000 notified structuralbreaks

log-linear;translog

elasticity estimates when flows from un-employment to non-employment are ig-nored, which we discuss when we lookat the estimates in more detail.

The aggregate estimates also find thatthere are other variables that influencematching in a systematic way. Disaggre-gate estimates, summarized in table 2,have not contradicted the aggregate es-timates but concentrated instead onfinding out what are those other vari-ables, and whether aggregation intro-

duces biases that can be estimated.Hazard studies also focus on identifyingother influences on transitions, espe-cially those related to individual charac-teristics. With the number of estimatesgrowing significantly in recent years, itis natural that there are estimates ofboth increasing and decreasing returnsto scale. But such divergencies fromconstant returns are only mild and rare.The stylized fact that emerges from theempirical literature is that there is a

TABLE 2SECTORAL MATCHING FUNCTION STUDIES

AuthorCountry and

coveragePeriod andfrequency

Level ofdisaggregation

Dependentvariable

Burda 1993 Czech Rep.Slovakia

1990–92monthly

76 districts38 districts

hires from U

Bennet andPinto 1994

Britain, men 1967–83quarterly

104 localdistricts

unempl.outflow

van Ours 1995 Netherlands 1981–83annual

8 regions hires from U;hires from N

Coles andSmith 1996

Englandand Wales

1987 257 TTWAs filledvacancies

Boeri andBurda 1996;Profit 1997

CzechRepublic

1992–94quarterly

76 districts hires from U

Burda andProfit 1996

CzechRepublic

1990–94monthly

76 districts hires from U

Burgess andProfit 1998

U.K. 1985–95monthly

303 TTWAs unempl. outflow;filled vacancies

Profit andSperlich 1998

CzechRepublic

1992–96monthly

76 districts hires from U

Broersma andvan Ours 1999

Netherlands 1988–94quarterly

6 industries hires from U;filled vacancies

Münich, Svejnar,and Terrel 1999

Czech Rep.Slovakia

1991–96monthly

76 districts38 districts

hires from U;hires from N

Anderson andBurgess 2000

U.S. 1979–84quarterly

4 states ×20 industries

all new hires;hires from non-empl.;

hires from empl.

396 Journal of Economic Literature, Vol. XXXIX (June 2001)

stable aggregate matching function of afew variables that satisfies the Cobb-Douglas restrictions with constant returnsto scale in vacancies and unemployment.

Table 3 summarizes the results ofstudies that tested for constant returns.The estimates of Burda and Wyplosz(1994) for some European countriesshow decreasing returns to scale. Thoseof Blanchard and Diamond (1990b) andthe translog specifications of RonaldWarren (1996) for U.S. manufacturing;

Eran Yashiv (2000) for Israel; andDaniel Münich, Jan Svejnar, and Kath-erine Terrell (1999) for the Czech Re-public (and in some cases Slovakia) showincreasing returns. All other estimatessupport constant returns.

3. Microfoundations

What are the reasons for the exis-tence of a well-behaved matching func-tion and what are the other variables

TABLE 2 (Cont.)

AuthorJob

seekersJob

vacanciesOther

variables Specification

Burda 1993 unemployed notified log-linear

Bennet andPinto 1994

unempl.rate

vacancyrate

log-linear

van Ours 1995 unempl. +empl. seekers

all vacancies reg. dummies non-linear

Coles andSmith 1996

unemployed notified wages, size of TTWA,demographic

variables

log-linear

Boeri andBurda 1996;Profit 1997

unemployed notified time dummies,area dummies,lagged dep. var.

log-linear

Burda andProfit 1996

unemployed notified time dummies,spillover effects

across areas

log-linear

Burgess andProfit 1998

unemployed notified time trends,spillover effects

across areas

log-linear

Profit andSperlich 1998

STU and LTU notified area dummies,lagged dep. var.

log-linear;nonparametic

Broersma andvan Ours 1999

unemployed;U + non-U seekers

notified industrydummies

log-linear

Münich, Svejnar,and Terrel 1999

STU and LTU notified human capital,output per head,

demographicvariables

translog

Anderson andBurgess 2000

unempl.rate

help-wantedrate

∆ employm. in ind.,replacement ratio,

demographicvariables

log-linear

Petrongolo and Pissarides: A Survey of the Matching Function 397

TABLE 3EVIDENCE ON THE RETURNS TO SCALE IN THE MATCHING FUNCTION

Authorcoef[ln U](t − stat.)

coef[ln V](t − stat.)

CRS test(if any) R2

Log-linear specifications

Pissarides 1986 U.K. 0.70 0.30(unrestricted estimates not reported) p = 0.74

Blanchard andDiamond1990b

OLS

IV (1)

IV (2)

OLS (Manuf.)

0.35(3.9)0.60(2.4)0.42(2.3)0.67(7.2)

0.54(6.9)0.75(4.0)0.62(4.3)0.71

(14.7)

0.47

0.43

0.41

0.92

Layard et al.1991

U.K. 0.81(6.6)

0.19(6.6)

(unrestricted estimates not reported) p = 0.24

van Ours 1991 Netherlands 0.48(2.8)

0.67(7.4)

0.93

Burda 1993 Czech Rep.

Slovakia

0.42(3.5)0.61(3.0)

0.44(3.3)0.10(0.9)

p > 0.05

p > 0.05

Burda andWyplosz 1994

France

Germany

Spain

U.K.

0.52(13.7)0.68

(25.1)0.12

(1.67)0.67

(21.1)

0.09(2.59)0.27

(11.3)0.14(3.0)0.22

(6.78)

p = 0.00

p = 0.06

p = 0.00

p = 0.02

0.95

0.97

0.97

0.93

Bennet andPinto 1994

Britain sum of elasticities: 0.65 − 1.15(90% of 104 regressions)

Coles andSmith 1996

England and Wales 0.34(10.6)

0.66(16.0)

0.91

Berman 1997 Israel 0.29(3.22)

0.39(4.33)

p = 0.07 0.67

Anderson andBurgess 2000

All new hires

New hires from NE

New hires from E

0.43(2.4)0.39(2.2)0.54(1.1)

0.81(4.0)0.75(3.8)0.87(1.6)

p = 0.49

p = 0.67

p = 0.67

0.86

0.86

0.61

Yashiv 2000 0.49(6.1)

0.87(14.5)

p = 0.00 0.97

398 Journal of Economic Literature, Vol. XXXIX (June 2001)

that influence the matching rate? In or-der to answer these questions we needto look at the microfoundations behindthe aggregate matching function. Theliterature has done that; but althoughthere are several microeconomic mod-els that can be used to justify the exis-tence of an aggregate matching func-tion, none commands universal supportand none convincingly says why the ag-gregate matching function should be ofthe Cobb-Douglas form. The literaturehas had more success, however, insuggesting what should be the othervariables that influence the matching rate.

The other variables can be classifiedinto two groups. The first group in-cludes everything that individuals doduring search, such as choosing howmany applications to make, changingtheir advertising methods, etc. The sec-ond includes shifts unrelated to individ-ual search decisions. We take up thesecond group first. Most of the theo-retical work on matching functions stud-ies individual behavior and is discussedin the subsections that follow.

3.1 Mismatch

The shifts in the matching functionthat are unrelated to search decisions

are due to technological advances inmatching and to aggregation issues.Technological advances include reformssuch as the computerization of employ-ment offices, job advertising on the in-ternet, an increase in the resources thatgovernments put into subsidized match-ing, and other similar changes. Al-though changes of this type have beenobserved recently in most industrialcountries (see OECD 1994, ch. 6; 1999)and they have influenced the matchingprocess to the extent that the OECDrecommends them to its membersas the most cost-effective “active” la-bor market policies, they have attractedlittle formal theoretical or empiricalwork.

Aggregation issues have attractedmore attention from labor economists,often disguised under the label “mis-match.” Mismatch is an empirical con-cept that measures the degree of het-erogeneity in the labor market across anumber of dimensions, usually re-stricted to skills, industrial sector, andlocation. Large differences in the skillspossessed by workers and those re-quired by firms would lengthen thetime that it takes to match a givengroup of workers to a given group of

TABLE 3 (Cont.)

Authorcoef[ln U](t − stat.)

coef[ln V](t − stat.)

CRS test(if any) R2

Translog specifications

Warren 1996 sum of elasticities: 1.33(sample mean)

p = 0.03 0.56

Münich et al.1999

Czech Rep.

Slovakia

1.54 − 2.51(sample mean)

0.34 − 2.62(sample mean)

0.65 − 1.00(sample mean)

0.17 − 0.25

Yashiv 2000 0.28(sample mean)

0.80(sample mean)

p = 0.00 0.97

Notes: p denotes the p–value for the rejection of constant returns to scale.

Petrongolo and Pissarides: A Survey of the Matching Function 399

firms, as agents search for a good matchamong the heterogeneous group. Indus-trial sector matters in matching becauseof industry-specific skills that may notbe picked up by generally availablemeasures of skills. Finally, location in-fluences matching because of imperfectlabor mobility. Although the term “mis-match” has been used in the literatureto describe all three dimensions (seeRichard Layard, Stephen Nickell, andRichard Jackman 1991), the term “im-balance” in numbers in the local markethas been used before to describe dif-ferences in the distribution of loca-tions (see, e.g., Charles Holt 1970b)and is a useful way of distinguishingbetween skill mismatch and differences inlocation.

If mismatch and imbalance in aneconomy were identically zero in alltheir dimensions, the matching functionwould not exist and jobs and workerswould match instantaneously. It is be-cause of the existence of some mis-match that meetings take place only af-ter a search and application process. Ifthere is an exogenous rise in mismatch,the rate of job matches at given inputsmust fall, implying a shift in the aggregatematching function.

Of course, if empirically mismatchchanges frequently in ways that cannotbe accurately measured, the usefulnessof the concept of the matching functionis reduced. But this requirement is notdifferent from the one on other aggre-gate functions in the macroeconomist’stool kit. Some of the early controversiesin production theory (like the capitalcontroversy of the two Cambridges)were about the question whether fac-tors of production could be aggregatedinto two or three composites that entera single-valued differentiable produc-tion function. Whether in practice ag-gregation problems are serious enoughto question the usefulness of the match-

ing function is an empirical question.The available evidence does not sup-port serious aggregation problems thatcannot be dealt with empirically.

In the empirical literature, mismatch,or imbalance, bears some relationshipto the frequently discussed “sectoralshifts hypothesis,” and to the older viewof “structural” unemployment, whichwas thought to be unemployment aris-ing from fast structural change in theeconomy as a whole. For example, it hasbeen argued that the oil, technology,and other supply shocks of the 1970sand 1980s increased the speed withwhich unemployed workers needed toadapt to the changing requirements ofemployers. This led to increased mis-match between the skills possessed byworkers and the skill requirements ofemployers, which increased the durationof unemployment (and hence the stockof unemployment) at given vacancies.

David Lilien (1982) argues that im-balance in the distribution of jobs andworkers changes over the business cy-cle, to the extent that it can adequatelyexplain the observed fluctuations in ag-gregate employment. Although he findsthat his sectoral shifts hypothesis hassome success in explaining U.S. employ-ment data, his findings have been effec-tively criticized by Katharine Abrahamand Lawrence Katz (1986) andBlanchard and Diamond (1989). Theircritiques point to the fact that the ob-served positive correlation between thedispersion of employment growth andthe unemployment rate can be pro-duced either by sectoral shifts or by ag-gregate demand fluctuations. Informa-tion on job vacancies allows one todistinguish between the two explana-tions. The strong negative correlationbetween unemployment and vacanciessupports an aggregate-demand interpre-tation of U.S. employment fluctuationsrather than one based on sectoral shifts.

400 Journal of Economic Literature, Vol. XXXIX (June 2001)

Similar conclusions can be reachedfrom the observation that job creationand job destruction rates across sectorsare negatively correlated over the cycle(see Davis, Haltiwanger, and Schuh1996).

Layard, Nickell, and Jackman (1991,ch. 6) follow a different approach andmeasure mismatch by the variance ofsectoral unemployment rates. Theyshow, however, that their measure ofmismatch cannot account for the shiftsin the aggregate matching function orthe variance in U.K. unemployment.More recently, Marco Manacorda andPetrongolo (1999) propose a measure ofskill mismatch that makes use of infor-mation about the demand and the sup-ply of skills, represented respectively byproductivity parameters and labor forceshares. This leads them to the conclu-sion that the unbalanced evolution ofthe demand and the supply of skills canexplain some of the rise in unemploy-ment in Britain, and hence some of theobserved shifts in the matching function,but still not all.

On balance, neither the sectoralshifts hypothesis nor mismatch has hadmuch success in accounting for a largefraction of fluctuations in employmentor for the secular rise in unemploymentin some countries. So although empiri-cal mismatch variables can account forsome of the shifts in the aggregatematching function, we should look else-where for the main shift variables. Butsome authors (for example Horst Entorf1998) argue that the measurement ofmismatch in aggregate studies of match-ing functions still suffers from manyproblems, and may be able to accountfor more of the unexplained variance inmatchings than is currently found in theliterature.

If aggregation problems are not an is-sue, what can account for the matchingfunction and what else can shift it?

3.2 Coordination Failures

The first matching function owes itsorigins to a well-known problem ana-lyzed by probability theorists, that ofrandomly placing balls in urns (GerardButters 1977; Robert Hall 1979; Pis-sarides 1979; Kevin Lang 1991; JamesMontgomery 1991; and Blanchard andDiamond 1994). Firms play the role ofurns and workers the role of balls. Anurn becomes “productive” when it has aball in it. Even with exactly the samenumber of urns and balls, a randomplacing of the balls in the urns will notmatch all the pairs exactly, because of acoordination failure by those placingthe balls in the urns. Some urns willend up with more than one ball andsome with none. In the context of thelabor market, if only one worker couldoccupy each job, an uncoordinated ap-plication process by workers will lead toovercrowding in some jobs and to noapplications in others. The imperfectionthat leads to unemployment here is thelack of information about other work-ers’ actions, though simple extensionscould enrich the source of frictions.

In the simplest version of this processU workers know exactly the location ofV job vacancies and send one applica-tion each. If a vacancy receives one ormore applications it selects an applicantat random and forms a match. Theother applicants are returned to thepool of unemployed workers to applyagain. The matching function is derivedby writing down an expression for thenumber of vacancies that do not receiveany applications. Given that each va-cancy receives a worker’s applicationwith probability 1/V, and there are Uapplicants, there is a probability (1 –1/V)U that a given vacancy will not re-ceive any applications at all. Therefore,the number of matches that take placeat each application round is

Petrongolo and Pissarides: A Survey of the Matching Function 401

M = V[1 − (1 − 1/V)U]. (2)

For a large V a good approximation to (1 –1/V)U is the exponential e−U/V, giving thematching function

M = V(1 − e− U/V). (3)

This matching function clearly satisfiesthe properties satisfied by the generalfunction in (1), and in addition it satis-fies constant returns to scale. It is, how-ever, too naive to be empirically a goodapproximation to matching in real labormarkets. For example, it implies an im-plausible combination of levels and du-rations of unemployment. If the level ofunemployment and vacancies is the same,the mean duration of unemployment is1.58 periods, and if the level of unemploy-ment is three times as high as that ofvacancies, mean duration is 3.16. In ac-tual labor markets duration would rise bymore than the function (3) implies whenthe level of unemployment is higher.

The introduction of small additionalfrictions to the urn-ball framework canenrich the matching function consider-ably. We consider three related exten-sions. In the first, workers do not knowthe firms with the vacancies and chooseat random one firm to apply. Then theprobability that a vacancy receives noapplications is (1 – 1/(N + V))U, where Nis the level of employment. If in addi-tion the labor force size is L, N = L − U,the matching function becomes

M = V(1 − e−U/(L − U + V)). (4)

This matching function exhibits increas-ing returns to scale in U and V and mayeven fail the assumption of diminishingreturns to unemployment, though thiswould require more vacancies than em-ployment. But it satisfies constant re-turns to L, U, and V, so it avoids thecounterfactual implication that largercountries should have lower equilibriumunemployment rates than otherwiseidentical smaller countries.

In the second extension, not all work-ers are suitable for the vacancies avail-able but the worker does not knowwhich vacancies are suitable. Let K bethe fraction of workers who are suitableemployees for a randomly selected va-cancy. The probability that a vacancywill not be visited by a worker is still1/V but only KU workers can now takethe job. The matching function thereforegeneralizes to

M = V(1 − e−KU/V), (5)with the inverse of K standing as an in-dex of mismatch between the availablejobs and workers.

Our third extension gives a similarmatching function but the new parame-ter is associated with search intensity.Each period a fraction 1 – s of the un-employed do not apply for a job. Thisfraction rotates, so each unemployedworker misses one application roundout of every 1/(1 – s) rounds. Then, theprobability that a given vacancy re-ceives no applications during a givenapplication round is (1 – 1/V)sU, givingthe matching function

M = V(1 − e−sU/V). (6)Both (5) and (6) satisfy all the prop-

erties of (1) for given K and s, but inaddition open up the possibility of mod-eling mismatch and the frequency ofapplications, and so bringing the simpleform (3) closer to the data. The meanduration of unemployment for thesefunctions is again U/M and so more im-balance or a lower application fre-quency gives the longer mean durationsfor given vacancy-to-unemployment ra-tio that the data suggest. We take upthe question of what might determine snext.

3.3 Worker Heterogeneity: SearchIntensity and Reservation Wages

The hazard rates (or unemploymentdurations) derived in the preceding

402 Journal of Economic Literature, Vol. XXXIX (June 2001)

section were for “representative” indi-viduals, without explicit dependence onindividual characteristics. Yet, in em-pirical estimates, it is found that indi-vidual characteristics play an importantrole in accounting for differences inhazard rates across individuals. In thissubsection and the next, we suggest twoways of introducing the influence of in-dividual characteristics in the matchingtechnology and show what this does tothe aggregate matching function.

Worker heterogeneity is most con-veniently introduced into the matchingfunction by making the assumption thatthe intensity of search is a choice vari-able. We define intensity of search asthe number of “units” of search sup-plied by a given individual. Units aredefined as follows. If individual i sup-plies si units of search and individual jsupplies sj units, then in a small timeinterval individual i is si/sj times morelikely than individual j is to find amatch. Search units are supplied at acost, which is normally increasing, andthey are chosen optimally to maximizethe net returns from search (Pissarides2000, ch. 5). Therefore, different indi-viduals will choose a different number ofsearch units, depending on their searchcosts, the cost of unemployment, andthe expected returns from employment.

To derive the matching function im-plied by this extension, let s be the av-erage number of search units suppliedby an unemployed person.5 Then, thetotal number of search units suppliedis sU, and so the aggregate matchingfunction is

M = m(sU,V), (7)

a more general form of (6). Of course,varying intensity could also be intro-duced for job vacancies, in symmetric

fashion. The hazard rate for an individ-ual who supplies si units of search issim(sU,V)/sU. The fact that this functiondepends on individual characteristicsthrough the optimal choice of intensityof search justifies the econometric esti-mates of hazard functions that make useof individual survey data. On average,the representative individual will chooseintensity s, so the average transition ratefor unemployed workers, which can beused in macro modeling, is m(sU,V)/U.

Another channel through which het-erogeneity can influence the matchingfunction and market outcomes ariseswhen there is a distribution of wage of-fers. The distribution may be due eitherto identical firms offering differentwages, as in the model of Kenneth Bur-dett and Mortensen (1998), or to matchheterogeneity, as in the model of BoyanJovanovic (1979). The individual choosesa reservation wage and rejects all wageoffers below the reservation. In equilib-rium models the reservation wage foreach job that the worker encounters issuch that neither the firm nor theworker will want to form a match if thewage is below reservation (Pissarides2000, ch. 6). Of course, if individualcharacteristics differ, workers maychoose different reservation wages.

Let m(U,V) be the technology thatbrings vacant jobs and unemployedworkers together. When a pair meets, itis faced with a wage offer w, which isassumed to be a drawing from a prob-ability distribution G(w). If the prob-ability distribution is known to jobseekers, the optimal policy of individuali is characterized by a reservationwage Ri, such that the job is acceptedif w ≥ Ri, and rejected otherwise. Thehazard rate for this individual is [1 –G(Ri)]m(U,V)/U. Aggregation over allindividuals gives the average transitionrate, and from there, multiplication by theunemployment rate gives the aggregate

5 This s bears a close resemblance to the s of thepreceding section, which explains the use of acommon symbol.

Petrongolo and Pissarides: A Survey of the Matching Function 403

matching function. Clearly, given thatin general the probability G(Ri) is non-linear, the aggregate function takes arather complicated form, but to a firstapproximation we can define R as theaverage reservation wage and write theaggregate matching function as

M = [1 − G(R)]m(U,V). (8)

As with the function derived for vari-able search intensity, (7), this functionjustifies the introduction of aggregatevariables that influence individual deci-sions during search into estimatedmatching functions. The variables canbe demographic variables that influencethe intensity of search—for example, ifyouths search with lower intensity thanadults, the youth share in the popula-tion should be a shift variable. Or theycan be variables that influence the costof search and moving, such as unem-ployment insurance and housing trans-actions. The list of variables that can in-fluence search intensity and reservationwages has been a fertile ground forsearching for statistically significantshift variables in empirical matchingfunctions, an issue discussed in theempirical sections that follow.

3.4 Ranking

Blanchard and Diamond (1994) con-sider the alternative assumption thatfirms receive many applications at atime and have preferences over job ap-plicants. They rank applicants and offerthe job to the person first in the rank.Their motivation for studying thisprocess is a feature of European labormarkets, that with the rise in unemploy-ment durations, the long-term unem-ployed became “disenfranchised” andless desirable employees than thosewith more recent work experience.

The matching function used byBlanchard and Diamond (1994) is simi-

lar to the urn-ball function (3), but theimplications of the ranking principlecan be illustrated more generally. Sup-pose the unemployed are divided intotwo groups, the short-term unemployedand the long-term unemployed. Let thenumber of short-term unemployed beUS and the number of long-term unem-ployed be UL. Then, if a short-term anda long-term unemployed compete forthe same job, the short-term unem-ployed always gets it. Therefore, thelong-term unemployed do not causecongestion for the short-term unem-ployed during search, and the long-term unemployed get only jobs forwhich there are no short-term appli-cants. The implication of the first claimis that the matching function for theshort-term unemployed is mS(US,V),where V are all the vacancies, and thematching function satisfies all the prop-erties of (1). If the long-term unem-ployed knew which vacancies are nowbeing taken by the short-term unem-ployed, their matching function wouldbe mL(UL,V – MS). But more generally,if there is a coordination failure be-tween short-term and long-term unem-ployed, we write as usual m(US + UL,V)for total matches and then attributethe difference between M and MSto matches involving long-term unem-ployed. That is, the aggregate matchingfunction is

M = m(US + UL,V) (9)

but the hazard rate for the short-termunemployed is mS(US,V)/US and for thelong-term unemployed m(US + UL,V)/UL –mS(US,V)/UL.6 Simple calculations showthat if the matching functions are iden-tical, the hazard rate of the short-term

6 Note that the expected duration of unemploy-ment of the long-term unemployed is the inverseof their hazard rate, but for the short-term unem-ployed account has to be taken of the fact that ifthey survive to long-term unemployment, theirhazard rate will fall.

404 Journal of Economic Literature, Vol. XXXIX (June 2001)

unemployed is always higher than thehazard rate of the long-term unemployed.

Blanchard and Diamond (1989) esti-mate a specification similar to (9) andimpose that the short- and long-termunemployed are perfect substitutes upto a scale parameter. If the estimatedvalue of this parameter is below one itis evidence in favor of the ranking hy-pothesis. Their point estimate of the scaleparameter, however, slightly exceedsone, but is not significantly differentfrom zero.

3.5 Stock–Flow Matching

The matching functions discussed sofar were derived under the assumptionthat job seekers take a vacant job at ran-dom and apply for it. This assumption isconvenient and realistic in many situ-ations, given that there is an element ofluck in hearing about job offers. Butthere is also a systematic element insearch. This subsection and the nextdiscuss the derivation of an aggregatematching function from assumptionsthat go to the other extreme of norandomness in job applications.

Melvyn Coles (1994) and Coles andEric Smith (1998) consider the implica-tions of the assumption that job seekershave complete information about theavailable job vacancies and apply simul-taneously to all the ones that they thinkare likely to be acceptable. Let thisnumber be the entire universe of jobson offer. But because of heterogeneity,not all job matches turn out to be ac-ceptable. Let a constant α be the prob-ability that a job match is unacceptableto the pair. A matching round then be-gins in a “marketplace.” Job–workerpairs that made contact and are unac-ceptable are rejected. The remainingacceptable ones are sorted out so thatno firm and worker who could form anacceptable match remain unmatched.Thus, unlike the urn-ball process of the

preceding example, there is no coordi-nation failure in this case. Those work-ers who remain unmatched do so be-cause there are no vacancies suitablefor them among the existing pool.

It follows that no job vacancy or un-employed worker who has been throughone round of matching will attempt tomatch again with a pre-existing jobseeker or vacancy. Of course, the as-sumption that the length of time whenjob seekers and vacant jobs get to knoweach other is one matching period is asimplifying one. Coles and Smith’s as-sumption captures a realistic feature ofsearch markets, that a job seeker scansa lot of advertisements before decidingwhere to apply, and once an advertise-ment has been scanned and rejected,return to it is less likely than applicationto a new advertisement.

Under Coles and Smith’s assumptionthere is a sharp distinction between thestocks of unemployed workers and va-cant jobs and the new inflows. Thestock of unemployed workers at the be-ginning of the period will not matchwith the stock of vacant jobs also at thebeginning of the period, because theywere both participants in the matchinground in the previous period. The re-sulting matching process is thereforeone where the unmatched stock of trad-ers on one side of the market is tryingto match with the flow of traders on theother side. This is often referred to as“stock–flow” matching.7

Let the stocks at the beginning of theperiod be U and V. If the flow of newunemployed workers and new job va-cancies into the respective pools duringthe period are u and v, the U initialworkers match with the new inflow vonly, whereas the inflow u matches withboth V and v. Coles and Smith consider

7 Coles (1999) discusses the turnover externali-ties implied by stock–flow matching.

Petrongolo and Pissarides: A Survey of the Matching Function 405

a period of infinitesimal length and soignore the probability of a newly unem-ployed worker matching with a newlycreated vacant job. In this case, theprobability that a new vacancy ismatched on entry is 1 – αU, so thematches due to new vacancy creationare v(1 – αU). Recall that α is the prob-ability that a random pairing is unac-ceptable. The probability that a newworker is matched on entry is 1 – αVand so the new matches due to the newentry of workers is u(1 – αV). Sincethere are no matches between old un-employed and old vacancies, the sum ofthe two matches gives the entire match-ing rate in the economy. That is, thematching function is

M = v(1 − αU) + u(1 − αV), (10)

with 1 > α > 0.The hazard rate for workers who are

unemployed at the beginning of the pe-riod is v(1 – αU)/U and for the new in-flow 1 – αV. The latter is likely to bebigger because for the short period un-der analysis, the stock of jobs and work-ers is likely to be much bigger than thenew flow, i.e., v is likely to be muchsmaller than U. In the data usually haz-ard rates for recently unemployed work-ers are much bigger than those whohave been unemployed longer, althoughmany other reasons can contribute tothis difference.

The matching function in (10) exhib-its increasing returns to scale in thestocks and the flows, although it is nothomogeneous. The reason is that jobseekers apply to all the available job va-cancies simultaneously. If we doublethe number of job vacancies and unem-ployed workers, the applications of eachand every job seeker double. This con-trasts with the matching function in (3),where each job seeker applies only toone job and so doubling the number ofjobs doubles the number of applica-

tions. Applying to more than one va-cancy at a time is a realistic feature ofthe application process but it dependson a constant rejection probability α.When the rejection probability is endog-enized, we would expect it to increasewhen the matching probability in-creases. Intuitively, the model capturesthe fact that in a large market job seek-ers have more options but not the factthat they would be more choosy as aresult.

The model implies that the matchingprobability for the unemployment in-flow does not suffer from congestion,whereas the pre-existing unemployedsuffer congestion from each other. Thisresult derives from the assumption thatnewcomers flow into the market indi-vidually, given the continuous timestructure of the matching process thattakes place across time periods of infini-tesimal length. If instead we considertime periods of discrete length, a newlyunemployed can match with a new va-cancy, and at the same time all thenewly unemployed can cause conges-tion to one another when trying tomatch with existing vacancies. The extracongestion externalities generated inthis case are shown by Paul Gregg andPetrongolo (1997) to rule out increasingreturns to scale.

Stock–flow matching has receivedsome empirical support. Coles andSmith (1998) argue that, due to stock–flow matching, exit rates are higherwhen traders first enter the labor mar-ket, and drop sharply thereafter. Thissuggests that traders who are unlucky attheir first round of search need to waitand queue for new entrants in order tofind a suitable match. There are, how-ever, many other reasons for the fall inunemployment exit rates, which includeranking, discouragement, and loss of skillsduring unemployment. But more detailedevidence on matching combinations

406 Journal of Economic Literature, Vol. XXXIX (June 2001)

among labor market participants showsthat stock–flow matching plays a sig-nificant role in raising the matchingprobabilities of recently unemployedworkers.

Coles and Smith estimate a log-linearmatching function dividing the outflowfrom unemployment into durationclasses. They find that both the stockand the inflow of vacancies increase theunemployment outflow at short dura-tions of search but at longer durationsonly the inflow of new vacancies in-creases significantly the job-findingrates of the unemployed. Qualitativelysimilar results are also found by Greggand Petrongolo (1997), who estimatequasi-structural outflow equations forunemployment and vacancies derivedfrom a stock–flow matching model indiscrete time.

3.6 Aggregation over Distinct Markets

We finally discuss a derivation of theaggregate matching function that relieson the existence of disequilibrium inmicro markets and limited mobility oflabor. The assumption is that the econ-omy is divided into micro markets thatdo not suffer from frictions but sufferfrom a disequilibrium in the sense thatthe demand for labor in each market isnot equal to the supply. There is no mo-bility of labor or capital between mar-kets. This assumption can be inter-preted as the source of the friction thatgives rise to the aggregate matchingfunction. It implies that markets withunemployment can coexist with marketswith job vacancies, although no markethas both. Aggregation over all marketsgives an aggregate function that containsboth vacancies and unemployment.With perfect mobility workers wouldmove until the short side of the aggre-gate economy cleared and no aggregatematching function would exist.

A model of this form was first used by

Bent Hansen (1970) to derive theBeveridge curve and by Holt (1970b) toderive an expression for structural un-employment. Other studies that followthis approach are Jacques Drèze andCharles Bean (1990), Samuel Bentolilaand Juan Dolado (1991), and WolfgangFranz (1991). Borrowing results dis-cussed by Drèze and Bean (1990, p.14), who credit Jean Paul Lambert(1988) for the derivations, let the ratioof inputs of firms and workers intosearch (say the number of vacancies andunemployment that initially enter themarket) in each micro market be log-normally distributed. Then, if the shortside of each market clears, namely, ifthe matching function in each market isMi = min(Ui,Vi), and U and V are the ag-gregate quantities, there is a CES-typerelationship that could be interpretedas an aggregate matching function

M = (U−ρ + V−ρ)−1/ρ, (11)

where ρ > 0 is related to the variance ofthe ratio of unemployment to vacanciesacross micro markets.

The derivation of this matching func-tion needs the assumptions of exoge-nous distributions of unemploymentand vacancies across space. RicardoLagos (2000) derives instead optimalrules for the allocation of agents acrossspace, under the assumption that thereis uncertainty about the number ofagents at each location. He shows thatthe resulting matching equilibrium isone where the short side of the marketclears—but now the number of agentson one side is optimally selected (seealso Lagos and Gianluca Violante 1998).

As far as we are aware, there are notests of this microfoundation for the ag-gregate matching function. A key prob-lem here is to define the unit of themicro market. If a micro market isinfinitesimally small, and consists of atmost one job, the assumption is trivially

Petrongolo and Pissarides: A Survey of the Matching Function 407

correct. If it is large and equal to theeconomy as a whole, the assumption isincorrect, since at the aggregate levelvacancies and unemployment coexist. Atravel-to-work area would appear to bethe most appropriate disaggregationlevel, but no tests have been conductedat this level. Another difficulty with theCES form is that it relies on distri-butional assumptions about unemploy-ment and vacancies, and a test of theCES restrictions (e.g., versus Cobb-Douglas) would need to test the validityof the distributional assumptions aswell.8

4. Empirical Methods and Findings

In the matching framework the equi-librium levels of unemployment and jobvacancies that persist in steady state arethe result of the intensity of the job re-allocation process and of the matchingeffectiveness of the labor market. Oneway of making inferences about the em-pirical properties of the matching func-tion is to estimate such a long-runvacancy–unemployment relationship,the UV or Beveridge curve. The advan-tage from taking this indirect route isthat estimation of the Beveridge curverequires only data on stock variables,not flows, which are more readily avail-able. The early literature on matchingfollowed mainly this approach. Butpartly because of the difficulty of mak-ing accurate inferences about thematching function from estimatedBeveridge curves (outlined below) andpartly because the connection betweenthe matching function and the Beveridgecurve became better understood, mostof the empirical literature since the late1980s and early 1990s estimated di-

rectly the matching function. As moredata became available, estimated match-ing functions appeared in the literaturemaking use of aggregate time-series forthe whole economy or for some sector(most frequently manufacturing), paneldata for regions or districts, and data onindividual re-employment hazards. Wereview the main results of each ap-proach with focus on the results notpreviously discussed.

4.1 Beveridge Curves

A steady-state relationship betweenthe unemployment rate and the vacancyrate can be derived from the simplematching function (1). Let U and V bethe number of unemployed workers andjob vacancies respectively, and N and Lthe level of employment and the laborforce (so L = N + U). Define the unem-ployment rate u = U/L and let the va-cancy rate be v = V/N (an inconsequen-tial change from the alternative v =V/L). Assume also that the job separa-tion rate is λ, so total separations areS = λN. Then, imposing constant re-turns to scale on m(.) and noting that insteady state the number of matches Mequals the number of job separations S,we get the Beveridge curve,9

λ = m

UL

LN

,VN

= m

u1 − u

,v

. (12)

Given the separation rate λ, our assump-tions on m(.) imply a negative steady-staterelationship between the unemploymentrate and the vacancy rate.

An aggregate Beveridge curve of theform of equation (12) was estimated bya number of authors for the aggregatestocks of vacancies and unemployment

8 Graph theory can also potentially be used toderive results about the interaction of agents inmarkets with frictions, although there are as yetno clear-cut implications for the aggregate match-ing function. See Yannis Ioannides (1997).

9 Note that constant returns in U and V are notneeded here. Suppose for example that the match-ing function has constant returns in U, V, and N,as in (4) but increasing returns in U and V. Thendividing through by N gives an expression withproperties similar to (12).

408 Journal of Economic Literature, Vol. XXXIX (June 2001)

(see Jackman and Stephen Roper 1987;Alan Budd, Paul Levine, and PeterSmith 1988; Jackman, Layard, and Pis-sarides 1989, and Howard Wall andGylfi Zoega 1997 for Britain; Abraham1987 for the United States; Franz 1991for Germany; Per-Anders Edin and Ber-til Holmlund 1991 for Sweden; GiorgioBrunello 1991 for Japan; and Jackman,Pissarides, and Savvas Savouri 1990 fora multicountry study). The form pre-ferred is usually log-linear, which im-plies a Cobb-Douglas matching func-tion if the foundation for the Beveridgecurve is the aggregate matching func-tion. All studies establish the existenceof a negative long-run relationship be-tween the vacancy rate and the un-employment rate, as implied by (12).But virtually all studies also identifysome shift variables not present yet in(12).

Of course, (12) is consistent with manydifferent micro frameworks, some per-haps unrelated to the matching frame-work. But if we posit that there is anaggregate matching function underlying(12), some lessons immediately emergefrom the Beveridge curve studies aboutthe properties of this matching function.

First, there is support for the restric-tions on the simple two-variable match-ing function, including some tentativeevidence for constant returns. Thenegative convex-to-the-origin shapepredicted by the model fits the datawell and in the cross-country regres-sions country size does not appear to bean influence on the position of theBeveridge curve, something that wouldbe implied by some models of increas-ing or decreasing returns to scale. Butno study conducts a careful test of in-creasing returns to scale by testing, forexample, whether the matching rate im-proves when the total number of par-ticipants increases for a given ratio ofvacancies to unemployment, or whether

there are increasing returns to U and Vbut constant returns to U, V, and L, asimplied for example by (4).

Second, there have been shifts in therelationship, especially in Europeancountries. These shifts coincide withthe secular rise in European unemploy-ment, which started in the mid-1970s.The unemployment rate has increaseddespite the fact that the separation rateand the vacancy rate, λ and v in (12),have not shown any trend. The implica-tion for the matching function is thatthere are variables besides u and v thathave played an important role in match-ing in the last two decades and thesevariables contributed to a deteriorationof the matching rate.

Reasons that have been suggested inthe literature include mismatch (Jack-man, Layard, and Pissarides 1989)—which, as we have seen, may explainsome but not much of the shift—thegrowth in long-term unemployment,which reduces both the search intensityof the unemployed and their employ-ability through loss of skill (Budd et al.1988), the generosity of the unemploy-ment insurance system (Jackman et al.1989) and active labor market policy(Jackman, Pissarides, and Savouri1990). Jackman and Roper (1987) haveshown that in Britain the shifts in theregional Beveridge curves were of thesame order of magnitude as the aggre-gate curve, casting doubt on the powerof regional mismatch to explain theshift in the aggregate curve. On a morepositive note, Jackman et al. (1990)show that the different position of theestimated Beveridge curves in Europeis positively correlated with theirspending on active labor market poli-cies. Countries with more spending onpolicies that aid matching haveBeveridge curves closer to the origin.

But on average, no single or combina-tion of variable(s) can account for the

Petrongolo and Pissarides: A Survey of the Matching Function 409

deterioration of the matching rate sincethe mid-1970s, and the literature oftenattributes it to unmeasured elements ofthe unemployment insurance systemand mismatch. It is interesting thatmeasured components of the unemploy-ment insurance system do not play arole in the deterioration of the match-ing rate. Unmeasured elements men-tioned in the literature are usuallystatements about the leniency of thesystem and its coverage. In the estima-tion, such measures are usually pickedup by time trends, which could ofcourse account for many other unob-served or unidentified influences onmatching.

Estimation of log-linear UV curves,along the lines followed by most of thestudies mentioned, suffers from someproblems connected with the assump-tion of flow equilibrium, the endo-geneity of the separation rate, and thefact that inferences about the microprocess underlying matching cannot beeasily made from such an aggregateframework. More recent studies esti-mate matching functions by making useof flow data, which are more disaggre-gated and do not have to rely on either aconstant (or exogenous) job separationrate or flow equilibrium.

4.2 Aggregate Studies

Table 1 gives a summary of the speci-fications and the results of studies thathave estimated aggregate matchingfunctions. Pissarides (1986) estimates anaggregate matching function for Britainover the period 1967–83. The specifica-tion uses quarterly data, with the aver-age monthly outflow rate from male un-employment during the quarter as thedependent variable. The unemploymentseries used is for registered male unem-ployment and the series for vacancies isnotified vacancies adjusted upwards for

incomplete coverage.10 Results withboth linear and log-linear specificationsare reported. The estimated log-linearspecification is

ln

MU

t

= α0 + α1 ln

VU

t

+ α2t + α3t2

+ lags + structural variables.(13)

Both the linear and log-linear specifica-tion strongly support constant returns toscale in U and V (see table 3). The esti-mated elasticity of matching with respectto vacancies is 0.3 with an implied elas-ticity with respect to unemployment 0.7.No other variables were found to be sig-nificant except for the time trends,which indicate a large fall in the rate ofjob matching at given unemployment andvacancy rates during the sample period.

Later estimation of a similar regres-sion by Layard, Nickell, and Jackman(1991, ch. 5) for 1968–88 found similarelasticity estimates but also found thatthe rise in long-term unemployment re-duces the matching rate at a given un-employment rate. But the time trendremains significant in their regression.Also, the authors do not deal with theendogeneity of long-term unemploy-ment but measure its impact by com-puting an index for duration effects.This index is a weighted average of

10 Reported vacancy data are generally unreli-able. In several countries (including the UnitedKingdom, France, Germany, and Israel), data onjob vacancies are collected on a regular basis. Thedata, however, are for vacancies notified to stateemployment agencies and they suffer from under-reporting, with the exception of some rare in-stances where reporting is mandatory (see, e.g.,Eran Yashiv 2000). In addition, the proportion ofvacancies notified varies with general economicconditions, both aggregate and sectoral (see Jack-man et al. 1989). Jackman et al. suggest an adjust-ment method to correct for the underreporting,which makes use of information contained in thefraction of job matches realized through state em-ployment agencies. In the United States there isno comparable vacancy series. The proxy most fre-quently used is the help-wanted index, which isbased on the counts of job advertisements in ma-jor metropolitan newspapers (see Abraham 1987).

410 Journal of Economic Literature, Vol. XXXIX (June 2001)

duration with fixed weights that areproportional to the outflow rates fromeach category in a base year. The factthat outflow rates fall with duration andduration increases during the samplegives an upward trend to the index,which is positively correlated with thetrend in unemployment.

Long-term unemployment has been afrequent candidate for shifts in the ag-gregate matching function (see Budd,Levine, and Smith 1988). Although thisis related to Blanchard and Diamond’s(1994) idea of ranking, it is more gen-eral, in the sense that the claim beingmade is that the average matching rateshould be higher the lower the inci-dence of long-term unemployment.11Denoting again the stock of short-termunemployed by US and the stock oflong-term unemployed by UL, this im-plies that the aggregate matchingfunction takes the form

M = mUS + UL,V,

UL

US + UL

(14)

where the last variable included shouldhave a negative impact on the matchingrate. This prediction is confirmed by Si-mon Burgess (1993) for Britain, KarenMumford and Peter Smith (1997) forAustralia, and Una-Louise Bell (1997) forBritain, France, and Spain. The specifi-cation adopted by Layard et al. (1991)is similar to the one in (14) but withthe alternative measure of long-termunemployment described above.

Blanchard and Diamond (1989, 1990b)estimate a matching function for theUnited States over the period 1968–81.The estimated equation is a log-linearspecification in levels:

lnMt = α0 + α1 lnUt + α2 lnVt + α3t, (15)

where the log of monthly national hir-ings is used as the dependent variable,unemployment is interpreted as a proxyfor all job seekers (including employedand out-of-the-labor force) and the va-cancy series was constructed from thehelp-wanted index. The estimated elas-ticities of matches with respect to vacan-cies and unemployment are positive andsignificant, and the time trend generallycomes in with a negative and significantcoefficient (but smaller than in Britainor other large European countries), im-plying a deterioration in the matching ef-fectiveness of the labor market since thelate 1960s. They find clear evidence ofthe existence of an aggregate matchingfunction with constant or mildly increas-ing returns to scale, unit elasticity ofsubstitution, and weights of 0.4 and 0.6on unemployment and vacancies respec-tively. But the weight 0.4 is found whenthe unemployment rate is used as aproxy for all job seekers, which may notbe appropriate when the number of em-ployed job seekers is pro-cyclical; whenthe left-hand side variable is restricted toinclude only job matches from unem-ployment, the weight on unemploymentrises to 0.6.

The higher unemployment elasticityof matching found in the British studiescan be the result of the different depen-dent variable used. Pissarides (1986)and Layard et al. (1991) use the totaloutflow from unemployment whereasBlanchard and Diamond (1989, 1990b)construct a flow variable that approxi-mates the total number of hires (includ-ing job-to-job moves and flows from in-activity directly into employment, apoint that is not relevant here but ad-dressed in the next section). Burda andWyplosz (1994), who estimate log-linearmatching functions for France, Ger-many, and the United Kingdom by

11 The way that we formalized the ranking ideain equation (9) does not justify the claim made inthe text about average matching rates. Rankingaffects only the distribution of matches acrossthe unemployed. But the frequently made assump-tion that the long-term unemployed reduce theirsearch intensity or lose their skills would justify it.

Petrongolo and Pissarides: A Survey of the Matching Function 411

regressing total exits from unemploy-ment on vacancy and unemploymentstocks, also found high elasticities ofmatches with respect to unemployment,in the range 0.5–0.7.

To see more formally our point, let Xdenote total exits from unemploymentand M denote total hires, again fromunemployment. Let also D denote exitsfrom unemployment to out-of-the-laborforce, a combination of “discouraged”worker effects, early retirement and go-ing back to school. Let M be a log-linear constant returns to scale functionof the type estimated by Blanchard andDiamond and D depend on vacancieswith elasticity −α and on unemploy-ment with elasticity β. If the tightnessof the market V/U is a good measure ofthe cycle (as it is likely to be under con-stant returns; see Pissarides 2000), andmovements from unemployment to in-activity depend only on the cycle, weexpect α = β. But if the experience ofunemployment has additional influ-ences on retirement and dropping out,we should expect on a priori grounds β≥ α. The function estimated by theEuropean studies is (with constantsomitted)

X = M + D= UηV1 − η + UβV−α.

(16)

Studies that use a measure of M in theirregressions estimate η directly. Blanchardand Diamond’s estimate for this numberis 0.6 and similar estimates (in the range0.55–0.70) are found by Jan van Ours(1995), Tito Boeri and Burda (1996), andBurda and Stefan Profit (1996) for othercountries.

Studies that use X as dependentvariable in a log-linear regressionapproximately estimate

∂X∂U

UX

= ηMX

+ βDX

= η + (β − η)DX

(17)

and∂X∂V

VX

= (1 − η)MX

− αDX

= 1 − η − (1 − η + α)DX

.

(18)

The elasticity estimate obtained by stud-ies that use the total exit as dependentvariable should be lower for vacanciesand, if β > η, higher for unemployment.Moreover, given that both sets of studiesfind constant returns to scale, theparameters must be such that

β = 1 + α. (19)

This necessarily implies that β > α, thatis, the experience of unemployment hasan independent influence on droppingout of the labor force, in addition to itscyclical influence.

Blanchard’s and Diamond’s estimateof 0.6 for η, the Pissarides–Layard et al.estimate of 0.7 for the unemploymentelasticity of total exits, and a plausiblemean value for the ratio D/X (the frac-tion of unemployment exits that leavethe labor force) of 0.412 give β = 0.85and a negative value for α. These num-bers, however, are derived from the dif-ference between a point estimate of 0.6and one of 0.7 and they are sensitive tosmall changes in these estimates. Giventhe estimates, a useful approximationthat is well within the confidence inter-val of the elasticity estimates is onewhere α = 0, i.e., one with implied totalexit from unemployment of

X = UηV1 − η + γU, η,γ ∈ (0,1). (20)

Blanchard and Diamond (1989, 1990b)also estimate equation (15) for the U.S.manufacturing sector alone. The resultsthat they obtain in this case are broadlyconsistent with the aggregate ones, withthe important qualification that themanufacturing matching function dis-plays increasing rather than constant

12 This is obtained from the data on workerflows reported by Burda and Wyplosz (1994).

412 Journal of Economic Literature, Vol. XXXIX (June 2001)

returns to scale. The estimated sum ofthe elasticity of matches with respect tovacancies and unemployment is now 1.4(see table 3). Estimates for U.S. manu-facturing are also reported by RonaldWarren (1996), who estimates a moreflexible translog function for all manu-facturing for 1969–73, when a vacancyseries for this sector was available. Thetranslog specification gives a more accu-rate estimate of the returns to scale of atechnology than the Cobb-Douglas form(see David Guilkey, C. A. Knox Lovell,and Robin Sickles 1983). The dependentvariable in Warren’s study is total hiresin manufacturing and the unemploy-ment variable consists of all those cur-rently unemployed who previously heldjobs in the manufacturing sector. Thecorrespondence of the flow variablethat is used as dependent variable withthe stock on the right-hand side is poor,but almost inevitable when hires in onlyone sector are used. He finds statisti-cally significant increasing returns toscale with sum of coefficients on vacan-cies and unemployment of 1.33. Similarresults are found by Yashiv (2000), onboth a log-linear and a translog match-ing function for the whole Israeli econ-omy over the period 1975–89. The esti-mated returns to scale in his matchingfunction lie in the range 1.20–1.36.

The other studies summarized in table1 generally confirm the results of theearlier studies discussed in this sectionfor different countries and time periods.

4.3 Sectoral Studies

The difficulty with making inferencesabout labor market matching from ag-gregate time series beyond the initialresults of the studies discussed in thepreceding section led many authors toswitch to more disaggregate specifica-tions, either in panel or single cross-sections. Table 2 summarizes results fora number of sectoral studies.

Anderson and Burgess (2000) estimate astate-industry panel for the United Statesover the period 1978–84, using a similarspecification to Blanchard and Diamond’s(1989) aggregate study. They also includevariables for sex and age composition ofthe labor force and the degree ofunionization, and distinguish new hiresby origin, namely whether they comefrom employment or non-employment.Although the sum of estimated elastici-ties is well above one when hires fromemployment are used as the dependentvariable, in neither case can the con-stant returns hypothesis be rejected atthe conventional significance levels.

In an attempt to apply the matchingfunction analysis to local labor markets,Melvyn Coles and Eric Smith (1996)and Robert Bennet and Ricardo Pinto(1994) both provide cross-section esti-mates of the matching function for locallabor markets in Britain. Local labormarkets are represented in Coles andSmith (1996) by travel-to-work areas.They use data for 257 areas in 1987 andestimate a regression for total hirings.As in the U.S. studies they find an elas-ticity of 0.7 on vacancies and 0.3 on un-employment. Their study also shows theimportance of the geographic density ofunemployment and vacancies in the hir-ing process, with more concentrated la-bor markets having higher matchingrates. The analysis of Bennet and Pinto(1994) uses instead data from Trainingand Enterprise Councils, estimating atime series for each (Britain is dividedinto about 100 such areas). They findthat the parameters of the matchingtechnology do not vary substantiallyacross districts, the elasticities beingwithin a narrow range of 0.5 for bothunemployment and vacancies, andtherefore confirm that there are noserious problems of aggregation.

The Coles–Smith and the Bennet–Pinto studies treat local labor markets

Petrongolo and Pissarides: A Survey of the Matching Function 413

as isolated marketplaces. Interactionsamong neighboring districts are mod-eled by Burda and Profit (1996), Bur-gess and Profit (1998), and Petrongoloand Étienne Wasmer (1999), who findevidence of matching spillovers acrossspace but with smaller coefficients forneighboring districts. This finding high-lights the importance of moving costs inmatching and is consistent with Colesand Smith’s finding that populationdensity matters in local matching rates.We take up the issue of spatial aggregationbelow.

Most of the available evidence onmatching functions at the sectoral levelis based on log-linear specifications. Anotable exception is the work byMünich, Svejnar, and Terrell (1999),who estimate a translog matching func-tion on 76 Czech and 38 Slovak dis-tricts, which includes, apart from vacan-cies and unemployment, a number oflocal economic and demographic indica-tors. They test and reject both theCobb-Douglas and constant returns re-strictions for both countries. They findthat both the elasticity estimates andthe return to scale parameters vary overtime and between the two sets of re-gions, with the Czech Republic showinga more dynamic, increasing returnseconomy and Slovakia starting off witha week matching process, characterizedby diminishing returns, but picking upto reach increasing returns by 1994.The study by Münich et al. (1999), andother studies summarized by Svejnar(1999), make a new use of the matchingfunction, to study the labor marketresponses to the transition shocks incentral and eastern Europe.

4.4 Micro Studies

The estimation of re-employmentprobabilities for unemployed individu-als has the potential of distinguishing

between the determinants of the prob-ability of receiving a job offer and thatof accepting it. The former depends onthe set of characteristics that influencea worker’s productivity (such as age,education, and experience) and on locallabor demand conditions, which is theeffect captured by aggregate matchingfunctions. The second probability de-pends on a worker’s reservation wage, andtherefore on the expected distributionof wages, the cost of search, unemploy-ment income, and the probability ofreceiving a job offer.

Structural studies (see Nicholas Kie-fer and George Neumann 1979a,b, 1981;Christopher Flinn and James Heckman1982; Wiji Narendranathan and StephenNickell 1985; Kenneth Wolpin 1987; andZvi Eckstein and Wolpin 1995) identifyseparately an accepted wage equationand a wage offer equation, and so theycan distinguish between the determi-nants of each of these probabilities.Reduced-form or hazard function stud-ies estimate instead the factors affectingthe product of the two probabilities,namely the transition of workers fromunemployment to employment, and aretherefore more directly comparablewith matching function studies.

Despite this connection, however,micro studies have not been used in theempirical search literature to make in-ferences about the properties of the ag-gregate matching function, with veryfew exceptions. Their contribution canbe twofold. Micro studies control for anumber of individual characteristicswhich can be aggregated to give shiftvariables in the aggregate matchingfunction besides U and V. They can alsobe used to test for the effect of local labormarket conditions on re-employmentprobabilities and from there aggregateto make inferences about the influence oflocal conditions on aggregate matching.

The early study by Tony Lancaster

414 Journal of Economic Literature, Vol. XXXIX (June 2001)

(1979) uses a sample of British un-skilled male workers to show that exitrates from unemployment are nega-tively affected by age, the duration ofsearch, and the local unemploymentrate. The age effect in the job-findinghazard implies that the age compositionof the labor force should play a role inaggregate matching function estimates,with a younger pool of job-seekers de-livering higher exit rates from unem-ployment (see for example Coles andSmith 1996, and Anderson and Burgess2000). Negative duration dependencein job search implies that the incidenceof long-term unemployment should re-duce the unemployment outflow in ag-gregate specifications, which is con-firmed by, among others, Layard et al.(1991) and Burgess (1993). Finally, thenegative effect of unemployment cap-tures the congestion effect of a largerpool of job-seekers on individual job-finding rates. This should translate intoan aggregate elasticity of matches withrespect to unemployment less than 1,which is the case in all aggregate studies.

Following Lancaster’s application ofduration models to re-employmentprobabilities, a large number of papershave studied the determinants of exitrates from unemployment, looking at avariety of specifications and controlvariables.13 Perhaps surprisingly, a re-sult that frequently appears in the mi-cro studies but not in aggregate studiesis the influence of the unemploymentinsurance system (see Nickell 1979, andNarendranathan, Nickell, and Stern1985). Although there are dissentingvoices (e.g. Anthony Atkinson, JoannaGomulka, John Micklewright, and

Nicholas Rau 1984), on balance microstudies find a (small) influence of un-employment insurance on re-employmentprobabilities. Aggregate studies havefailed to find a robust effect, perhapsbecause of the complexity of the systemand the difficulty of measuring accu-rately its dimensions in a time series.For example, it has been claimed thatthe duration of unemployment benefitsis the most important dimension of thesystem that influences matching. Butbecause there is very little time-seriesvariation in the duration of entitle-ments, only cross-country data can beused to test for this effect. Yet, in cross-country regressions variations in dura-tions are also limited, with some coun-tries having unlimited durations andsome restricting it to six or twelvemonths (see OECD 1994, and Pis-sarides 1999). Another dimension of theunemployment insurance system thathas been emphasized in descriptivework is the leniency of the system. In atime series it is difficult to get a goodmeasure of leniency.

When conditioning on the state ofthe local labor market, only a few microstudies (Nickell 1979; Atkinson et al.1984; Maarten Lindeboom, van Ours,and Gusta Renes 1994; and Petrongolo2001) take into account the demand sideof the labor market and employers’ search,by controlling for the local vacancy-to-unemployment ratio. A higher labor mar-ket tightness, represented by the V/Uratio, significantly increases the job-finding hazard in these studies, con-firming the results of aggregate studies.Petrongolo (2001) also tests for theinfluence of the size of the localmarket on re-employment probabilities.Re-employment probabilities are condi-tioned on the number of unemployedworkers and vacancies within the travel-to-work area of each worker. The coeffi-cients on lnUt and lnVt are estimated

13 See Theresa Devine and Kiefer (1991) for asurvey of hazard studies. Despite a rich literatureon the study of unemployment exit rates, littlework has been done so far on vacancy durations.Notable exceptions are van Ours and Geert Ridder(1992, 1993) and, more recently, Burdett andElizabeth Cunningham (1998).

Petrongolo and Pissarides: A Survey of the Matching Function 415

separately and found to be not signifi-cantly different from each other acrossa number of different specifications,which confirms constant returns toscale in matching.

Lindeboom et al. (1994) go one stepfurther than the other studies, by mak-ing use of the link between the aggre-gate matching function and hazard ratespecifications for evaluating the relativeeffectiveness of alternative search chan-nels. They find that in the Netherlandsemployment offices are most effectivein matching unemployed job seekersand vacancies, while newspaper adver-tisements are most effective in match-ing employed job-seekers and vacan-cies. Informal channels appear to beeffective in both cases.

5. Search On the Job and Out of theLabor Force

A large number of job matches inmodern labor markets are transitionsfrom other jobs or directly from out ofthe labor force to employment. The for-mer has an unambiguous theoretical in-terpretation: some employed workersare active job seekers. The latter ismore vague. Since anyone without a joband actively searching for one is classi-fied as unemployed, the workers whomove directly from out of the laborforce to employment are most likely theresult of inadequate measuring, due forexample to the length of time betweensurvey points. A worker previously outof the labor force may become an activesearcher and get a job within a week,and so miss the classification of unem-ployment in a monthly survey. In coun-tries where labor force surveys arequarterly this problem can lead to largeinflows of workers from out of the laborforce to employment.

In principle, there is no difficulty in-troducing employed job seekers in the

models underlying the matching func-tion (in theoretical work on matching,those out of the labor force who transitto employment directly do not have aseparate status from the unemployed, asthe period of analysis can be made suf-ficiently short to ensure that all thosewho enter employment pass first fromthe pool of job seekers). The way inwhich employed job seekers enter thematching function depends on the as-sumptions that one makes about theirsearch behavior and its relation to thatof the unemployed job seekers (Burgess1993, Pissarides 1994). For example, ifemployers prefer employed job seekersto the unemployed, a ranking modelcould be used to arrive at (9), but withthe number of employed job seekerstaking the place of the short-term un-employed in the expression and the to-tal number of unemployed workersranking below them in the applicationqueue.14 If, on the other hand, it is be-lieved that the main difference betweenemployed and unemployed job seekersis in the choice of search intensity orreservation wage, a function like (7) or(8) would be more appropriate. We de-rive the matching function (8) whenthere are employed job seekers as an il-lustration, under the reasonable as-sumption that employed job seekers havea different (usually higher) reservationwage than unemployed job seekers.

Let RE be the mean reservation wage ofemployed job seekers and RU the meanreservation wage of the unemployed. Thenumber of unemployed seekers is, asbefore, U and the number of employedjob seekers E. The number of job

14 This would be the most appropriate frame-work for the analysis of “vacancy chains,” wherebythe employed take the new and better vacanciesfirst, vacating jobs down the line, and the unem-ployed get pushed to the bottom of the vacancychain. See Contini and Riccardo Revelli (1997)and George Akerlof, Andrew Rose, and Janet Yel-len (1988).

416 Journal of Economic Literature, Vol. XXXIX (June 2001)

vacancies is V and all workers qualifyfor all vacancies. If the unemployed searchwith intensity s and the employed withintensity normalized to unity, the con-tact technology is m(sU + E,V) and theprobability that an employed worker meetsa job vacancy is m(sU + E,V)/(sU + E).The probability that this vacancy is ac-ceptable is 1 – G(RE), so the hazard ratefor the employed is [1 – G(RE)]m(sU +E,V)/(sU + E). The hazard rate for theunemployed satisfies a similar expres-sion, [1 – G(RU)]sm(sU + E,V)/(sU + E).Therefore, the aggregate matchingfunction is

M =[1 − G(RE)]E + [1 − G(RU)]sU

E + sUm(sU + E,V).

(21)

The introduction of employed jobseekers opens up two em